....v.s .v......k .:...lfp.va. . r .L: :7. . .... . ..av‘...,. ... .V... .x. . ,I..h.,z . . ‘l. . V V V H _ _ _ V .a.,.,.,.._wu.w. . : , . .. ...VV ,. . . A . .. . . . ._. ... G, Chant...“ 7' V t v V. I. . . rin-c’u . .. H V . , ,. V _ .A gravy» 1.2,; V V . V V . V. _ V ,.,.,.,..,VV.:.. ”W . hm . . P. m R M E J. V n v G' arm-rouzvlunfll fl we E m n _ w Hm VS V D V 9 An E W . . f. nu. B . mm. MW. m R. .P .71 m 5...... Enos..:::2...~ 133.7 V. V 1 IKE}: .., Hutu" 433.3 rar... #62113 721.3}... 2:. 1.35272»:..»vrv.v..v.n....c.3'. Isa—I; A. Iv. . x 5.— .r..\..._,. V}... i. «Bag ryrrlnzfié :vanwrilr .. .9 Fr...” :13??? .1 _ «TIM. _... nun“. _. LIBRARY Michigan State University ABSTRACT STATE SPACE MODELS FOR ECOLOGICAL SYSTEMS By Robert H. Boling, Jr. To assist in the analysis, design, and management of ecological systems, this thesis develops basic pOpulation and interspecific coupling components needed to assemble a dynamic ecosystem model based on state space concepts of system theory. A generalized model for the pOpulation of a developmental stage in an organism's life cycle is developed as a component for a dynamic model of a population. The formulation of the develop- mental stage component combines biological functions with the structural requisites of state-space theory to produce a realistic quantitative model description. The state variables and behavioral features of the developmental stage component model are formulated for living organisms in general, and are applicable to almost all species papulations. The developmental stage population components are coupled to assemble a multi-species segment of an ecosystem. The inter- acting consumptive activities of competing species are expressed by an abstract consumption component which is modeled to optimize the feeding behaviors of the various competitors. The behavior of the consumption component is formulated as the solution to a Robert H. Boling, Jr. mathematical programming prdblem.with linear constraints and a quadratic objective function. Many of the parameters needed to describe the behavior of the consumption component are best treated as random variables, and a generalized approach is developed to solve mathematical programming problems involving random.variab1es. Biological research results from the literature are used to determine quantitative formulations of the physiological func- tions needed to describe the system behavior of the developmental stage component. The behavioral parameters for the consumption component are described as functions of characteristic parameters of the consumer and consumable pOpulations. STATE SPACE MODELS FOR ECOLOGICAL SYSTEMS By ‘11 ‘2‘“; Robert HgEBoling, Jr. A THESIS 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 1971 To Smokey ii ACKNOWLEDGMENTS I would like to acknowledge the guidance of Dr. William E. Kilmer during the early stages of this thesis and I am grateful to Dr. Robert O. Barr for his assistance and support during the preparation of this thesis. I would like to thank Dr. Herman E. Koenig for introducing me to the study of ecological systems and for his encouragement on this project. My thanks also to Dr. Donald J. Hall, Dr. Richard C. Dubes, and Dr. Carl V. Page for their interest and advice. iii TABLE OF CONTENTS Page I. INTRODUCTIW OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 1 II. POPULATION COMPONENT FOR ECOSYSTEM MODELING ....... 4 2.1 Objectives for ecosystem model ............... 4 2.2 Component description ........................ 7 Objectives of component model ................ 7 Identification of behavioral features ........ 8 Identification of stimulus and response variables OOOOCCCOOCOCOCOOOOOCCCOO00.0.00...O. 10 2.3 Coupling considerations ...................... 13 Intraspecific couplings ...................... 13 Interspecific couplings ...................... 14 III. CONSMTION COWONENT O...OOOOOCOCOCOOOOOOOOOOOOOOO 16 3.1 Functions and behaviors of the consumption component O00....OOOOOOOOOOOCOOOOOOOOOOO00.... 16 3.2 Discussion of consumption strategies ......... 18 3.3 Mathematical formulation of the Strategy Space 21 3.4 Implementation of optimum feeding strategies . 24 IV. OPTIMIZATION WITH RANDOM.PARAMETERS ............... 29 4.1 Incorporation of random variables ....... ..... 31 4.2 Optimization criteria under uncertainty ...... 35 4.3 A priori optimization under uncertainty ...... 37 V. FUNCTIONAL FORMULATION OF COMPONENT BEHAVIOR ...... 44 5.1 Basic developmental stage .................... 44 5.2 Consumption component ........................ 52 5.3 Summary of required parameters ............... 60 5.4 An example of simulation results ............. 60 VI. SUMMARY.AND OPEN QUESTIONS ........................ 66 REFERENCES 0..O...O0..O...00......OOIOOOOOOOOOOOOOO 71 iv Table 4.1.1 4.3.1 5.3.1 5.3.2 5.3.3 5.4.1 LIST OF TABIES Page Approximate density function description ........... 33 Optimization results for example ................... 37 Parameters required for each population stage ...... 61 Additional parameters required for each consumer population stage ................................... 62 Additional parameters required for each consumable papulation stage ................................... 62 Parameters for example simulation .................. 64 LIST OF FIGURES Figure Page 2.2.1 Basic developmental stage model ................... 10 2.3.1 A complete species population model ............... 13 2.3.2 Interspecific couplings in a community ............ 14 3.1.1 Nutrient flows coupled through a consumption comment OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. 17 4.1.1 Density function for a normal N(0,1) random variate ........................................... 33 4.1.2 Approximate discrete density function ............. 34 4.1.3 Compared cumulative distribution functions ........ 34 4.3.1 Example strategy Space ............................ 37 5.2.1 Linear search area ................................ 55 5.2.2 Curved search area ................................ 56 6.1.1 Example ecosystem model ........................... 68 vi I INTRODUCTION For years, many scientists from varied disciplines have been concerned with quantifying their understanding of biological phenomena. Results abound in the mathematical modeling of such ecological functions as population growth, predator-prey relation- ships, hunger, reproduction rates, and ecosystem energy budgets. Some of these results are purely empirical, others totally hypothetical, with most models representing a mixture of theo- retical and experimental results. Much progress has been made in the modeling of single-species functional behavior, but models dealing with an entire ecological community are rare. Progress in understanding and managing ecosystems has been slow indeed, primarily because of the enormous number of factors interacting in an ecological community. The concept of an ecosystem as a set of interacting components is a potent aid in the analysis, control, and understanding of that system. For such a representation to have any value, the components must be carefully defined and modeled consistent with the objectives of the system model. Extensions of single-species behaviors to an ecosystem model can be valid only if the species models have been developed with behavioral variables properly chosen. The isolation of functional components amenable to valid mathematical modeling is necessarily the first step in assembling a reliable quantitative model of any system. A generalized model of a population developmental stage is perhaps the most fundamental biological component needed in an ecosystem model. .Almost all existing ecosystem models are an extension of the milestone work on population dynamics done by Lotka (1925) and Volterra (1928), and the trophic-dynamic aspects of ecosystems recognized by Lindeman (1942). The population equations of Lotka are truncated Taylor series expansions of the time derivative of a species pOpulation about an assumed constant equilibrium point, which is based on an absolute population level that is environ- mentally supportable in the absence of interspecific interactions. Lotka's truncated series approximation eliminated all terms other than first-order interactions. While this approximation yields satisfactory correlations with observed population dynamics under sufficiently controlled conditions, it should not be expected to accurately describe population dynamics in the general case. Other researchers, notably Sack (1964), Garfinkel (1967), and MacArthur (1964), have included more terms in the Taylor series expansion of the population derivative in an effort to fit the equations more closely to observed behaviors. The Lotka4Volterra formulation is a classical case of a tractable solution that was fitted to a new problem. Chapter II of this thesis attempts to coalesce biological theory and applied mathematics into a state-space component model for the population of a developmental stage in a species life cycle. Such a component has the attributes of a fundamental "building block" component for modeling ecosystems. The interconnection of these fundamental components to model a multiple-species ecosystem indicates the need for an additional component to effect the consumption preferences and interactions of the community. This system level component is developed in Chapter III. Since the function of the consumption component is to quantify the active consumption strategies employed within the ecosystem, alternative species-related and community level strategies are discussed. The functional implementation of these strategies is also incorporated in the mathematical formula- tion of the consumption component. The factors affecting the exact behavior of any natural ecosystem are so numerous that a realistic model must treat many parameters as random variables. The characterization of the consumption component in terms of a behavioral model with random parameters is discussed in Chapter IV. II POPULATION COMPONENT FOR ECOSYSTEM MODELING Formal techniques for modeling discrete, inanimate physical systems have become well developed over the last few decades. Al- most all physical components and processesof this type have been sufficiently described to permit the construction of a reliable predictive mathematical model. This elevated state of the art is partially dependent, however, on the availability of mathematical models of the interconnected components. Universally accepted component models for the elements and processes in an ecosystem are just now being developed; researchers are defining potential components and attempting to describe them mathematically. Systems science has developed criteria for isolating a particular process or sub-system, defining it as a component of lthe system, and describing the necessary features of its behavior. In the early stages of describing any systems-theoretic modeling component, the behavioral features of interest must be selected to guarantee consistency and completeness when the component is coupled into the complete ecosystem model. Further, the behavioral features chosen must be sufficient to accomplish the overall objectives of the modeling project. 