Modeling and Differential Equations in Biology

T. A. Burton

• Publisher: Routledge
• Year: 2017
• eBook ISBN: 9781351431026
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

First published in 1980. CRC Press is an imprint of Taylor & Francis.

STOCHASTIC PREY-PREDATOR RELATIONSHIPS

Georges A. Bécus

Engineering Science Department
University of Cincinnati
Cincinnati, Ohio

INTRODUCTION

Half a century ago, Lotka [24] and Volterra [40] initiated independently the deterministic theory of population dynamics. In the following decennium Kostitzin [20] and Kolmogorov [19] furthered the study of the nonlinear Lotka-Volterra model to the point that little has been added to the deterministic theory in the ensuing forty years. For an excellent account of the deterministic theory the reader should consult [31].

The stochastic theory of population dynamics was initiated by Feller almost forty years ago. Since then a vast amount of works dealing with stochastic models for population studies has appeared. The justification for such stochastic models can be found in the books by Bartlett [3], Iosifescu and Tautu [17], May [25], and Goel and Richter-Dyn [13].

This article presents a survey of various stochastic models (and their analysis) of interacting populations in a prey-predator relationship. These various works have been regrouped under six general headings. (A seventh one regroups hard-to-classify works.) However the reader will soon discover that the demarcation line between these six groups is not as clearly defined as the table of contents would indicate.

Caveat lector! This survey is restricted to prey-predator interactions. A number of works dealing with competitiv...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Contributors
• Persistence in Lotka-Volterra Models of Food Chains and Competition
• Mathematical Models of Humoral Immune Response
• Mathematical Models of Dose and Cell Cycle Effects in Multifraction Radiotherapy
• Theoretical and Experimental Investigations of Microbial Competition in Continuous Culture
• A Liapunov Functional for a Class of Reaction-Diffusion Systems
• Stochastic Prey-Predator Relationships
• Coexistence in Predator-Prey Systems
• Stability of Some Multispecies Population Models
• Population Dynamics in Patchy Environments
• Limit Cycles in a Model of R-Cell Stimulation
• Optimal Age-Specific Harvesting Policy for a Continuous Time-Population Model
• Models Involving Differential and Integral Equations Appropriate for Describing a Temperature Dependent Predator-Prey Mite Ecosystem on Apples