W U163; 59:! new “.0“?th LIBRARY Michigan State University W This is to certify that the thesis entitled A Matrix Structure for Modeling Population Dynamics presented by A.R. Tipton has been accepted towards fulfillment of the requirements for Ph.D. Fisheries & Wildlife degree in prof Date September 19, 1975 0-7 639 q Q44 \ a;\\ ABSTRACT A MATRIX STRUCTURE FOR MODELING POPULATION DYNAMICS By Alan Ray Tipton The use of mathematical models in population ecology has shown a marked increase in recent years. For the most part these models have been of two types, general population models whose predictions are qualitative in nature, and more detailed models for dealing with quite Specific questions. The purpose of this study was to develop a model with a wide range of generality but capable of including the essential biology that pertains to specific population phenomena. A matrix structure was selected as the basic model structure because most empirical data are collected over discrete intervals. Most species exhibit age- and sex-specific survival and reproductive rates and therefore these characteristics were included as parameters in the model. In addition, few species exhibit a continuous breeding pattern so provisions were made in the model to separate the breeding and nonbreeding components of the population. The model allows for seasonal changes in survival and reproduction and can mimic population response to changing environmental parameters. Because the values for most demographic parameters are estimated means, stochastic functions of survival and reproduction were included in the model. The straightforward representation of demographic machinery provides a means to establish functional dependency of regulatory mechanisms thought to impinge on the population. Alan Ray Tipton Simulations were run on data for several species to illustrate the generality and capabilities of the model, and to provide realistic estimates of population response. A MATRIX STRUCTURE FOR MODELING POPULATION DYNAMICS By Alan Ray Tipton A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Fisheries and Wildlife 1975 ACKNOWLEDGMENTS I wish to thank Walt Conley, chairman of my Doctoral committee for his guidance and patience during this work. I am also grateful to the other members of my committee, Drs. L. W. Gysel, E. D. Goodman, W. E. Cooper, for their many suggestions and editorial assistance. Special appreciation is extended to F. C. Elliott, who graciously provided data; M. J. Kutilek and T. M. Butynski who provided assistance and references; and J. D. Nichols and J. B. Hestbeck who provided essential discussions, evaluations and intellectual support. I am also indebted to J. J. Church and J. A. Boger for technical assistance and moral support. Use of the Michigan State University Computing Facilities was made possible through support, in part, from the National Science Foundation. ii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . LIST OF FIGURES . INTRODUCTION METHODS AND RESULTS . . . . . . . A Basic Model Structure . Sex Ratio Stochastic Survival Function . Breeding Frequency . Expected Reproductive Effort Variable Time Intervals . . . Density Dependence . DISCUSSION . . . . . . APPENDIX . . . . . . . . . . . . . BIBLIOGRAPHY iii 12 16 18 25 27 31 ho an A8 Table A1 A2 LIST OF TABLES Demographic parameters for Capra aegagrus. (From Papageorgiou, IQTA). Equations used for generating protein—energy requirements for Capra aegagrus. (Modified from Moen, 1973; Agricultural Research Council, 1965; and French et al., l9hh). Demographic parameters for Microtus pennsylvanicus. (From Krebs and Myers, 197A: Table IX; Myers and Krebs, 1971; and data made available by Dr. F. C. Elliott, Crops and Soil Science Department, Michigan State University). Proportion of Microtus pennsylvanicus females in reproductive condition. (From Krebs et al., 1969: Fig. 12). iv 3h AS A? Fi 7a LIST OF FIGURES Estimated density of M, pennsylvanicus population predicted by the nonstochastic model using demographic data from Appendix (Table l) and proportion of females in breeding condition (Appendix, Table 2). Comparison of predicted population density from two simulation runs using stochastic function for survival (P ) and demographic data for M, pennsyl- vanicus (Apgendix, Tables 1 and 2). Comparison of predicted age structure at t = Ah for the two stochastic simulation runs in Figure 2 (A and C) and the nonstochastic simulation runs in Figure 1 (B). Comparison of predicted stable age distributions for different proportions of females in breeding condition (A = 1.0; B = .80; C = .39; D = .20). Demographic data for M, pennsylvanicus (Appendix, Table 1). Pro- portion of females in reproductive condition is equal for all age classes and constant for all time periods. Comparison of population density for M, pennsylvanicus population with the proportion of females in breeding condition in the second age class equivalent to that of the older age classes (A) and population with reduced proportion of breeding females in the second age class (B). Data for both simulations in Appendix (Tables 1 and 2). Empirical and predicted frequency distributions of litter sizes of M, pennsylvanicus. See text for discussion. Comparison of predicted population density for M, pennsylvanicus using stochastic functions for reproduction and survival. Two selected simulation runs (A and C) are compared to the nonstochastic matrix model (B). Survival data and proportion of females in reproductive condition for all three runs in Appendix (Tables 1 and 2). Litter sizes for nonstochastic run (Appendix, Table 1). See text for derivation of litter sizes for stochastic runs. l7 19 22 2h 26 28 LIST OF FIGURES (Cont‘d): Number 7b 10 ll MeanS, 95% confidence intervals and coefficients of variation for 10 simulation runs using stochastic functions for reproduction and survival. Demographic data for M, pennsylvanicus (Appendix, Tables 1 and 2). Comparison of predicted population density for M, pennsylvanicus using model containing 2A (A) vs. 11 (B) age classes. Data for 2h age class run from Appendix (Tables I and 2). See text for discussion of derivation for 11 age class run. Means, 95% confidence intervals, and coefficients of variation for 10 simulation runs using density- dependent functions for survival and reproduction and stochastic function for survival. Demographic data for Capra aegagrus (Table l). Predicted age structure and protein—energy require- ments for Capra aegagrus for 1953 and 1957 selected from one simulation run Figure 9. Demographic data for Capra aegagrus (Table l). Comparison of predicted age structure for Capra aegagrus for 1973 for nonstochastic matrix model (A) and stochastic model (C) using density-dependent functions of survival and reproduction, and the observed age structure (B), of the population. Demographic data Capra aegagrus (Table l). vi 3O 36 37 39 INTRODUCTION In recent years there has been an increase in the use of mathematical models in population ecology. These models have helped to improve understanding of population processes and interactions between a population and its environment (Watt, 1956; Watt, 1968; Gross, 1972). Because of the richness and complexity of ecological systems, such models by their nature, must be an abstraction of natural phenomena. The construction of a model that contains all processes that pertain to a population, existing in a multispecies ecosystem, requires a complexity exceeding our current understanding; in addition, the size of such a model would exceed the capacity of even the largest computers. Even if the technology were available for such an effort many of the necessary relationships have not been adequately