E I'd. THEmI LIBRARY Michigan State University This is to certify that the thesis entitled Applications of Modern Control Theory to the Management of Pest Ecosystems presented by Roger V. Varadarajan has been accepted towards fulfillment of the requirements for Doctorate degree in Electrical; Engineering & Systems Science Major professor Date August 3, 1979 0-7639 OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. ‘ah © 1979 ROGER V. VARADARAJAN ALLRICHI'S RESERVED APPLICATIONS OF MODERN CONTROL THEORY TO THE MANAGEMENT OF PEST ECOSYSTEMS BY Roger V. Varadarajan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1979 ABSTRACT As of the present, very little has been accomplished in terms of utilizing the available information on pest ecosystems to arrive at an optimum combination of control strategies that can be implemented in the field. Conventional pest management strategies based on field ex- perience tend to be ad hoc and do not necessarily lead to satisfactory results. From the systems theoretic point of view, this problem can be interpreted as the determination of the "optimal control" strategies for the management of the ecosystem. The present research focuses on the management of the cereal leaf beetle (CLB), Oulema melanopus, a key economic pest of cereal grains in the United States and Canada. A comprehensive state space model con- sisting of 33 state variables is developed for the CLB ecosystem, which includes the CLB, its larval parasite, T, lulis, and a host plant com- ponent represented by oats. Both chemical and biological control as- pects are incorporated into the model so that the model can be tested within the framework of Integrated Pest Management (IPM). An economic optimization problem is formulated in which we seek to maximize the profit earned by the farmer. The optimal control prob- lem is solved for both single season and multiple seasons. Optimal control strategies are characterized by emphasizing biological control and reducing chemical control usage, and are compared with conventional spraying schemes currently used. A sensitivity analysis is carried out with reference to the timing and amount of pesticide sprayed. In gen- eral, the optimal strategies are at least as good as, and often times better than, the conventional schemes. It is found that the conventional spray is timed earlier in the growing season and is aimed at the CLB spring adults and eggs, while the optimal spray occurs a little later in the season and is targeted on the early larval instars of the CLB. "Externality costs" are included to reflect the penalty imposed for environmental pollution caused by pesticide usage. Two different approaches are analyzed with reference to the externality problem. One is a regulatory approach, in which pesticide use is limited by absolute restrictions on the amount that can be used. Taxation is the other approach considered, in which case the performance measure is augmented with a tax. The discrete-time optimal control problems are solved using a first order successive approximation technique. The necessity for stochastic estimation schemes in connection with pest management problems is pointed out. The Linear-Quadratic-Gaussian (LQG) approach is proposed for the combined stochastic estimation and control problem, leading to On-Line Pest Management (OLPM) systems. The overall approach to the pest management problem adopted in this work is general enough to be extended to a wide range of problems in biological resource management. To My Wonderful Parents ACKNOWLEDGEMENTS I have a lot of people to thank since so many individuals have helped me shape this research work. Of these, I'd like to single out Dr. Lal Tummala who has been more than an advisor to me. I am grateful to him for suggesting this research problem, as well as the guidance and advice that he has provided all these years. I'd like to thank Dr. Dean Haynes and Dr. Lal Tummala for the financial support, encouragement, and for providing an environment conducive to research. I wish to ex- tend my thanks to Dr. Thomas Edens for the many valuable discussions and also for the guidance on the economic aspects of the problem. In brief, it has been a great pleasure to have been associated with the "trio"--Drs. Edens, Haynes and Tummala. I am grateful to Drs. Schlueter and Barr for their inspiring teaching, helpful comments, and encourage- ment. I thank Bill Ravlin and Ray Carruthers for the many valuable dis- cussions, and, more importantly, for allaying the fears of this engineer toward working with biologists and turning it into a pleasant encounter. Most of what I have learned about entomology has come from informal dis- cussions with Bill and Ray. I would like to extend my thanks to Dr. Stuart Gage for his help with the oats plant model and for his many use- ful comments. I'd like to thank Drs. Alan Sawyer and Winston Fulton for their valuable discussions on the modeling aspects of the CLB problem. I am thankful, also, to Emmett Lampert for providing me with the field- data for the oats plant, and to Steven Kraus for his valuable help with the software implementation. I am fortunate to have been associated with three outstanding women--Dorothy, Joanna, and Kim--who have helped me in many ways, and more importantly by being good friends. They will always remain special to me. I would like to thank Rosie for all her help. My profound thanks and appreciation go to Kim and Susan, both of whom have done an excellent job of typing this thesis and cheerfully complied with my rather pestering requests for editorial help. I'd also like to thank my friends, Krish, Sajjan, and Naveen, fbr boosting my spirits from time to time. My parents and my immediate family continue to remain the greatest source of inspiration and encouragement to me--I do not want to make a vain attempt of expressing in words the gratitude that I owe them. TABLE OF CONTENTS LIST OF TABLES O o o o o o o o o o o o o o o o o o o o 0 vi LIST OF FIGURES o o 0 o o o o o o o o o o o o O o o o o o Vii INTRODUCTION . . . . . . . o o o o o o o o o o o o o o o 1 MODELING AND OPTIMIZATION IN THE CONTEXT OF ECOSYSTEM 3 MANAGMNT o o O o o o o o o o c o a o I o o o o o o 0 Simulation Models . . . . . . . . . . . . . . . . . . 3 Mathematical Models . . . . . . . . . . . . . . . . . 5 The Need for Optimization Schemes . . . . . . . . . . 6 LITERATURE REVIEW OF OPTIMIZATION MODELS IN PEST MANAGEMENT AND RELATED AREAS . . . . . . . . . . . . . 8 Drawbacks in Past Efforts . . . . . . . . . . . . . . 18 Drawbacks Related to the Biological Model Representation . . . . . . . . . . . . . . . . . . 19 Inadequacies Related to Economic Considerations . . . 21 Drawbacks Related to Optimization Schemes . . . . . . 25 PROBLEM DESCRIPTION . . . . . . . . . . . . . . . . . . . 33 Description of the CLB Ecosystem . . . . . . . . . . 35 Modeling Aspects . . . . . . . . . . . . . . . . . . 37 OATS PLANT MODEL . . . . . . . . . . . . . . . . . . . . 43 Estimation of Parameters for the Oats Plant Model Using Time-Series Analysis . . . . . . . . . . . . 45 CLB-Oats Plant Interactions . . . . . . . . . . . . . 48 SYSTEM MODEL FOR THE CLB ECOSYSTEM . . . . . . . . . . . 53 Dictionary of State Variables . . . . . . . . . . . . 53 System Parameters . . . . . . . . . . . . . . . . . . 55 System Model . . . . . . . . . . . . . . . . . . . . 56 Attack Equation . . . . . . . . . . . . . . . . . . . 58 iv TABLE OF CONTENTS (continued) Density Dependent Mortality of I and IV Instars . . . 58 Mortalities Induced By Pesticide . . . . . . . . . . 58 OPTIMIZATION SCHEME . . . . . . . . . . . . . . . . . . . 59 A FIRST ORDER SUCCESSIVE APPROXIMATION TECHNIQUE: THE GRADIENT WTHOD O O C O O O O O O O O O O I O O O O O O 62 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . . . . 71 Sensitivity of Analysis . . . . . . . . . . . . . . . 88 Analysis of Control Strategies for the Multi- Season . . . . . . . . . . . . . . . . . . . . . . 93 Environmental Considerations . . . . . . . . . . . . 104 Effects of Change in Crop Price on Pesticide Use . . 111 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . 113 APPENDIX A O 0 O C 0 O O I O O O O O O O O O O O O O O O 119 Program Structure . . . . . . . . . . . . . . . . . . 119 Convergence Properties . . . . . . . . . . . . . . . 119 FORTRAN Listing . . . . . . . . . . . . . . . . . . . 124 APPENDIX B C O O O O O O O O O O O O O O O O O O O O O O 145 L-Q-G Design for On-Line Control . . . . . . . . . . 14S Techniques for Implementing the L-Q-G Algorithm . . . 149 LITERATURE CITED . . . . . . . . . . . . . . . . . . . . 157 LIST OF TABLES Review of optimization models in pest management and related areas 0 O O O O O O O O O O O I O O O 0 Optimization problems in related areas . . . . . . Comparison of the optimal control policy with conventional spray and no-spray schemes for a single season problem . . . . . . . . . . . . . . . Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 2.000/sq ft and TJ = 0.001/sq ft Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 2.0/sq ft and TJ = 0.1/sq ft . . Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 1.000/sq ft and TJ = 0.001/sq ft vi 14 74 96 97 98 10. 11. 12. 13. 14. LIST OF FIGURES Flow diagram illustrating the methodology for ecosystem management using quantitative models . . . . . . . . . . Dosage response characteristics of CLB larva, CLB adult, and T, julis to a pesticide spray of malathion . . . . . Diapause functions for T, julis-—observed field data and fitted curve 0 O O O O O O O O O O O I O O O O O O O O 0 Block diagram illustrating the parameter estimation for oats plant model using time-series analysis . . . . . . Weight of plant and surface area of grainhead of the oats p1ants--observed and estimated . . . . . . . . . . . . . Weight of grainhead and surface area of grainhead of the oats plants--observed and estimated . . . . . . . . . . Relationship between spring-adult CLB density and yield from oats plant under no-spray conditions (plotted on a semi-logarithmic scale) . . . . . . . . . . . . . . . . Comparison of the timing and amount of pesticide spray under optimal and conventional spraying strategies . . . Leaf surface area of oats plant under optimal, conven- tional, and no-spray schemes . . . . . . . . . . . . . Weight of oats plant under optimal, conventional, and no-spray schemes . . . . . . . . . . . . . . . . . Surface area of grainhead of the oats plant under optimal, conventional, and no-spray schemes . . . . . . Weight of grainhead of the oats plant under optimal, conventional and no-spray sthemes . . . . . . . . . . . CLB-spring adult density under optimal, conventional, and nO‘Spray SChemeS o o o o o o o o o o o o o o o o o o CLB-summer adult density under optimal, conventional, and no-spray schemes . . . . . . . . . . . . . . . . . . vii 42 43 47 50 51 73 79 81 81 82 82 83 83 LIST OF FIGURES (Continued) 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. Density of adult T, julis under optima, conventional, and nO'Spray SChemes o o o o o o o o o o o o o o o o o 0 Density of diapausing T, julis under optimal, conven- tional, and no-spray schemes . . . . . . . . . ; . . . . CLB-egg density under optimal, conventional, and no- Spray 5 Cheme S O I O I O C O O O O O I O O O O O C O O O CLB-first instar density under optimal, conventional, and no-spray schemes . . . . . . . . . . . . . . . . . . CLB-second instar density under optimal, conventional, and {IO-Spray SChemeS o o o o o o o o o o o o o o o o o o CLB-third instar density under optimal, conventional, and {IO-Spray SChemeS . o o o o o o o o o o o o o o o o o CLB feeding under optimal, conventional, and no—spray schemes . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity of cats yield (bushels/acre) with reference to changes in the timing and quantity of pesticide Spray 0 O O O O O O O O O C O O C O O O O C O O O O O 0 Sensitivity of profit (dollars/acre) with reference to changes in the timing and quantity of pesticide SP ray 0 O O O O O O O O O O O O O O O O O O O O O O O 0 Sensitivity of the density of spring adult CLB (of the next season) with reference to changes in the timing and quantity of pesticide spray . . . . . . . . . Sensitivity of the density of adult T, julis (of the next season) with reference to changes in the timing and quantity of pesticide spray . . . . . . . . . . . . Sensitivity of the CLB egg density with reference to changes in the timing and quantity of pesticide spray . Sensitivity of the total CLB third instar density with reference to changes in the timing and quantity of pesticide spray . . . . . . . . . . . . . . . . . . . . Quantity of pesticide sprayed for the multi-season problem under different initial densities for the CIJB and TJ C O O O O C O O O O O O O O O O I O O O O O 0 Yield from the oats plant for the multi-season problem under different initial densities for the CLB and TJ . . viii 84 84 85 85 86 86 87 9O 9O 91 91 92 92 99 100 LIST OF FIGURES (continued) 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. Profit obtained for the multi-season problem under different initial densities for the CLB and TJ . . . Density of spring adult CLB for the multi-season problem under different initial densities for the cm and TJ O O O O O O O O 0 O O O O O O O O O O O 0 Density of adult T, julis for the multi-season problem under different initial densities for the CLB and TJ O O I O O O O O O I I O O I O O O O O O O Oats yield and profit under different regulatory managements of the quantity and pesticide sprayed . Quantity of pesticide sprayed by optimal and non- optimal users as a function of tax imposed . . . . . Oats yield obtained by optimal and non-optimal users as a function of tax imposed . . . . . . . . . . . . Profit obtained by optimal and non-optimal users as a function of tax imposed . . . . . . . . . . . . . Sensitivity of cats yield and amount of pesticide sprayed with reference to changes in the price of oats C C C O I O O O C O O O O O O O C O O O O O O 0 Block diagram illustrating L-Q-G design for on—line control of a pest ecosystem . . . . . . . . . . . . Block diagram illustrating the structure of the com- puter program . . . . . . . . . . . . . . . . . . . Convergence of the optimization algorithm . . . . . Computer flow chart for the optimization algorithm . Schematic illustrating a generalized control problem Internal structure of a compensator . . . . . . . . Schematic for L-Q-G design . . . . . . . . . . . . . ix 101 102 103 108 110 110 110 112 117 120 121 123 147 153 154 INTRODUCTION The survival of human society requires that it exert some form of control over some of the other existing systems. In the case of pest- crop ecosystems there exists a competition between human and insect com- munities for resources like vegetation. Thus, exerting control on these systems is dictated more out of necessity than choice. However, control of natural systems is by no means trivial, in spite of our so-called technological and scientific progress. Insect populations no longer ap- pear to be inert masses passively responding to changing environmental pressures (Wellington 1977). Oftentimes the consequences of our control actions have been counter-productive. Heavy crop losses in spite of tremendous application of pesticides, resistance to pesticides developed by pests, the adverse environmental effects of pesticides, and a low success rate with biocontrol attempts all provide corroborating evidence. It appears that we have been in search of a panacea, and somewhere along the way, have grossly underestimated the intricacies that typify biologi- cal systems. A logical outgrowth of these turns of events are the in- creasing demands for pest management programs that are not only economic- ally feasible and profitable, but also ecologically compatible and ac- ceptable. Toward this end two significant ideologies have emerged: (l) the concept of "Integrated Pest Management" (Stern 1959, Smith 1962), incor- porating strategies that attempt to utilize an optimal combination of all known pest control techniques including biological, cultural, and chemical approaches, and (2) the important concept of "On-Line Pest Management" (Tummala and Haynes 1977) which, in addition to being integrated in scope, provides for periodic updating of control strategies in light of changing meteorological conditions (hence changing ecological states) and the relative effectiveness of previous control strategies. However, very little has been accomplished in terms of utilizing the available information on the pest ecosystems to arrive at an optimum combination of control strategies that can be implemented in the field. From the systems theoretic point of view this can be interpreted as the determination of "optimal control" strategies for the management of the ecosystem. The major objective of this research is to provide the theoretical foundations for the design of control systems that will lead to optimal control strategies for on-line pest management. In general, the optimal control strategies will be characterized by efforts to emphasize biotic control while minimizing the use of pesticides. Economic and environ- mental trade-offs that are inherent in pest management problems will be discussed. Our research efforts will be directed toward the Cereal Leaf Beetle (CLB) (Oulema melanopus (L.)) ecosystem. However, the over- all problem-solving methodology developed in this research will be gen- eral enough to be extended to a wide range of problems in biological resource management. MODELING AND OPTIMIZATION IN THE CONTEXT OF ECOSYSTEM MANAGEMENT Most biological systems (and many other real-world systems as well) are far too complex to be understood in all their details at any level, and far too intricate to be