Population Dynamics and the Tribolium Model: Genetics and Demography

The study of populations is becoming increasingly focused on dynamics. We believe there are two reasons for this trend. The ftrst is the impactof nonlinear dynamics with its exciting ideas and colorful language: bifurcations, domains of attraction, chaos, fractals, strange attractors. Complexity, which is so very much a part of biology, now seems to be also a part of mathematics. A second trend is the accessibility of the new concepts. Thebarriers tocommunicationbetween theoristandexperimentalistseemless impenetrable. The active participationofthe experimentalist means that the theory will obtain substance. Our role is the application of the theory of dynamics to the analysis ofbiological populations. We began our work early in 1979 by writing an ordinary differential equation for the rateofchange in adult numbers which was based on an equilibrium model proposed adecadeearlier. Duringthenextfewmonths weftlledournotebookswithstraightforward deductions from the model and its associated biological implications. Slowly, some of the biological observations were explained and papers followed on a variety of topics: genetic and demographic stability, stationary probability distributions for population size,population growth asabirth-deathprocess, natural selectionanddensity-dependent population growth, genetic disequilibrium, and the stationary stochastic dynamics of adult numbers.

1 Introduction.- 1.1 A Brief History of the Mathematics of Tribolium.- 1.1.1 The Beginning.- 1.1.2 Models of Adult Numbers.- 1.1.3 Models of Age Structure.- 1.1.4 Beetle Movement, Food Preference and Tunnels.- 1.1.5 Species Interactions.- 1.1.6 Population Genetics.- 1.2 Topics in the Monograph.- 1.2.1 Life Stage Interactions.- 1.2.2 Stochastic Population Growth.- 1.2.3 Natural Selection.- 1.2.4 Species Interactions.- 2 Biology and Dynamics of Age Structure.- 2.1 Models for Age-Structured Populations.- 2.1.1 McKendrick-von Foerster Equation.- 2.1.2 The Leslie Matrix Model.- 2.2 Life Stage Interactions.- 2.2.1 Life Cycle.- 2.2.2 Mortality.- 2.2.3 Fecundity.- 2.2.4 Equilibrium.- 2.3 Parameterization.- 2.3.1 Life Cycle.- 2.3.2 Fecundity.- 2.3.3 Natural Mortality.- 2.3.4 Cannibalism.- 2.4 Experimental Data and Computer Simulations.- 2.4.1 Experimental Design.- 2.4.2 Discretization.- 2.4.3 Equilibrium and Stability.- 2.4.4 Demographic Data and Simulations.- 2.4.5 Perturbations.- 2.4.6 Stable Limit Cycle.- 2.4.7 Bifurcation Points.- 2.5 Summary and Conclusions.- 2.6 Appendix.- 3 Demographic Oscillations.- 3.1 Egg-Larval Submodel.- 3.1.1 Fast and Slow Time Scales.- 3.1.2 Some Simplifying Assumptions.- 3.1.3 Integral Equations.- 3.2 Analytical Results.- 3.2.1 Equilibrium.- 3.2.2 Local Stability.- 3.2.3 Nonlinear Analysis.- 3.3 Numerical Results.- 3.3.1 Difference Equations.- 3.3.2 Multiple Attractors.- 3.3.3 Bifurcation Portrait.- 3.4 Biological Relevance.- 3.4.1 Age-Dependent Cannibalism.- 3.4.2 Complete Life Cycle.- 3.4.3 Populations in Parameter Space.- 3.4.4 Experimental Possibilities.- 3.4.5 Theoretical Questions and Extensions.- 3.5 Summary and Conclusions.- 3.6 Appendix.- 4 Continuous Stochastic Models.- 4.1 Deterministic Model for Adult Numbers.- 4.2 Stochastic Model for Adult Numbers.- 4.2.1 Stochastic Differential Equation.- 4.2.2 Stationary Distribution.- 4.2.3 Properties of the Stationary Distribution.- 4.3 Statistical Procedures.- 4.3.1 Data Organization.- 4.3.2 Parameter Estimation.- 4.3.3 Testing the Gamma Hypothesis.- 4.3.4 Comparing Two Distributions.- 4.3.5 The Problem of Time-Dependence.- 4.4 Applications.- 4.4.1 Different Environments.- 4.4.2 Genetic Strains.- 4.4.3 Geographical Strains.- 4.4.4 Populations at Different Times.- 4.4.5 Population Outbreaks and Crashes.- 4.5 Stationary Time Series.