defined, much less measured with the accuracy that would permit their use. The results from a model that is a 1:1 representation of the natural world would provide little insight into pertinent processes of population dynamics because of its complexity. Clearly, if a model is to be of benefit it must be a simplification of the systems it presumes to describe. Given that a model must be a simplification of reality two schools have developed concerning the purpose of models, which in turn determines their complexity and structure. On the one hand workers have been concerned with models of a qualitative nature, sacrificing precision and realism for generality when dealing with the complexities of multispecies ecosystems (e.g. Levins, 1966; May, 1973; Maynard Smith, l97h). Conversely, there is a class of "tactical" models that pertain to specific situations and contain a detailed description of the subject being modeled (e.g. Holling, 1966, 1968). A continuous spectrum of possible models can be envisioned ranging from empirical models to abstract qualitative models. Qualitative models contain a wide range of generality, and provide a conceptual frame for analyzing broad classes of phenomena as well as providing theoretical principles. They do not, however, provide answers or predictions concerning specific questions. Empirical models are specific in purpose and provide a means for answering specific questions about population phenomena. Presumably, the role of qualitative models is to establish a firm basis from which "ecological engineers" can construct specific models, thus combining theoretical concepts and the detailed information that has been collected on particular population processes (May, 1973; Maynard Smith, l97h). Ecology is now at the stage where there are some general theoretical "beliefs", such as the niche concept, r—k selection and trophic structure, which are analogous to the ideal gas and friction- less plane laws of the physical sciences. However, as a result of extensive laboratory and field studies, questions arise concerning the utility of some of these general rules when applied to real populations (Conley, ms; Conley et al. 1975; Neill, 197h; Nichols et al. ms). It is apparent as more detailed knowledge of population dynamics becomes available, models that can include this information are necessary. There is a need for qualitative models that provide a theoretical framework about phenomena for which sufficient information is not available. There is also, however, an increasing need for models that are structured for situations where understanding is such that specific questions arise and can be answered. The question that must be asked is "What is the purpose of such a model?" Contrary to Levins (1966) a model should be considered a hypothesis and therefore subject to criteria of testability via Popper (1959, 1963). Excluding certain aspects leaves a model in- complete in its representation of biological processes. However, the selection of processes to be included in the model, their mathematical representation, and the means in which they interact with other components satisfies all requirements for construction of scientific hypotheses. The model can then be tested using existing data or through experimentation that provides such data. If it cannot provide information that pertains to its purpose then rejection must follow. Certainly one result of using models is to generate hypotheses and pose questions about phenomena that are incompletely understood, but the concept that a model is not verifiable by experimentation begs the question of its original purpose. It was the purpose of this study to construct a single—species population model that interfaces population theory and empirical data. The goal was a model with a wide range of generality, but precise enough to make definitive statements about populations and their interactions with their environment. The generality goal is a difficult one to attain because of the complexity and diversity of the environment in which a population exists. Generality should not be confused with simplicity. A model which is general in nature must encompass various complexities that are exhibited by different species as they deal with their environments. A compromise was made in this respect by providing a framework of demographic parameters that can vary according to temporal and spatial changes in the environment, but without attempting to provide actual mechanisms that are responsible for these changes. Demographic processes on which regulatory forces are presumed to impinge were included. The straightforward representation in the model provides the means for better interpretation of results, and leads the way to meaningful application and testing of hypotheses suggested. Additionally, the model structure allows inclusion of mechanisms that are known to affect demographic parameters of particular species. Thus, the model can cope with quite specific biological problems while maintaining general applicability. METHODS AND RESULTS A Basic Model Structure Early population models attempted to describe the population with a single variable, population density. The simplest of these is the exponential growth model that assumes the rate of change of the popula- tion is dependent on the size of the population. Modifications have produced models that are capable of including asymptotic limits to growth, time delays for population response and the interaction of two or more species. Inherent in all of these "whole population models" is the assumption of fixed age structure or that all animals are qualitatively identical. Use of these models provides little insight into population processes, nor can such models be used to predict more than broad trends and responses. Because most animal species exhibit age-specific survival and reproductive rates any realistic population model must be capable of distinguishing animals of different ages. By separating the population into distinct age classes and considering discrete time intervals, the number of new females that will enter the population at the next time interval is given by m n = 2 F n l 0,t+1 x=0 x x,t ( ) where nx,t is the number of individuals in age class x at time t, m is the last age class that contains reproductive individuals, and Ex is a function of fecundity defined as Ex = the number of daughters born in the interval t to t+1 per female alive aged x to x+1 at time t who will be alive in the age class 0 at time t+1 (Leslie, 19u5). The number of individuals in age class x at time t who will survive to enter the x+1 age class at time t+1 is obtained by "x+1,t+1 = Ex "x,t (2) where Rx = the probability that a female aged x to x+1 at time t will be alive in the age class x+1 to x+2 at time t+1 (Leslie, l9h5). These equations form a system of first—order linear difference equations. In matrix notation this is Nt+1=MWt where Nt is a column vector with elements representing animals per age class I \ n0,t "1,: Nt = . (3) n m,t k and M has the form F0 F1 F2 Sm—J F5 p - _ _ - 0 M = - p1 _ . - - (u) u__- - _ - - Em-J - _m_t Systems of difference equations have long been used by human demographers (Cannon, 1895; Bowley, 192A; Whelpton, I936). The matrix form of this system was first used by Bernardelli (19M) and Lewis (l9h2), and was thoroughly investigated by Leslie (l9h5, l9h8). The P; values in the matrix were derived by Leslie (l9h5) as L x+1 Pm: L (5) x where x+1 L =f 2 d (6) or approximately Z + Z _ x x+1 Lx — ————————2 (7) Lx is the average number of individuals in age class x to x+1 and Zx is the proportion of animals from a cohort of lo animals that survive to enter the xth age class. For the Ex values, Leslie (l9h5) considered the survival of the daughters within the time interval t to t+1, and the average number of females in age class x to x+1, as being exposed to the reproductive rate mx+%,x+1 for the time interval t to t+%, and those that survived this period as being exposed to a reproductive rate of m . . } x+1,x+1% for the time interval t+2 to t+1. Values for Ex are therefore a combination of a reproductive rate and survival rates for the reproducing females and the offspring. The above derivation is complicated by the discrete approximation of reproduction and mortality that actually occurs continuously during the time interval. Recently, to simplify the calculations for these functions, reproduction and mortality were considered to occur at the first of each time interval (Leslie, 1966; Smith, 1973). Other derivations have used expected production of daughters, mm, for the Ex values (Usher, I972, I973; Emlen, 1973, p. 251—256). These various derivations for the elements of the projection matrix result from the attempt to fit continuously occurring biological processes into the discrete framework of the matrix model. The choice of a discrete structure over a continuous one was made because the discrete model is better suited for empirical data collected at discrete intervals. Mathematical approximations for processes of reproduction and survival depend on the biology of the species being considered; however, by judicious choice of time units used in the model structure discussed below, these approximations can include essential biology with a minimum of mathematical difficulty. By excluding females in post—reproductive age classes, the Leslie matrix is a non-negative irreducible matrix. The matrix will always have a positive eigenvalue, A0, such that the moduli of all other eigenvalues of the matrix will not exceed A0 (Gantmacher, 1959, p. 58). Corresponding to A0 is an eigenvector, To, with all non-negative elements. Biologically, given assumptions of constancy, A0 is the finite rate of increase per time interval, and To is the stable age distribution which will ultimately be attained by the population. The existence of two consecutive non-zero elements in the top row of the matrix insures that the modulus of A0 will be strictly greater than the moduli of all the other eigenvalues and the population will reach a stable age distribution. Recently there has been increased interest in population stability theory. Theoretical aspects of the existence of a dominant eigenvalue and attainment of a stable age distribution have been discussed by Lopez (1961), Skellam (1967), Sykes (1969), MacFarland (1969) and Cull and Voght (1973, l97h). The concept of a stable age distribution is of questionable value when considering populations whose demographic parameters are changing in response to a changing environment. The basic projection matrix contains age—specific reproduction and survival. Biologically these are necessary parameters for describing population processes. However, these parameters are not sufficient to fully describe complex demographic patterns that most species exhibit. The following assumptions, inherent in the basic Leslie model, limit its applicability and detract from the biological realism of resulting simulations. 1. Survival and reproductive rates are constant over time and are therefore independent of changes in the population or in the environment. 2. The inclusion of only females in the population assumes a 1:1 sex ratio and equal survival rates for both sexes. 3. The size of the age classes x to x+1 must be equal for all x and must also be equal to the time intervals t to t+1. This imposes a restriction on the number of age classes used to describe the population since the age classes must be small enough to permit an animal to move into the next age class during each time interval. Also, the time intervals must be large enough to permit only one reproductive effort. 10 A. All females in the population reproduce during each time interval and post—reproductive animals are not included. 5. The response of the population structure is instantaneous. 6. The elements of the matrix are constant and independent of population density, therefore population growth is deterministic and exponential. These assumptions greatly reduce applicability, in the choice of species that have life histories that conform to the biological restrictions, and in the results of the simulations. Since the initial development various studies have utilized the matrix structure for population projections. Several modifications to the basic matrix have been directed at some of the above restric— tions. Leslie (19A8, 1959) discussed the consequences of using matrix elements that were density-dependent, therefore producing population growth asymptotic to some upper limit. Pennycuick et a1. (1968), Pennycuick (1969) and Fowler and Smith (1973) developed density- dependent functions for survival and reproduction. Leslie (1959) used a stochastic function to increase or reduce survival and repro- duction in response to weather conditions. Skellam (1966) divided t (in years) into n different periods using separate matrices to include seasonal variation in reproduction and survival. Williamson (1959) developed a matrix that included both sexes to describe species with sex specific survival and reproduction. Lefkovitch (1965) constructed a projection matrix that contained development time of the organisms by grouping the population into size classes rather than age classes. He utilized non-zero elements 11 throughout the matrix to represent biological dependence of one stage on all previous stages. Under certain conditions an animal either remains in the same stage or passes through one or more stages in one time interval. A special case of this general form has non-zero diagonal elements representing the probability that an animal will remain in the same age class for the next time interval (Usher, 1972). This modification is appropriate for species where survival and reproductive rates are constant for older animals, or for species that have several adjacent age classes with similar survival and reproductive rates. Usher (1972) calculated the reproductive rates (F5) for a red deer population by combining the number of females in breeding condition with an individual reproductive effort of one calf per effort. In essence he utilized the proportion of the population in breeding condition but only in the initial determination of the matrix elements and not as a part of the model structure subject to change over time. Stochastic forms of the basic matrix have been developed. Pollard (1966) selected the number of animals in each age class at time t from a distribution determined from the expected mean and variance of age class size. Skellam (1967) developed a stochastic analogue to the Leslie matrix, with the matrix elements expressed as probability density functions of the number of females in each age class, and the joint distribution of the number of offspring and the number of survivors arising from a single individual in an age class. 