broken up into components without destroying the integrity of the system. Hence, models are used extensively in the representation of these systems. Modeling allows us, in principle, to isolate the components of a system and to study their interactions. This helps us recognize some important relationships that exist in the real- world system but are normally masked by complexities and interactions. More importantly models provide a framework for analyzing the system under various hypothetical situations. Especially for ecological prob- lems these analyses can proceed well beyond what is experimentally pos- sible to demonstrate under field conditions. For our purposes, the models of ecosystems are classified under the broad categories of simulation models and mathematical models (Figure 1). SIMULATION MODELS Simulation models (sometimes referred to as descriptive models) describe ecosystem interactions usually in terms of a set of computer instructions. With the advent of modern digital computers, simulation models have found widespread use in the ecosystem analysis (see Patten 1970-1976). Simulation models can handle a great deal of detail and therefore tend to be very large. By their very nature, these simulation swwvvzxro \ uni-1E V :__.. srutrslrs I / ‘\_/ 90L ITICAL \cous'mm \ FIGURE 1 . REM. ”LO "IDLE! CNCEPTUAL NOEL smuunou ”ML CWUTEI IULEKITATIW BEST STRATEGY AM THE USER-SUPPLIED occxsxou Mllm PROCESS EIVIMITN. CONSTRAINTS ”TRENT! CAL NOEL OPTIllllTlm loan COPUTEI INLEKITATXM OPTIML (2me STRATEGIES orcxsxou Mllm ’ROCESS MEIEIT 0' ECOSYSTEH Flow diagram illustrating the management using quantitative models. PERFUME CRITERIA MO COISTMIITS EXTEM EWIMITM CWSTMIITS methodology for ecosystem models do not lend themselves to the analysis of alternative management options. Each one of the management options requires a large scale simu- lation, and the cost associated with the analysis of a large number of Options is prohibitive. The main drawback is that the simulation ap- proach does not provide the means of eliminating options that are not "optimal" in an efficient and systematic manner. Thus, in the case of simulation models, the search fbr optimal policy is usually restricted to management strategies supplied apriori (i.e. user suppled policies). MATHEMATICAL MODELS The shortcomings of the simulation models lead us to the second class of models, namely, "mathematical models." These models reflect the dominant features of the system, and their concise representation provides us with an alternative to the detailed (and large) simulation models. Mathematical models are generally expressed in "state-space" form (Ogata 1967). The state equations can be expressed in differential, dif- ference, or partial differential forms. The major advantage of using state-space models is that it is a very effective approach to mathematical representation of systems. Also, several important analytical tools like control theory, optimization theory, estimation theory, etc. are almost exclusively based on state-space models. Hence, for decision an- alysis involving optimization, mathematical models are generally prefer- red over simulation models. An excellent background on mathematical models for pest ecosystems can be found in Tummala et a1 (1975, 1976), Tummala (1974), Barr et a1 (1973), Shoemaker (1973, 1974), Kowal (1971) and Ruesink (1975) among others. THE NEED FOR OPTIMIZATION SCHEMES The need for optimization schemes arises in the context of goal- seeking in ecosystems or the so-called teleological approach to ecosys- tems. The controversy over the role and acceptability of the teleolog- ical approach to biological systems is both very old and still outstand- ing (Davis 1961). However, for many important classes of biological situations, only by using a goal-seeking description (Mesarovdr:l968) an effective and constructive specification of the system can be developed. Indeed, the whole area of pest management, or in general, biological resource management, is basically a goal-seeking endeavor. Essential to the management of any system is the inherent assumption that certain performance goals be defined--we would like to manage the system in such a manner that our performance goals are achieved. From the system theoretic point of view this can be interpreted as the deter- mination of "Optimal control" strategies for the management of the sys- tem. Though optimization techniques have found widespread use and suc- cess in engineering and physical systems there is only a limited amount of literature on the use of optimal control theory for ecosystem manage- ment. However, the value of optimization has been recognized by biolo- gists. Patten (1971) stated: "The whole area of Optimization theory is certainly pertinent to renewable resource management and could be used for...management schemes." Watt (1963) pointed out that "...many prob- lems in the management of renewable natural resources are extremal prob- lems: we would like to maximize fish yield* from a lake, lumber yield *However, in the real world, one strives for profit maximization, not yield, because maximization of yield does not necessarily lead to maxi- mum profit due to the existence of price elasticity. from a farm, or minimize survival of a pest." Optimization schemes are most often used in conjunction with mathematical models in state-space form and are generally referred to as "optimization models" (refer to Figure l) . LITERATURE REVIEW OF OPTIMIZATION MODELS IN PEST MANAGEMENT AND RELATED AREAS In recent years several studies (Tables 1 & 2) have appeared in the literature related to optimization schemes for pest management. Watt (1963) was among the first to point out the potential use of optimization procedures like dynamic programming in pest management problems. The approach of Watt (1964) was based on an essentially brute-force technique. Several predetermined policies were tried on a spruce budworm model, and optimal policies were chosen on the basis of minimum total cost that included timber loss and control costs. Such an approach is limited by the number of policies to be considered--it only searches over a set of predetermined (i.e., user supplied) control policies and not over the entire policy space. This approach, though useful when used with simu- lation models, is clearly inefficient when used with mathematical models. Jacquette (1970), Mann (1971), Becker (1970) have developed simple mathematical models in which pest populations are described by Markov processes, either continuous-time or discrete-time birth and death processes. They used dynamic inventory theory (similar to the principle of optimality) and calculus of variations to derive some necessary con- ditions. Jacquette (1972) pointed out that these are elementary models and have little practical use. Goh (1970, 1972) and Vincent (1975) have discussed the application of the maximum principle to population models described by simple Lotka-VOlterra type equations. 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Vincent et a1 (1977) introduced the concept of isochronal systems and derived necessary conditions for periodic optimization of a scalar system representing biological populations. Hueth and Regev (1974) proposed a hypothetical model with pest, plant, and pesticide-resistance components and derived necessary condi- tions for optimality using the maximum principle for a profit maximiza— tion problem. Marsolan and Rudd (1976) developed a distributed parameter model for the Southern Green Stink Bug, a major pest of soybeans, and used the maximum principle to derive optimal control strategies. Rorres and Fair (1975) considered an age-specific population and derived condi- tions for optimal harvesting subject to linear ecological and economic constraints. Peder and Regev (1975) pointed out the importance of exter- nality costs and analyzed the effects of user-cost on optimal policy. Mitchiner et al (1975) discussed the application of optimal linear regu- lator theory to the pest management problem, and used the maximum prin- ciple to solve a hypothetical problem. Taylor and Headley (1975) pre- sented a model with genotype classes in which physiological resistance to insecticides is incorporated, and suggested the use of dynamic pro- gramming to solve a simplified version of the problem. Headley (1971) in his elegant, yet simple, work reintroduced the concept of "economic threshold" that provided significant insights into the economic consid- erations that are inherent in pest management decision-making. Shoemaker (1973) demonstrated the application of dynamic programming to arrive at optimal pest management strategies by considering a semi- realistic pest-parasite model for the Mediterranean Flour Moth. Hall l6 and Norgaard (1973) derived an optimal quantity of pesticide spray under the assumption that there exists a single optimal time for the applica- tion of pesticides. Talpaz and Borosh (1974) derived optimal frequencies and quantities per application of pesticide spray for a cotton crop eco- system consisting of a single pest population. Regev et a1 (1976) util- ized non-linear programming to solve an economic optimization problem for the alfalfa weevil. Use of dynamic programming for the determination of optimal sterile male release strategies for a hypothetical pest pop- ulation is discussed by Taylor (1976). Carlos Ford-Livene (1972, 1973) used dynamic programming to solve a hypothetical pest management problem described by a linear model and suggests the use of stochastic dynamic programming to solve the stochastic optimal control problem. Birley (1977) proposed a transfer function approach to modeling pest ecosystems and used a modified dynamic programming technique to solve a linear optimal control problem with binary valued controls (i.e., spray/no spray scheme). Dantzig (1974) provided a Markov-chain inter- pretation to the dynamic programming approach by incorporating state- transitional probabilities. Winkler (1977) utilized a modified dynamic programming technique based on Dantzig's approach to solve a fairly realistic pest management problem for the spruce budworm. Talpaz et al (1978) discussed a simulation model for the boll weevil-cotton ecosystem, and used a modified version of Fletcher-Powell-Davidson's non-linear programming algorithm to derive optimal pesticide spraying schemes. Gutierrez et al (1979) presented a simplified model of the alfalfa weevil ecosystem with four components: (1) population dynamics of the l7 alfalfa weevil, (2) dynamics of the alfalfa crop, (3) mortality induced by pesticides, and (4) evolution of pesticide resistance in the weevil population, and used non-linear programming techniques to derive optimal spraying strategies for two cases--with and without information on the development of resistance by the weevil population. Shoemaker (1977) employed dynamic programming to successfully solve a comprehensive model for the alfalfa weevil. The approach was based on decomposing the original model into two coupled models in order to reduce the dimensionality. However, the model has some simplifying assumptions: for example, it was assumed that pest control measures are applied only once per generation, and the pest population has discrete (non-overlapping) generations. Further, the approach lacks generality because the assump- tions made toward simplification of the problem are very specific to the pest ecosystem under consideration. Such inadequacies are likely to preclude its applications in many problems of practical interest. Never- theless, the work represents one of the few notable exceptions that are oriented toward the solution of a realistic (and invariably complex) pest management problem. In related areas, Walter (1975, 1976) utilized stochastic dynamic programming in conjunction with a scalar model to arrive at optimal catching policies for salmon. Goh (1970), Sancho and Mitchell (1975) proposed optimal management schemes for fisheries based on rudimentary models. Rauch et a1 (1975) used the maximum principle to determine temperature control schemes for lobster growth in a controlled environ- ment. Katz (1978) explored the use of the maximum principle to gain insight into optimal feeding strategies for African weaver birds. 18 Hutchinson and Fischer (1979) discussed the application of stochastic control theory to fishery management, and solved a simple logistic model of the Atlantic sea scallop fishery using stochastic dynamic programming. Dynamic programming for deer management (Davis 1967) and non-linear pro- gramming to game management (Swartzman 1970) have been attempted in the area of wild life management. Optimal control approaches are also being explored for improving existing control schemes in medical and biomedical problems related to population control, health care, nourishment, etc. For example, Baharami and Kim (1975) presented optimal schemes for chemo- therapy. They used a control vector interaction scheme based on the gradient projection method to solve a bilinear model with a binary val- ued {0,1} control. Detailed surveys of such problems in medical, bio- medical, and related areas are presented by Jacquette (1972) and Swan (1973). Generally speaking, these models, like their counterparts in pest management, tend to be hypothetical and highly simplified, while dynamic programming remains the tool that is widely used with these simplified models. DRAWBACKS IN PAST EFFORTS The past works contributed immensely to the understanding of optimal control schemes for pest management and related problems. Nevertheless, several drawbacks exist. Generally speaking, the shortcomings associated with past works are broadly classified into three categories: (1) drawbacks related to the model-representation of the biological system, (2) inadequacies in dealing with economic considerations, and (3) drawbacks related to the choice and use of mathematical representation and optimization schemes. l9 Drawbacks related 32 the biological model representation The major shortcoming of most of these models is that they are over simplified from a biological point of view. Generally, the dimensions of the models were 3 or 4, with most of them scalar. Such highly sime plified models could not capture most of the biological details. Ideally, the models should include the three major biological components of the ecosystem: pest, plant, and parasite. Several state variables may be needed to represent each one of these three components. Tummala (1977) pointed out that the advisability of implementing any control measure and the effectiveness of the control scheme depends on many factors, such as the age distribution of the pest population, the maturity and the vigor of the plant, the size of beneficial insect (parasite) popu- lations, and weather. The age distribution of the pest populations is very important be- cause the damage an insect inflicts, and its susceptibility to insecti- cides, predators, and parasites depends on its stage of development. For example, some parasites attack only eggs, others attack only larvae of a specific size. The damage inflicted on crops also varies as an insect develops. For example, fourth instar larvae of the cereal leaf beetle (hereafter referred to as CLB) cause about 24 times the damage caused by the first instar larvae. Another related factor that should be taken into consideration in determining the effectiveness of control measures is plant vigor and maturity. Vigorous plants can often com- pensate for moderate damage so that pest infestations have little effect on yield. A plant's susceptibility to damage also varies as it matures. 