Population Dynamics and the Tribolium Model

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.