12 For the most part these modifications have been concerned with a particular species and specific environmental conditions. There has not been an attempt to construct a comprehensive model by relaxing all of the above restrictions. This paper presents a matrix structure that has a wide range of applicability and includes those demographic processes necessary to describe the dynamics of real populations. To illustrate the general applicability of the model, simulations were run on several species for which adequate data were available. For comparative purposes and to maintain continuity between sections pertaining to specific processes, a majority of the simulation runs utilized data on the meadow vole, Microtus pennsylvanicus (Krebs et al., 1969; Keller and Krebs, 1970; Myers and Krebs, 1971; Krebs and Myers, 197A). A description of these data is presented in the Appendix. The model was written in Fortran IV and simulations were run on a CDC 6500 computer at the Michigan State University Computer Center. The following sections pertain to selection and inclusion of necessary demographic processes. The mathematical representation of these processes is described and the results and implications of their inclusion are discussed. Sex Ratio By assuming that the sex ratio, the proportion of males to females, for a population is 1:1, and therefore considering only females, the applicability of a population model is greatly reduced. The inclusion of males allows for sex specific survival and reproductive rates. Information concerning age—specific sex ratios is useful in determining 13 causal mechanisms for variation in sex ratios. Age-specific sex ratios can be categorized as ratio at conception, at birth, at the end of the period of parental care, for newly independent nonbreeding animals and for the older breeding animals. The effect of sex specific survival rates has been well documented and is the main cause for a sex ratio other than 1 (Moen, 1973, p. 391—395; Leigh, 1970; Smith, 197A). However, recent studies indicate that unequal sex-specific reproductive rates may also be a cause of sex ratios that are not equal to 1. Fisher (1930, p. lh2-lh3) has shown that natural selec- tion will act to produce a 1:1 sex ratio at the end of the period of parental care, for those animals that expend energy on the young after birth, or at birth for those species that do not provide parental care. Fisher's argument is that the reproductive values for males and females must be equal in sexually reproducing diploid species. Energy expended through reproduction and parental care therefore must also be equal for each sex. Kolman (1960) and Willson and Pianka (1963) discuss sex ratios other than 1 at birth resulting from differential energy expenditure by parents in sexually dimorphic species. Shaw and Mohler (1953) discuss the possibility of sex ratio genes and their effect on a population's sex ratio. Hamilton (1967) considers effects of local competition for mates within a population and sex linked genetic effects on the population's sex ratio. Myers and Krebs (1971) discuss the possibility of a slight excess, (3%) in the production of male M, pennsylvanicus in wild populations. From genetic studies (Tamarin and Krebs, 1969) it was determined that there could be a relationship between genetic structure and sex ratio. Obviously the question of 1h causes of sex ratios other than 1 have not been completely deter- mined. Williamson (1959) developed a single matrix that included both sexes. In this model two separate matrices and age structure vectors are used. Mathematically, this extension to include both sexes is: m 2 F n n = . . x,t x,1,t t = 0,i,t+1 38:0 (8) ([1 for females [2 for males nx+1,i,t+1 = Bx,i nx,i,t x = 0’1’2' ' .,m-1 (9) with elements of the total population age vector given by 2 — Z nx,t+1 — n i=1 x,t,t+] Usher (1972) utilized Williamson's single matrix two-sex model to simulate a red deer population using data from Lowe (1969). Simulations using the two-sex two—matrix model described above produced identical population estimates. The use of a separate matrix for each sex is conceptually easier to visualize and is more amenable to other functional inclusions. Figure 1 illustrates population growth for M, pennsylvanicus using the reproductive and sex-specific survival rates and proportion of females in reproductive condition given in the Appendix (Tables 1 and 2). Assuming a 1:1 sex ratio at birth, population density at t = A8 was Al composed of 53% females and h7% males. An additional simulation was run using the same reproductive and survival rates but with a 53zh7 15 .Am tapes .xaesmmmahmccum um mo hpflmcoo popwaflpmm .H enemas t5eL Oo'o Am4<>mmhz. xmw>> NV mi; mm mm 1 1 4 1 cm 1 o. 1 1 1 COO mans .OON LOOn AlISNBO 16 sex ratio at birth as suggested by Myers and Krebs (1971). Here the population density at t = A8 was 32 composed of 1:1 sex ratio. This illustrates the effects of sex ratio of offspring on the ultimate sex ratio and growth of the population. Stochastic Survival Function For most discussions of stable age distributions the first age classes contain as many as 100 to 1000 animals depending on the number of age classes in the population and the stable age distribu- tion. The large number is necessary to include at least one animal in older age classes. The projective nature of the original Leslie matrix, multiplying survival values less than one by the number of animals in each age class, creates age classes with fractional parts of animals unless population size is kept unrealistically high. For simulating low density populations this inclusion of fractional parts of animals is important. To eliminate this artifact of the model a random number is drawn for each animal in an age class and compared with the Ex value for that age class. If the number is greater than the Ex value the animal is eliminated from the population; otherwise, the animal moves into the next age class. Thus, older animals appear in the population and individual animals can remain in the popula— tion for more than one time interval. Figure 2 illustrates two simulation runs on the M, pennsylvanicus population using this stochastic survival function. The variation in population response at t = AA is the result of an unequal proportion of females and the 17 .Am one H moapwe .xflpcmmmamwccmm um pom memo oflsmoawoemp one Aamv Hm>fl>sdm Mom GOHpocSm oflpmmnoOpm wcflmz Am .Hdvo mmoHo owe pcooom map cH COHpHocoo wcHooopQ cH mOHoaom wo coHpnomosQ one 39H: COHpoHsmom mSOHco>Hmmccom am now thmsoo GOHpmHsmom mo comHsmmaoo .m ossmam Z‘fxl $32132. xmmafimsze mm mm cm ON 0.. N_ m o o 4 [I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I o— 3'2 AllSNBO . on i Oh i Om voo— .. CON _ Loon 25 Expected Reproductive Effort Expected age—specific reproductive performance (mm) is a combina— tion of individual reproductive efforts and the proportion of the age class in breeding condition. Additionally, reproductive perfor— mance in the field is affected by the intrinsic status of the population (e.g. density, genetic structure, sex ratio, age distribution, and intraspecific competition); and extrinsic factors such as temperature, precipitation, predation, and interspecific competition for space or food, nutrition, etc. In the event these factors can be ignored, or their effect determined and approximated, the fact that mx values are an average of individual efforts of all reproductively active females in an age class still detracts from the realism of simulations using such data. Given sufficient reproductive data, a frequency distribution of reproductive efforts (litter size) for each age class provides full information on expected performance. Such data bases are presently available from laboratory colony studies and represent a theoretical maximum for reproductive efforts. This procedure was developed by Conley et al. (ms) using data from three taxa of Peromyscus. A binomial distribution (Pi + Qi)k was chosen, where Pi represents the probability of an ovum being fertilized and producing a live offspring for the ith age class and k is the maximum litter size (Conley et al. ms). A similar analysis on the M, pennsylvanicus data was conducted as a part of this study. Figure 6 represents the empirical and theoretical distributions for the M, pennsylvanicus litter data. 26 com .QOHmmsOme Mom vamp .mSOHom>Hhmccom 2% mo mONHm soppHH mo chHpanspmHo zocosvoae oOPOHUOMm one HoOHHHmsm .m whdem 26a. ouhuaug ... 