20 Susceptibility also depends on the temporal relationship between the susceptible stages of the plant and the damaging stages of the insect population. The rates at which pest populations deve10p from one life stage to the next depend on temperature, humidity, and other climatic factors. Finally, the potential damage to the crop can be reduced if the pests are controlled by beneficial insects. However, the size of this reduction depends on the sizes of the pest population and the bene- ficial insects, the time synchrony between the two populations. and the age distribution of the pest when it is killed by its natural enemies. It is obvious that most, if not all, of these factors have to be included in any real-world pest management problem, whereas the effect of including them will be that of tremendously increasing the number of state variables used in the model. It should be emphasized here that we do not advocate unnecessarily increasing the complexity of the models. As Bolling (1977) points out, ...a major effort in modeling should be directed toward achieving a minimal representation of the system and one has to be ruthlessly parsimonious in selecting the state variables. However, we do wish to emphasize the fact that most of the optimization models found in ecological literature are so simplified that they cease to represent even the dominant features of the ecosystem, and can no longer be identified with the real-world problem. As one would suspect, this is one of the prime reasons why biologists have chosen to largely ignore optimization models while opting for complex simulation models. Clearly there exists trade-offs between the simplicity desired for the 21 purposes of optimization and mathematical analysis, and the complexity needed to capture the biological details--a meaningful and acceptable (to biologists) compromise must be found. In addition to being simplified, most of the optimization models found in pest management literature are entirely hypothetical and are not based on biological studies. This is because functional forms for the pest ecosystem models are either not available, or (for the most part) are not sought after. Incidentally, this points out the signifi- cance and the need for an interdisciplinary approach to pest management and related problems. Furthermore, past efforts have mostly ignored the parasite or the beneficial insect component of the ecosystem. This is a major drawback since the parasite component represents the biotic control of the pest population. Thus, optimal control schemes that have ignored the biotic control aspect of pest management and have concentrated only on chemical control do not represent an "integrated" control approach to pest manage- ment. Inadeguacies related to economic considerations Another area which has not been adequately explored is that of the economic considerations involved in the pest management decision-making process. The science of economics enters into the design of pest man- agement strategies primarily because the goals of pest management are mostly economic in nature. Pest management problems are often posed as economic optimization problems (i.e., profit maximization or cost minimization). Economic 22 theory, as it has been applied to pest management problems, is usually presented in the form of a threshold analysis. Such an analysis is popularly termed as the "economic threshold" in pest management problems (See Edens (1977) for an excellent critique of economic threshold, and Tummala and Varadarajan (1976) for an extensive bibliography). The concept of economic threshold evolved as a direct application of microeconomic optimization techniques to agricultural management. In its most simplified form it is merely a restatement of the economic cost minimization criterion--undertake an additional expenditure only when the incremental increase in revenue which occurs as a result of the effort is greater than (or equal) to the incremental cost (Edens 1977). In the context of pest management, the lowest pest density that can cause economic damage to the crop is often referred to as the Eco- nomic Injury Level. Based on this notion, the economic threshold is defined as the pest density at which control measures should be deter- mined in order to prevent the pest population from reaching the economic injury level. It is obvious that the economic damage to any crop is dependent on a variety of factors, including the specific crop, the particular growing season, the prevailing crop price, the pesticide cost, etc. Hence, pest density represented by the economic threshold may vary through time, or vary from crop to crop, region to region, and season to season with society's changing scale of economic values. However, for the present, economic thresholds specified by entomologists tend to remain as static levels. For example, the economic threshold for the CLB is currently specified as 3 eggs and larva/stem (Ruppel 1974). 23 From the control theoretic point of view, the economic threshold can be interpreted as the optimal state (usually representing a specific lifestage with an associated age class) trajectory for the pest density obtained as a solution to the economic optimization problem. The more realistic the model of the pest ecosystem the more meaningful will be the resulting economic threshold. Several agricultural economists (Headley 1971 & 1975, Hall and Nor- gaard 1973, Hilderbrant 1960, and Hueth and Regev 1974) considered eco- nomic optimization problems in pest management leading to useful theo- retical interpretations of economic threshold. However, these works have serious drawbacks in terms of the ecosystem model considered. Generally speaking, the models were purely hypothetical and highly sim— plified, and lacked the biological control component. Except for a few isolated cases, e.g. Shoemaker (1977) attempts to solve meaningful eco- nomic Optimization problems related to pest management are clearly lack- ing. As Edens (1977) rightly points out: The main impediment to the more generalized utilization of the threshold principle is the level of abstraction at which the concept is generally presented...Even though conventional optimization techniques are not all applicable to all pest management problems, they are currently under- utilized, largely because of the difficulty involved in operationalizing them. As the increased cost of the chem- ical control becomes more apparent, it will be clearly recognized that complex optimization techniques based on the dynamic interactions of the agroecosystem have to be used in order to arrive at viable management strategies. A closely related topic of interest that is often overlooked is that of “externality costs" associated with the chemical controls. Ex- ternalities arise due to the tacit assumption--that there is no difference 24 between private and social benefits or between private and social costs--does not hold good in several instances. In the economic jargon, externality due to the environmental pollution caused by chemical con- trols will be classified as the "external diseconomy of production" (Mansfield 1975). An external diseconomy occurs when an action taken by an economic unit results in uncompensated costs to others. When such costs are due to increases in a firm's production, they are termed external diseconomies of production. Most of the environmental pollu- tion problems fall into this category of external diseconomies of pro- duction. In such cases, the private costs do not reflect the full social costs since the firms responsible for the pollution are not charged for their actions that lead to environmental degradation. In recent years, however, consumer groups have become increasingly vocal in protesting against environmental pollution--and rightly so. It is conceivable that legal enforcements will become widespread in the years to come. Therefore, it is worth our efforts to consider the prob- lem of externality as it is present in pest management problems. This has to be carried out within the framework of hypothetical enforcement criteria, leading to potentially useful policies. Unfortunately, a very limited amount of literature exists in this area, especially with refer— ence to quantitative analysis. Peder and Regev (1975), Regev et al (1976), Brook (1972, 1973) have made some initial attempts in this direction by considering hypothetical pest ecosystems. Obviously more research is needed. It is the belief of this author that it has to come from econo- mists. Nevertheless, an attempt will be made in this research to gain 25 insight into potential enforcement policies that take into account ex- ternalities, and more importantly, to provide control theoretic inter- pretations for such policies. Drawbacks related Eg_optimization schemes One of the major drawbacks in past efforts is associated with the choice, and consequently, the limitations of the specific optimization schemes used. We are concerned here with discrete-time Optimization techniques since most of the pest ecosystem and other biological systems are conceptually modeled as discrete approximation to continuous-time systems. Also, as Innis (1974) points out in his excellent paper "Dy- namic Analysis of Soft-Science Studies: In Defense of Difference Equa- tions," the use of difference equations is more appropriate in modeling biological systems. This is because insects have several distinct life stages, with time delays associated with maturation in each stage, which are modeled easily with difference equations. Further, it is more mean- ingful to model the control variables, such as pesticides, as discrete variables, since they are usually applied at certain discrete levels and not at a continuously varying level. Alternative modeling schemes for pest ecosystems include differential equations involving time delays, and partial differential equations (Barr et al 1973) that treat each in- dividual's maturity or physiological age as a point in a continuum. In general, such modeling approaches are cumbersome (when compared with dis— screte-time models) from the standpoint of optimization. Thus, we are interested in optimization schemes that can handle discrete-time problems as well as rather large dimensional problems (since most of the real world pest management problems are large scale). 26 Basically there are three different optimization approaches that have been tried in the past: (1) dynamic programming, (2) maximum prin- ciple, and (3) non-linear programming. Among the optimization techniques that were used, "dynamic pro- gramming“ due to Bellman, is very appealing because a feedback solution is obtained. Further, hard constraints on state and control variables (which are very difficult to incorporate in most of the optimal control schemes) are very easy to handle with the dynamic programming approach. In fact, the presence of constraints on admissable state and/or control variables actually simplify the dynamic programming procedure by reduc- ing the size of the region over which the search for optimal solution is made. Furthermore, extension of dynamic programming to stochastic areas is fairly straightforward. However, the straightforward dynamic pro- gramming technique is hampered by the "curse of dimensionality" (Kirk 1970). Thus, for a system with just three state variables and 100 quanti- zation levels for each state, we will require (100)3 = 106 storage loca- tions which is enormous even for modern day computers to handle. As a result, several techniques have been developed that attempt to reduce the amount of storage locations required to implement the dynamic pro- gramming algorithm. State increment dynamic programmingtxfiLarson (1968) provides considerable savings in high speed storage requirements. Bell- man and others have suggested polynomial approximations for the return function in order to reduce dimensionality requirements. Nevertheless, the computer solution of a dynamic programming problem still remains a formidable task when the dimension of the problem is greater than, say, 3 or 4 (Jacobson and Mayne 1970). 27 Yet another approach that has been utilized is that of transforming the dynamic optimization problem into an equivalent static optimization (i.e., mathematical programming) problem, and then employing mathematical programming techniques to arrive at optimal solutions (Pearson and Srid- har 1966, Cannon, Cullum and Polak 1969, and Tabak and Kuo 1971). This method has the advantage that constraints are easily handled and aperi- odic problems can be considered. However, it is unwieldy in most cases, especially when the grid points in the discretized time horizon are large, resulting in an extremely large number of variables. In general, a dynamic optimization problem with N state variables, M control varia- bles, and K grid points in the discretized time horizon will be trans- formed into an equivalent static optimization problem with (N + M) * K variables. Thus, a problem of the size considered in this research will result in (33 + 1) * 30 variables--over 1000 variables. Also, some computational simplifications that are possible with this approach, when the system equation is linear and time-invariant, cannot be extended to the case of biological systems that are generally non—linear and time- varying. In recent years, the maximum principle has been applied to discrete- time problems. In reality, the (continuous) maximum principle is not universally valid for the case of discrete systems. Due to restrictions on possible variations of the control signal, the continuous maximum principle must be modified for the general discrete case (Sage and White 1977). Athans (1966, 1972) discusses the restrictions of the discrete maximum principle with reference to the convexity/directional convexity 28 requirement on the reachable sets. Even though the discrete maximum principle has been used in deriving necessary conditions for optimality, very little computing experience with this approach is reported in lit- erature (Kleinman and Athans 1966, Athans 1972, and Jacobson and Mayne 1970) in contrast to the continuous case. The dimensions of real-world pest management thus dictate the use of a more suitable discrete-time optimization algorithm. However, it should be emphasized that the choice of the optimization algorithm de- pends on a variety of factors, including the specific problem on hand, the computational aspects of the algorithm, and the individual's own preference for any particular algorithm. In short, there are no general rules for choosing between optimization schemes. Another major drawback associated with the currently available op- timization models in pest management is the fact that they have completely ignored the stochastic aspects of the control problem and concentrated instead on the deterministic problem. In the deterministic optimal con- troller design, one assumes that gxagt_measurement of all_state variables are available. This is seldom the case in practical applications and especially so in pest management problems. For example, while it is generally easier to take measurement of larval stages of an insect, it is difficult to measure densities of pupa and adult. The problem is fur- ther compounded by the fact that certain age-classes (within life stages of an insect) introduced for modeling purposes, cannot be distinguished in the field, and therefore, cannot be measured. Even if one could mea- sure all the state variables, there would be measurement errors intro- duced by physical sensors (or human errors) in carrying out the 29 measurements. This measurement uncertainty should be taken into account in the design of the optimal controller. Also, in real world situations there is likely to be disturbance inputs acting on the physical process described by the system model, e.g. climatological changes affecting an ecosystem. It is obvious that a deterministic optimal controller will not be optimal in a real world stochastic situation. In order that we may take into account the stochastic aspects of the problem, the design of the optimal controller should include a stochastic estimator/filter and a scheme for stochastic feedback control (Athans 1971). Very few people have discussed or attempted the stochastic aspects of pest management problems. Logan (1977) came up with an elementary form of filter based on regression equations to provide improved density estimations for the larvae of the CLB. Hildebrand and Haddad (1977) considered the estimation problem for insect populations and derived a parametric filter based on a distributed parameter model for the alfalfa weevil. These two approaches, however, are confined to the filtering problem and do not deal with the control problem. Ford-Livene (1972) is the only one to have touched on the topic of stochastic estimation and optimal control for pest management. However, his approach appears short-sighted: he assumed a linear system dynamics and suggested the use of stochastic dynamic programming. As mentioned earlier, it will be a mistake to assume linear system dynamics for the generally non—linear pest ecosystem problems. Starting with a linear dynamics for the system (as Ford-Livene did) is markedly different from considering a linearized version of a non-linear system about the Opti- mal trajectory. Further, Ford-Livene suggests that the use of stochastic 3O dynamic programming for solving the stochastic optimal control problem. As Athans (1972) rightly points out, this approach is entirely impracti- cal for most of the real world problems since the curse of dimensionality associated with dynamic programming is far more severe in the stochastic case as compared to the deterministic case. It is also worth noting in passing that while Ford-Livene outlined the stochastic optimal control problem, he has not provided the design of such a controller, nor ex- tended it to a realistic pest management problem. In any case, the sto- chastic dynamic programming approach will be unsuitable to large scale pest management problems due to the curse of dimensionality. Another approach, popular among statisticians, for handling some stochastic aspects is the use of stochastic models incorporating "birth and death processes" (Jacquette 1970, Mann 1971, and Chatterjee 1973). However, this approach is restricted to very simple applications, and caters mostly to theoretical interest. Also, this approach lacks the generality, usefulness, and the computational advantages of the state- space approach traditionally used in the engineering disciplines. Another important aspect in the design of the optimal controller for pest management systems that has not been explored in the past works is the on-line capability of the controllers. Tummala and Haynes (1977) in their paper "On-Line Pest Management Systems" give a lucid account of the need and desirability of on-line features in pest management systems. They point out that pest management systems should have provisions for periodic updating of control strategies in light of changing meteorolog- ical conditions, ecological states of the ecosystem, and effectiveness 31 of previous control strategies. This important feature of on-line cap- ability has not been addressed in past works. A complete survey of the algorithm and computational techniques for optimal control and estimation problems is beyond the scope of this writing due to space limitations. Besides, several excellent survey papers are available on these topics. For example: Survey papers on optimal control-~Fuller (1962), Paiewonsky (1965), Athans (1966), Bryson (1967), Larson (1967), Athans (1971), Mendel and Gieseking (1971), Athans (1972), and Polak (1973). Survey papers on estimation techniques--Rhodes (1971), Athans (1971), Mendel and Gieseking (1971), Athans (1972), and Leondes (1970). In addition, there are several well-written texts in these areas-- Bryson and Ho (1975), Meditch (1969), Jacobson and Mayne (1970), Schweppe (1973), Dyer and McReynolds (1970), Saridis (l970)--to name just a few. Recapitulating, the drawbacks in past approaches (with a few ex- ceptions in each case) are summarized as follows: 1. Most of the models are based on hypothetical ecosystems. 