2.1 Objectives for Ecosystem Model Let the objectives of an ecosystem model be defined as: a) b) e) Predicting the population levels of all biological components in the modeled ecosystem for gny_permissible set of stimuli; Yielding further understanding of the cause-effect couplings with the system; and Providing a realistic model for simulating the ecosystem response to control efforts over time. A generalized approach to the mathematical modeling of an arbitrary system component may be deduced from Zadeh and DeSoer (1963), G8knar (1969), and others. The basic procedure may be outlined as follows: a) b) Identify all interactions between the object and its environ- ment which are relevant to the behavior of the object to be modeled. Define the abstract "boundary" through which all such couplings between the object and its environment mst pass. Each of the couplings to the isolated object which intersect the boundary are known as behavioral features. Orient the component by classifying each behavioral feature as a stimulus or response feature for the component. The assignment of behavioral features must not preclude the satisfaction of the following inter-consistency requirements: 1) Each component orientation must be causal (non- anticipative); no response feature may predict the value of a stimulus feature at a later time. 2) Each component orientation must be deterministic; the values of the response features at any time t are uniquely determined by the initial state of the component at time t0 and the stimuli from to to t. Note that "deterministic" as used here does not mean non-random. 3) Define a direct connection as a junction that equates all attached behavioral features. [Among all behavioral features coupled to a single direct connection, there must be exactly one response feature. The dichotomy of features into stimulus and response features is a critical step in the derivation of a tractable formulation of a component model. Some behavioral features are easily labeled stimulus or response variables; for example, light_impinging upon retinal cones is Obviously a stimulus to the ocular nervous system. In biological systems, a distinct time lag usually occurs between stimulation and response, and behavioral features exhibiting a delayed reaction to another feature must be classified as responses to fulfill the causality requirement of the model. Designate the value of the vector of stimulus variables, a, at any time t as a(t), and the value of the vector of response variables, r, as r(t). c) Now, if the chosen orientation of the object permits a causal description, the value of the response vector r at time t may be written as r(t) = flirt (t), o(t)] (2.1.1) 0 where 3 is some function. Additionally there should be some function .9 that "updates" the vector Yt : i.e., 0 Yt (t) =£[Yt (E), 0(2)], to s E < c . (2.1.2) o o The vector variable Yto(t) incorporates all necessary memory from tO to the present time t and is known as the state vector for the object. For a memoryless component, no state variable is needed. While there is no general rule to determine the elements of Y, it may be stated that whatever features other than current stimuli and fixed parameters needed to formulate equation (2.1.1) are state variables. The cartesian product of the range spaces of the state variables constitute the state space. Moreover, 2.1.2 is called the state equation and 2.1.1 is called the response equation; the pair of equations is called the state model. 2.2 Component Description A likely candidate for an ecosystem component is a model for the papulation of a developmental stage in the life cycle for an arbitrary species. The life cycles of many biological organisms are character- ized by a similar sequence of epochs over a wide variety of species; the birth of an organism is an easily recognized event in time, followed by a succession of the organism through separable periods of growth until it reaches its terminal state. These developmental stages in the life cycle may be easily distinguishable, as are instars for insect larvae. In other organisms, the transition point between stages may be less naturally defined, although it will usually be possible to separate the birth-to-termdnal state life span into disjoint segments identified by the presence, absence, or size of characteristic features of the organism. A modeling component of such a deveIOpmental stage could be generalized, of course, to model a developmental stage (or even an entire popula- tion) of a set of similarly-behaved species. Read and.Ashford (1968) formulated a population model using such life cycle stages, although they considered the survival, growth, and death of the organisms to be purely probabilistic and independent of exogenous influences. Their formulation virtually ignores the wealth of knowledge of biological organisms accumulated by natural scientists. In this thesis, a model for a generalized life stage is developed which incorporates a great deal of the scientific knowledge accumulated by biologists and ecologists. The following couplings between the component and its environment may be identified as affecting the population of the component: a) Recruitment of members from the environment through: (1) Individuals entering from a different stage in the life cycle because of growth, aging, or certain physiological changes, (2) Migration from other communities of the environment, (3) Births; b) Loss of members to the environment through: (1) Natural death, (2) Predation or other harvesting, (3) Exiting to the next life cycle stage, (4) Migration from the modeled ecosystem; c) 'Ingestion (nutritive, inert, and toxic intake); d) Egestion and excretion; e) Reproduction; and f) Natural abiotic stimuli such as temperature, humidity, and solar radiation. Consider the ten-terminal component diagrammed in Figure 2.2.1. The arrows on the schematic representation indicate direc- tion of biomass flow where approPriate. The model may be considered a biomass and energy converter representation for a pOpulation of functionally similar organisms. All features except possibly 6, yf, and ye can be char- acterized by a single number at each time t. These exceptions may require a vector representation; for example, 6 may be a vector of abiotic environmental variables such as ambient temper- ature, humidity, wind velocity, intensity of solar radiation, and atmospheric levels of CO2 and $02. The schematic representation of Figure 2.2.1 implies that the terminal variables shown comprise the necessary behavioral features for a sufficient characterization of the developmental stage pOpulation. They will be functionally related by the state and response equations, which must be formulated to be consistent with the units apprOpriate to the behavioral features involved. Although the yh variable, representing external harvesting of the pOpulation, is an outward flow of biomass from the component, the actual quantity harvested per unit time is primarily determined by the component's environment, such as the density and activity levels of predators and other harvesting agents. Hence, yh is classified as a stimulus for the component model. 10 yf yg yb Ymi .Yd ym e yh ye Figure 2.2.1 Basic Developmental Stage Model where a - Abiotic stimuli (temperature, humidity, solar radia- tion intensity, etc.) yf - flow rate of nutrients per unit time into this stage (consumption rate). yb - entries per unit time from the previous stage of the life cycle (organisms growing into this stage from the coupled preceeding stage, or births for the initial stage.) y - number of individuals immigrating from outside the 1 modeled ecosystem per unit time. y - number of individuals emmigrating to different eco- e system models per unit time. yh - number of organisms harvested per unit time from this stage. y - number of births (seeds, eggs, etc.) per unit time pro- duced by the population of this stage. y - number of organisms exiting per unit time from this stage into the following stage. yd - number of deaths per unit time of organisms from this stage. y - discharge rate of unassimilated nutrients and by-pro- ducts of assimilation and respiration (excretion and egestion). 