Stochastic Population Models: A Compartmental PerspectiveThis monograph has been heavily influenced by two books. One is Ren shaw's [82] work on modeling biological populations in space and time. It was published as we were busily engaged in modeling African bee dispersal, and provided strong affirmation for the stochastic basis for our ecological modeling efforts. The other is the third edition of Jacquez' [28] classic book on compartmental analysis. He reviews stochastic compartmental analysis and utilizes generating functions in this edition to derive many useful re sults. We interpreted Jacquez' use of generating functions as a message that the day had come for modeling practioners to consider using this powerful approach as a model-building tool. We were inspired by the idea of using generating functions and related methods for two purposes. The first is to integrate seamlessly our previous research centering in stochastic com partmental modeling with our more recent research focusing on stochastic population modeling. The second, related purpose is to present some key research results of practical application in a natural, user-friendly way to the large user communities of compartmental and biological population modelers. One general goal of this monograph is to make a case for the practical utility of the various stochastic population models. In accordance with this objective, we have chosen to illustrate the various stochastic models, using four primary applications described in Chapter 2. In so doing, this mono graph is based largely on our own published work. |
Contents
Overview of Models | 2 |
Basic Methodology for Multiple Population Stochastic Models | 9 |
Models for a Single Population | 16 |
Linear ImmigrationDeath Models | 30 |
Linear BirthImmigrationDeath Models | 40 |
Nonlinear BirthDeath Models | 49 |
Models for Multiple Populations | 72 |
Standard Multiple Compartment Analysis | 101 |
Linear DeathMigration Models | 119 |
Linear ImmigrationDeathMigration Models | 137 |
Nonlinear BirthDeathMigration Models | 161 |
Nonlinear HostParasite Models | 172 |
| 189 | |
| 199 | |
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Stochastic Population Models: A Compartmental Perspective James H. Matis,Thomas R. Kiffe Limited preview - 2012 |
Common terms and phrases
a₁ a₂ African bee AHB population analytical solutions Application approximation errors assumed assumption b₁ b₂ bioaccumulation birth rate birthrate bivariate calcium carrying capacity coefficient matrix compartment Consider corresponding cumulant approximations cumulant functions denoted deterministic model deterministic solutions Drenthe equilibrium distribution Erlang Erlang distributions estimated exact cumulants example fish follows gelderl Gelderland given hence illustrated in Figure immigration-death model intensity functions K₁ K₁(t Kolmogorov equations Kolmogorov forward equations LBID levels of immigration LID model linear mean value functions mercury methodology migration mite muskrat NBID model nonlinear numerical solutions obtained order cumulants overijl Overijssel parameter values partial differential equations particles pi(t po(t Poisson distribution population growth models population models population size distribution previously probability distribution probability functions quasi-equilibrium quasi-equilibrium distribution rate functions saddlepoint approximation Section skewness functions solving standard stochastic model stochastic solutions three cumulant functions tions transient distributions truncating variable variance and skewness variance functions vector әк ӘР
Popular passages
Page 191 - MA Johnson and MR Taaffe. Matching moments to phase distributions: Mixtures of Erlang distributions of common order.
Page 193 - Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering. Bull. Math. Biol. 43.
Page 193 - Modeling pharmacokinetic variability on the molecular level with stochastic compartmental systems, in M. Rowland, LB Sheiner and JL Steimer (eds), Variability in Drug Therapy: Description, Estimation, and Control, Raven, New York, 1985.
Page 193 - On approximating the moments of the equilibrium distribution of a stochastic logistic model. Biometrics 52.
Page 189 - A note on relationships between moments, central moments and cumulants from multivariate distributions, Stat & Prob.