2. Most are overly simplified from a biological point of view. 3. The parasite component (which represents the biotic control component) has been largely ignored. 4. Age-distribution of the biological populations has not been taken into account in many cases. 5. Economic considerations have not been adequately addressed. 6. Most of the optimization schemes employed cannot handle more than 3 or 4 state variables. 32 Most of the approaches were deterministic. The need for sto- chastic control and estimation schemes has been mostly ignored. On-line capability for the optimal controller has not been attempted. PROBLEM DESCRI PT ION Briefly, the research problem can be stated as the determination of optimal decision rules for the "integrated control" of the CLB eco- system. This involves the determination of both the timing and the amount of pesticide spray to be used in the field. In addition, the Optimal decision rules to manage the CLB will be spearheaded by efforts to take maximum advantage of the beneficial effect of T, julis, a larval parasite of the CLB. Obviously there exists trade-offs between the use of biocontrol and chemical control approaches--especially with ref- erence to revenue from the crop which is of great economic importance to the farmer. For example, chemical controls lead to short-term eco- nomic benefits. On the other hand, biocontrol attempts do not give instant pay-offs, but, over a long run, are likely to provide stable economic gains. As such the optimization attempts will be aimed at striking a reasonable balance between biocontrol and chemical control with minimal sacrifices in profit. Within the framework of our model, optimal decision rules will be evaluated against current control practices (which are based on the re- commendations of economic entomologists) in order to gain insight into potentially useful, and conceivably better, control strategies. Inci- dentally , this will also allow us to view in proper perspective the current spraying recommendations that are based on valuable field 33 34 experience of entomologists but have never been quantitatively evaluated for either the timing or the amount of spray. The individual farmer, who is mainly concerned about the ultimate revenue from the crop, has a tendency to emphasize chemical control and make some immediate monetary gains. This frequently leads to excessive spraying, and consequently to environmental pollution. In this context, there exist economic and environmental trade-Offs in all pest manage- ment problems. In this work these trade-offs will be discussed within the framework of optimal decision rules for pest management. For a single growing season, the larval parasite T, juli§_virtually plays no role, but its effect will be felt in the subsequent growing season. Thus, control policies have to be evaluated over multiple seasons in order to determine the effect Of biocontrol. Repeated appli- cation Of conventional control policies season after season, as well as repeated use of optimal policies (on a season by season basis) will be evaluated within the framework of the CLB ecosystem model. As part of our approach to determine the optimal control strategies for the CLB problem, we will develop a discrete-time optimal control technique based on the successive approximation algorithm of Dyer and McReynolds (1970) that is similar in scope to the differential dynamic programming approach of Jacobson and Mayne (1970). The algorithm is utilized within a deterministic framework to solve several types of Op- timal problems associated with the integrated control approach to the CLB problem. Extension of the deterministic approach to the stochastic case is discussed, and the Linear-Quadratic-Gaussion (L-Q—G) methodology 35 is proposed for the design Of an on-line control system for pest man- agement. In the following sections, we will discuss at length, all of the aforementioned aspects of our approach to the research problem. DESCRIPTION OF THE CLB ECOSYSTEM The pest management problem considered in the present work is that of the CLB with its larval parasite, T, julis (TJ) and a crop com- ponent represented by oats. The rationale behind the choice of the CLB ecosystem is two-fold: 1. The CLB is a key economic pest of the cereal grains in Mich- igan and several other states in the United States and Canada. 2. A large amount Of data is available on a number of aspects of the CLB ecosystem from the research studies conducted at the Michigan State University over the years (Castro 1964, Yun 1967, Helgesen 1969, Ruesink 1972, Gage 1972 & 1974, Casa- grande 1975, Jackman 1976, Logan 1977, Fulton 1975 & 1978, and Sawyer 1978). The CLB, Oulema melanopus (L.) is native to Europe and Central Asia. The first reliable indentification Of this pest was made in south- western Michigan in 1962. Since then it has rapidly spread and has established itself as an economic pest in an area ranging from Pennsyl- vania to Wisconsin and from Kentucky to Michigan and Ontario. The CLB attacks small grains, mainly wheat, oats and barley. The annual com- bined acreage of these crops in the United States is close to 100 million acres (Cooper and Edens 1974). Radiation methods for sterilization of 36 the CLB have proved to be ineffective because the dosage required to sterilize an adult is almost lethal. Studies of plant resistance have not provided a readily available method Of control. Chemical control is the only viable control that is used extensively. There is a general acceptance among entomologists that satisfactory control of the CLB can be achieved only by a pest management program based on thorough ecolog- ical research (Haynes 1973). The CLB overwinters as an adult in forest litter, grass, tree bark, or in small crevices protected from heat and cold. In Michigan, adults become active in April and feed on grasses and winter wheat prior to oviposition. The oviposition activity continues for 45-60 days; during this period each female lays an average of 50-150 eggs. The spring adult population declines to a negligible level due to natural mortality in about 60 days. The eggs hatch in a few days and the larval instars feed extensively on succulent leaves of wheat and (preferably) oats. There are 4 larval instars. New adults emerge in a few days, feed in- tensively on any available green grass, and disperse to overwintering sites. These adults diapause and do not lay eggs until the following spring. Most of the damage to the crop is caused by larval feeding during the early stages Of plant development. The CLB was introduced into North American with few, if any, of its natural control agents. Important biocontrol agents include the imported parasites: Anaphes flavipes, and egg parasite, and the three larval parasites--Tetrastichus julis, Diaparsis carinifer, and Lemophagus curtus. It appears that the egg parasite, A, flavipes will not have a 37 major influence on the CLB population since the mortalities of the first and fourth instar CLB are density dependent; thus, as egg density is reduced, survival of larval instars will increase (Helgesen and Haynes 1972). Preliminary studies indicate that the larval parasite 3, julis is better synchronized than the other larval parasites. Moreover, it has two generations per year, very high reproductive potential and a relatively low dispersal quality. The mathematical model considered focuses on this parasite. More detailed descriptions concerning the biology of the CLB can be found in Haynes (1973), Barr et a1 (1973), Tummala et al (1975), Lee et a1 (1976) and several theses cited in the literature. MODELING ASPECTS Considerable modeling work has been done on the CLB ecosystem. Presently four models are available on various aspects Of the CLB eco- system. Gutierrez et a1 (1974) provided a simulation model for the within field dynamics of the CLB in wheat and oats. Fulton (1978) developed a detailed simulation model for the CLB that can be used in an on-line fashion. Both models are aimed at providing detailed descrip- tions of the pest population dynamics through time. However, they can not includethe parasite component represented by T, julis, There- fore these models are not suited for analyzing the biolgical control of the CLB. Furthermore, both models lack a dynamic host-crOp component; hence the economic impact of the CLB feeding on the host plant cannot be evaluated. In addition, the models are based on extensive simulations, 38 and are prone to all the drawbacks associated with the simulation ap- proach to pest management (refer to earlier discussion). In short, these models are best suited for analyzing population dynamics Of the CLB, but are not useful in evaluating a large number of management strategies. Lee et a1 (1976) presented a comprehensive model of the CLB-T, julis ecosystem based on partial differential equations and an ordinary dif- ferential equation model for the host plant. The model, however, did not include the chemical control component. The model is used in a simula- tion mode to describe the maturity distribution Of the CLB and its ef- fect on the host plant through time. Since partial differential equations are rather cumbersome when used in conjunction with optimization schemes, the model of Lee et al is not particularly attractive for management purposes. Tummala et a1 (1975) developed a detailed model of the CLB-T, juli§_ dynamics based on a discrete component approach. The discrete-time state-space model was utilized to illustrate the effect of 2, igli§_on the CLB under varying densities. Since the major Objective of their work was to highlight the beneficial effect of biological control, they did not incorporate in their model the host plant component and the im- pact Of chemical control on the pest-parasite complex. In order to be useful in analyzing integrated control strategies, the models should include both biological and chemical control, and dynamic descriptions of the economic yield from the crop, and lend them- selves suitable for use with Optimization schemes. The state-space model Of Tummala et a1 (1975) is particularly attractive for optimization 39 purposes. In the present research, we will develop an updated version of the model Of Tummala et al (1975) in such a manner that the final version of the model encompasses all the features necessary for analy- zing integrated pest management Options. A chemical control component is added to the earlier version of the model; thus, mortality functions that account for the mortality caused by the application of pesticides are introduced. The pesticides used have impact on both the pest and the parasite. In addition, the pesticides used have different impacts on different life-stages of the insect. These factors should be incorporated into the model. Generally, insect mortality is described in terms of dosage response characteristics that give the relationship between the amount of pesticide applied and the corresponding mortality (expressed as a percentage) induced. Sevin (carbaryl) and malathion are the pesticides that are exten- sively used in controlling the CLB. Both carbaryl and malathion are ef- fective against the CLB larvae and adults, the larvae being more suscep- tible than the adults. In addition, carbaryl is a powerful ovicide (i.e., kills eggs), has a prolonged residual effect (compared to malathion), and is known to cause adverse side effects. The dosage response characteris- tics Of the CLB to these pesticides are discussed in the literature (Yun and Ruppel 1965, Ruppel 1977, and Casagrande 1975). Our model will focus on the pesticide malathion for the following reasons: 1. Malathion is widely used (states like New York recommend only malathion for CLB control). 2. More data is available on this pesticide. 4O 3. Malathion has very little residual effectiveness, while Sevin has a prolonged residual effectiveness that is rather diffi- cult tO model and is likely to add quite a bit of complexity to the present model. The dosage response curves used in the model are illustrated in Figure 2. These are based on published data and discussions with Dr. Ruppel. Currently there is no data available on the impact of malathion on the larval parasite T, julis, However, according to entomologists (Dr. Ruppel personal communication, Michigan State University) the effect of malathion on T, jugi§_is likely to be very similar to that of malathion on the CLB larvae. Due to the lack Of availability of data, Tummala et a1 (1975) as- sumed a hypothetical function for the T, julis diapause. In the present model it is replaced by one that is based on field studies conducted by Gage (1974). The field data and the functional approximation (an eighth degree polynomial fit) used in the present model are illustrated in Figure 3. The development of a model for the oats plant component is discussed in detail in the following section. (Note: The threshold tempera- ture for oats is 42°F, whereas it is 48°F for the CLB and TJ. Hence, a transformation from 42°F to 48°F is used in calculating the cumulative de- gree days for the plant model. This transformation is required for opti- mization purposes. The error associated with the transformation is mini- mal because of the proximity of the thresholds.) 41 DOSRGE RESPONSE CURVES - —‘—— “ CLB LRRVR RND TJ PERCENT HORTRLITY or x ....... CLB HDULT N I ’I . ’1’ ofi I ' V T I r T I V I l’ I I 1 0.0 0.3 0.6 0.9 1.2 1.5 1.8 PESTICIDE SPRRY - LBS / RCRE FIGURE 2. Dosage response characteristics Of CLB larva, CLB adult, and T, julis to a pesticide spray of malathion. 42 DIRPRUSE FUNCTION FOR T.JULIS 100 .4 -- FITTED CURVE D - OBSERVED FIELD DRTR PERCENT T.JULIS IN DIRPRUSE f 1 I I I I I o 300 600 950 11200' DEGREE DRYS (BRSE 48) j I jfi U r T l 1500 1800 FIGURE 3. Diapause functions for T, julis--Observed field data and fitted curve. OATS PLANT MODEL Since the goals of pest management are, to a large extent, economic in nature, the economic yield from the host plant is of utmost signifi- cance. For a realistic characterization of the oats plant as a compo- nent Of the CLB ecosystem, it is necessary to identify the interactions between the CLB and the plant (for more details see Barr et al 1973, Lee et a1 1976, and Gage 1972). The feeding caused by the CLB population results in a reduced leaf surface area of the plant. The reduced photosynthetic capability in turn affects the final yield. Most Of the CLB feeding occurs on the top three leaves, which are responsible for over 85 percent of the net photo- synthetic activity (Gage 1972). Plant growth is dependent on a variety of factors, including mois- ture, soil chemicals, light exposure, etc. However, in our model it is assumed that all these factors are prevalent in a non-stressed or "stan- dard" condition. The key variables that are chosen to represent plant growth are the total weight of the plant W, the leaf surface area S, and the weight WH and surface area SH of the grain seeds as functions of degree days. The selection of these quantities as state variables is based on the following characteristic mechanism. The biomass generated through photosynthesis by the leaves is accumulated as plant biomass and is converted into seed when the plant matures. The weight of the heads directly reflects the quantity of seed produced. Moreover, as the heads 43 44 develop, their surface area should also be considered as an active photosynthetic component. The metabolic processes of plants are determined by the relation- ship between the active mass which undergoes catabolism (respiration) and the necessary surface to support anabolism (photosynthesis). The anabolism is expressed as the product of the rate k1 at which mass is produced per unit area and the effective surface S through which ex- changes take place. Similarly, catabolism is proportional to the entire bulk W of living material. Thus: V where: k1 0 k < O 2 t physiological time for the plant. This equation was first proposed by von Bertalanffy (1957). The growth of surface area S also can be represented with a similar equation. Thus, a linear approximation for the plant dynamics can, in general, be ex- pected to have the form: "w T ('w T d S S _ = K dt WH WH s L8H. L HJ where: K = 4 x 4 matrix. The above equation can be written in the difference equation form as follows: -W (n + 17 1” (Dr S (n + l) S (n) wH (n+1) = P me L5H (n + 1).' L8H (n) where: P = 4 x 4 matrix :3 ll discretized physiological time steps. 45 The elements of the matrix P are in general functions of soil fer- tility, moisture, light intensity, etc. The form of these functions is unknown at present. Evaluation of possible control strategies such as fertilization, irrigation, etc., so as to minimize crop damage from the CLB by manipulating plant growth would require extensive study to deter- mine these functional forms. Nevertheless, development of management policies focusing on the manipulation of CLB and parasite densities under "standard" cultural conditions and practices for the crop compo- nent can proceed using the P matrix with constant parameters. ESTIMATION OF PARAMETERS FOR THE OATS PLANT MODEL USING TIME-SERIES ANALYSIS The essential features of our approach to parameter estimation for the oats plant model is illustrated in Figure 4. As discussed earlier, the structure Of the plant model can be expressed as follows: 1(k+1) =_P_¥_(k) or: "yl (k + 17 'P(1,1) p<1,4fl ’ylm‘ y (k + l) - ° y (k) 2 z 2 y3 (k + l) - - y3(k) _y“ (k + 15 3(4'1) p(4,4)‘ 3am?