11 To Obtain a causal orientation of the population component, the yf, yb, 6, ymi, ymé, and yh features will be considered to be stimuli. Behavioral features yg, yr, yd, and ye are viewed as responses of the population component. Migration occurs primarily as an external influence, and Forrester (1971) has suggested that net migration between any two communities may be given as a function of the difference between the "quality of life", as defined by an appropriate metric, in one community and that in the other. Considering migration in this fashion implies a migra- tion component to connect like species models in different communities. Such a migration component would have the quality of life values of each community as stimuli and the resultant migration as a response feature. Hence, the migration features ym1 and yme will be classified as stimuli to the pOpulation model to permit deterministic coupling to a migration component such as discussed above. Since nutrient input flow, entering organisms, and environmental factors are exogenous to the component, yf’ yb, and 6 are considered to be stimuli. It can be noted that stimuli yh, yb, yf, ymi, and yme may be influenced by abiotic factors; however, 6 as used in the model reflects the direct stimulation of the pOpulation component and does not pre- clude indirect stimulation through these other behavioral features. The causal orientation of the pOpulation component is emphasized schematically in Figure 2.2.1 by placement of stimuli on the left side and responses on the right side. In Chapter V, the state variables are chosen to be population, hunger, energy reserve, and the developmental distribution 12 of the population. To complete the functional description of the component, each response variable must be expressed as a function of the state variables and the values of the stimulus features. This formulation must be done for each of the response variables, and will be dependent upon the criteria for distinguishing separate stages, and the particular species being modeled. Examples of such formulations will be given in Chapter V of this thesis. Levine (1966) has stated three basic requirements for any valid ecosystem model as generality, precision, and realism. The component described may be considered as a fundamental building block for all ecosystem models; this population sub- section may be defined for all forms of living organisms, from plants and animals to bacteria and other micro-organisms. Any quantitative definition of precision must be in terms of the model's maximum error, or variance from actual behavior. ‘Most errors resulting from computations with this component will arise, not from the component definition, but rather from an "imprecise" mathematical formulation of the functional behaviors modeled or errors in parameter estimation. Such errors are inevitable, al- though the sources of error may be identified more clearly within a model such as the one proposed. The behavior of an ecosystem model built with these components should closely simulate observed behaviors, since experimental data is used to evaluate the func- tional parameters. 13 2.3 Coupling Considerations A birth-to-terminal-state population model for a single species, or a functionally similar set of species, may be directly assembled by connecting the required number of the develOpmental stage modeling components described in the previous section. Obviously, the yb terminal of each component stage is coupled directly to the yg terminal of the previous stage, except for the first (youngest, smallest) stage, where the yb is connected to the sum of all yr terminals of the total Species model. The intraspecific connections are illustrated in Figure 2.3.1. Also, the yg terminal of the last stage (oldest, largest) is identically zero a lst stage 2nd stage nth stage Figure 2.3.1 A complete species model In ecological literature, the set of interspecific couplings of yf terminals to yh terminals of other species is known as the "food web" of the community. A subset of these couplings might appear as in Figure 2.3.2. l4 Array of consumables Consumption (Predation) Interactions Array of Consumers Figure 2.3.2 Interspecific couplings in a community However, a closer look at the implications of such con- nections reveals that stimulus terminals are connected only to stimulus terminals. Such a junction leads to indeterminate system behavior, as it conflicts with one of the consistency re- quirements stated earlier. There are two solutions to this dilemma: (I) expand the component model to structure yf as a response to external stimuli from the array of available consumables and competitors present, or (2) incorporate the prOperties of the consumption strategies and activities into an additional component representing the environment for the feeding activities of all consumers in each subsector of the community. In (2), all the yf and yh behavioral features continue to be interpreted as stimuli to their respective species models. However, they will be coupled to response terminals of an interaction component whose stimuli are the response and state variables of all population components to which it is coupled, along with the environmental stimulus set. 15 In a no-competition environment, i.e., only one consumer species per prey species, the first solution is indicated. How- ever, in the general case, the latter approach is preferred for computational and modeling convenience, and will be labeled the consugption cogponent. In this chapter, the requisites of a consistent state- space component formulation have been reviewed, and these con- siderations were used in the formulation of a component repre- senting a deve10pmenta1 stage of a single-species pOpulation. Aspects of coupling the developmental stage components to form complete single-species population models were discussed as well as the couplings occasioned by the formation of a multi-species ecosystem model. A consumption component to effect the competitive behaviors in a general multi-specific ecosystem model will be developed in the next chapter. III THE CONSUMPTION COMPONENT The consumption component developed in this chapter must embody the feeding preferences and activities of all species competing for the same dietary items. After a discussion of modeling considerations, various consumption strategies are compared, and a scheme of optimal feeding behavior for a segment of an ecological community is mathematically formulated as a standard static optimization problem, for which solution techniques are available. 3.1 Functions and behavior of the consumption component The food web potentially encompasses all species within an ecosystem, and the consumption component develOped in this chapter is formulated with the intent of coupling sub-sectors of the community, and as such, several consumption components will be required within an extensive model. Since each species popula- tion functions simultaneously as both a consumer and a consumable relative to other populations, an apprOpriate computational pro- cedure must be determined that effects behaviors consistent with both roles. A complete model of an ecological community of multiple simultaneous consumer/consumable groups may be effectively simulated by evaluating the consumptive results of each linked group sequentially with a small time lag. l6 17 This abstract consumption component of the ecosystem must incorporate all behavioral features necessary to effectively couple the individuals harvested from some components with the proper food input terminals of the feeding species' components. To permit determinate orientations of all ecosystem components, the consump- tion component's behavioral features that couple to yh and yf terminals of the population components must be reSponse variables of the consumption component. The stimuli for the consumption component are potentially all the state variables, which can be viewed as additional response variables, and the response variables of other components in the system plus the stimulus vector 6, which is a response of the environment. The consumption component appears in the system graph for a subset of the ecosystem as indicated in Figure 3.1.1. : Set of stages subject to consumption by elements of CB Response and state information of coupled stages Consumption Component ——?Z__________.— Set of stages feeding on elements of QA Figure 3.1.1 Nutrient flows coupled through a consumption component 18 The behavior of the consumption component is actually a composite of the behaviors of the interacting consumers and consumables within the modeled ecosystem, and must be consistent with environmentally imposed limits. It is becoming more evident that natural ecosystems are guided by near-Optimal behaviors and strategies employed by the constituent organisms. Cohen (1970), Tullock (1971), Schoener (1969), Hamilton and Watt (1970), Emlin (1966 and 1968), and others have addressed this apparent phenomenon in recent years. This is not to suggest that optimal strategies are intelligently determined and carried out as situations arise, but rather that the viability of species whose behaviors are near-optimal is equivalent to the concept of natural selection. This concept permits the investigator to partly answer the question "Why do animals behave as they do?" and begin to address questions such as "What will animals do in a certain environment?" by determining what behaviors will enhance the viability of the organism and the species...the Optimal behavior. 3.2 Discussion of consumption strategies The behavior of the consumption component must incorporate the competitive feeding and avoidance strategies of the inter- acting pOpulations. While feeding strategies have been studied for individuals, few population-level strategies have been investigated. Emlin (1966) advanced the hypothesis that natural selection has favored the development of feeding preferences and strategies that, within the physical, sensory, and nervous l9 limitations of a species, optimize the net energy gain per indi- vidual of that species per unit time. Hamilton and Watt and Schoener have also postulated a similar rationale for a food preference and acquisition strategy. Such a strategy may be easily incorporated into the framework of an optimization prob- lem, where optimum is defined relative to the energy exchange from the consumption activity of a species. The net energy gain of the consumption activity is the difference between the benefit in energy units gained from digestion of ingested consumables, and the 29;; is similar energy units expended in the searching out, pursuit, capture, and digestion of food. Hamilton pp pl_list four potential objectives in the exploitation of energy by predators as: a) Minimize the rate of decline of negative net energy exchange. (This objective is utilized by animals in hibernation or other state of diapause. It becomes a strategy for resource acquisition when the environment does not permit positive energy exchanges due to lack of sufficient trOphic capacity.) b) Maintain net energy at zero. c) Increase net energy to some fixed amount above zero. d) Maximize net energy gain without bound. Note that a) will be adopted under limited food conditions if any of the other three strategies are employed. Objective 3) is just objective d) applied under sub-maintenance trOphic con- ditions. Objective b) is just a special case of objective c). Hence, these four objectives represent only two distinct 20 strategies: the maximizing of net energy gain without bound and the maintaining of some constant energy gain. Hamilton pp pl_suggest that, since size may have a positive selective advantage to a consumer, it may be advantageous for immature individuals of a species to grow large as rapidly as possible to maximize the energy efficiency involved in food capture. This reasoning implies that the feeding behavior of the young of a species is likely to be governed by a strategy of maximization of energy exchange from consumption activities, while the adults strive to maintain an efficient size. Also, Schoener's arguments supporting an optimal size for predators suggests the adaption of the zero net energy gain strategy by adults of a species. These two strategies for the young and the adults are assumed to be the objectives utilized by each population component. . In all cases, however, the environment as it appears to each consumer group places numerous restrictions on the actual implementation of the optimization policy. For example, some desired consumables, although present in the community, may not be really accessible to a potential consumer pOpulation, perhaps because of avoidance strategies employed by the prey, or because another consumer species is a more efficient competitor for some particular prey species. Such factors must influence the resultant feeding behavior for all consumers in the community. What is needed is a consumption component with a behavioral description that determines the constrained optimal feeding behavior of the community's population components. 