- 4.5.1 Linear Approximation for the Stochastic Model.- 4.5.2 Autocorrelation Function.- 4.6 Time-Dependent Gamma Distribution.- 4.7 Summary and Conclusions.- 5 Discrete Stochastic Models.- 5.1 Population Growth as a Birth-Death Process.- 5.1.1 Kolmogorov Forward Equations.- 5.1.2 Stationary Probabilities.- 5.1.3 Extinction Times and the Quasi-Steady State.- 5.2 Nonlinear Birth-Death Process for Tribolium.- 5.2.1 Recruitment and Mortality Rates.- 5.2.2 Extinction Times and the Quasi-Steady State.- 5.2.3 Approximations Based on the Normal Distribution.- 5.2.4 Cannibalism Rates and Extinction Times for Eight Genetic Strains.- 5.3 Linear Birth-Death Approximation.- 5.3.1 Linearized Recruitment Rate.- 5.3.2 Time-Dependent Mean and Variance.- 5.3.3 Transient and Stationary Probability Distribution.- 5.4 Application to Tribolium brevicornis.- 5.5 Summary and Conclusions.- 5.5.1 Deterministic versus Stochastic Models.- 5.5.2 Continuous versus Discrete Stochastic Models.- 5.5.3 Environmental versus Demographic Stochasticity.- 5.6 Appendix.- 6 Natural Selection.- 6.1 Ecological-Genetic Perspective.- 6.2 Density-Regulated Natural Selection.- 6.2.1 Density-Dependent Fitness.- 6.2.2 Selection on Recruitment.- 6.2.3 Selection on Mortality.- 6.2.4 Selection on Both Recruitment and Mortality.- 6.3 Experimental Results on the Corn Oil Sensitive Mutant.- 6.3.1 Predictions.- 6.3.2 Genetic Polymorphism Experiment.- 6.3.3 Genetic Perturbation Experiment.- 6.3.4 Stationary Distributions of Population Numbers.- 6.3.5 Autocorrelation of Population Numbers.- 6.3.6 Maximization of Population Numbers.- 6.4 Summary and Conclusions.- 7 Genetic Disequilibrium.- 7.1 An Entropy Measure of Genetic Disequilibrium.- 7.1.1 Ginzburg’s “Fitness Entropy”.- 7.1.2 Haldane’s “Cost of Natural Selection”.- 7.2 Rate of Genetic Adaptation.- 7.2.1 Entropy and the Cost of Selection.- 7.2.2 Entropy and the Adaptation Rate.- 7.2.3 An Experimental Test.- 7.3 Density-Independent Population Growth.- 7.3.1 The Reference Population.- 7.3.2 Entropy and Selective Delay.- 7.3.3 An Experimental Test.- 7.4 Density-Dependent Population Growth.- 7.4.1 Logistic Population Growth.- 7.4.2 Entropy and Life Not Lived.- 7.4.3 Properties of Life Not Lived.- 7.4.4 An Experimental Test.- 7.5 Extensions to the Theory.- 7.5.1 A Generalization for Weak Selection.- 7.5.2 Several Interacting Species.- 7.5.3 Asexual Age-Structured Populations.- 7.6 Summary and Conclusions.- 8 Two Interacting Species.- 8.1 Competition as an Adult-Immature Interaction.- 8.1.1 Two Species Equations.- 8.1.2 Analysis of the Model.- 8.2 Two Cannibalistic Species.- 8.2.1 T. castaneum versus T. confusum.- 8.2.2 The Fan-Shaped Outcome Pattern.- 8.2.3 Indiscriminate Cannibalism.- 8.2.4 Discriminate Cannibalism.- 8.2.5 Stochastic Effects: Demographic or Genetic?.- 8.3 An Inhibiting Species with a Cannibalistic Species.- 8.3.1 T. castaneum versus T. brevicornis.- 8.3.2 Parameter Estimates and Model Predictions.- 8.4 Two Inhibiting Species.- 8.4.1 T. madens versus T. brevicornis.- 8.4.2 A Hypothesis.- 8.4.3 Developmental Inhibition.- 8.5 Summary and Conclusions.- 9 Dynamics and the Tribolium Model.- 9.1 Demography.- 9.1.1 Exploring Parameter Space.- 9.1.2 Strange Attractors and Chaos.- 9.1.3 Fast versus Slow Time Scales.- 9.2 Stochasticity.- 9.2.1 Parameters of the Gamma Distribution.- 9.2.2 Stochasticity and Age-structure.- 9.2.3 Extinction.- 9.2.4 Scale of Stochastic Forces.- 9.2.5 Stochastic Disequilibrium.- 9.3 Natural Selection.- 9.3.1 Natural Selection and Age Structure.- 9.3.2 Joint Distributions of Population Numbers and Allele Frequencies.- 9.3.3 Estimating the Life Not Lived.- 9.4 Species Interactions.- 9.4.1 Complex Demographic Models.- 9.4.2 Stochastic Dynamics of Coexistence.- 9.5 Summary and Conclusions.- References.- Author Index.