- where: y1(k) = weight of oats plant/sq ft y2(k) = leaf surface area/sq ft y3(k) = weight of head/sq ft y“(k) = surface area of head/sq ft X_(o) = X_initial = initial conditions for the state variables X, 46 .mflm>amcm moanmw Iwfifiu chms Hopes UCOHQ mumo new :ofiumeumo HouoEmHma ozu mafiumuumsaaw Emummflp xuofim .v mmDUHm Ly 2:58: 225on :xmndlxoug V mamhmgg m5. mo... 1, mass 22:: A. 5252.: w 3» m .. 2 + 5» E53: 2: «E 3:228“: as $225 :8: 55.. 33 85:3 SEES uses... - + :8: £25.. $3 «8 fig 3:: a m as. _ 33% Oz: 33: as. -- so» 95:329. 25.: 47 The parameters of the P matrix remain to be estimated. These par- ameters have to be chosen in such a manner that the outputs Of the plant model match closely the field data for the corresponding variables. The optimization algorithm Of Box* (also known as the COMPLEX algorithm) is used in conjunction with the model as shown in Figure 4. An initial guess, as well as a lower and upper bound, are supplied for each one Of the parameters. Based on the initial guesses for the parameters and the given initial condition, X_initia1, for the state variables, the model generates the state variables ka) through time steps k, from initial time to to final time t The outputs Of the plant model are compared f. with the corresponding real world time-series (i.e., field data) at selected points in time. A weighted least squares criterion is used as the performance index (PI) in the optimization algorithm: 4 n yi(k) - yobsi(k) PI= iZl k=1 yobsavgi where: yi(k) = data generated by the plant model yobsi(k) = field data yobsavgi = average values of the observed field data n = final time step. The COMPLEX optimization algorithm finds the values of the unknown parameters of the P matrix so that the performance index is minimized. The parameter values that are generated by the Optimization procedure are fed back to the plant model, and the model is run with these new parameter values. When this process is repeated, a specified conver- gence is Obtained with the Optimization procedure. The aforementioned *See Kuester and Mize (1973) for details of the algorithm. 48 procedure is repeated for different initial guesses of the parameters in order to make sure the global Optimum is Obtained with the optimiza- tion algorithm. The unknown parameters of the P matrix estimated using the approach discussed here are given below. P MATRIX: "0.803 0.517 -0.056 0.000' -0.112 1.240 -0.018 0.065 0.000 0.036 0.765 0.614 b0.000 0.047 -0.172 1.303 Convergence criterion employed: [(PI/PILAST) - 1| fi'lE - 5 for 3 consecutive iterations. The trajectories of the plant variables generated by the "best- fit" model, and the corresponding real world time-series form field data are illustrated in Figures 5 and 6. CLB-OATS PLANT INTERACTIONS The CLB-plant interactions are caused by the CLB feeding on the leaves of the host plant. The CLB-plant surface area coupling may be described by a simple form in view of the work Of Gage (1972). He Ob— served that leaf consumption by the four CLB larval instars is in the ratio 1.00, 2.87, 5.97, and 24.23, and that feeding by the adult CLB is negligible when compared to larval feeding. Thus, the total larval feeding in terms of the first instar feed- ing equivalents can be written as: LlEQ(n) = FEEDQ[Ll(n) + 2.87 L2(n) + 5.97 L3(n) + 24.23 L4(n)] - OHS / SQ FT DRY HEIGHT 49 .. actour or PLRNT - oosenveo . o HEIGHT or rum - Esnnmo .. sunrac: uses or Leaves - onsenveo , . _ . p on -,.’ ‘ .l m m warns: men or Leaves - Esrxnnreo .2 ’ r—fi N -¢D d ’0 FIGURE 5. r T I j I f Y 600 900 IEODv DEGREE nnvs (BRSE 48) V T I 1 1500 1803 Weight of plant and surface area Of leaves of the oats plants--Observed and estimated. — SQ DH / SQ FT SURFRCE RRER 50 O -O ”I m J 1. HEIGHT or 8580 - onscnvco _ 0 11:10:11 or 11:80 - £31111an0 t; L: ..3 o sunracc am: or 11:80 - oasanveo 1... ca N N c: m sunrnce FREE or HERD - ESTIHRTED 0') 0‘) 4 \ ‘\ z 2‘" "<3 c3 co )— CE I Nd DN LLJ C) '7 .. 0: -‘ C: u . 3 LAJ L.) )- CE 0: -‘ '- LL— E: (D (D m D J U") 600 800 1000 1200 1400 1800 1803 DEGREE DRYS (BRSE 48) FIGURE 6. Weight of grainhead and surface area of grainhead of the oats plants--Observed and estimated. 51 where: Ll(n)‘ . CLB first instar through fourth instar larval . > density at time n L4(n)j LlEQ(n) = first instar feeding equivalents at time n FEEDQ = feeding coefficient (0.002029) that represents the leaf surface area (in sq dm) consumed by a first instar CLB larva in a time duration 60DD48 n = physiological time (in units of 60DD ). 48 Further, it is known that the biomass corresponding to 1 dm2 of leaf surface area is 0.25 gm in oven dry weight (Gage 1972). Hence, the CLB-plant coupling can be expressed as: 'w (n + 1)" "w (n7 ”0.257 s (n + 1) =3 8 (n) _ 1.00 LlEQ(n) WH(n + 1) WH(n) 0.00 LDSH(n + l)‘ EST-I‘m! b0.00.i The ultimate yield (i.e., seed weight) from the oats plant is di- rectly related to the weight of the grain head at harvest time, WH(N), through the following equation (Reference: Lampert, unpublished data): YIELD gms/sq ft WH(N) * 0.9119 - 0.003035 YIELD bushels/acre YIELD gms/sq ft * 3.000992. It is worth emphasizing here that unlike most of the pest manage- ment models Of the past, the model developed in this research work is comprehensive (over 30 state variables), includes all the three major components of the CLB ecosystem (the CLB, T, julis, and oats plant), and provides dynamic relationships for calculating the crop yield. It has both biotic and chemical control components incorporated into it so that integrated control strategies can be evaluated. Most importantly, 52 the model, to a large extent, is based On actual field studies, and not on hypothetical relationships. It should be Obvious that models are the critical links in developing management strategies for complex pest ecosystems. The more realistic the model, the more meaningful will be the resulting management strategies. A complete mathematical description of a system model for the CLB ecosystem is presented in the following pages. SYSTEM MODEL FOR THE CLB ECOSYSTEM NAME X1 X2 X3 X4. X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X21 X22 X23 X24 X20 X25 X26 X27 X28 DICTIONARY OF STATE VARIABLES DESCRIPTION Spring Adult CLB Density CLB Egg Density First Instar CLB Density Second Instar CLB Density Third Instar CLB Density Unparasitized Fourth Instar CLB Density Unparasitized CLB Pupa Density Summer Adult CLB Dinsity Diapausing TJ Density Adult TJ Density Parasitized Fourth Instar CLB Density Parasitized CLB Pupa Density Parasite Per Pest Individual in Different Stages 53 54 DICTIONARY OF STATE VARIABLES (continued) NAME X29 X30 X31 X32 X33 DESCRIPTION CLB Larval Feeding Weight Of Oats Plant Leaf Surface Area Of Oats Plant Weight Of Grain Head Surface Area of Grain Head FEEDQ T S)! DF(n) P(1,1) P(4,4) SYSTEM PARAMETERS DESCRIPTION Spring Adult Survival CLB Eggs/CLB Female/60DD Summer Adult Survival 2, jgli§_Adult Survival Max Eggs/TJ Adult/60DD Max TJ Eggs/CLB Larva/60DD TJ Searching Constant TJ Mortality Inside CLB Mortality of CLB Eggs Mortality Of Mortality of Mortality of Mortality of CLB L1 CLB L2 CLB L3 CLB L4 Mortality of CLB Pupae Mortality Of Overwintering CLB Mortality of Overwintering TJ Exponent in Parasitism Equation Feeding Function Coefficient Time When TJ First Shows Time When CLB Leaves Oats Diapause Function for TJ VALUE 0.70 22.00 1.00 0.60 20.00 5.00 100.00 0.00 0.10 Variable 0.30 0.45 Variable 0.40 0.77 0.50 0.75 0.002029 6.00 (i.e. 360DD Base 48) 16.00 (i.e. 960DD Base 48) -"fin eighth degree polynomial fitted to field data from Gage (1974) Parameters for the Oats Plant Model Obtained Through Time- Series Analysis 55 Refer to pg. 48 Xl (n X2 (n X3 (n X4 (n X5 (n X6 (n X7 (n X8 (n X9 (n X10(n Xll(n X12(n Xl3(n Xl4(n X15(n X16(n Xl7(n X18(n Xl9(n X20(n X21(n X22(n l) l) l) 1) l) l) l) l) 1) 1) l) l) l) l) l) l) l) 1) l) 1) 1) l) SYSTEM MODEL X4(n) X5(n) X6(n) X7(n) a Xl(n) SA b Xl(n) X2(n) X3(n) (1 - K1) (1 - K2) (1 - K3) (1 - K4) (1 - K5) X8(n) X9(n) XlO(n) Xll(n) X12(n) x13(n) Xl4(n) SL SL SL [1 - f2] SL (c X16(n) + (1 - K6) X15(n)) SA Xl7(n) + DF(n) X24(n) X28(n) (d X18(n) + (1 - DF(n)) X24(n) X28(n)) SL (1 - K4)f2 X7(n) SL fl/(f2 X7(n)) (1 - K5) Xl9(n) 56 X21(n) 57 X23(n + 1) = X22(n) X24(n + 1) = X23(n) X25(n + 1) = (1 - rp) X20(n) x26(n + 1) = x25(n) X27(n + l) = x26(n) X28(n + l) = X27(n) x29(n + 1) = FEEDQ [x5(n) + 2.87 X6(n) + 5.97 (X7(n) + 24.53 (X8(n) + Xl9(n))] X30(n + l) = P(l,l) X30(n) + P(l,2) x31(n) + P(l,3) X32(n) + P(l,4))(33(n) - 0.25 x29(n) X31(n 1) P(2,l) X30(n) + P(2,2) x31(n) + P(2,3) X32(n) + P(2,4) X33(n) - x29(n) X32(n 1) P(3,l) X30(n) + P(3,2) x31(n) + P(3,3) I X32(n) + P(3,4) X33(n) X33(n 1) P(4,l) x30(n) + P(4,2) x31(n) + P(4,3) X32(n) + P(4,4) X33(n) ATTACK EQUATION 58 X7(n) X18(n) fl = X7(n) el _ £1 £2 ’[xumez] X18(n) ‘£_ e2 + e3 DENSITY DEPENDENT MORTALITIES OF I AND IV INSTARS* K2 = 0.46 logE - 0.85 K5 = 0.28 logE - 0.18 where: K2, K5 = E = season. MORTALITIES INDUCED BY PESTICIDE** Dosage response for CLB larva Dosage response for CLB adult where: SL SA {.1 ll *See Helgesen and Haynes 1972. ogxz < 0.99 0_<_1<5 <0.99 lst and 4th instar mortalities, respectively. total number of eggs laid per sq ft for the entire and TJ = (1 - SL) _ l l + e-(10*u - 6.0) = (1 - SA) 1 1 + -(10*u - 8.5) e survival based on pesticide spray for CLB larva and TJ survival based on pesticide spray for CLB adults pesticide spray of Malathion lb/acre **Dr. Ruppel, Michigan State University--personal communication. OPTIMIZATION SCHEME The optimization procedure utilized (Dyer and McReynOlds 1970) in the present work is derived from dynamic programming. It is a succes- sive approximation technique, based on dynamic programming instead of the calculus of variations, for determining optimal controls of non- linear dynamic (or static) systems. The method is motivated from a consideration of the first and second order expansion Of the return function about some nominal control variable sequence. In each itera- tion, the system equations are integrated forward using the current nominal control, and the accessory equations (which yield the coeffi- cients of a first or second order expansion of the cost function in the neighborhood of the nominal state trajectory) are integrated backward, thus yielding an improved control sequence. Iteratively, this method results in control sequences that successively approximate the optimal control sequence. The first order technique of Dyer and McReynOlds (1970) is known as the successive approximation by gradient method. The first order methods are characterized by slow convergence near optimum, but are simpler to compute and are very useful in getting starting solutions. The second order approach of Dyer and McReynOlds (1970) is called the successive sweep method, while a similar second order approach Of Jacob- son and Mayne (1970) is popularly known as the differential dynamic pro- gramming. The second order methods have faster convergence than the 59 60 first order schemes, but are computationally more involved and are susceptible to nominal controls. Thus, hybrid schemes, in which first order methods are used to start the Optimization procedure and get an improved nominal control sequence, while second order methods are uti- lized later on to improve convergence, are more appealing. Since these optimization techniques are based on successive approximation schemes, the storage and computational time requirements are very small, com- pared to dynamic programming and possibly several other Optimization schemes as well. However, it is to be noted that, unlike dynamic pro— gramming, a true feedback solution is not Obtained with these approaches, although it is possible to compute Optimal feedback control in the neigh- borhood of the Optimal trajectory. In the present work, the first order successive approximation al- gorithm is used in conjunction with the Optimization model of the CLB ecosystem consisting of 33 state variables. This is a marked improve- ment considering the fact that almost all the optimization approaches employed in the past (in connection with pest management and related problems) are confined to dimensions of 3 or 4. Furthermore, the method is general enough to be extended to other problems in pest management and many other areas as well: the major requirements being an available state-space model of the system under consideration and a properly forms ulated optimization problem. In addition, the Optimization approach fits in nicely with the overall methodology of Linear-Quadratic-Gaussian (L-Q-G) design (refer to Appendix B) proposed for on-line pest manage- ment. 61 We would like to point out here that the successive approximation algorithm and differential dynamic programming are increasingly used in solving non-linear Optimal control problems, especially of the dis- crete-time type (Gershwin and Jacobson 1970, Iyer and Cory 1972, Jamshidi and Heidari 1977, to name just a few). Most recently, Profes- sor Ohno of Kyoto University, Japan, has come up with-a new approach to differential dynamic programming that can directly solve optimal control problems with hard constraints on state and/or control variables without adjoining them (Ohno 1978, ,_Ohno--personal communication 1978) . As Of the moment, Ohno's approach is restricted to rather small dimen- sional problems. However, it is worth noting that efficient ways of handling constraints is one of the most difficult problems encountered in the computation of optimal controls. We envisage more widespread use of differential dynamic programming (and variations thereof) in the future, especially in large dimensional, discrete-time Optimal con- trol problems arising in pest management and related areas. A detailed derivation of the first order successive approximation technique used in our work is given in the following pages. In addition, a flow chart for computer implementation of the technique and a listing of the FORTRAN program are included in Appendix A. A FIRST ORDER SUCCESSIVE APPROXIMATION TECHNIQUE: THE GRADIENT METHOD A brief description leading to the gradient algorithm is given here. For a detailed description of the gradient (first order) algorithm, the successive sweep (second order) algorithm, and other related second order algorithms, the interested reader can refer to Dyer and McReynOlds (1970), and Jacobson and Mayne (1970). System dynamics: The dynamics of the system are expressed in terms of a set of discrete equations. The process is assumed to have N stages and the state of the system through these stages is governed by a dif- ference equation of the form: x(i + 1) = F(x(i),u(i),01), i = 0,1, , N - 1 (1) where, T F: (F (F p 000 I F) 1 2 n is an n - dimensional vector Of functions, that are in general non-linear. "x (if 1 x (i) 2 x(i) = ° x (i) n 1. .J is an n-dimensional column vector of the state variables. 62 63 31(1)) u (i) 2 u(i) = ' Emu) d is an m - dimensional column vector Of control variables which normally vary from stage to stage. is a p - dimensional vector of control parameters that are constant and thus do not vary from stage to stage. The performance index considered is of the form: N-l J = 00:00.0) + 2 L1x1i).u(i).a) (2) i=0 where L and 0 are scalar, single-valued functions of their respective arguments, and 0 represents the performance attached to the final state of the system. The second term represents the summation (discrete-time counterpart Of integration) Of performance over the stages. Further, constraints can be imposed on the system. The most gen- eral form of constraint is given by: N-l o = 10:08.0) + Z M(x 0, compute the new control from uj+1(k) = uj(k) + €[3VI3uj(k)]T. ’+ 5. At the initial stage compute the new control parameter a] 1 .+ . - _ from a] 1 = a] + evaT(0), where 8 > O. '+ '+1 6. Use the new control variables uJ l[0,N - l], 013 I and the system equations to construct a new trajectory xj+l[0,N] and compute the new cost Jj+l. 7. If Jj+1 > Jj, continue with steps 2 and 3. If Jj+l < Jj, reduce the step size parameters E, E; for example, set E = 8/2, E = E/Z, and repeat steps 4, 5 and 6 etc. 8. The iterations are continued until either no further increase in the cost is possible, or until the desired accuracy is attained. Side constraints can be handled by appropriate modifications to this basic algorithm (Dyer and McReynOlds 1970). Another alternative to a certain class of problems with side constraints, is the well-known pen- alty function technique. However, this technique can lead to very slow convergence in certain cases. RESULTS AND DISCUSSIONS As discussed in earlier sections, the major Objective of this research is the determination of optimal control strategies for the integrated control of the CLB. In the systems terminology, the afore- mentioned Objective can be transformed into an Optimal control problem --derive the Optimal timing and quantity of pesticide spray, given the state-space model Of the CLB ecosystem, a desired performance measure to be Optimized, and other constraints imposed on the problem to reflect the real-world situation. We will compare the optimal decision rules determined through the use Of Optimal control techniques with conventional spraying schemes currently used in practice, and with the strategy of spraying no pesti- cide at all (which may be a viable option under certain circumstances). All three of the aforementioned strategies will be evaluated within a certain framework of the CLB ecosystem model under identical initial conditions. Reasonable starting densities for the CLB and T, jgli§_have to be chosen for the problem as the Optimal policies will be strongly de- pendent on the initial conditions for the biological variables. Figure 7 describes the relationship between the spring adult CLB and the yield from oats plant under no-spray conditions. This relationship is Ob- tained through computer runs Of the CLB ecosystem model under no-spray conditions. It can be Observed from the figure that spring—adult-CLB 71 DRTS YIELD - BUSHELS / RCRE I) FIGURE 7. 