70> 50> 40> 30> 20> JLJJLLJIJJJAJJJJLJJAJJJJIJJLIJLJJLLJJJJJLJJJLJ 4 8 l2 IS 20 26 28 32 35 4O 44 ALLAJJAJJL‘JJJJJ44ALJJJAAAAIAJJJJJJJJAJJJ 8 l2 IS 20 24 28 ' 32 36 4O 44 TIME (2WEEK INTERVALS) 48 A L A A 29 and long life span increases the size of the resultant projection matrix. An artifact of using a large number of age classes is the time the population takes to reach a stable age distribution from some initial population size, and because of an increase in response time. Simulations such as the introduction of animals into new areas or harvesting animals from existing populations are thus less applicable to real populations. Leslie (l9A5) and Keyfitz (1968) provide methods for combining age classes and approximating survival and reproductive rates for these larger age classes. However, time intervals must also be increased with a resultant loss in resolution concerning seasonal survival and reproduction. For species with constant survival and reproductive rates over several age classes it is possible to use non—zero diagonal elements to reduce the number of age classes in the projection matrix (Lefkovitch, 1965; Usher, 1972). An average survival value for these age classes is P; = J/' m ' (12) H Pi 7L=k+1 where m = total number of age classes, k = the last age class to be included in the reduced matrix and j = m-(k+1). Setting this survival value as Ph,n (i.e. the lower right diagonal element) in the reduced projection matrix, the population can be described by n age classes. P; is the probability that an animal will remain in the last age class for the next time interval. Reproductive and survival rates for M, pennsylvanicus are fairly constant for age class 10 and older (Appendix, Table 1). Use of this reduced matrix produces results that are similar to the full projection matrix (Figure 8). 30 .QSM mmoHo owe HH hoe cOHpo>Hpoo mo QOHmm5omHo Mom vamp mom .Am pom H mOHmev xHocomm< Some cop mmoHo owe am now mama .mommoHo own Amv HH .m> A¢V :m mchHopcoo Hoooa mcst mSOHcm>HHmdcwm am now thmcoo QOHpoHdmom popOHoopm mo QOmeomEoo .w OHSme 5.22132. xmmafi m2: 0v ¢¢ 0? mm mm mm 0N ON 0. N. m v I I I I I I I II I I I I I I I I I I I I I I I I I It I I I I I I I I I I I 308, 0. .00 .100 i0» 100 v00. AllS N30 31 This procedure is also effective for species with fairly constant survival and reproductive rates in any sequence of age classes. If P&_1,x is the probability that an animal moves into the xth age class at time t + 1 then P; can be obtained as above and is the probability that an animal remains in the x—J to x age class at time t + 1. Density Dependence Thus far we have been concerned with changes in population parameters without considering the mechanisms that are responsible for these changes. This model can realistically mimic population dynamics once appropriate values have been selected. For most species the mechanisms causing changes in the demographic parameters of a population are still unknown. For some species, proposed mechanisms have been determined and these mechanisms can be incorpor- ated into the model. When considering populations for which the regulatory mechanism is density—dependent, the exclusion of post- reproductive animals (as is the case in the basic Leslie Model) is unrealistic. Although these animals do not affect the population reproductively unless they contribute social labor, they may affect the population by utilizing a limiting resource. The role of the post-reproductive animals through social interaction has not been well documented but several studies indicate that their presence can affect the total population (Mech, 1966; Croze, 197A; Park et. a1. 196A). This model is capable of including post—reproductive animals. To illustrate the inclusion of a regulatory mechanism in the model, data for a population of wild agrimi, Capra aegagrus were used. 32 The agrimi is the only large mammal on Theodorou Island off the coast of Crete. This situation is thus well suited to test the predictive capabilities of the model using a functional interaction between the population and specific environmental components. Because the popula— tion is protected from outside interference, and there are no known predators on the island, it is reasonable to hypothesize that density is being regulated by food and space limitations. Data for this example were taken from Papageorgiou (197A). Table 1 contains the survival and reproductive estimates for the population. Survival rates were estimated from annual growth rings of skulls found on the island. Direct observation during the reproductive period provided estimated mx values. Daily protein-energy require- ments and preferred forage species were determined through feeding trials. Vegetative analyses provided productivity estimates for the nine preferred species. These estimates were used to determine the total protein-energy available to the population. It was estimated that the population of 97 animals was utilizing 70% of the available forage. Equations for determining protein-energy requirements for each of 12 age classes were adapted from Moen (1973) and Agricultural Research Council (1965). Coefficients for these equations were modified to agree with estimated requirements from the feeding trials (Table 2). Yearly protein—energy requirements for the total population were calculated from the individual daily requirements. Requirements for pregnancy and lactation for reproductively active females were added to the total population requirements. A one year time-lag between 33 .Azme .onwpoowomom thmv .mdpmmmoo opmmo pom nymposdpom OHzmoumoeoQ .H oHDmB oo. 00. OH. on. mv. mv. mm. mv. mv. mm. no. 00. RE ocHudouuo no :oHu0500um po>Huoo u me. oo. 1 non. non. 00. non. coo. hHm. OH. coo. non. vno. on. can. om. coo. ov. How. com. mam. mv. omm. va. men. so. nnm. ave. «be. me. mes. mom. mom. cc. com. nnm. who. mm. amm. Hum. own. mo. «mm. who. coo.H co. omm. we a» as an ocHumouuo no coHuosvoum po>Huoo mqux mmA vmn. mvn. non. vHo. amo. mHh. awn. can. fine. vow. com. ooo.H an H co. m oo. HH oo. v mvH. ow. mu. MHH mm. ID on. on. om. FNVD oo. 5 oo. ouauoauuo as can mcHuauuuo no coHuospOuA oo>uouno mmH<2 ono. H oo. «50. HH an. H oo. moH. OH 5mm. « oo. nnn. a wow. u «vH. vvv. a com. H ow. aHn. s can. v mu. man. u ~hm. n no. «on. m «on. m cm. one. v now. o mm. poo. n th. m on. cos. a ems. o oo. th. H ooo.H s cc. coo.H o a» ousuonuu- an a» oaaHo can con mcHuaouuo uo cOHuosvoum oo>uoono mquxuh mmsq<> AIDBUI 3A Table 2. Equations used for generating protein-energy requirements for Capra aegagrus. (Modified from Moen, 1973; Agricul— ture Research Council, 1965; and French et a1., 19AA). utll P9 PP pa 0 ¢P DH m DH¢ DH p Protein Requirement. Endogenous uniary nitrogen in 9/6.! .7 _ (10s on,“ 1.53 a, 1000 Protein loss is (0“.‘6.25' 12%) where 6.25 - conversion factor from nitrogen to protein 3V, biological value. - .9 Quantity of protein required for daily gain in gldsy I 0 100 Total protein required for z'egnancy (grams per day per kg fetus weight at birth). 165 -2.36 + 04:7 0 - l 0 ° I t l ) '.1los PP I {—173 t-l ' where .1805 - scale factor for weight of fetus at birth Total protein required for lactation °p¢ . §° l (r-csi’ e zoo ) 0 .0091 0 6.3a x-l metabolic fecal nitrogen 0"“ '(l%% -l)° 50H where DH - dryweight intake (kg/day) Protein loss - 0"“ ' 6.25 Energy Requirements Energy required for basic maintenance ’73,) . . 