This book demonstrates that population structure and dynamics can be reconstructed by stochastic analysis. Population projection is usually based on age-structured population models. These models consist of age-dependent fertility and mortality, whereas birth and death processes generally arise from states of individuals. For example, a number of seeds are proportional to tree size, and amount of income and savings are the basis of decision making for birth behavior in human beings. Thus, even though individuals belong to an identical cohort, they have different fertility and mortality. To treat this kind of individual heterogeneity, stochastic state transitions are reasonable rather than the deterministic states. This book extends deterministic systems to stochastic systems specifically, constructing a state transition model represented by stochastic differential equations. The diffusion process generated by stochastic differential equations provides statistics determining population dynamics, i.e., heterogeneity is incorporated in population dynamics as its statistics. Applying this perspective to demography and evolutionary biology, we can consider the role of heterogeneity in life history or evolution. These concepts are provided to readers with explanations of stochastic analysis.
Author(s): Ryo Oizumi
Series: SpringerBriefs in Population Studies: Population Studies in Japan
Publisher: Springer
Year: 2022
Language: English
Pages: 106
City: Singapore
Preface
Introduction: Stochasticity in Demography
Contents
1 Deterministic and Stochastic Population Models
1.1 Malthus Equation and Stochastic Malthus Equations
1.2 Basic Properties for SDE Models
1.3 SDEs and Partial Differential Equations
References
2 Linear Structured Population Based on SDE
2.1 Age-Structured Population Model with Individual Stochasticity
2.2 Path Integral Representation and Classical Life Schedule
2.2.1 Path Integral Representation and Analytical Mechanics in Demography
2.2.2 Lagrangian and Hamiltonian Expressions in Path Integral
2.3 Characteristic Function and Eigenfunctions
2.3.1 Characteristic Function and Asymptotic Population Behavior
2.3.2 Eigenfunctions and Feynman–Kac Formula
2.4 Demographic Indicators
2.5 Application to a Semelparity Model
References
3 Non-linear Structured Population Models
3.1 r/K Selection Theory
3.2 Non-linear Multi-state Population Model
3.3 Stability of Stationary Population
3.3.1 Stationary Population
3.3.2 Characteristic Equation for Local Stability
3.4 Concrete Example
3.4.1 Resource Acquisition Competition Model and Stationary Population
3.4.2 Local Stability of Stationary Population in Semelparous Species
References
4 Life History Evolution and Adaptive Stochastic Controls
4.1 Basic Theorem for Adaptive Life History
4.2 Optimal Stopping Problem in Life Histories
4.3 Adaptive Reproductive Timing for Semelparous Species
4.4 Optimal Life Schedule Problem and Hamilton–Jacobi–Bellman Equation
4.4.1 Stationary Control
4.5 Reconsideration of r/K Selection Theory
4.5.1 Adaptive Resource Utilization Model in Optimal Foraging Problem
4.5.2 Complicated Relationship among Demographic Indices under the Density Effect
4.5.3 Summary
References
5 Application to External Stochasticity
5.1 External Stochasticity and Perturbation Method
References
Appendix Epilogue: Population Dynamics from the Perspective of Individual Stochasticity
Reference