21 3.3 Mathematical formulation of the strategy space Define an m-n-dimensional Euclidean space ST by S.r =‘Y11 x le X...X Ymn’ where Yij is the range of the number of consumable species type i eaten by consumer Species type j per time period 7. Then any community consumption activity between n consumer groups and m consummable groups may be represented by a point y 6 ST. Choice of the period T is arbitrary but, of course, should be consistent with the natural cycling period of the phenomena being studied. The actual subset of ST containing all possible results of consumption activities may be restricted by theoretical and physical constraints. Some obvious constraints imposed on the community strategy space ST may be listed as follows: a) No more individuals of any particular consumable population component may be taken than exist. b) No consumer population may expend more energy in pursuit than is available to it as a result of previous feeding and respiratory activity. c) No consumer pOpulation may Spend more time than the specified period T in pursuit, handling, and swallowing. d) No consumer pOpulation may ingest more biomass than the total available (unfilled) gut capacity of that consumer component. e) Since yi.1 6 Y1 is always non-negative, only the non- 1 negative orthant of ST need be considered. The application of such constraints to ST defines S , the subset of ST containing the range of behaviors of the 22 consumption component. The optimization process may then be applied to ST to determine the current behavior of the con- sumption component. A linear constraint matrix equation incorporating the restrictions above may be formulated as follows: Define yL1 as the number of elements of consumable species 1 eaten by consumer type j during time T; c as the energy cost to consumer species j for the location, capture, and digestion of each element of consumable species 1; 11 E as the expendable energy available within the population of consumer component j; N as the pOpulation of consumable component i; s as the volume per element of consumable species i; H as the available (empty) gut capacity of the population of consumer type j; t. as the time required by an element of consumer group j to pursue, capture, and swallow an individual of consumable group 1. Then, the constraint equations for a ecological community of n consumer population groups and m consumable population components may be expressed in the following linear form: 23 — q D T cll c21 ... cm1 0 0 E1 O ...-0 c12 c22 ...cu12 0—-" '——O E2 0 ... . C ... c - E 1n 2n y' 1 n tn t21 tml 0 8“ 11 T 0__. .‘0 1:12 t22 coo tmz 0 ° . 0 y21 T . ' . I - (3.3.1) 0 ' ' ' t ... t y S '1' S s ... s 0 1n 2n 8“ ml H lv 2v mv y12 H1 0 ’0 S 8 cos 8 O . . O 2 l 2 m . V V V o 0 ' 0 °: 0 s s . . s l ' . 1v 2v mv Lymn Hn I I l ‘ N1 ' l m : m ' 2 m “2 I . I : - -° , I : I ' . . : I l N L .. . “E or, ,Ay s d (3.3.2) and y 2 0 . (3.3.3) The constrained set ST of feasible y is the set of all y satisfying 3.3.2 and 3.3.3. Referring to Figure 2.3.2, it may be seen that n Y = E y. (3.3.4) hi j=l 13 Define a 5 X 1 vector S1 to describe the composition of an organism of consummable species 1 as 24 ”hi .1 Fprotein biomass/type i organisun F P 31 carbohydrate biomass/type i organism c 31 = Si Q fat biomass/type i organism (3.3.5) f 31 inert biomass/type i organism 1 Si volume/type i organism Lv-J L a Then, if yf is defined similarly as 'yf ‘1 "protein biomass ingested F P yf carbohydrate biomass ingested c yf = yf Q fat biomass ingested , (3.3.6) f yf inert biomass ingested i yf volume of food ingested a 'V.J a a the nutrient flow into the jth consumer population, yf , may be 3 computed as m yf = z sy.. - (3.3.7) 3.4 Implementation of Optimum Feeding Strategies The objective function which maps points in the constrained set to a scalar index of optimality is critical to the behavior of the consumption component. The objective function will be designated by p(y,p), where p is a vector of parameters and variables other than y that are used in the formulation of m. The different consumption strategies employed by young and adult organisms require different objective functions, which will be 25 formulated separately as my and q3, respectively, and then combined to produce the generalized objective function, m, for the consumption component involving both young and adult consumers. In the case of the young populations where the objective of the consumption process is to maximize the net energy benefit to the set of consumers, p is composed of cost and benefit vectors, c and b, and the Optimization problem may be stated as :31 --W 11 c11 A b A C Maximize py Q (b-c)'y, where b = 12 , c = 12 . (3.4.1) ye 'r I I b c L“: L“: Here, bij is the energy derivable by an element of consumer group j from the consumption of an element of prey component i and the prime denotes matrix transpose. Note that the formulation of 3.4.1, with the constraint set of 3.3.2 and 3.3.3, is in the form of a standard linear programming problem. While the young of a species are likely to adopt a linear maximization objective, the adult stages are probably employing a strategy of near zero net energy exchange. This strategy for adult organisms is consistent with the objective of maximizing reproduction of breeding organisms for some species, since the energy costs of physically maximum fecundity may be considered as part of the maintenance costs to the organism. By introducing a parameter vector u to account for maintenance activities including reproduction, the objective function for adult consumers, ma, may be formulated as a quadratic function. Define pi as the average maintenance energy requirement for consumer Species i 26 for all activities other than feeding per T units of time. To Simplify the notation of the construction, let F l P ‘1 P I ylk b1k °1k sz b2k c2k yk = : bk = I ck = : (3.4.2) Y b c mk mk mk L .J L .a L .1 Also, for convenience, define f = (b-c); and r- 1 r- 1 flk b1k ' °1k f2k b2k ' °2k fk = : = : (3.4.3) fmk bmk ' cmk L .J L (- Now, the goal of net zero energy exchange for adult con- I sumer species 1 may be interpreted as making f1 y1 equal to pi. This objective may be formulated as I 2 Maximige - 2 (f1 y1 - pi) y E ST i Since the maximum value of this expression is zero and is I achieved iff fi y1 = pi for each adult consumer species i. Therefore, let n .1 A 1 i 2 c9, = - E (f y - pi) (3.4.4) i=1 n n i' i i' i i' i 2 = - 2 [(f y )'f y - Zuif y ] - 2 pi (3.4-5) i=1 i=1 n 2 The last term, - 2 p1, may be drOpped since it is 3 i=1 constant and hence does not affect the optimization over y. 27 I. oI- I I Also, Since f1 y1 is a scalar, f1 y1 = (fi yi)' = y1 f1. There- fore, an equivalent objective function may be written as n u . u . . i 1 1 (pa = - z: (y flf y‘ - zuifl y) (3.4.6) i=1 = 2f'Uy - y'Fy (3-4-7) where P H m I “1 O A m u = “21 (3.4.3) ‘ m 0 pnI r- , ‘1 and flf1 O I A f2f2 F ', (3.4.9) 0 ° £“fn' L 3 The objective functions my of 3.4.1 and pa of 3.4.7 may be incorporated into a Single quadratic objective function p which can be used to simultaneously Optimize the feeding be- havior for all elements of the consumption component. To construct the generalized objective function m, the young and adult con- sumers must be distinguished for each yij. Hence, let the vector y be expressed as the sum of independent mn-dimensional vectors yy and ya, reflecting the feeding activities for young and adult consumers, respectively; i.e., A = 3. O y yy+ya ( 410) where yy = (I - J) y and y = Jy 28 and (J)i = 1 if i = j, and j is a subscript j for an adult stage (3.4.11) 0 otherwise. NOW, CP = (py(yy:p) + (Pa(ya:p) (304°12) I um I I = f (I - J)y +'2f UJy - (Jy) FJy = g'y - y'Ly (3.4.13) where g = (Itun - J +-2UJ)'f and L = J'FJ . (3.4.14) An optimal solution is guaranteed and solution algorithms are available to solve the quadratic programming problem defined by 3.3.2, 3.3.3, and 3.4.13 if L in 3.4.13 is positive definite (see Aoki (1971)). From the definitions 3.4.9 and 3.4.14, it is obvious that L is positive semi-definite. Charnes (1961) suggests the addition of some small number a to the zero diagonal elements of a semi-definite matrix to satisfy this requirement. This manipulation constructs a positive definite matrix L with negligible effect on the solution vector y. These comments, of course, refer to the case where parameters A, d, and p are deterministic. In summary, the consumption component response is char- acterized by y, which must lie in a linear constraint set ST defined by 3.3.2 and 3.3.3 and which maximizes an objective func- tion m(y,p) given by 3.4.13. The relationship between y and the population component stimuli yh and yf is given by 3.3.4 and 3.3.5. Realistically, the parameters A, d, and 9 should be treated as random variables, which is the subject of the next chapter. IV OPTIMIZATION WITH RANDOM PARAMETERS Obviously, many of the parameters in the constraints 3.3.1 and the parameter set 9 of the objective function 3.4.12 cannot be precisely known. Even if exact information were known, it is totally unrealistic to assume that