88.8 for males 60.0 for females 1.4 - conversion factor for free ranging animal Energy required for growth 09 - bouts where b is a function of weight (wig, and ranges from 1384 heal/hg/dag to 2092 heal/hg/day Total energy required for pregnancy per kg fetus weight at birth 0 _ 02‘ lg’ e2.99+.0295t for gestation time of 165 days ‘9 {-1 Total energy required for lactation so 0 - 1.6 i (so IH+AH) '73‘ i-l + 110) where 1.6 - cost to female above energy contained in the milk and lactation period is 90 days. Dry Height Requirements Total forage intake for maintenance in g/dag DH _ . "t .734 . 1" . _ 41.7 for males ' 9 31.0 for females 1.4 - conversion factor for free ranging animal Forage intake necessary to supply entire energy for growth, pregnancy and lactation in g/day 0' Due - DE/Z.16 where D: - digestible energy for growth, pregnancy and lactation. forage intake necessary to supply extra protein for growth, pregnancy and lactation in g/dag D"p - by where DP - digestible protein, 4.6 is percent protein in forage .046'.65'.9 _ . .65 is digestability coefficient and .9 is biological value coeffrcxent 35 population requirements and population response was utilized to approximate effects of grazing and subsequent vegetative recovery. Survival data suggested that density was not affecting survival except in the first and latter age classes. Because of the small number of young produced (1A young for 29 females) density appeared to affect reproduction. Maximum expected reproductive rates were estimated by back calculation from the present 70% resource utiliza- tion rate to an assumed density-independent situation. Additionally, survival in the first and latter age classes was modified to approxi- mate increased survival at low density (Table 1). From estimates of population requirements and available protein- energy, an index of utilization was established. The index, a ratio of population requirements divided by available protein—energy, provides a value that ranges between zero and one, thus representing the extremes of zero utilization and 100% utilization. Reproduction and survival rates for the first and latter age classes were reduced as the index approached one. When the utilization index exceeded .9 all reproduction ceased and survival for all age classes was reduced. Simulations were run matching an initial population of two animals as introduced on the island in 1928, with subsequent intro- ductions of two animals in 1937 and again in 19A5. Figure 9 represents a statistical analysis of population growth from 10 simulation runs using density—dependent reproduction and survival and a stochastic function for survival. Figure 10 shows the contrasting age structure and protein-energy requirements for the population at time periods 1953 and 1957. The population at these two time periods, taken from a run that was close to the means in Figure 9, show equal density, and 36 .AH OHQoBv mnnwowoo onmwo now mono OHnmonwoemQ .Ho>H>nnm now noHponnw OHpmonOOpm one noHponoonmon one Ho>H>nnm now mnOHponnm Pnoonogoouthwnoo mnHmn mnnn noHpmHnnHm OH non nOHpoHnm> mo mPnOHOHmmooo ono .me>MOan oonmonnoo Rmm .mnnoz .m onanm ,3 a a I l omMOtosr IO lLLl l S as L l 8 8 0n if; as haseeoebeeaeeeen...obese;meagreehbns llLLLl O [s L on. AJJSN'EIC] 37 .AH OHQva mnnwmmom mammo mom memo OHnmmnmoEom .m mnanm non noHpmHnnHm ono Bone oOPOOHom pmmH onm mmmH now mnhmmmom mammo pom mpnoeoandon hwnonolnHoponm onm onnvonnpm owm oOPOHoonm .OH beamem b zofimoaoma . 8. 0m. 9. 0.. no. 0 m 0.. 0.. 0m. I I I 4/ 53.0. x tmmoua 568. o. x commie I I I l I_ —O OlCDNCDIOQ'ION :9 b zocaoaoma 0N. m.. 0.. no. 0 q I a .3... so. x Show... acts. to_ x mmooe um IL; —0 mmNQW¢MN :9 881710 39V 38 emphasizes the importance of the age structure of the population in relation to its requirements. Figure 11 compares the observed structure of the population in 1973 with that predicted from simulations using the stochastic and nonstochastic survival forms of the model. The use of the stochastic function for survival rates produces age structures that coincide with empirical information. Because of the high population density, and resultant overgrazing, 17 animals were removed in the summer of 1973. Most of these were older animals. Simulations were run using age- and sex-specific cropping procedures. It was determined that two efficient procedures for reducing the population with a single cropping were either to select all females of random ages or to select all young animals of either sex. Alternative procedures for cropping the population are currently being evaluated. 39 .AH meva mnnwmmom mnmmo memo OHnmmnmoSoQ .nOHHmHnQom onp no .Am. onnponnpm owm oo>nompo onp onm .nOHponoonmon onm Hm>H>hnm mo mnoHponnm pnoonomoolthmnoo wnHmn ADV Hoooa oHpmmnoopm onm An. Hoooa anpmS OHpmmSOOPmnon Mom mme pom mnnmmwom mummo Mom manponnpm omm oOHOHoonm no nomHnmmaoo .HH onanm 3 9 dr .o. zopcomomm mo. 0 8. o I -QQON@0'nN—O . o 0.. zo.hm0a0me 8. o 00. 0.. a I -omov~onen~—o b zofimoaoma 0 0.. 00 0 3.0.. I -OOO¢)'~OOQION—O amu- SSV'IO 39V DISCUSSION The analysis of population regulatory mechanisms has played a central role in ecology. Obviously population size is limited, and for most populations, density fluctuates about some relatively constant value. Because of the richness and diversity of the natural world, species have evolved great variation in life histories that dictate the manner in which they deal with the complexities of their environment. In the model developed here, simulations utilizing density- dependent functions of survival and reproduction illustrate that density alone cannot adequately describe population requirements or responses. The inclusion of sex and age structure is essential. The effects of mechanisms limiting population growth are ultimately exhibited through changes in net reproduction and survival, however, these basic parameters are functions of many biological processes acting within the population. Net reproduction is affected by changes in age-specific reproductive rates, age at sexual maturity, the proportion of the population in breeding condition and immigration. Survival at any time t is a function of age- and sex-specific survival rates and emigration. Each of these components of survival and reproduction is in turn determined by many ecological factors. The value of a model is to provide understanding about population processes. It is meaningless to pose questions about population A0 A1 phenomena if the model does not contain parameters that pertain to those phenomena. For example, it would serve no purpose to use a model that contains only density to examine the effects of predation if predation is age—specific. Nor would a model whose parameters are not functions of genetics be appropriate to study the effects of genetic variation in a population. The model must be capable of adequately describing demographic processes inherent in the popula- tion and, in addition, provide a means to distinguish those processes that are being studied. The basic structure of the model developed here contains those demographic parameters necessary to describe demographic processes. Most species exhibit age specific reproduction and age- and sex— specific survival: therefore, these characteristics must be included in any population model whose goal is a contribution to an understanding of demographic phenomena. In addition, few species exhibit a continuous breeding pattern in nature. The general pattern is one of seasonal variability in the proportion of reproductively active individuals. The model developed here includes age- and sex-specific survival and reproductive rates and possesses the capability to separate breeding and nonbreeding components. The model also allows for seasonal changes in survival and reproduction and can mimic population response to changing environmental parameters. Because the values for most demographic parameters