0.001 0:01 0:1 110 1610 160.0 SPRING RDULT CLB / 30 FT Relationship between spring-adult CLB density and yield from oats plant under no-spray conditions (plotted on a semi-log- arithmic scale). 72 73 densities up to 0.1/sq ft make little impact on the ultimate yield from the oats plant which remains around 60 bushels/acre. At a CLB-spring- adult density of 1.0/sq ft the yield is fairly low--down to about 36 bushels/acre, and at a CLB density of 2.0/sq ft the yield is very low --about 14.7 bushels/acre. At spring adult densities of 3.0/sq ft and above the CLB almost completely destroys the oats crop. It is Obvious that CLB densities below 0.1/sq ft require no chemical control at all. On the other hand, CLB spring adult densities of, say, 1.0/sq ft and above warrant chemical control measures so as to prevent economic dam- age to the crop. Based on the above information, and in order to i1- lustrate vividly the differences between the optimal control strategies and conventional spraying schemes currently in use, the CLB starting density of 2.0/sq ft is chosen for our example, together with an initial density of 0.001/sq ft for the T, jgli§, The conventional control scheme currently in use is to spray 1 1b/ acre of the pesticide malathion when there are more than three eggs and larvae per stem of the oats plant (Michigan State University Cooperative Extension Service Bulletin E-829 by Dr. Ruppel, February 1977; Dr. Rup- pel personal communication 1978). This criterion for conventional spraying was incorporated into the CLB ecosystem model and it was found that the spray of l lb/acre occurs at the third time step of the9model (i.e., 180 DD48) and results in 23.85 bushels/acre of oats yield along with a corresponding profit of $26.20/acre. In contrast, the no-spray scheme leads to 14.70 bushels/acre of yield from the oats plant and a profit Of $16.85/acre (Table 3). 74 TABLE 3. Comparison of the Optimal control policy with conventional spray and no-spray schemes for a single season problem. 2.000/sq ft Initial densities: CLB TJ = 0.001/sq ft Price Of oats = $1.35/bushel Cost of malathion = $3.00/lb Cost of pesticide application = $3.00/acre TYPE OF TIME OF AMOUNT OATS PROFIT OVER DECISION SPRAY IN 01“ SPRAY YIELD #/acre WINTERING RULE 60DD48 UNITS lb/acre Ibu/acre TJ/sq ft no-spray - 0.00 14.70 16.85 0.040 °°nventl°nal 3 1.00 23.85 26.20 0.032 spray °Pt1m°1 6 0.88 43.56 53.11 0.031 spray 75 We will now focus on the determination Of the optimal control strategy using Optimization techniques. Optimization problems are characterized by the performance measure utilized in the formulation problem. Within the framework of pest management, the most important Optimization problem that will be considered is the so-called profit maximization problem: Max[revenue - control cost]. This is probably the most realistic type of problem with regard to the prevailing real world situation: the individual farmer's choice of pest control schemes is generally motivated by the profit maximization criterion, and, as of the present, the farmers are not required to bear the externality costs associated with the pesticide usage. In view of the aforementioned reason, the profit maximization problem is chosen as a typical example for the comparison of Optimal strategies with con- trol strategies currently in use. The key state variable that is directly related to the yield from the oats plant is the weight of the grain head at harvest time (X32(N) in the system model, where N is the final time). The control costs consist of the cost of pesticides and the application costs. Further, the performance measure utilized in the profit maximization problem must reflect the integrated approach to pest management. In other words, the performance measure should be aimed at reducing pesticide use while at the same time, enhancing biological control of the CLB popula- tion. With reference to the optimal control problem, the pesticide spray is the only control variable that can be directly manipulated for 76 timing and amount. The biological control manifested by the parasite population is only an indirect form of control. Further, there is a dynamic interaction between the CLB and T, iglithhroughout the season. Thus, the parasite populations are represented by state variables in- stead of control variables. Furthermore, for a single season Optimi- zation problem, the parasite, T, jglis, virtually plays no role, but its effect will be felt in the subsequent growing seasons. The key entomological variable that captures the essence of this situation is the overwintering T, jggi§_density at the end Of the season. This can be directly transformed into a constraint on the terminal state (of the appropriate state variable) in the Optimization problem. In light of the above-mentioned attributes of the pest management problem, the following performance measure (referred to as the minimum control effort problem in the control literature) is found to be the most appropriate for characterizing the economic optimization problem. n-l Maximize: J:g x2 (N)P Q + Z x2 (1)9 Q - 02(1)P R 32 1 i=1 32 1 2 subject to the terminal constraint on diapausing T, julis: x18(N) = TJLAST where: P = price of oats P = costs associated with chemical control Q,R = weighting factors N = terminal time 3 desired density of overwintering T, julis at the terminal time. 77 Obviously there exists trade-Offs between minimizing pesticide use and maximizing revenue with reference to the profit maximization problem. Thus, the weighting factors can be adjusted so as to modify the rela- tive emphasis on pesticide use and revenue. The terminal constraint for the overwintering T, jgli§_is set so that the density of diapausing T, julis will be the same as that Obtained using the conventional spray— ing scheme. The profit maximization problem discussed above is solved using the Optimization technique based on the successive approximation algorithm (refer to earlier discussions). The Optimal control strategy is found to be a single spray of 0.88 lb/acre timed at 360DD (i.e. the sixth 48 time step in the model) (Figure 8). Strictly speaking, the Optimal strategy consists of a spray of 0.88 lb/acre at the sixth time step, along with sprays Of 0.0001 lb/acre, or less, at several other time steps from 1 through 18. Since these sprays are totally insignificant when compared to the spray of 0.88 lb/acre, they are ignored. In this sense, the single spray of 0.88 lb/acre is suboptimal. However, the computer runs made with the Optimal and suboptimal strategies lead to essentially identical results with an accuracy Of 4 decimal places. In view of this situation, the suboptimal strategy is substituted in place of the optimal strategy. Table 3 illustrates a comparison Of the optimal control policy, with conventional spray and no-spray schemes for the single season of Opti- mization problems considered above. It can be Observed that the optimal strategy results in a 12% reduction in pesticide use--0.88 lb/acre as 78 U) ‘2- INITIAL DENSITIES " CLB = 2.000 / so FT TJ = 0.001 I so FT :3 c." ‘ CONVENTIDNFIL SPRRY PESTICIDE SPRRY - LBS / RCRE D T.) o T OPT IHRL SPRRY : ID I «2.. : 5’ 1 I I I I I o 1 “3 l c) r I T T m 0 90 180 270 360 460 540 DEGREE DRYS (888E 48) FIGURE 8. Comparison of the timing and amount of pesticide spray under Optimal and conventional spraying strategies. 79 compared to 1.00 lb/acre with the conventional spray. More importantly, the density of overwintering T, jETT§_is almost identical in both cases (0.031/sq ft in the optimal case as compared to 0.032/sq ft with the conventional spray) as required by the terminal constraint (on the overwintering T, ngTg) included in the optimal control problem. It is also worth noting that in comparison to the conventional strategy the Optimal strategy leads to significant gains in oats yield (43.56 bush- els/acre as compared to 23.85 bushels/acre with the standard policy) and almost doubles the profit ($53.11 as compared to $26.20 with the conventional spray). From a biological point of view, there is a marked difference be- tween the conventional spraying scheme and the optimal strategy--the conventional spray is carried out early in the season (180DD48) and is aimed at CLB spring adults and eggs; whereas the Optimal spray is timed later in the season (360DD48) and is aimed at early larval in- stars Of the CLB. It is also worth noting here that the CLB larvae are more susceptible to malathion, as compared to CLB adults. Figures 9-21 illustrate the evolution. of several important varia- bles of the CLB ecosystem when subjected to different control strategies, namely, the conventional strategy, the Optimal strategy, and the strategy of no spray at all. Figures 9 through 12 illustrate the state variables related to the oats plant, namely the weight of the plant, the leaf surface area, the weight Of the grain head, and the surface area of the heads. It can be easily Observed that, for the problem under consider- ation, the Optimal strategy is far superior to the conventional strategy, 80 a- r; " Dmmapsram a 4 .. ........ m 0 \ cl 0 z o 1 G a: ; . ...“ 500191101181. SPRAY ‘5 1 11.1 m C: DJ 0‘ U a: u_ 4 a: :3 ‘0 4 N 0.. C; u.) 1 .J o 1 v I r v I‘r v I f f I T v I T v 1 0 300 000 000 1200 1500 1000 DEGREE DRYS (BRSE 48) FIGURE 9. Leaf surface area of oats plant under optimal, conventional and no-spray schemes. . 0fI_Lt1fit. SPRAY ,,§MY.§NTIONRL SPRAY ’ D SPRn?‘~~.\ HEIGHT 0F PLRNT - 0H5 / SQ FT 0 v ' 330 T Y 800 ' ' 900 v 1200' v 1600' ' 1800 DEGREE DAYS (BRSE 48) FIGURE 10. Weight of oats plant under Optimal, conventional and no-spray schemes. Y SURFRCE RRER 0F HERD - 50 DH / SQ FT 81 0PTIHBL,§PRRY EDDNVENLIDNAL SPRAY 0 o I’ ,.v’14() fl fir fi r fifi ' V fi fl 7 350 600 900 1200' DEGREE DAYS (BASE 48) T ' I 1500 1800 FIGURE 11. Surface area Of grainhead of the oats plant under HEIGHT 0F HERD - OHS / SQ FT optimal, conventional and no-spray schemes. OPTIHRL SPRRY CONVENTIONBL SPRAY 0 I ' I O ' ’ ,I’ND A Y T f r V T y Y T V I Y— 300 800 ' 960 1200r DEGREE DAYS (BASE 48) I 1 1500 1000 FIGURE 12. Weight of grainhead of the oats plant under opti- mal, conventional and no-spray schemes. CLB - SPRING RDULTS / SQ FT 1.8 1 1 1 1.2 0-8 1 l 0.4 0-0 82 - N0 SPRRY - CONVENTIDNRL SPRRY I l | I I 1 | \ ‘3- OPTIHRL SPRRY | | I 1 | | I I I | ........ - u ............... 1er 300 480 600 750 900 DEGREE DRYS (885E 48) FIGURE 13. CLB-spring adult density under optimal, CLB - SUHHER RDULTS / SQ FT conventional and no-spray schemes. N0 SPRRY CONVENTIDNRL SPRHY ’———'--------- a a " Y 300 I r 800 960 1200' DEGREE DAYS (BASE 48) f i V v f I I ' I 1500 1800 FIGURE 14. CLB-summer adult density under optimal, conventional and no-spray schemes. 83 O 3 O :57 E ,r-- N0 SPRAY __9. ' ‘-- CDNVENIIDNAL SPRAY 0 IL 5; .41 .-- DPIInAL SPRAY ~\ 55 E 2 «25‘ 5 P : saw EES 03°.“ :3 . D 0 P 8 O 0" F. D o‘ C O D 5 '* ' I I ' I . 5’ r ' ' r’ r’ I I 0 300 600 900 1200 1500 1000 DEGREE DRYS (888E 48) FIGURE 15. Density of adult T, julis under optimal, conventional and no-spray schemes. 0.05 0.04 L CQNMENLLQNRL SPRAY 3"DPIInAL SPRAY 0.03 0.02 1 L l T.JULIS IN DIRPRUSE / 50 FT 0.01 T ' fi T v r v v r r 0 300 8307 ' 800 1200' DEGREE DAYS (BASE 48) 0.00 I . ' I 1500 1800 FIGURE 16. Density of diapausing T, julis under optimal, conventional and no-spray schemes. CLB - EGGS / 50 FT 84 - N0 SPRHY - CONVENTIONRL SPRRY \ \ \ 1 \ OPTIHRL SPRRY ~~ ' O ~ 0 o -‘----- ....... - a ---------..o r Y 150 4550 450 500 750 800 DEGREE DAYS (BASE 48) FIGURE 17. CLB-egg density under optimal, conventional CLB - lST INSTRR / SQ FT 1 32 L 24 I, 16 1 L and no-spray schemes. s '0 . ‘s 7‘. ... -- ‘ o . -------.-.-. . 200 ' v 400 800 800 1000 1200 DEGREE DAYS (BASE 48) FIGURE 18. CLB-first instar density under optimal, conventional and no-spray schemes. 85 CONVENTIDNRL SPRRY CLB - 2ND INSTRR / 50 FT r V V Y 0 ‘ 200 400 v 800 800 ' 1000 1200 DEGREE DAYS (BASE 48) FIGURE 19. CLB-second instar density under Optimal, conventional and nO-spray schemes. CLB - 3RD INSTRR / SQ FT ----- . O o -----.. o 200 ' 400 ' 800 ' . 800 1000 1200 DEGREE DRYS (BRSE 48) FIGURE 20. CLB-third instar density under optimal, conventional and no-spray schemes. 86 .0 1 0.8 1 80 DH / 50 FT 0:6 1 l c) 't- 2: C) D E - (:DNVENIIDNAL SPRAY 1:. :3 3‘ DPIInAL SPRAY 0 . N0 SPRAY 0. ~~:1.-"-'.‘.-. c, T I’ T I I ”T I I I . I 0 300 800 900 1200 1500 1800 DEGREE DRYS (BRSE 48) FIGURE 21. CLB feeding under Optimal, conventional and no-spray schemes. 87' which as can be expected, is better than no control at all. It can be seen from Figure 13 that the conventional strategy is more effective in controlling the CLB spring adults as compared to the Optimal strate- gy; whereas the Optimal strategy is more effective against CLB summer adults (Figure 14). The effect of these different control strategies on the parasite population is depicted in Figures 15 and 16. It is interesting to note the density of T, jETT§_in diapause at the end of the season is approximately the same for both the Optimal and con- ventional strategies as necessitated by the terminal constraint imposed on the overwintering T, ngTg, The small difference between the two can be attributed to the fact that constraints are only approximately (and not exactly) satisfied in the computational implementation of the opti- mization algorithm. The major difference between the optimal and con- ventional strategies is highlighted in Figures 17 and 18, which illus- trate the CLB egg and first instar densities under different control strategies; the conventional strategy is aimed at the CLB spring adults (Figure 13) and CLB eggs, while the Optimal control strategy is directed toward the early larval instars of the CLB. The impact of the three different control strategies on the CLB second and third larval instars are portrayed in Figure 19 and 20. Figure 21 illustrates the CLB feed- ing on the oats plant for the three different control schemes, and clearly brings forth the effectiveness Of the optimal spray in reducing the CLB feeding. Optimal strategies may not always lead to such spectacular gains over conventional policies; however, Optimal strategies will always be 88 as good as, and often times better than, the conventional schemes. This is due to the fact that the optimization algorithm routinely searches numerous policy options, with reference to the timing and amount of pesticide spray, and chooses the one that Optimizes the de- sired performance measure specified for the problem. In the event the conventional spraying scheme happens to be the optimal, the optimiza- tion scheme will automatically choose it. SENSITIVITY ANALYSIS In order that we may fully appreciate the effect of the timing and amount of pesticide spray on the economic yield from the crop and sev- eral other important variables characterizing the CLB ecosystem, a sensitivity analysis is carried out with reference to the timing and amount of pesticide spray. Thus, several computer runs of the CLB eoc- system model (initial conditions remaining as before at CLB = 2.000/sq ft and TJ = 0.001/sq ft) are carried out with the timing of pesticide spray kept fixed at 180DD48, while the amount of spray is varied from 0.1 to 2.0 lbs/acre. This process is repeated for several other spray times-- 240DD 300DD 8' 360DD 48’ 4 8' and 420DD The results are illustrated in 4 48' Figures 22 through 27. It can be seen from Figure 22 that the conventional spray timed at 180DD48 leads to very poor yield from the oats plant; whereas the opti- mal spray timed at 360DD results in the maximum yield. A similar 48 situation exists in the case Of profit (Figure 23). It can also be seen from Figures 22 and 23 that a pesticide spray of slightly over 1 lb/acre 89 ’ ‘ 1 DRTS YIELD - BUSHELS / RCRE 8.1 4 ‘ SPRAY IInE IDD DD 0... O EDD DD ... . :00 W 4 O m N X OED DD 0 ' I T I ' I r I ' I . I 0.0 0.4 0.8 1.2 1.0 2.0 2.4 PESTICIDE SPRRY - LBS / RCRE FIGURE 22. Sensitivity of oats yield (bushels/acre) with reference to changes in the timing and quantity of pesticide spray. Du ° 11¢ AAAA - - - : - 1 4 “'-' a.) ,. ' / u) m: ‘ / U I C 0# \ n o» i l I" .4 - § u. o J SPRHY TIHE m: _ m. ‘* a (nun NJ 0 EDD N d A) anon . o anon 3 02° 00 a ‘r I T’ I r* T v I f, r v I 0.0 0-4 0.8 1-2 1-8 2-0 2-4 PESTICIDE SPRHY - L85 / RCRE FIGURE 23. Sensitivity of profit (dollars/acre) with refer- ence to changes in the timing and quantity of pesticide spray. 90 SPRRY TIME 100” 24000 MOD 30000 42000 3 l 04.93 fl NEXT SERSDN’S SPRING RDULT CLB / 50 FT '*5 I I ' TI ' I ' I I I 0.0 0.4 0.8 1.2 1.8 2.0 2.4 PESTICIDE SPRRY - LBS / RCRE FIGURE 24. Sensitivity of the density of spring adult CLB (of the next season) with reference to changes in the timing.and quantity of pesti- cide spray. O S SPRRY TIHE P— 45- “ ‘3‘ u. a 1mm» :3 . O 2min (02 A SMDD \9.) o mmuooo O (n H .4 4 :3 '1 2 0° F- :57 .— 5] 4 a: (I 5* a) \ g; . m. 88-. (0° :2 . I“ 8 =5 c: ‘5 . T r47 r 0.0 0.4 0.8 1-2 1.8 2.0 2.4 PESTICIDE SPRRY - L85 / RCRE FIGURE 25. Sensitivity of the density of adult T, julis (of the next season) with reference to changes in the timing and quantity of pesticide spray. 91 o SPRRY TIHE ‘91 7 I “000 . 0 (“MD A 8”!” E34 0 0'300 I: " SfR. a «non c3 . a) \'8- m d CD CD 4 u) an a: .4 :- L) J 1 a: .— C31: h-g- '” ' I ' I ' I ' I ' I . ’I “5.0 0.4 0.8 1.2 1.8 2.0 2.4 PESTICIDE SPRRY - LBS / RCRE FIGURE 26. Sensitivity of the CLB egg density with reference to changes in the timing and quantity of pesticide spray. 8 SPRRY TIME 7. I 100!” 