the consumer community has access to such information or is able to compute it. The consumer species act on imprecise notions and probabilities, whether they are called instinct or intelligence. Hence, the Optimization by the mathematical model of the behavior of the consumption com- ponent Should be carried out with imprecise parameters expressed as random variables. Moreover, it is more intuitively acceptable to represent parameters such as "searching velocities" and "handling and swallowing times" by probability density functions rather than by constants such as the estimated means of the random'variables. Three basic approaches have been develOped for handling random variables in mathematical programming problems, each with different assumptions and goals. Unfortunately, each has been concerned only with problems having linear objective functions. An excellent overview of the literature on probabilistic pro- gramming is given in Sengupta and Fox (1969). Chance-constrained programming, as developed by Charnes and Cooper (1961), seeks an optimal solution subject to a 29 A ‘4‘ A Alli-III 3O probabilistic constraint set which is to be satisfied at some pre- determined probability level, i.e., the constraining equations be- come P(1th constraining equation in Ay S d is satisfied) 2 “1’ where n = (“l’°"’n3n+m) is a vector of desired levels, and P is the probability measure. Two-stage programming, prOposed by Dantzig (1955), solves for the solution vector which minimizes the costs of adjusting to the actual constraints as they are realized. In spite of efforts to generalize the two-Stage programming technique, it is presently limited to problems in which random variables appear only in the constraint vector d of 3.3.2. Tintner's (1955) stochastic programming approach evaluates the distribution function of the objective function, which is expressed as a function of the random variables in the problem. In the most general form of Tintner's algorithm, the optimal solution vector is derived under rather severe simplifying assumptions about the random errors around the optimal basis. For the most part, the random variables appearing in the constraining equations 3.3.1 and the parameter set p of 3.4.12 may be considered independent. However, the solution technique to the mathematical programming problem involving random variables that is developed here is applicable regardless of the dependence between the random variables involved or the form of the objective function p(y,p). The main difficulty with a generalized approach such as Tintner's has been the apparent necessity to evaluate an n-fold convolution to obtain the joint distribution function in a problem 31 involving n random variables. This barrier is sidestepped in the approach taken here by constructing a discrete approximation to the joint distribution. One feature of this approach is the necessity to select a "best" solution among many potential Optima. Statistical decision theory is applied to this sub-problem. The approach taken here reduces a stochastic programming problem to a series of deterministic problems of the same form. Hence any programming prOblem for which a solution algorithm exists may be solved even though it contains random variables. The development here is not particularized to the consumption component, but is carried out for the more general problem. 4.1 Incorporation of random variables Consider the matrix A, the vector d, and the parameter vector p of Chapter III as containing M. random variables, which are assembled into an M-dimensional random vector X, de- fined on sample points w E n, the sample space. Now, the dis- tribution function of the random variable X may be characterized by defining a probability measure P on 0 and a cumulative dis- tribution function P such that: FX(CY) = PEG): X10») 5 a1,...,XM(w) S QM] . (4.1.1) If the distribution function Fx is a step function, or if it can be approximated arbitrarily closely by "discretizing" the random variables X1, and if the "discretized" values of X are finite in number, the vectors, OR = (a:,...,q;), with k = l,...,K, may be defined such that 32 P{m: X1(w) = o:,...,XM(w) = mg} = pk (k = l,2,...,K) (4.1.2) K k k and 2 pk = 1. Define wk = {(1): X1(w) = a1,...,XM(u)) = QM}. k=1 If A(k), d(k), and p(k) are defined as the values of A, d, and p when the kth environment prevails, P{[A,d,p] = [A(k), d(k), p(k)]} = pk (k = l,2,...,K). (4.1.3) One key requirement in the development here is the need for all random variates to assume only discrete values. This re- quirement is already fulfilled in many applications since the parameters are described by a histogram of actual observations, which are necessarily discrete and finite in number. However, in some cases, observations have already been transformed from multiple data points to an alternative form, by generalizing to a continuous descriptive function or by dividing the range of observations into finite width bins, each characterized by a relative frequency. The latter descriptive form is most advan- tageous for the procedures develOped here. Continuous distributions may be approximated by dividing the range of the random variable into, say, R bins and repre- senting the probability mass of each bin by a mass point located at the expected value of the random variable over the bin. This location is the x-coordinate of the centroid of the probability mass of each bin. The following example illustrates this approximation technique. EXAMPLE: Let X be a standard normal random variable N(0,1), with the probability density function sketched in 33 Figure 4.1.1. LPX(X) Hut» N -2 -l 0 Figure 4.1.1 Density function for a normal N(0,1) random variate Now, if R is chosen to be nine and the bins selected to be % standard deviation wide, except for the end intervals, the approximating distribution is defined by Table 4.1.1 and Figures 4.1.2 and 4.1.3. Table 4.1.1 Approximate density function description Interval P x I. E(xlx E I.) = y. ng < 1.) I1 = (~m,-l.75) 0.04006 -l.876 0.0 12 = (-l.75,-1.25) 0.06559 -l.439 0.04006 I3 = (-1.25,-0.75) 0.12098 -0.959 0.10565 I4 = (-0.75,-0.25) 0.17466 -0.479 0.22663 I5 = (-0.25, 0.25) 0.19742 0 0.40129 I6 = ( 0.25, 0.75) 0.17466 0.479 0.59871 I7 = ( 0.75, 1.25) 0.12098 0.959 0.77337 18 = ( 1.25, 1.75) 0.06559 1.439 0.89435 I = ( 1.75, +m) 0.04006 1.876 0.95994 34 px(X) 0.2 .“. I A A 0.1 ._ I} h 0 LA I a? ‘“H’ A A: x -2 -1 0 1 2 Figure 4.1.2 Approximate discrete density function Further, the goodness of fit may be estimated by comparing the cumulative distributions of the continuous and the discrete random.variables as in Figure 4.1.3. P(X S x) l. - .5 J. #— 1 H1)- i -2 - 0 Figure 4.1.3 Compared cumulative distribution functions It is assumed henceforth that all random variates appear- ing in the triple 0A,d,p) are either discrete or are approximated by discrete distributions. For each possible state of nature, wk’ occurring with a non-zero probability pk’ the set Y(k) Q {y\A(k)y S d(k),y 2 0} (4.1.4) 35 defines a convex set of possible strategies (or feasible points) in the strategy space ST. Further, for each state of nature, it is assumed that there exists at least one vector y(k) E'Y(k) that maximizes the Objective function p(y,p(k)), where p(k) is the set of parameters of the function p when the kth state of nature prevails. Define 2 as the value of the objective func- k tion p evaluated for y(k) and 9(k). 4.2 General optimization criteria under uncertainty If 2 k is evaluated for each Y(k), some rationale must be provided to determine the "best" y(k). Therefore, a decision function must be selected to choose the optimum y(k). The particular decision rule to be used depends upon the nature of the problem under investigation. The goal in modeling the consumption component's behavior is to estimate a priori the consumption activity that most enhances the viability of consumer species. Obviously, with naturally occurring random phenomena, the model is not expected to predict the consumers' activities exactly, but the actual and modeled behaviors should exhibit similar trajectories. The Bayes decision function implemented here is a reasonable choice for desirable model characteristics. To derive an Optimal Bayes decision rule, a loss function L(y(k),w ) is defined which assigns a loss or risk to the decision J that y(k) is best when, in fact, wj is the true state of nature. Considering loss as "negative gain", let L(y(k),w.) 3 (4.2.1) 3 o;k#j. Now the Bayes optimal decision rule may be stated as: Choose y = y(k) if En[L(y(k),w)] S En[L(y(j).w)] j = 1.2.---.K . (4-2-2) Evaluating the expectation, 4.2.2 becomes K K 2 me(y s 2 me(y(J).wm) (4.2.3) m=l m=l or, -zkpk S -szj . That is, choose y = y(k) if k is the smallest integer for which pkzk 2 pjzj ; j = l,2,...,K . (4.2.4) Thus the decision rule reduces to the maximization of pi°m(y(i),pi), where i = l,2,...K. The application of this decision rule may be combined with the determination of the y(i) vectors. Let A[y,a(i)] = pip[y,p(i)], where 3(i) is an augmented parameter set which includes pi' The optimization prob- lem may then be expressed as: find max max A[y,3(k)] . (4.2.5) lsksk y€Y(k) The existence of a vector 9 which achieves this maximum follows from the assumption in Section 4.1 that y(k) exists, k = l,2,...,K. 37 4.3 A priori Optimization under uncertainty Each possible choice y(k) submitted to the solution tech- nique developed in Section 4.2 is an absolute optimum for the objective function p if wk occurs. Since these solution vectors are computed as optimum given the environmental state m they k’ will be categorized as g posteriori solutions. An additional set of solutions may be obtained by considering sets of outcomes, and these will be labeled 9 priori solutions; that is, there is possibly a solution 7 # y(k), k = l,2,...,K, which is a preferred solution. Although y may not be the optimum a posteriori decision for any wk, it may produce a greater expected gain than any of the "pure" Optimum y(k) where k = l,2,...,Ki AS an example of a sub-Optimal choice being preferred, consider the constraint sets Y(l) and Y(2) shown in Figure 4.3.1 for the case of K = 2, p1 = p2 = 0.5, IID and 21 = A[y(i),§(i)] Figure 4.3.1 Example strategy space 1 = 0.5(4 + 2) = 3 and 22 = 0.5(3 + 3.5) = 3.25, and the decision rule of 4.2.5 would imply a choice of 9 = y(2) = (3,3.5), Now, 2 38 with expected gain A[y(2),a(2)] = 3.25. However, consider the set Y(3) = Y(l) F]Y(2), as indicated by cross-hatching in Figure 4.3.1, as a new set of possible strategies. Also consider w3 as the event that defines Y(3) as the set of possible strategies. Noting that w 3 all states of nature, p3 = P(w3) = 1. Since p(-) is the same occurs under for all m, take p(3) = 9(1). With the event w3 the maximum of A[y,§(3)] for y E Y(3) is attained by y(3) = (3,2), for which 23 = A[y(3),§(3)] = 1(3 +'2) = 5. If K is incremented to 3, the decision rule of 4.2.5 dictates a choice of y = y(3) 23 >22 >21. The solution technique of Section 4.2 will now be extended as optimal, Since to include the sub-optimal choices. The example above showed that although the events wk are mutually exclusive, the sets Y(k) are not necessarily disjoint for 1 s k s K. There are K _ K! (.1) i!