are estimated means, the use of stochastic functions for reproduction and survival adds more realism to the model. The straight—forward representation of demo— graphic machinery provides a means to establish functional dependency A2 of selected parameters according to regulatory mechanisms hypothesized by the researcher. This model can play an integral part in research programs whose goals are to determine specific mechanisms affecting populations. By using data from laboratory studies, which may represent extreme values, specific questions can be asked about factors that produce the values observed in field populations. The proposed mechanism can be incorporated into the model to predict population response. For example, the effects of hypothesized levels of predation can be examined by separating survival into components of predation and natural survival. Specific questions can then be posed concerning the effects of predation on specific age classes, and appropriate experiments can then be designed. Similarly the effects of dispersal, a proposed mechanism for regulating microtine populations (Krebs et al., 1969), can be examined by adding a component that selectively removes animals from the population in response to changes in age structure and density. Alternately, the effect of immigration can be deter- mined by adding an immigration component to the age vectors, thus allowing chosen age and sex categories to receive migrating animals. Qualitative changes in individuals can also be explored. For those populations that are thought to be limited by available food or space, density—dependent functions for survival and reproduc- tion, directed to specific sex and age classes, can be established and allowed to vary over time. The demographic role of genetic variation in the population can be explored by adding a function Of genetic information. Reproduction and survival could be functionally A3 related to specific genotypes and thus used to determine population response with different genetic structures. The value of the model developed here is the ability to realistically describe demographic processes common to most species and the capability for inclusion of specific mechanisms that affect particular species. The flexibility of the model to incorporate causal mechanisms provides the means to evaluate the effects of such mechanisms. APPENDIX APPENDIX The M, pennsylvanicus population was divided into 2A two week age classes. Table 1 contains basic demographic parameters for this population. Survival rates for males and females were calculated by combining estimates for juvenile survival with the survival rates for animals after they enter the trappable population. Juvenile survival rates from birth to four weeks were obtained using an early juvenile survival index of .96 (Krebs and Myers, 197A: Table IX). Survival rates from age at first capture were taken from Myers and Krebs (1971: Fig. 3). The probabilities of survival Pm, were calcu— lated using Eq. 5 and the linear approximations for Ex, Eq. 7. Age— specific litter sizes for M, pennsylvanicus were obtained from data provided by Dr. F. C. Elliott, Crops and Soil Science Department, Michigan State University. These data were collected in a laboratory colony from April 1970 to April 1975 and represent 3550 litters containing 19,919 young. Ninety—nine percent of the litter sizes were 15 or less and 99 percent of the litters were produced by females less than 3A weeks old. This is because very few of the females remained in the colony past this age. Myers and Krebs (1971) state that the females are sexually mature at 25 grams, or approxi- mately four to six weeks of age. Given a 21 day gestation period females can produce their first litter at about six to seven weeks of age and subsequent litters every four weeks. Data from Elliott's AA A5 . thmno>HnD mpmpm nmeQOHE .pnoapnmmom . . oesonom Hnom boa anono .epoHHHm .o .e .so so oHosHHmeo been some ass .HemH .mposx boa anon: mxH OHnt ":PmH .mnmhz onm moonm nonmv .mnOHnm>Hhmnnom mSpOMOHz now mnoposmnmm OHnmmnwoaoa .H¢ mHQmB v lo (3'0 :3 o :3 lo «9 ricn «ten «new Oltfl r104 pied Pica r470 caio c>Io ‘0 In MH NH uflHflz QOHIEOW QCOHUIHHHEHM uouoo> one HaHuHsH mm.m mm.m mm.m wo.m mm.m om.m mm.m mm.m mm.m mm.m om.m Nm.m mm.m Hw.m om.m vo.w mm.m mm.m om.m sv.m oo.m oo.m o o now been nouqu no no no we no co co Hv mm Hv mm mm ov mm mm o o .>.U .0.m HH - om ea me man men oaH ham o~m see eme can see me o o abouuan mo Honfioz mv.w mm.m mw.m vH.w mm.m mm.m Hm.m mm.m v0.0 mm.m mm.m om.m hv.m Ho.m m~.m o o beam uouuaq new: «on. mmn. mmm. mmh. mmm. Hmm. mom. com. com. mmm. onm. va. onm. onm. mmm. mmm. «mm. mew. mew. vmm. Hmb. onm. mmv. we nOHmz woo. hoo. moo. 0H0. MHo. vHo. mHo. hHo. mac. bmo. mmo. mmo. moo. vmo. Hmo. >00. >50. «mo. vOH. mHH. moH. maH. mmo. ooo.H xw I moo. mmm. HHo. mos. MHo. bmm. mHo. mom. mHo. mmm. Hmo. Huh. hmo. mum. Hmo. mum. mmo. vom. mmo. mmm. woo. onm. mmo. mbm. omo. mam. moo. «mm. bro. ohm. omo. mmm. mmo. boa. mOH. Hmm. HNH. mom. va. mom. va. «no. mmH. mom. mmv. one. ooo.H we xw nOHgEom mm mm Hm om mH mH hH wH mH VH mH NH HH 0H HNMVIDWI‘Q O nmmHU mod A6 colony indicate that females were capable of bearing young in the second age class (four to six weeks). Because of the small number of litters produced by the second age class and for the age classes 13 and older, values for the simulations were modified for these age classes. These data indicate that there was not a decline in litter size for older animals which corresponds to results obtained by Keller and Krebs (1970). Since the females are not reproductively mature until the second age class and therefore do not influence the population in a reproductive capacity for the first two age classes no attempt was made to include a survival rate for the animals within the zero age class in deriving the Ex values. Rather, the expected reproductive effort, mm, was used for these values. Because two week time intervals were used for the simulations the total litter size for the second age class was used and litter sizes for subsequent age classes were halved approximating one litter every A weeks. Age vectors were obtained from Zx values for an initial population density of 100 animals. To illustrate population response and growth rates at different densities the proportion of the females in reproductive condition was varied. These proportions are given in Table 2 and were obtained from Krebs et al. (1969: Fig. 12). Because of the small number of females that are sexually mature in the second age class (Table 1), the proportion of females in reproduc- tive condition for this age class was reduced by 50% for all time periods. 1.7 hmlmmmH How mnunoz d mm on mH mm mH oo oo oo om om mm mo om mm om ov om mo mm om mm om hm om .nOHuHonoo o>Huonoonmmu nH monEom mSOHnm>H>wnnom wouOHOHz mo nOHuHomonm ..~H .oae .aoaH .Hm no moose some. .N onbae BIBLIOGRAPHY BIBLIOGRAPHY Agricultural Research Council. 1965. The nutrient requirements of farm livestock. No. 2. Ruminants. London: Agricultural Research Council, xi+ 26A pp. Beland, P. 197A. 0n predicting the yield from brook trout popula- tions. Trans. Amer. Fish. Soc. 2: 353-355. Bernardelli, H. 19A1. Population waves. J. of Burma Res. Soc. 31(1): 1-18. Bowley, A. L. 192A. Births and population of Great Britain. J. of the Royal Economic Society 3A: 188-192. Cannon, E. 1895. The probability of a cessation of the growth of population in England and Wales during the next century. The Economic Journal 5: 505—515. Conley, Walt. (In press). Competition between Microtus: a behavioral hypothesis. Ecol. Monogr. Conley, W., J. D. Nichols, and A. R. Tipton. 1975. Reproductive strategies in desert rodents. 