4 O 2«I00 O SGDDD *- LL 5:- O SIUDD C3 8 4&000 “3 - ‘x :3“ p. 03 4 2: H 23‘ - a) 41 C: .— CD Cid ...N 4 a fi I T r V 1 V l V ‘l V " 0.0 0.4 0.8 1.2 1.8 2.0 2.4 PESTICIDE SPRRY - L85 / RCRE FIGURE 27. Sensitivity Of the total CLB third instar density with reference to changes in the timing and quantity of pesticide spray 92 is the probable upper limit on the amount of a single spray--in other words, at any one particular time period, any amount of spray over and above this upper limit Of l lb/acre will not improve the yield, but will only lead to a reduction in profits due to unnecessary expenses incurred on pesticide-related costs. It is also interesting to note that the amount of spray currently recommended, namely 1 lb/acre, also happens to be the upper limit on the amount of spray. Figures 24 and 25 illustrate the effect of timing and the amount of the pesticide spray on the spring adult CLB and the adult T, igTTg of the subsequent season (i.e., the impact of spraying in the current season on the starting densities Of CLB and TJ for the next season). With regard to the timing Of the spray, early applications of pesticide are less effective in reducing the CLB population. However, early applications of pesticide are more beneficial to T, jETT§_as compared to the sprays at a later point in the season. The Optimal policy chose a spraying amount of 0.88 lb/acre (in con- trast to 1.0 lb/acre of the conventional scheme) such that the densities of overwintering T, ngTg at the end of the season are the same under the conventional and the Optimal schemes, as required by the constraints we imposed on the Optimization problem. Thus, the optimization algorithm simultaneously chooses the timing and the amount of spray so that the de- sired performance measure is optimized. Figures 26 and 27 illustrate the results of the sensitivity analysis in the total CLB eggs and the total third larval instar for the entire season. It is obvious that the conventional spray is most effective against the CLB eggs while the optimal spray is most effective against 93 the CLB larvae. Incidentally, this highlights the biological implica- tions Of the control schemes-~conventional policy being aimed at CLB spring adults and eggs, while the optimal spray is aimed at the early larval instars of the CLB as described earlier. Sensitivity analysis with reference to the timing and the amount of pesticide spray can be very useful in analyzing various control op- tions and quickly narrowing the Options to a few good (not necessarily Optimal) strategies. Optimization techniques will still be needed to determine the best control strategy. Further, sensitivity analyses with reference to timing and amount of pesticide spray will be more cumbersome when there is a need to spray several times during the growing season as Opposed to the single spray strategy that we have considered for the CLB ecosystem. ANALYSIS OF CONTROL STRATEGIES FOR THE MULTISEASON PROBLEM As discussed earlier, the single season optimization problem could not vividly illustrate the beneficial effect of the parasite population, because for any given season the impact Of T, ngT§_cannot be perceived during the current growing season, but the beneficial effect of the parasite manifests itself in subsequent growing seasons. In the single season optimization problem, this situation is implicitly taken into account by imposing a terminal constraint on the density of the diapausing T§ ingglat the end of the season. Nevertheless, in order to fully cap- ture the beneficial impact Of T, igTTg on the CLB ecosystem, it is necessary to consider multiple-season problems. 94 The multiple season Optimization problem is solved as a series of single season optimization problems. In this sense, the Optimal policy will be only suboptimal over the time horizon comprising the multiple season as a whole. Such multiple season problems extendingwover a four year period are solved for several different combinations of starting densities for the spring adult CLB and the adult T, ngTg: 1. Initial densities: CLB = 2.000/sq ft TJ = 0.001/sq ft 2. Initial densities: CLB = 2.000/sq ft TJ = 0.100/sq ft 3. Initial densities: CLB = 1.000/sq ft TJ 0.001/sq ft Furthermore, these initial densities are chosen in such a manner that comparison of cases 1 and 2 will illustrate the effect of a change in the initial density of T, jETT§_(for the same initial density Of CLB) on the evolution of the CLB ecosystem. Likewise, the comparison of cases 1 and 3 will exemplify the impact on the CLB ecosystem due to a change in the starting density of CLB spring adults. In the Optimization problem used with the multiseason analysis, the terminal constraint imposed on the overwintering T, jETT§_is set in such a manner that the overwintering T, ngT§_density under the optimal policy will be about 80% of that Obtained under the no-spray scheme. Briefly, this implies a further increase of T, jETT§_0verwintering density as compared to the Optimization problem considered before in which the con— straint level of overwintering T, ngT§_is set to be the same as that obtained with the conventional spraying scheme. Such an increase in the 95 level Of the terminal constraint on overwintering T, jETT§_is incorpor- ated in the optimization problem in order to fully capture the beneficial effect of the parasite pOpulation. The results of the multiseason analysis of the repeated application of conventional and optimal policies over a four year period are given in Tables 4, 5 and 6 and are graphically illustrated in Figures 28 through 32. Figures 28 through 32 clearly illustrate the enormous advantages associated with the Optimal strategy--of great significance is the amount Of pesticide used, which is much less with the Optimal scheme as com- pared tO the conventional practice, for §TT_of the three cases considered (see Figure 28). Furthermore, in comparison to the conventional policy, the optimal strategy always leads to higher yield from the oats plant (see Figure 29), correspondingly higher profits (Figure 30) and is su- perior in terms of controlling the CLB (Figure 31). In addition, the Optimal strategy is more conducive to the parasite T, jETT§_as compared to the control policy currently in use (Figure 32). The beneficial effect of T, ngTgDin controlling the CLB population and the manner in which the optimal strategy exploits this beneficial aspect to reduce the use of pesticides are clearly brought forth in the multiseason analysis. Thus, it can be seen that, for the same density (2.0/sq ft) Of spring adult CLB, case 2 with a higher (as compared to case 1) T, ngTg population not only uses much less pesticide (Figure 28 a and b) but also results in higher yield (Figures 29 a and b) and correspondingly higher profit (Figures 30 a and b). This beneficial effect Of T, julis can be observed in both the conventionaland optimal 96 TABLE 4. Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 2.000/ sq ft and TJ = 0.001/sq ft. Table 4a. Repeated application of conventional policy over a 4 year period. PESTICIDE TIME OF OATS SPRING YEAR SPRAY S PRAY IN YIELD $3223: ADULT T £DULTf t lb/acre 60DD UNITS bu/acre CLB/sq ft sq l 1.0 3 23.85 26.20 2.00 0.001 2 1.0 3 11.45 9.45 2.83 0.016 3 1.0 3 3.24 - 3.42 0.224 4 1.0 3 0.55 - 3.62 1.708 Table 4b. Repeated application of optimal policy over a 4 year period. YEAR “SEES“ 5123312 51?; 335;: Sin??? nggmft lb/acre 60DD UNITS bu/acre CLB/sq ft q l 0.96 5 42.58 53.08 2.00 0.001 2 0.90 5 39.39 48.92 2.18 0.029 3 0.95 5 37.41 46.20 2.46 0.902 4 0.92 5 39.08 48.49 2.25 5.288 97 TABLE 5. Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 2.0/sq ft and TJ = 0.1/sq ft. Table 5a. Repeated application of conventional policy over a 4 year period. PESTICIDE TIME OF OATS SPRING YEAR SPRAY SPRAY IN YIELD 532::2 ADULT nggLTft lb/acre 60DD UNITS bu/acre CLB/sq ft q l 1.0 3 23.85 26.20 2.00 0.10 2 1.0 3 13.15 11.76 2.71 0.96 3 1.0 3 10.79 8.56 2.87 4.11 4 1.0 3 18.86 19.47 2.32 10.88 Table 5b. Repeated application of optimal policy over a 4 year period. PESTICIDE TIME OF OATS SPRING YEAR SPRAY SPRAY IN YIELD 232::2 ADULT nggant lb/acre 60DD UNITS bu/acre CLB/sq ft q 1 0.96 5 42.58 53.08 2.00 0.10 2 0.91 5 40.97 51.05 2.04 1.56 3 0.91 5 44.60 55.85 1.80 7.15 4 0.90 5 51.36 65.08 1.08 11.78 98 TABLE 6. Comparison of optimal and conventional spraying schemes for a multiseason problem with initial densities of CLB = 1.000/ sq ft and TJ = 0.001/sq ft. Table 6a. Repeated application of conventional policy over a 4 year period. PESTICIDE TIME OF OATS SPRING YEAR SPRAY SPRAY IN YIELD 272::2 ADULT TJ?:UL:t lb/acre 60DD UNITS bu/acre CLB/sq ft q 1 1.0 4 48.79 59.86 1.00 0.001 2 1.0 4 45.97 56.06 1.19 0.011 3 1.0 4 43.88 53.24 1.37 0.127 4 1.0 4 43.27 52.42 1.43 0.818 Table 6b. Repeated application of optimal policy over a 4 year period. PESTICIDE TIME OF OATS SPRING YEAR SPRAY S PRAY IN YIELD 232::1' ADULT T JI/XISDUL': t lb/acre 60DD UNITS bu/acre e CLB/sq ft q l 0.88 5 51.94 65.86 1.00 0.001 2 0.85 5 48.42 61.20 1.25 0.028 3 0.91 5 46.83 58.89 1.57 0.822 4 0.86 5 46.70 58.84 1.43 3.664 9S) '1 E‘r H E) CONVENTIDNRL SPRRY DPT I HRL SPRRY PESTICIDE SPRRY - LBS / RCRE 0 80 3. INITIAL DENSITIES ° CLB = 2.000 / so FT . TJ = 0.001 / 50 FT a. s ' T I r I ”00 1.0 2.0 3.0 41.0 5T0 0.0 YEAR ID 9. w 4 a: O 8 3* x————E———E——-£ coNvENTIDNAL SPRAY \ w 1 (n \D _J a) I 0" >' 4 C m o :5 §« 0PTInAL SPRAY 1..) D 1 U D :2 1:. INITIAL DENSITIES {3° CLB=2.D/SD FT m . TJ = 0.1 / 50 FT 13' 8 I I °0.o 17.0 210 31.0 4.0 5.0 0.0 YEAR 3 W O a: o g .1: o———-o———o—o CDNYENTIDNAL SPRAY \ J m -J 3 .3 INITIAL DENSITIES ' ° CLB = 1.000 / SD FT E 1 TJ = 0.001 / SD FT C m S U D O-0 4 U : 8 DPTIHAL SPRAY 0) .5," U D. C. 8 °0.0 11.0 210 3:0 410 51.0 01.0 YEAR FIGURE 28. Quantity of pesticide sprayed for the multi- season problem under different initial den- sities for the CLB and TJ. 100 31 J :2 ‘3‘ \v 0PTINAL SPRAY U a: J \ 83‘ . . INITIAL 0ENSITIES c: CLB = 2.000 / so FT :19; T0 = 0.001 / so FT : m 4 p— C O 2.. J a. a T T r 0.0 1.0 2.0 3.0 4.0 8.0 8.0 YEAR 81 0PTINAL SPRAY Lu 3‘ K U C 1 \ 3 .. INITIAL OENSlTIES m” CL8=2.0/SO FT ' TJ = 001 I so FT D ..J tiz‘ ; CONVENTIoNAL SPRAY E D 5:4 J I). o T T T T T 1 0-0 1.0 2.0 3.0 0.0 6.0 6.0 YEAR 81 g e- % 0PTINAL SPRAY 3 . CONVENTIONAL SPRAY \ E 3‘ I D .1 213‘ )— ‘n 4 .— 8 N- INITIAL 0ENSITIES ‘ CLB = I.000 / so FT . TJ = 0.001 / so FT c. °o.0 tI-O 2:0 310 01.0 510 010 YEAR FIGURE 29. Yield from the oats plant for the multi-season problem under different initial densities for the CLB and TJ. l()l 8W :- G*\“\‘15~..“19’,,,.4D OPTINAL SPRAY U 4 G U 1“ ‘9'! “ . INITIAL OENSITIES ' CLB = 2.000 / so FT : .. TJ = 0.001 / so FT L“ O K . O. 24 a. NVENT 0NAL SPRAY 1 I °oo Io as so am so so YEAR 2 1 OPTInAL SPRAY :4 w J 5 G N. INITIAL OENSITIES ; ' CLB : 2.0 / so FT .4 TJ=0.1/SOFT ::J is m * CONVENTIONAL SPRAY b. V T T I °00 L0 20 30 do so so YEAR °T ::::::::::::::::::::: OPTIHRL SPRAY $4 CONVENTIONAL SPRAY g . U ¢ Nd \v . { I h“ “N {L O x 4 Q. .. INITIAL OENSITIES ‘ CLO = 1.000 / so FT . TJ = 0.001 / so FT C. I I I °0o Io am so so so so YEAR FIGURE 30. Profit obtained for the multi-season problem under different initial densities for the CLB and TJ. 102 9. 4 CONVENTIONAL SPRAY N- I.“ U- n O {D 1 2.94 INITIAL OENSITIEs -a” CLB = 2.000 I so FT :3 . TJ = 0.001 I so FT .J D I. 8'9‘ 2 I OPTINAL SPRAY 3.. 0. ~- w N a. I9 _, T T v T I 0.0 I 0 2 0 3.0 6.0 8 0 0 0 YEAR 0 ,‘,'1 N- 3‘ “- N a CONVENTIONAL SPRAY 0') \ O a ,1" .J U 0— .1 D N D _:‘ Z OPTIIIAL SPRAY 2. I! u a.) so INITIA 0ENSITIES CLa = 2.0 I so FT . TJ = 0.I / so FT b. 9 ”On 110 2'0 :10 c'o 5'0 :0 YEAR 9 N'- .— O t :4 ----- OPTINAL SPRAY 0 ”’ ---- CONVENTIONAL SPRAY .22. ..J U :3 D 0 8‘5“ o 1 3. 3:3. INITIAL OENSITIES CLB = I.000 / so FT . TJ : 0.00I I so FT C. 9 ‘00 1‘0 2 o s'.o «to s'o To YEAR FIGURE 31. Density of spring adult CLB for the multi- season problem under different initial den- sities for the CLB and TJ. RDULT - TJ/ 50 FT 103 OPT IHRL SPRFIY TNITIFIL DENSITIES CLB = 2.000 I 50 FT TJ = 0.001 I 50 FT CONVENT IONRL SPRFIY a. I °IT.0 I .0 2.0 310 JO 5.0 s .o YER g- 2" OPTIHRL SPRRY b ‘ CONVENTIONAL SPRAY LL 0 U) a," \ _, 1 p— : “A D D . C .. INITIHL DENSITIES ” CLB = 2.0 / so FT TJ : 0.I / 50 FT b. I T I I I °o.o 1.0 2.0 3.0 0.0 5.0 0.0 YER a. d "1 z 4 OFTIHRL SFRHY a In no \ j . INTTIRL OENSITIES "' CLB : 1.000 / 50 FT 5". TJ = 0.001/ 50 FT 3 D C _‘1 CONVENTIONRL SPRRY C. J O U I I 1 0.0 I.o 2.0 3.0 4.0 s.o 0.0 YEAR . - FIGURE 32.’ Density of adult I: julis for the multi- season problem under different initial densities for the CLB and TJ. 104 schemes; however, it is more pronounced in the optimal case. An anal- ogous situation exists between case 1 and case 3, both of which have the same starting density for the 2, igli§_(0.001/sq ft) but different starting densities for the CLB (2.0/sq ft in case 1 as opposed to 1.0/ sq ft in case 3) (Figures 28-32). It can also be observed (Figures 29c and 30c) that for case 2 with a low CLB density (1.0/sq ft) the conventional spray leads to high crop yield and large profit only slightly less than the corresponding ones obtained through the optimal strategy. However, bear in mind that these spectacular gains with the conventional spraying scheme have been achieved at the expense of using a much larger quantity of pesticide (Figure 28 c) as compared to the optimal strategy. Further, with the conventional spray, 2, jgli§_density through the years is much lower than the optimal case (Figure 32c) even though the spring adult CLB density is just about the same in both cases (Figure 31c). It is remarkable that the optimal scheme is superior to the conven- tional scheme in all 3 cases, representing different combinations of CLB-TJ densities. The optimal policy leads to higher yields and greater profits, is more effective in suppressing CLB densities, and reinforces the increase of g, juli§_popu1ations. More importantly, all of these gains are obtained with a much smaller pesticide use when compared to the standard practice. ENVIRONMENTAL CONSIDERATIONS The pest management problems discussed thus far have focused on the profit maximization criterion without due consideration to the 105 externality costs associated with environmental pollution. More often than not, farmers are not held liable for most of the envirommental dam- age caused by pesticide use. In the case of agricultural pest manage- ment problems we are concerned with negative externalities (i.e., external diseconomies of production) that result in uncompensated costs to the society. In this sense, there exists a divergence between private pro- fits and social benefits. The major part of these externalities falls outside the scope of the market system and is not reflected in relative market prices (Kneese 1971, Kneese and Schultze 1975). Externality problems have two important characteristics: (1) there exists an element of interdependency—-interactions between the decisions of economic agents (e.g. the decisions of the individual farmer and those of neighboring farmers concerned with market prices for the crop, pesti- cide costs determined by the chemical companies, etc.) and (2) there exists no compensation; therefore, the one creating the externality costs (e.g. the farmer) is not legally or socially liable to pay for it. Another, but less important property of externalities, is emphasized by Mishan (1976) who points out that the environmental spill-over should be unintentional . 33:: C O °0 - I0 - 20 ' :5 '7 «3 . 50 V 00 TEX - 0 / L8 OF PESTICIDE USED FIGURE 35. Oats yield obtained by optimal and non-optimal users as a function of tax imposed. OPTIflRL """" NOT OPTIHRL PROFIT - 3 / HERE fl T ' T T 0 IO 20 30 '«S'sé'w TAx - o I La 0F PESTICIDE USEO FIGURE 36. Profit obtained by optimal and non-optimal users as a function of tax imposed. 