(K-i)! ways to choose iK sets from a collection of K sets. Hence there are at most 21(1) compound events that may i=2 be formed by forming unions of the events w1,w2,...,wK. Let K K A K' = K + 2 K = 2 K = 2K - l and identify each of the i=2 1 i=1 1 compound events by a symbol wj for K.< j S K'. Moreover, de- fine Ij as the collection of indices from among {l,2,...,K} used in forming wj; that is, m U wk, j = K+1,...,K'. j kEIj Since the events wk for l s k s K are independent and mutually exclusive, ij) =P(U m1) = z P(wk) = 2 Pk- (4.3.1) kEIj kEIj kEIj 39 Now, the set of feasible points obtained with the occurrence of the joint event w for K.< j S K' must be a set of points J that are feasible whenever any one of the set of (wk: k E Ij} occurs. The set of feasible points satisfying this requirement is O ‘Y(k). Define Y(j) = n ‘Y(k) and k I E J REIj ) = 2 pk. (4.3.2) P = P(m J J REIJ where l S k s K.< j s K'. The optimization problem may be defined for the jointly feasible sets Y(j) for KI< j s K' as finding a point y E Y(j) that maximizes ACY33(3)] for some suitable §(j). This 3(j) may be taken to be the expected value of the random vector p over the set {wk}k€Ij’ augmented by pj as computed in 4.3.2. The expected value of p(j), K.< j s K', may be evaluated as follows: K A 90) =E Mm.) = 2 p(k) PM = 9001...] 0 J k=1 3 K = E 9(k) PEQ = 9(k).w.]/P(w.) . (4.3-3) k=1 3 3 Since the compound event wj occurs iff some wk occurs such that k E Ij’ pk ; k 6 I3 P[p = p(k),m ] = , for k = l,2,...,K . j 0 ; k é I 40 Therefore, p(J) - ml) 2 poopk + 1321 y(k) (o) > 3 kelj j l- : p(k)pk . (4.3.4) 9 j kte The set Y(j) for K.< j s K' may be computed from the following construction: Pun)? F-c1(11)1 MiZ) d(12) Y(J) = y = y S , y 2 0 (4.3.5) uA(iq)_ Ld(iq)d where the set of indices {il,iZ,...,iq} = Ij' The extended optimization problem may now be stated as: find a solution y that achieves max max A[y,fi(k)] . (4.3.6) ISkSK' y€Y(k) The following example illustrates the techniques developed in this chapter. Given: K = 5, m = l, n = 2, A[y(k).9(k)] = pk[pl(k)y1(k) + 92(k)y2(k)] , and 41 k A(k) d(k) 900 1 0.2 [J 1 0.2 [2] 0.2 l 1] 0.2 [2] [. 2 1 0.2 [2] [.1 5 Now, there are K' = 2(5) = 5 + 10 + 10 + 5 + l = 31 sets Y(k) i=1 to optimize over. The results of these Optimization problems are U1 b L» N H ‘NHHHOHOHOH HHl-‘Or-‘HI-‘OHO 6:3 r--1 at: given in Table 4.3.1. The parameter vectors y(k) for 6 s k s 31 are computed from 4.3.4; for example, 1 f, 1 + 3 + 4 p1 ... p3 ... p4 {91M} p3p() pap()} __ [3+ [3+ “[3] $322] = [i] ' The solutions y(k) were obtained by standard programming tech- 9(19) niques. It may be seen from Table 4.3.1 that the optimum a posteriori solution is 9(2) = (§,2) and A[y(2),§(2)] = 0.9, while the optimum a priori solution is y(28) = (%,l) and A[y(28),§(28)] = 1.1. In the example here, the computational complexity jumps by a factor of 6.2 when a priori solutions are considered. However, note that 5 1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 # Optimum a priori solution X NXNN NNNNNN NNNN X Table 4.3.1 Optimization results for example Intersected Sets 5.5113(2) 4(3) “4141(5) 9100 92(k) X X X X NN NNNN >< xxxxx 42 HHj—IHp—IHt—IHHHHHHHHHp—IHHHHHHHHHHHHHH * Optimum a posteriori solution 1 2 a o 3 1.5 .75 k 1.25 1/3 7/8 7/8 .75 .8 o o o o o o 0 "U a\ ¢~ e~ c~ c~ c~ c~ c~ ¢~ c~ c~ a: h: h) n: halx‘ 00000-0000 mama‘osoaxo‘o‘ox 0 00000000 H 3' (k) (1.1) (5.2) (1.0) (1.0) (1.0) (3.1) (1.0) (1.1) (5.1) (k.%) (5.1) (0.2) (1.0) (1.0) (1.0) (1.1) (%.1) (5.1) (1.0) (1,0) (1.0) (5.5) (5,5) (5.1) (1.0) (%.%) (5.3) (%.1) (5,1) (5,5) (5.5) AB" (R) .6 (10] 0.4 0.9 0.2 0.2 0.2 0.8 0.4 0.6 0.5 0.5 0.6 1.0 0.4 ‘ 0.4 0.4 0.867 0.9 1.0 0.6 0.6 0.6 0.675 0.6 1.05 0.6 0.75 0.8 1.1 0.8 0.7 0.9 # 43 the a priori solution achieves a 22% improvement in the objective function over that obtained for the a posteriori optimum solution. This chapter has discussed and develOped techniques for solving mathematical programming problems involving random vari- ables. In addition to the readily computable a posteriori solution vectors, sub-optimal solutions are obtained from each non-empty intersection of different collections of the possible feasible sets Y(k). These intersected sets satisfy the con- vexity requirement of mathematical programming since the inter- section of convex sets, such as the Y(k) for k = l,2,...,K, is also convex. The optimal and sub-optimal solution techniques were unified in transforming the basic deterministic problem statement to the general form of 4.3.6. Using the techniques developed in this chapter, the behavior of the consumption component, as characterized by a vector y that maximizes the objective function of 3.4.13 subject to the constraints of 3.3.1, may be determined even when the parameters of 3.3.1 and 3.4.13 are random variables. The behavioral realism of the consumption component is significantly enhanced by allowing the behavioral parameters to reflect the random variations of the natural environment. V FUNCTIONAL FORMUIATION OF COMPONENT BEHAVIOR The preceeding chapters were devoted to the macroscOpic formulation and interconnection of state space models for ecological systems. This chapter will suggest Specific formula- tions for the response and state equations of the population develOpmental stage component and for the parameters of the con- sumption component. 5.1 Basic developmental stage The set of response variables to be incorporated into a state-space description are yg, yd, ye, and yr. As a first approximation to the general case, define the following variables as potential state variables for the model of a pOpulation developmental stage. N, the population of the developmental stage; expressed in numbers of individuals, surface area, or biomass, whichever is most appropriate for the modeled organisms. An approximate transformation to convert from one unit to another is assumed known. H, the hunger level of the population; e.g., the empty gut Space in animal populations, and E, the "accessible" energy level of the entire population N; the caloric value of assimilated biomass that is 44 45 readily and safely catabolizable. E may be viewed as an energy pool that accumulates energy from assimilation. It is the source of all energy used by the organism for growth, respiration, and reproduction. G, a variable that characterizes the size or maturity distribution of the N organisms of the developmental stage. G may be a vector. State variables N, H, and E are used in the constraint equations 3.3.1. Designate the state vector Y = 0N,H,E,G)'. The following general expressions relate the response variables to the states and stimuli as in 2.1.1. yg = $803.3.6) = 380156), where 1% is that subset (5.1.1) of the total pOpulation N that is available to exit the stage. N is determined by N and G. yd = sdmfius) -- 3,101.6) (5.1.2) ye = 9e (‘1’.yf) (5.1.3) y1r = around) = stow) (5.1.4) The state equations will be formulated tentatively as difference equations for the period T. 9.1---- Define 3 yb ym. yg yh yd ym , then 1 e N(t + T) N(t) +fi = 3N(Y369ybsyh3ym ,ym ): (50105) i e as: + T) = assay. (5.1.6) 46 E(t + T) = $E(Y,yf,c), where c is the energy (5.1.7) expended in feeding activities, as determined by the consumption com- ponent, and 60: + T) = FGWdaybgyhflm sym) 0 (501-8) i e Biological research has not investigated these relations sufficiently to permit any ready formulation. Hence, approximate relations will be suggested, which should not be taken as neces- sarily accurate, but illustrative of function construction. The hunger and energy levels of individual organisms entering and exiting the developmental stage are assumed tobe approximately equal to the hunger and energy levels in the stage. One key factor in the develOpment is a mathematical formulation of the processing functions 3 E material. First define ye, the material egested per time period and ?e of ingested T, as a vector of the fundamental composition variables of food such as protein, carbohydrates, fats, and inert or unusable material similarly to the ingestion vector yf defined in 3.3.6. That is, let P H P ye protein biomass egested P ye carbohydrate biomass egested c A Y = y = fat biomass egested . (5.1.9) e ef ye inert biomass egested i ye _J volume of egested material L V L .a 47 Assimilation efficiency will be assumed to be a constant for any given organism for eaCh form of food processed. Define Bp’ BC, and Bf as the fraction of ingested biomass that is assimilated for protein, carbohydrate, and fat biomass, respectively. Now, an anabolic efficiency vector B will be defined using estimated conversion factors tabulated by MacFadyen (1963). Let B e (4.025p 5.653c 9.355f O 0)‘ with units of small calories per milligram. Then the energy gained from consumption may be found as B'yf. Hence 5.1.7 may be written as E(t + T) =[E(t) + B'yf .. IJ- - c] o[l+ 111%; (5.1.10) where u, as defined in Section 3.4, may be viewed as the basal metabolic rate times T-N(t) plus the energy used in reproduc- tion. The BMR is tabulated for many species in the literature. The consumption cost c is determined by the consumption com- ponent and is supplied as a stimulus feature to the pOpulation component. The factor 1 +-fi%zi] adjusts the energy level for the change in population, in accordance with the earlier assump- tion about the energy levels of entering and exiting organisms. Holling (1966) has given the following expression for hunger H, for no ingestion during [t, t + T]: H(t + T) = Hk - [Hk - 11(t;)]e'°‘T (5.1.11) where Hk is maximum gut capacity in volumetric units H(t) is hunger at time t and a is a gut evacuation rate coefficient (assumed constant). 