225 R. H. Wauer and D. H. Riskind eds. Transactions Symp. on the Biol. Resources of the Chihuahuan Desert Region, U. S. and Mexico. (In press). Conley, W., R. Boling, and L. Ames. (In prep.). Response surfaces for natality as an age specific function. Ms. Croze, H. 197A. The Seronera Bull problem I. The Elephant. E. Afr. Wildl. J. 12: 1-27. Cull, P., and A. Vogt. 197A. The periodic limit for the Leslie model. Math. Biosci. 21: 39-5A. Cull, P., and A. Vogt. 1973. Mathematical analysis of the asymptotic behavior of the Leslie population matrix model. Bull. Math. Biol. 35: 6A5-661. Emlen, J. 1973. Ecology: an evolutionary approach. Addison—Wesley, Reading, Mass. xiv+ A93 pp. Fisher, R. A. 1930. The genetical theory of natural selection. Clarendon Press, Oxford. 272 pp. A8 A9 Fowler, C. W., and T. Smith. 1973. Characterizing stable populations: an application to the African elephant population. J. Wildl. Mgmt. 37(A): 513-523. French, M. H. l9AA. The feeding of goats. E. Afr. Agr. J. 10: 66—71. Gantmacher, F. R. 1959. The theory of matrices. Vol. 2. Chelsea, New York. ix+ 276 pp. Gross, J. E. 1972. Criteria for big game planning: performance vs. intuition. Trans. N. Am. Wildl. Nat. Resour. Conf. 37: 2A6- 257. Hamilton, W. D. 1967. Extraordinary sex ratios. Science 156: A77— A88. Hamilton, W. J., Jr. 1937. The biology of microtine cycles. J. Agr. Res. 5A: 779-790. Hoffmann, R. S. 1958. The role of reproduction and mortality in population fluctuations of voles (Microtus). Ecol. Monogr. 28(1): 79—108. Holling, C. S. 1966. The strategy of building models of complex ecological systems. P. 195-213. 225 K. E. F. Watt, ed. Systems analysis in ecology. Academic Press, New York. Holling, C. S. 1968. The tactics of a predator. P. A7-58. 225 T. R. E. Southwood, ed. Insect abundance. Blackwell Scientific Publications, Oxford. Jensen, A. L. 1971. The effect of increased mortality on the young in a population of brook trout, a Theoretical Analysis. Trans. Amer. Fish. Soc. 3: A56-A59. Keller, B. L., and C. J. Krebs. 1970. Microtus population biology. III. Reproductive changes in fluctuating populations of M, ochrogaster and M, pennsylvanicus in southern Indiana, 1965-67. Ecol. Monogr. A0: 263-29A: Keyfitz, N. 1968. Introduction to the mathematics of population. Addison-Wesley, Reading, Mass. xiv+ A50 pp. Kolman, W. A. 1960. The mechanisms of natural selection for the sex ratio. Amer. Natur. 9A: 373-377. Krebs, c. J., B. L. Keller, and R. H. Tamarin. 1969. Microtus population biology: Demographic changes in fluctuating popula- tions of M, ochrogaster and M, pennsylvanicus in southern Indiana. Ecology 50: 587—607. Krebs, C. J., and J. H. Myers. 197A. Population cycles in small mammals. Advances in Ecol. Res. 8: 267—399. 50 Lefkovitch, L. P. 1965. The study of population growth in organisms grouped by stages. Biometrics 21: 1-18. Leigh, E. G. 1970. Sex ratio and differential mortality between the sexes. Amer. Natur. 10A: 205-210. Leslie, P. H. 19A5. 0n the use of matrices in certain population mathematics. Biometrika 33: 183-212. Leslie, P. H. l9A8. Some further notes on the use of matrices in population mathematics. Biometrika 35: 213—2A5. Leslie, P. H. 1959. The properties of a certain lag type of popula- tion growth and the influence of an external random factor on a number of such pOpulations. Physiol. Zool. 32(3): 151-159. Leslie, P. H. 1966. The intrinsic rate of increase and the overlap of successive generations in a population of guillemots (Uria aalge pont). J. Anim. Ecol. 35: 291-301. Levins, R. 1966. The strategy of model building in population biology. Amer. Scient. 5A: A21—A3l. Lewis, E. G. 19A2. 0n the generation and growth of a population. Sankya. 6: 93—96. Lopez, A. 1961. Problems in stable population theory. Ph.D. Dissertation, Office of Population Research, Princeton, N. J. iii+ 110 pp. Lowe, V. P. W. 1969. Population dynamics of the red deer (Cervus elaphus L.) on Rhum. J. Anim. Ecol. 38: A25—A57. May, R. M. 1973. Stability and complexity in model ecosystems. Monographs in Pop. Biol., No. 6, Princeton Univ. Press, Princeton, N. J. ix+ 235 pp. Maynard Smith, J. 197A. Models in ecology. Cambridge Univ. Press. xi+ 1A6 pp. McFarland, D. D. 1969. On the theory of stable populations: A new and elementary proof of the theorems under weaker assumptions. Demography 6: 301-322. Mech, L. D. 1966. The wolves of Isle Royale. U. S. Natl. Park Serv., Fauna Ser. No. 7. xiv+ 210 pp. Moen, A. N. 1973. Wildlife ecology. W. H. Freeman, San Francisco. A58 pp. Myers, J. H., and C. J. Krebs. 1971. Sex ratios in open and enclosed vole populations: demographic implications. Amer. Natur. 105 (9AA): 325-3AA. 51 Neill, W. E. 197A. The community matrix and interdependence of the competition coefficients. Amer. Natur. 108(962): 399-A08. Nichols, J. D., W. Conley, B. Batt, and A. R. Tipton. 1975. Temporally dynamic reproductive strategies: an alternative to r or K selection. (Submitted to American Naturalist). Papageorgiou, N. 197A. Population energy relationships of the agrimi (Capra aegagrus cretica) on Theodorou Island, Greece. Ph.D. Dissertation, Michigan State Univ. viii+ 83 pp. Park, T., P. H. Leslie, and D. B. Mertz. 196A. Genetic strains and competition in populations of Tribolium. Physiol. Zool. 37: 97-162. Pennycuick, J. C., R. M. Compton, and Linda Beckinghamn 1968. A computer model for simulating the growth of a population of two interacting populations. J. Theoret. Biol. 18: 316—329. Pennycuick, L. 1969. A computer model of the Oxford Great Tit population. J. Theoret. Biol. 22: 38l-A00. Pollard, J. H. 1966. On the use of the direct matrix product in analysing certain stochastic population models. Biometrika 53: 397-A15. Popper, K. R. 1959. The logic of scientific discovery. Harper & Row, New York, N. Y. A68 pp. Popper, K. R. 1963. Conjectures and refutations: the growth of scientific knowledge. Harper & Row, New York, N. Y. xiii+ A17 PP. Shaw, R. F., and J. D. Mohler. 1953. The selective significance of the sex ratio. Amer. Natur. 87(837): 337-3A2. Skellam, J. C. 1967. Seasonal periodicity in theoretical population ecology. P. 79-205. 1p, L. M. LeCom and J. Neymen eds. Proc. Fifth Berkley Symp. on Math. Stat. Prob., Univ. Calif. Press, Vol. A. Smith, A. T. 197A. The distribution and dispersal of pikas: consequences of insular population structure. Science 55(5): 1112—1119. Smith, T. D. 1973. Variable population projection matrix models: Theory and application to the evaluation of harvesting strategy. Ph.D. Dissertation, Univ. Washington. Sykes, Z. M. 1969. On discrete stable population theory. Biometrics 25: 258-293. 52 Tamarin, R. T., and C. J. Krebs. 1969. Microtus population biology. II. Genetic changes at the transferrin locus in fluctuating populations of two vole species. Evol. 23: 183-211. Usher, M. B. 1972. Development of the Leslie matrix model. P. 29—60. IE, F. N. R. Jeffers, ed. Mathematical models in ecology. British Ecol. Soc. Symp. No. 12, Blackwell. Usher, M. B. 1973. Biological conservation and management: Ecological theory, application and planning. Chapman & Hall, London. xiii+ 39A pp. Watt, K. E. F. 1968. Ecology and resource management: A quantitative approach. McGraw-Hill, New York. xii+ A50 pp. Watt, K. E. F. 1956. The choice and solution of mathematical models for predicting and maximizing the yield of a fishery. J. Fish. Res. Ed. Can. 13: 613-6h5. Whelpton, P. K. 1936. An empirical method of calculating future population. J. Amer. Stat. Assoc. 31: A57-A73. Williamson, M. H. 1959. Some extensions of the use of matrices in population theory. Bull. Math. Biophysics. 21: 261-263. Willson, M. F. and E. R. Pianka. 1963. Sexual selection, sex ratio, and mating system. Amer. Natur. 95: A05-AO7. ”'IlIIIIlIILIflIjIIEIIMIIIIIIIIIIIIIII“ 7 8636