111 user decline at a faster rate as the taxes are increased. This is be- cause in the non-optimal case the farmer uses more pesticide in compar- ison to the optimal user. It is obvious that a large tax is required to reduce pesticide use, but in the problem under consideration, we have only two levels of pesticide use for the optimal case--O.88 lb/acre and no spray. This can be changed by using taxation schemes where the rate of taxation increases drastically with the increased use of pesti- cides. However, it will be extremely difficult to justify any one particular basis used in structuring the taxes. EFFECTS OF CHANGE IN CROP PRICE ON PESTICIDE USE As a matter of general interest, the effect of varying the price of the crop on the use of pesticide is also viewed within the framework of the profit maximization problem (results are illustrated in Figure 37). As expected, the increased crop price provides an incentive to the farmer to increase pesticide usage even when it is not warranted. Although the effective profit increases due to the increased crop price (and the increased pesticide use) it can be seen that the oats yield eventually stabilizes. We would like to emphasize once again that there is no easy solu- tion to the externality problems-all of the approaches, direct regula- tion, taxes and subsidies, pricing, etc., are complex and are difficult to implement. More research.is definitely needed in this area. 1.5 PESTICIDE SPRHYED - LB/HCRE 112 v —x OHTS YIELD PESTICIDE SPRHY OHTS YIELD - BUSHELS/HCRE 9 =2 f i I I r I V I fi T V V T V j “’0 so 120 :00 240 300 360° PRICE OF ORTS - CENTS / BUSHEL FIGURE 37. Sensitivity of oats yield and amount of pesticide sprayed with reference to changes in the price of oats. SUMMARY AND CONCLUS IONS In this research work, we have developed a comprehensive model of the CLB ecosystem with all its major components--the CLB, its larval parasite T, julis, and the oats plant. Both chemical control and bio- logical control aspects are incorporated into the model so that it can be tested within the framework of integrated pest management. A first order, successive approximation algorithm is utilized to develop optimal control strategies for the integrated control of the CLB ecosystem. The optimal control strategies are characterized by emphasis on biological control and reduction in the use of chemical control. The optimal strategies are compared with the conventional spraying schemes currently utilized. Such analysis are carried out for both single sea- son and multiple season pest management problems. The Optimal policy leads to higher yields and greater profits, reinforces the increase of T. jgli§_populations, and is more effective in suppressing the CLB dam- age; more significantly, all of the aforementioned gains are obtained with a reduced pesticide use as compared to the conventional spraying practice. Optimal strategies may not always lead to spectacular gains over conventional policies; however, Optimal strategies will always be at least as good as, and Often times better than, the conventional schemes. This is because the optimization algorithm routinely searches numerous policy options with reference to the timing and amount of pesticide 113 114 spray, and chooses the one that maximizes yield and profit with minimal pesticide use. In the event the conventional spraying schemes happen to be Optimal, the optimization scheme will automatically choose it. A sensitivity analysis is carried out with reference to timing and amount of pesticide spray, and it is found that the optimal timing is at Odds with the timing of spray under the conventional schemes: the conventional strategy is timed earlier in the growing season and is aimed at the CLB spring adults and the CLB eggs. On the other hand, the optimal spray is timed later in the season and is targeted for the early larval instars of the CLB. Regarding the amount of spray, the optimal scheme results in an average reduction of about 10% in the use Of pesticides as compared to the conventional strategy. This may not seem impressive at first glance, but considering the fact that the acre- age of oats harvested in the United States is close to 13.5 million acres (Michigan Agricultural Statistics, June 1978) and assuming 40% of these are infested by CLB, a 10% reduction will result in a reduction of 500,000 lbs of pesticide use annually--for oats alone. This will be an enormous reduction in terms of pesticide use and associated environmental pollution. Unlike the CLB infestation which usually requires just a single spray, optimal schemes will lead to greater savings in pesticide use when used with pest management problems in which frequent sprayings are common--like the onion maggot problem currently under investigation at Michigan State University. Furthermore, the Optimal scheme achieves such a reduction in pesticide usage with minimal reduction in crop yield and profits. A simple analysis is carried out, based on direct regulation and taxation approaches to regulating the environmental pollution caused by the pesticide sprays. It is found that, for the problem under con- sideration, the pesticide use is not very sensitive to taxes--heavy taxation is required to reduce pesticide use. The direct regulatory approach, in which absolute limits are prescribed as the level of pesti- cide use, is also discussed. In general, it is extremely difficult to formulate an equitable policy based on either direct regulation or taxation. The analyses carried out in this research point to several areas where improvements can be made by conducting more field (or laboratOry) experiments. Among them, the dosage response characteristics are the most important. The amount of spray used is directly dependent on the dosage response characteristics. Currently available data on dosage response are incomplete, especially with reference to low level dosages. Further, no experimental data is available on the effect of pesticides on T, julis, Field and laboratory experiments are needed to acquire the necessary data in these areas. Further, field experiments have to be carried out to evaluate conventional and Optimal spraying strategies with reference to their effect on crOp yield, profit, CLB feeding, parasite densities, etc. In the present research work we have designed an Optimal controller for pest management problems. However, it is confined to a deterministic framework, but the real world pest management problem is actually sto- chastic--due to variations in climatic factors, sampling errors, etc. 116 Thus, the deterministic optimal controller will not be optimal in real world stochastic situations. Further, we would like to have on-line features incorporated into the optimal controller so that the pest man- agement strategies can be implemented on-line. We propose the Linear- Quadratic-Gaussian (L-Q-G) methodology (Athans 1971, 1974) for the design of such an on-line controller. A brief description of the L-Q-G methodology is presented in Appendix B. Essentially the L-Q-G design consists of two components, (1) a deterministic optimal controller, and (2) a stochastic estimator. The complete design of the deterministic optimal controller has been accomplished in this research. The design of the stochastic estimator must await another work. The essential features of the on-line controller for pest manage- ment are illustrated in the block diagram of Figure 38. The determinis- tic optimal controller utilizes the ecosystem model to compute (off- line) the determinisitc Optimal control strategy. On the other hand, the stochastic estimator (filter) combines model estimates and actual field data (containing errors) of biological and climatological variables of the ecosystem, to give a new improved estimate of the states of the ecosystem. The deviations between the actual states of the ecosystem from its ideal, deterministic response generated by the model are used to generate the on-line correction strategy. One of the most signifi- cant features of the L-Q-G design is that it fits the general guidelines established for the on-line pest management systems (Haynes and Tummala 1977). Furthermore, the methodology developed in this research can be easily extended to the L-Q—G design. 117 .unfi I 8 gmzH NZHAIZO £092ng .Eoum>mooo umom m «o Houucoo ocwaIco uOm conOO OIOIA mewumuumsaafl Emumoflo xOon 13409—28 ugmso UHBWHZHtmmEm—D :Fu’: Among! §8m>m8m atamuaixmfi_w:am omuummo act aaaeao momzmm gamma Aromaéam dog—.200 an: . 489200 mzHAIn—mo .usm. musesm omz=m,Monvrnzcso),sunvru(3a),sunvr~2(3C) [BLOCK] SL1(30),SL6(3'),F1(3?),DIAFFN(3?),DIFMFN(37) IaanxI oxrwruz(31),r109(35),r2(36),05L1(30>,05La(3¢) IFxl onsa,onsoz,r1oncr,r1oicp1,x1acr,x1°cp1 [INPUT] lrLaa, NITER, EFSILON, STEFHIN, SIEFnAx, o, 9 IIVFUT/ usrapr, TVALUE, TDLER, £1, TLAST, CLBIN warutl JULISIN, FINALTJ, NYEAR, TSPRAY, SPRAY C"”‘(N IINPUTI PRICE, TAX crmno~ [MODEL] xner(35),oxru,sunu,sumuso,rxruso,r1 (DEMON IWODELI IEGG, 7J2, ILAR3, TUNPAR‘, TPAF‘ (SV‘ON IIlan TR, ”3, NSTEF, NX, NU DATA I F(1,I), I31,‘ ) I 8.035127 E'OT , 5.17'637 E'31, '5.6‘“611 E'OZ , {.0 I DATA ( P(2,I), I31,‘ ) I'1.127719 5'01 , 1.2‘0371 E‘OO, -108.6|-73 E-02 ' 605‘852‘ [-32 I DATA ( F(3,I), 131,‘ ) I (.0 , 3.638312 5'02, 7.656(6‘ 5-01 , 5.155583 5’01 I DATA I P(b,1), 181,4 ) I C.O , 4.77990C 5'02, '1.726377 5'71 , 1.3'3389 E‘TC I DATA K1,K3,KA,K6,K7,KP I v.1, 0.3, 0.45, v.4, n.5, 0.5 I DATA A,B,C,D,F I 3.7, 22.3, 1.:, ‘.6, 1.‘ I DATA FEEDQ,QP,CP I 0.002329, 0.0, 0.75 I DATA TD,IDIA,IFEED,NTOATS I 13, 1, 1, 13 I DATA C1,C2,C3,CA I¢.68b8‘8E-“1, 1.829393E-?3, '3.1C8558E-.5, 6.634597E-58 I DATA D1,D?,D3,DA,D5 I 9.402374E'01, 1.6832525' 2, Z. 76TSFE'36, -2.31A922E-vo, 7.3266325-‘° I DATA nr1,or2,or3,or4,ors,ora,or7,ore,or9 I 4. o7i31E-o1, -3.:939ce+ro, 3.9525950'0. '1.15387E*OD, 1.66972E-t1, '1.23781E-?2, 6.86E'3E‘Jb, -9.7172“E-té, 7.?6777E-'8 I DATA E1,EZ,E3 I 2’.‘, 5.0, 1(0.0 I DATA DELTAU I 1.0 I 124 fifififififihflfififinnnnfiflnnfififinfinfififififinfififififififififi 125 ......it....fifiififitfififlitfiifi...tifififitfifitfitifiii.0.... O t * DICTICNARY OF STATE VARIABLES * o a ttifitfififitttitfitfitttfiititfiiiiifitfifitfi**ififlfiittit... x1 SPRING ADULT CLB DENSITY x2 x3 CLB EGG DENSITY “I x5 FIRST INSTAR CU‘ DENSITY xe SECLND INSTAR CLB DENSITY x7 THIRD INSTAN CLE DENSITY X? UNFARASITIIED TPURTH INSTAR CLB DENSITY xc x1 x11 x12 UNPARASITIZED CLP FUPA DENSITY x13 x11. X15 X16 SUFMER ADULT CLB DENSITY X17 DIAPAUSING TJ DENSITY 11H ADULT TJ DENSITY X1? FARASITIZED FOURTH INSTAR CLB DENSITY 12‘ X22 PARASITIZED CLB PUPA DENSITY X23 X24 AP. X25 X26 PARASITE PER FEST INDIVIDUAL IN DIFFERENT STAGES 7?? AI? ’29 CLB LARVAL EEEDING X32 HEIGHT Of OATS PLANT 731 LEAF SURFACE AREA 0F OATS PLANT 33 NEIGHT CF GRAIN HEAD X33 SURFACE AREA OF GRAIN FEAD nnannannnnnnnnnnnnnnnnnnnnnnnnn 126 titttttfititiifiiifiitfififififititfifiififiiitfifitititittittt. e 9 A SYSTEF PARAMETERS * A i iii......90‘!fittititfiiifiiififittfiitfiitfifitfi.....fifti. NArE DESCRIPTION VALUE A SPRING ADULT SURVIVAL .7c 8 CLB EGGS I CLE FEMALE I 6‘ on 22.;3 c SUMNER ADULT SURVIVAL 1.-o o T.JULIS ADULT SURVIVAL '.Ec E1 MAX EGGS I TJ ADULT I an on 2 .co E2 MAx TJ EGGS I CLN LARVA I 69 on 5.'c E3 TJ SEARChING CONSTANT T-n.cr as TJ MORTALITY INSIDE (LB ".60 (1 MORTALITY or CLE EGGS . n.1c x2 ~|ORTALITV or CLE LT VARIAFLE x3 NORTALITY or CLE L2 2.30 :4 MORTALITY or CLB L? ‘.55 K5 NORTALITY or CLE L4 VARIAELE K6 ”FRTALITY or CLE FUFAE '.LE (7 MCRTALITY nr OVERUINTERING CLe ‘.77 K8 MrRTALITY or OVFRHINTERING TJ ‘.5c CF EXPCNENT IN PARASITISH EoUATIoN ‘.75 FEEDD FEEDING FUNCTION CUEEEICIENT .noz*2° F NATnIx DIMENSIDNED (b,£) CONTAINING FARAFETERS FOR THE OATS PLANT MODEL OFTAINED THROUGH TIHE'SERIES ANALYSIS fihfififififififififi fif‘ \ ,"I h 127 P MATRIX VALUES: 8.C35127E'81 5.173637E-01 “5.650611E-’Z 5.C -1.127719E'f1 1.24‘871E’C0 '1.866D73E-OZ 6.588521F-TZ (.0 3.638312E-02 7.656066E-01 6.1A5583E'01 6.0 6.779CJOE-OZ “1.726377E-01 1.3(33896000 REHIND ALL r‘UTFUT FILES REHIND 3 S REHIND A S REUIND 5 S REUIND 6 1 REUIND 7 REHIND 77 S REUIND 78 DEFINE ALL VARIABLES IN IINFUTI CONNON BLOCK IN ORDER OF DECLARATION. PROHFT USER FOR EACH, THEN READ VARIABLE FREE FORMAT, ONE NUMBER PER LINE. SEE SURROUTINE INPUT. CALL INFUT ECF 8 E2**CP ONECP = 1.0 - CP NUMBER OF TIME STEPS FROM START TO HARVEST. NSTEP827 NN=NSTEPA1 N=NSTEP NUHBER OF STATE VARIABLES. NX'BS NU81 ‘999 DO LOOP FOR HULTIFLE YEAR RUNS *’** no 1 YEAR a 1, NYEAP YR 3 YEAR CLEAR ARRAYS DC 5 I=1,NN or S J21,NX LI(I,J)=D.‘ VX(I,J)83.5 NXII,J)8C.O HXII,J)IO.0 X (I,J)80.0 CONTINUE DC 6 I=1,NN IREFTI) 8 ’00C.O IF (I .LE. TLAST) XREF(I) 8 ET CONTINUE DC 7 181,N$TEP DC 7 J81,NU MU¢I,J)8?.“ LUII,J)=D.¢ DUDUII,J)=F.O DVDU(I,J)=3.€ U(I,J)=USTART CONTINUE *Efifi USTART IS ADJUSTED TO BE SAFE AS CONVENTIONAL SPRAY 99*. 128 C .... P(3,1) s 1.0 UCTSFRAY,1) s SPRAY DEFINE NON-ZERO INITIAL CONDITIONS (IF ANY ) FOR THE C STATE VARIABLES. XC6,19)8JULISIN X(1,1 )8CLBIN x<6,3‘) = 3.:92963 E-‘T x(6,31) = L.T<3903 60'; C INITIALIZATIDN FOR SURVATION VARIAPLES TEGG 8 1.05-7 TLAR3 8 :.C TEARS = :.C TUNFARL: ?.0 TJ? 8 0.0 SU’U 8 0.0 SU‘USG 8 0.0 C DEFINE TERMINAL VALUES FOR I (If ANY), VI, ux. x(NN,TE) = FINALTJ c aetttconttootttatttetttnetttotottttotttttttttttttteaDetectotcet C tttttttiti NAXIHIZATION PROBLEM titttfltttttttAiitttttfiittttti vxtNN,32) = 1.3 C **** DC LOOP FOR OPTIMIZATION STARTS HERE **** S 10 ITER=1,NITE9 C HRITE THE TERFINAL CONSTRAINT EQUATION H C **** CALL SUPRTUTINE NCDEL T0 COFFUTE STATE VARIABLES .888 CALL MODEL ADD ANY TERMINAL (NON-INTEGRAL) TERM T0 FI. CHECK TCR VERY TIRST ITERATIRN IT (ITER .ED. 1) GO TO 62 IF C ABSCPI/PILAST - 1.0) .LT. TULER ) GO TO 5‘ IT CFI .LE. PILAST) Go To 57 C Fl 15 LESS THAN RILAST EFSILfN = EPSILON¢STEPNAX CC T0 62 C PI 15 GREATER THAN PILAST 57 ECSILON : EFSILON*STEFNIN G: T0 177 62 PILAST . F! C CCRRUTE PARTIAL DERIVATIVES FOR THE RETURN TVNCTICN av C BACKHARD INTEGRATION. C FIRST EVALUATE Fx,Lx,Mx MATRICES. c NOTE: Tx AND TU ARE EVALUATED AS TUNCTInN SURERUGRARS. DC 16 K=1,NSTEF Lx (K,32) = 2.0 . YCK,32) c C D . PRICE I 1.35 ) IF ( ITLAG .NE. 1 ) GO TO 16 LU(x,1) : - 2.n . (RoTAx) . UCK,1) 16 CONTINUE Dr 17 K81,NSTTP vUCx,1) a 2.% . (RoTAx) . U(x,1) 17 CCVTINUE C DETIVE TERYINAL VALUES FOP x (IT ANY), vx,ux nn 129 C NEYT EVALUATE Rx,vx RATRICES. C CLEA 18 22 21 T) C TC F R FRHXFX,FRVXFX ARRAYS. DO 19 I81,NSTEP DO 19 J81,NX PRVXFX(I,J)8D.O PRHXFX(I,J)8O.O CONTINUE DO 20 K81,NSTEP N1K8 NAT-K N2K8 N’Z‘K DRSO 8 F10RCN1K)**(28CP) DRSOZ 8 F1DR(N1K)**(2.G*ONECF) FIDRCF 8 F1DRIN1K)** CF FTDRCEI 8 1.3IIF1DRCN1K) *8 (1.9-CP) ) XISCF 8 X(N1K,18) ** CP X18CP1 8 1.0 I (XCNTK,18) 8* (1.C-CP) I IF ( XCN1K,1P) .EO. 0.0 ) X18CF 8 1E'5 IF I KCN1K,1!) .EO. 0.0 ) X1RCF18 1E‘C DC 21 I81,NX DC 22 J81,NX FRX8FXCJ,I,N1K) PRVXFX(N1K,I)8PRVXFX(N1K,I)9VX(N2K,J)8FXX FRdXFX(N1K,I)=FRHXFX(N1K,I)*UX(N2K,J)8FXX VA(N1K,I)8PRVXFX(N1K,I)*LX(N1K,I) HX(N1K,I)8PRHXFX(N1K,I)§HX(N1K,I) CONTINUE IND THE NULTIPLIER V FOR CONSTRAINED PROBLE”S. C FIRST COHPUTE THE SUN TERPS. Z8 32 33 DC 28 I81,NSTEP D? 28 J81,NU FRVXFU(I,J)=C.D FRHXFUCI,J) 8C.O TERH1(I,J) 8 (.0 TERH2(1,J)8C.G CONTINUE SU*1 8 0.3 3”,? 3 50'. DC 30 K81,NSTEP KF1=KAI D? 30 I=1,NU Dr 32 J81,Nx EUU8 FUCJ,I,K) PRVXFUCK,I)8 FRVXFUCK,I)* VX(KP1,J) 8FUU FRHXFU(K,I)8 PRHXFUIK,I)* HRCKF1,J) *FUU TERHT(K,I)8FRHXFUCK,I)*NU(K,I) TERHZ(K,I)8PRVXFU(K,I)YLUCK,I) SUV1 8 SUHI A CTERHICK,I) *8 2 ) SUEZ 8 SUN? 0 CTERHTCK,I) A TERHZCK,I) ) CONTINUE DO A1 I81,NSTEP DO ‘1 J81,NU TERPUCI,J)= UCI,J) 130 41 CCNTINUE IF (IFLAG .E0. 1) GO TO 177 INTER! 8 1.5 I (EFSILON 8 SUVT) V 8 INTERN 8 (DIFUSO ' EESILON 8 SUMZ ) C COMEUTE GRADIENT HRT CONTROL C NOTE: DVDUII,J)‘TERHZ(I,J); DUDUCI,J)'TER'1CI,J) C UPDATE CONTROL 177 DELTAU 8 C.O DC ‘0 I31,N$TEP DC ‘0 J31,NU UCI,J) 8 TEMPUCI,J) 8 EPSILON8TERMZII,J) IF (IFLAG .EO. O) UCI,J) 8 UCI,J) 8 EPSILON8V8TERHTCI,J) UII,J) 8 ABS‘UCI,J)) DELTAU 8 DELTAU 8 ( UCI,J) ' TEHPUCI,J) ) LT CONTINUE PRINT ‘7,ITER,RI,EILAST,EPSILON,$UHU,SUPUSO,XCNN,1E) 47 FORMAT C"CITER'",IZ,5X,"PI3",1EE1C.3,5X,”FILAST8",1PETCo3, 85X,"EPSIL0N",TRET0.3,5X,"SUMU",1PETO.3,5X,"SUFUSO”,1PE10.3, 8ZX,”X(NN,13)3",2X,1EE10.3) 10 CONTINUE C COMPUTE INCOME FROM OATS YIELD C REF EMMETT 888 SEED UT 8 HEAD DRY UT 8 3.9119 - O.C03€35 SO SEEDHT : X(NN,32) 8 0.9119 - C.003035 C YIELD IN BUSHELSIACRE 8 SEED HT IN GMSISOFT 8 3.003992 YIELD 8 SEEDHT 8 3.COO99Z C INCOME SIACRE : YIELD IN BUIACRE 8 PRICE OF OATS IN S C CURRENT BUYING PRICE OF OATS IN MICHIGAN AS BOUGHT FROM FARMERS C $1.35 I BUSHEL REF: MASON ELEVATOR COMPANY INCOME 8 YIELD 8 PRICE COMPUTE PESTICIDE COSTS, INCLUDING MATERIAL AND APPLICATION COST COST OF MALATHION : S 3.00 I LB COST OF APPLICATION : S 3.00 I ACRE TAXES CAN ALSO BE IMPOSED ON PESTICIDE USE COST 8 SUMU 8 C PRICE 8 TAX ) 8 3.30 C NET PROFIT/ACRE : INCOME - COST PROFIT 8 INCOME . COST C COMPUTE PERCENT PARASITISM PARA 8 ( TFARA I C TPARA 8 TUNPARA ) ) 8 100.0 nnnn C INITIALIZE CLB AND T JULIS DENSITIES FOR NEXT YEAR c NCTE : OVERNINTERING MORTALITY FOR CLB IS SET AT 77 BASED ON C REFERENCES C YUN, 196A; VELLSO ET AL., 1970 ) C OVERHINTERING MORTALITY 0F T.JULIS IS SET AT SC CLBIN . n.23 . x(NN,16) JULISIN8 T.so . XCNN,17) C NRITE YEARLY OUTPUTS IN A SEPERATE TAPE A 77 NRITE ( 77,44A).YEAR, x<1,1), TEGG, x(6,18), TJ2,TLAR3, PARA AIS FORMAT»("-",T11,12,T19,TRE10.3,TIC,TRETC.3,TA9,1RE10.3, . TSI,1RE1G.3,T79,TRE10.3,T95,1RETO.3 ) C URITE INCOME-RELATED nUTRUTS ON A SEPARATE TAPE A 73 URITE C 7E,A99) YEAR, YIELD, INCOME, SUMU, COST, PROFIT 499 TORMAT C"-",T11,12,T2A,TRET0.3,TIA,TRETG.3,TGA,TRE10.3, o T87,1PEIO.3,T109,1PE10.3 ) C fiflfifififlfifififi fififinfifi ..I 131 Ottittt FRINT TABLES Arte... SUBROUTINE OUTPUT URITES UNHEADERED TABLES 0N OUTPUT TAPES AND ALSC PRINTS HEADERED TABLES CN OUTPUT. TABLE 1 IS THE CONTROL VARIABLES "U”, PRINTED YEARLY. THE UNHEADERED CONTROL VARIABLES ARE URITTEN T0 TAFE3. TABLE 3 IS FOUR TABLES CF STATE VARIABLES, PRINTED YEARLY: 1: X1-X1H 2: XTT'XZO 3: X21'X3? A: X31-X34 ON TAPE TAPEA TAPES TAPE6 TAPE7 CALL OUTPUT (1) CALL OUTPUT (2) CONTINUE TABLE I IS A DIRECT COPY OF TAPE77, HHICH IS HRITTEN ABOVE. TABLE 4 IS A DIRECT COPY OF TAPE78, HHICH IS HRITTEN ABOVE. BOTH CF THESE ARE URITTEN AT THE END OF RUN ONLY. NOTE THAT TAPE7? AND TAPE7R ARE THE UNHEADERED VERSIONS OF THE PRINTED TABLES 3 AND 4 RESPECTIVELY. CALL OUTPUT (3) CALL OUTPUT (6) END FlV.CTI(N * t t nvfirsfirfiflsn-srpa FUNCTION rx COMPUTES THE MATRIX (DF/DX) HITH RFFERENCE TO STATE VARIABLES X. 132 rx(J,1,r) fififfitittit...itOffifififiititfifitfiifitttitifitiiiifiifitttt . FUNCTI'N FX t t fifiiitfifitittfifiitfiifiitfifi.09.9..*fififitt.....tfififii0t... PF PARTIAL DERIVATIVES REAL K1,r3,K¢,x6,K7,rc,uoatrn,nontru2 cannon x(35,35),u(3$,2) CC‘HON CCMNON CDPHON CPMMON CC'HON CCHNON CCHHON CRHHON CGNHON [FX] FX=Z.G C FUNCTIONS ARE DEFINED EELUH GO A IF 1: IF IF IF If If If JJVCFU‘FKNRLa IF 0 IF 10 IF 11 IF 2 IF 13 IF 1‘ IF 15 IF 16 IF IF 47 IF IF IF [PLANT] P(k,&) [ENOUGH] C1,CZ,C3,C‘,D1,DZ,D3,D&,DS [ENOUGH] DF1,DFZ,DF3,DF4,DFS,DF6,DF7,DF?,DFC [ENOUGH] A,B,C,D,F,K1,K3,K6,K6,K7,K?,FEEDG [ENOUGH]RP,CF,ET,EZ,E3,ECP,ONECP,TD,IDIA,IFFED,NTOATS [BLOCK] HORTFN(3”),NORTFNZC30),SURVFN(3.),SURVFNZ(3 ) [BLOCK] SL1(3;),SL4(30),F1(30),DIAFFNCSG),DIFMFN(3C) [BLOCK] DIFMFN2(3?),F10R(36),F2(30),DSL1(3!),DSLLCB I DRSB,DRSOZ,F1DRCP,FTDRCPT,XTBCP,XTPCP1 FOR NCNZERO FUNCTIONS ONLY. TC (1,2,3,6,S,6,7,c,9,1 ,11,12,13,14,15,16,17,18,1°,2P, 21'22'23'2"25’26'27'2fi'29,}3,31,32,33'3‘) J (I.EG.1) (I.E0.1) (I.EG.2) (I.EO.3) (I.E0.4) (I.E0.5) (I.E0.6) (I.EG.7) PX: A*$URVFN2(K) Fr: 8 FX81.“ rx=1.c FX=(1.'-K1)*SURVFN(K) Fx= DSL1(K)*SURVFN(K) 7x: (1.9-K3)*SURVFN(K) PX: (1.(-Kb)*SURVFN(K)*(1.G - (((1.']ECP) SRETURN SRETURN SRETURN SRETURN SRETURN IRETURN SRETUPN tv1oacrtx1ecp - x18CPfix(K,7)*CP*F1DRCP*(1.LIE1)) I 0280)) ( I.Ea.18 ) Fx 8 - (1.0-Kb) * SURVFN(K) . (1.:IFCP) * (( F1DRCP 0 CF . x18CF1 * x(x,7) - x1tCP . x