48 This expression may be modified to a difference equation form incorporating consumption over the difference period T by ] (5.1.12) writing 5.1.11 as: A _ _ -ar'r. _ . N H(t + 'r) ’{Hk e [11k H(t) + yfv]} [1 + N(t) A where l + -E-- is the papulation correcting factor. Now, N(t) - 1 Ii - H(t) from 5.1.11, 0 - T loge Hk _ H(t+w) or, a = - jrl'loge (fractional evacuation rate during T). A tractable function describing the dynamics of the pOpula- tion distribution is not easily formulated. The technique of char- acterizing the distribution by a log-normal, uniform, or other fixed density pattern is realistically applicable only to stable population structures. Coulman et al (1971) quantized the size or maturity range of the pOpulation stage into distinct sub-stages. The pOpulation constituents were then successively aged through the quantized levels. This developmental sub-stage approximation appears to be a reasonable approach and such a technique will be developed and adOpted here. Let G = (ylyz ... yw)' for a distribution quantized into W sub-stages of develOpment. Let the elements of G be normalized such that W 2 ¥i(t) = l . (5.1.13) Assuming that death, harvesting, feeding, migration and encironmental effects affect individuals of the total pOpula- tion developmental stage equally, the relative densities are not 49 changed by these population alterations. Except for the first sub-stage, the relative p0pulation density of the ith sub-stage Yi may be updated as Yi(t +-T) = 61Yi_1(t) + 62yi(t) (5.1.14) where 61 is proportional to the assimilated energy available for growth. Assume _ sgt) - n. 61 - 1g [NM (5.1.15) where xg is determined for each developmental stage of each species and may be a function of 6, although Kg is assumed constant here. Y1 is affected directly by yb as v10: + T) = 62v1(t).+ yb/(N(t) +19). (5.1.16) Therefore, 36 may be expressed as P I" q 62 W yb/[e + N(t)] .1 .2 O o 61 52 0 C(t + T) = . ° C(t) 4- . (5.1.17) 0 .6 O 0 L- 1 2.4 L .1 Now, 5.1.13 and 5.1.14 imply that W W .E yi(t + T) = E (51yi_1(t) + 62Yi(t)) + y1(t + T) = 1 (5.1.18) 1-l i-2 W W-l 0ft 1 ‘ Y1(t +'T) = 2 52Y1(t) +' 2 61Yi(t) (5.1.19) i=2 i=1 W-l = (51 + 52)122y1(t) + 62yw(t) + 51y1(t) (5.1.20) 50 W-l but 2 yi(t) = 1 - y1(t) - yw(t) . (5.1.21) i=2 Substituting 5.1.16 and 5.1.21 into 5.1.20 yields A (5.1.22) - 51Yw(t) + 52Yw(t) + 61v1(t) A . or, 1 - yb /[N(c) + N] = 52 + 61 - 61Yw(t) = 52 + 61(1 - yw(t))- (5.1.23) _ A Therefore, 52 - 1 - yb /[N(t) +‘N] - 51(1 - yw(t)). (5.1.24) Incorporating 5.1.15 for 61 into 5.1.24 yields 5=1-Kj°-—--),[1-y(t)]-Et- (5125) 2 N + N(t) 3 W N“) Equation 5.1.14 implies that the fraction 61 of the pOpulation of each sub-stage is aged forward during the period T. Since N = wa(t) is the number of individual organisms in the last sub-stage, 61$ will be aged forward into the next developmental stage, constituting the exiting organism yg. Therefore 3g may be written directly as yg = 3g(‘i’,6) = 5133 = élwat) . (5.1.26) The following factors are pertinent to the natural death of organisms, and are therefore necessary elements of 3d: a) a constant generic death rate, 6d, b) malnutrition, a function of accumulative energy loss, as reflected by the state variable E, c) abiotic environmental factors, 6. 51 Assuming the effects of these factors are additive and independent, define the intermediate variable v as v = 5d + (a - 3>'Kd(a - 3) + MW - E 1 v otherwise . The biomass of the material egested is obviously 1.0 minus the fraction assimilated times the ingested biomass. The volume egested by the N(t) organisms must be H(t + ¢) - H(t), or ye = Hk - e'“T[uk - H(t) + yfv] - H(t) = H - H(t) V k = [Hk - H(t)][1 - 6-0T] - yf e'aT . (5.1.28) V Therefore, the function ye may be assembled as 52 000 [Hk - H(t)]. (5.1.29) 3r may be constructed in a manner similar to that used to define 3d. The reproductive rate should be a function of 5b, the generic birth rate, 6, and E. Assuming the effects of these factors on 3r are additive and independent, 3r may be approximated as *4 = N{sb + (a -3)'1 = IAxe-Axdx = fits-Ax - — A 0 DIr—I 55 which comes as no surprise. This a may be used as the searched area in a deterministic evaluation of the consumption component's behavior. However, the random variable a as described by equations 5.2.3 and 5.2.4 is a more realistic characterization of searched area and would be used in the Optimization scheme developed in Chapter IV. Now, with the reasonable assumption that the consumer does not cross his previous path in search- ing, and neglecting correcting factors for the beginning and ending of the searching period, the area covered by the consumer is a function of the path taken during the search, which may be closely approximated by a series of straight and curved segments. The area covered over either the curved or straight paths is identically the same, namely d’= 2rd, as shown by the following paragraphs. (a) Straight-line search path: the area swept out during a search is illustrated in Figure 5.2.1. Searched .................. , A I ‘5. l rea 67 . ’(’(path) ' ;, —4 Ending Point Starting : . \ (senSing Of 14 d i ,J If *1 Figure 5.2.1 Linear search area Area d7: d-(width of search) = d-2r = 2rd (5.2.5) 56 (b) Curved searching path: the area swept out during a search is illustrated in Figure 5.2.2. path d Figure 5.2.2 Curved search area Now, the distance of path d = so, for e in radians. Therefore, the searched area 4 may be computed as: Searched area a - area Bvée = area and - area 0186 = §s+rzze _ s-r 2 2 2 =§ (4:3) = Zr (89) = 2rd (5.2.6) which is independent of the radius of curvature s and the angle 9, for s 2 r. Therefore, it appears reasonable to accept the searched area as 2rd, i.e., 0': 2rd 8 a. or d 2r . (5.2.7) Therefore, t8 = 5— = 2—;$- . (5.2.8) js js ii) tp, the pursuit time, as suggested by Schoener, may be expressed as: 57 t = ‘ (5.2.9) where r is the distance at sighting between the pursuer and the chosen consumable (assumed approximately equal to the sensory range r defined earlier); v is the escape velocity of the prey; and V is the pursuit velocity of the predator. iii) te is given by Holling (1966) and others as a function of the physical size of the organism being consumed and the eating apparatus of the consumer. Holling found for H, Crassa, a mantid, eating houseflies that the eating time is essentially independent of hunger and is well described as a linear function of prey biomass; t = B s, (5.2.10) where Be is a constant determined by the physical eating apparatus of the predator, and the physical structure of the prey; Be has units of time per unit weight, and is the wet weight of the organism U!) being consumed. Since the digestive pause observed in many animals is likely to be a linear function of the prey biomass 58 consumed, this time is incorporated into te by in- crementing Be by the conversion constant associated with the digestive pause. Hence, expressing t as t = tS +'tp + te and in- ij ij serting the results of (i), (ii), and (iii), above, I' = —L + tij 2rV +-(v _ v ) Besi . (5.2.11) J8 3p 1 _ B. Computation of cij The parameter cij’ the energy cost to a consumer of type j for each organism of consummable type 1 located, captured, and eaten, may also be separated into three components; i.e., c = c + c + c (5.2.12) 5 p e ij where cs is the energy cost of searching; c is the energy cost of pursuit; and c is the energy cost of handling and swallowing. Hamilton and Watt have given a quantitative estimate of locomotive costs per unit distance as +av E =55;ij c (5.2.13) where as is a conversion constant, §j is the wet weight of the organism, Bb is related to the basal metabolic rate (BMR) of the organism and may be a function of temperature. Typically, 59 .75 s 6b s .94, BC is a constant peculiar to the consumer, and V is the velocity of locomotion. Energy costs in calories per unit time may be derived from 5.2.13 by multiplying E by V, therefore L v as 3 ab+e°vjs - A Bb+BCij _ '18 b .1 cs — vj Basj x ts - 2 . r - V s js 8 +8 V = QQBbgj b c 18)/2r, (5.2.14) and . W535 — A 5b+95Vj - r-Vjpaasj P cp - Vj Basj P X tp - V - V, . (5.2.15) P j 1 P Assuming the consumer stops while eating, and that the only energy cost during consumption is the minimum metabolic cost, which has already been con- sidered in the parameter p, ce = O. Incorporating these expressions into equation 5.2.12 yields av a as vaj3 VJ rgj c jp cij=5a§1b —-J-§—r———+ v _v . (5.2.16) j i p Computation of bij The value of bij’ the energy benefit to a consumer of type j derived from the digestion of one consummable organism of type i, may be computed by using the composition vector 31 defined in 3.3.5 and the B 6O vector of Section 5.1 as = ' = O + O + O O O O bi]. B si 4 02(3ps1,lp 5 655csic 9 BSBfSif + 0 + 0 (5 2 17) 5.3 Summary of required parameters The parameters required in formulating the multiple p0pu- lation interaction model develOped here are tabulated in Tables 5.3.1, 5.3.2, and 5.3.3. The values of some of these parameters may be obtained from the literature, while others may need to be estimated experimentally, or indirectly approximated. The initial conditions of the state variables N, G, E, and H must be specified, as well as the interaction area a? and the period T. Additionally, the elements of the environmental parameter vector 6 must be specified. Multi-dimensional entries in Table 5.2.1 are designated by underlining and functions of time are given per unit of time. These time functions should be particularized to the period T for simulations. 5.4 An example of simulation results Due to the non-linearities inherent in this state model, a closed-form solution for the state space trajectory is not easily determined. However, several computer simulations of single species pOpulations have been carried out. These illustrate the behavior of population components coupled by the growth and reproductive mechanisms as in Figure 2.3.1. For example, a three-stage single species p0pu1ation model was constructed with three sub-stages of maturation in the infant 61 Table 5.3.1 Parameters required for each population stage Relevant Symbol Definition Units Equation BMR Basal metabolic rate calories/time 5.1.10 3 Assimilation efficiency calories assimilated 5.1.10 p for protein mg biomass (protein) Bc Assimilation efficiency calories assimilated 5.1.10 for carbohydrates mg biomass (carbohy.) Bf Assimilation efficiency calories assimilated 5.1.10 for fats mg biomass (fats) Metabolic costs for main- calories 5.1.10 tenance and reproduction per time period T m) k Maximum gut capacity milliliters 5.1.11 Gut evacuation rate time.1 5.1.11 3 Fractional growth rate calories-1 5.1.15 xd Energy-relative death 1 5.1.27 rate calorie-time Environmental factor 1 5.1.27 mortality coefficients (6 unit)2-time Reference level for 6 same units as 6 5.1.27 id Generic death rate time.1 5.1.27 b Generic reproduction rate time"1 5.1.29 Eb Environmental effect 1 5.1.29 birth rate coeff. time-(6 unit)2 Ab Energy-relative birth 1 5.1.29 rate . time-calorie 62 Table 5.3.2 Additional parameters population stage required for each consumer Symbol Definition Units r Sensory range meters V Searching (cruising) meters/time js velocity Vj Pursuit velocity meters/time P Be Ingesting rate time/mg/biomass Ba Metabolism conversion calories/mg-time factor 6b Basal metabolic rate ex- ponent of organism weight (may be function none of temperature) Bc Exponential coefficient of velocity for meta- time/meter bolic rate §j Wet weight of organism milligrams Relevant Eguation 5.2.8 5.2.8 5.2.9 5.2.10 5.2.13 5.2.13 5.2.13 5.2.13 Table 5.3.3 Additional parameters required for each consumable population stage Symbol Definition 31 Volume/organism siv Protein biomass/organism sip Carbohydrate biomass/ c organism sif Fat biomass/organism sii Inert biomass/organism vi Escape velocity s Wet weight of organism Units milliliters milligrams milligrams milligrams milligrams neters/time milligrams Relevant Eguation 3.3.1 3.3.5 3.3.5 3.3.5 3.3.5 5.2.9 5.2.10 63 population stage, five sub-stages in the adolescent stage, and a single adult develOpmental stage. One set of parameter values used in this simulation is given in Table 5.4.1. These para- meters were hypothesized by assuming a homeothermic mammalian species with approximate weights of 20 grams for adults, 10 grams for adolescents, and 4 grams for infants. Under this assumption, realistic estimates can be obtained for many parameters, in particular, those parameters marked by an asterisk in Table 5.4.1. The other parameters were assumed to be close to realistic values. Average growth rates were chosen to allow ten days spent in the infant stage and 20 days in the adolescent stage. The average adult life span was chosen to be 20 days. Harvesting, as represented by yh, was set at 2% of each stage's pOpulation daily. The abiotic environmental stimulus vector 6 was simulated as a one-dimensional stimulus consist- ing of ambient temperature in degrees centigrade, which varied sinusoidally from 5°C to 25°C with a one-year period. The uaximum volume of food ingested daily was, of course, bounded by the hunger levels, and the food intake was bounded above by a maximum food input yf made available to each stage daily. The yf's used in the simulation were F'4000 T 30001 F1 200 q 4000 2000 1000 y = 4000 , y = 1500 , and y = 1000 , f1 f2 f3 9000 5000 4000 _ 200.. L 150.. _ 100‘ 64 and food acquisition costs were 11.5, 6.32, and 7.4 calories/mg for the infant, adolescent, and adult stages, respectively. Table 5.4.1 Parameters for example simulation Parameter Stage 1 BMR * 150 0.2 a? BC 0.15 Bf 0.1 u * 3.6 * Hk 0.2 a * 0.99 0.03 x8 xd 0.01 Kd 0.002 3 * 15 * 6d 0 * 0b 0 Stage 2 Stage 3 235 376 0.2 0.2 0.15 0.15 0.1 0.1 5.64 9.02 9yr(3) 1.0 3.0 0.99 0.99 ‘0.0148 - 0.005 0.05 0.001 0.0 15 15 O 0.05 0 0.01 0 0.0005 0 0.10 Units calories/hr calories assimilated mg protein biomass calories assimilated mg carbohydrate biomass calories assimilated mg fat biomass Kcal/day milliliters day"1 Kcalm1 (Kcal-day)-1 (Kcal-dayf'1 Seventeen state variables are used to characterize this example population; N, H, and E for each of the three develOp- mental stages, plus G vectors of three and five elements for the infant and adolescent population stages, respectively. Observation of these states indicate that no equilibrium point 65 was reached after 1000 iterations, using a time increment of one day. It is encouraging that the populations in the simulation did not become extinct or unbounded since that would contradict biological intuition. The other simulations also exhibited realistic trajectories in the state space and no population explosions or crashes occurred. For the multi-species case, at least seven more parameters must be supplied for each developmental stage that is coupled to a consumption component. No biological simulation experience in this area has yet been obtained: VI sums)! AND OPEN QUESTIONS In this final chapter, a summary of the original contribu- tions is presented, followed by an indication of additional areas for study which were encountered during the development of this thesis. A generalized model for the population of a developmental stage in an organism's life cycle was develOped in Chapter II. This model was constructed as a component for a dynamic system model of a pOpulation. The formulation integrates the quali- tative knowledge of the biological sciences with the structural requisites of state-space theory to produce a quantitative model structure that is sufficiently general in its formulation to allow coupling into more complex ecosystem models. The population developmental stage components are coupled to assemble a multi-species interactive segment of an ecosystem. The competition activities of a group of consumer Species were condensed into an abstract consumption component in Chapter III. Previously suggested theories of consumptive behaviors were directly incorporated into the formulation of the consumption component, and appropriate strategies Of feeding were implemented for both immature and adult organisms. The optimal behavior of the consumption component is formulated in terms Of a mathematical programming problem with linear constraints and a quadratic Objective function. 66 67 The coupling Of the developmental and consumption components treated in Chapters II and III can perhaps be further illustrated by the construction Of a typical ecosystem.mOdel. Figure 6.1.1 depicts a three trophic-level ecosystem involving two species of plants P1 and P2, one herbivorous species H, one omnivorous species 0, and one species Of carnivores C. Each Species is characterized by three develOpmental stages, except for P2 and 0, which have four stages of develOpment. This hypothetical system illustrates the usage of four consumption components 0C1, CCZ: C03: and CC4. Two accumulative components labeled "Nutrient Pool" are included. Interspecific couplings are drawn with solid lines, and intraspecific couplings are shown by dashed lines. Component terminals are designated by the subscripts of the previously defined variables, i.e., the yr terminal is labeled r. To simplify the diagram, many behavioral features such as egestion, death, migration, and abiotic environmental stimuli are not shown in Figure 6.1.1. Of course, the omitted terminals must be included for the model to be considered complete. TO enrich the realism of the behavior of the consumption component, random variables are allowed in the functional formula- tion of the optimization criteria. Chapter IV describes a gener- alized approach to the solution of a mathematical Optimization problem involving random variates. The techniques develOped for this general problem represent another contribution Of this thesis, since previously-described approaches are quite particularized and permit only linear Objective functions. Nutrient Pool ---q CC 68 GEOSPHERIC INPUT Nutrient A Pool H . ‘ Species : ' f f r. f f P2 f r f :’1‘ L I - - - — u s s s s b h 6‘! h cc1 q I f r I g b --‘------- ---1 Figure 6.1.1 Example ecosystem model 69 The first four chapters Of this thesis were concerned with the construction Of a viable framework for ecosystem modeling. Chapter V quantitatively formulated many of the biological func- tions necessary to complete the behavioral descriptions of the population and consumption components. Only heuristic functions could be ascribed to some behavioral relations, while others were well defined, particularly for the consumption component. The primary objective of this thesis has been to develop a structure for the modeling of ecological systems and hence has been basically eXpository in nature. For each problem resolved, several questions have been brought into focus for additional research. Obviously, the most pressing open questions generated by this thesis are concerned with experimental studies directed at the formulation of growth, assimilation, death, and repro- ductive processes Of biological organisms as functions of various stimuli. These general studies need to be augmented in specific applications to derive realistic parameters and obtain simulated results for multi-species interacting systems. In addition to the gut capacity constraint on the feeding behavior of animal consumers, Mayer (1964) argues the existence of a glucostatic satiation level. This constraint is determined for the concentration Of blood glucose and acts to inhibit further consumption past some limit; that is, there is an upper bound on the energy level per individual. Gut evacuation rate and assimilation efficiency were assumed constants in the formulations of Chapter V. These two 7O factors are probably not constant and are likely to be related functions. The precise assimilative process is not well under- stood and further research on digestion is certainly indicated. The technique develOped for optimizing a probabilistic programming problem is quite tedious in the present form and perhaps could be made more tractable by directed or random search techniques. Also, the investigation of various decision rules in the algorithm may yield additional useful Optimal objectives. REFERENCES REFERENCES Aoki, M. 1971. Introduction £9_Optimi§ation Techniques, The Macmillan Company, New York, 335 p. Andrewartha, H.G. 1957. "The Use of Conceptual Models in POpulation Ecology." Cold Spring Harbor Symposium.gg_ Quantitative Biology 22: 219-236. Charnes, A. and Cooper, W.W. 1959. "Chance-Constrained Programming." Management Science 6: 73-79. . 1961. Management Models and Industria1.Applications 2£_Linear Pro rammin , vol. II, Wiley, New York: 682-687. Cockerman, C.C. and Burrows, P.M, 1971. "POpulationS Of Interacting.Autogenous Components." 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