Integrodifferential Equations and Delay Models in Population Dynamics: (20 Lecture Notes in Biomathematics)

These notes are, for the most part, the result of a course I taught at the University of Arizona during the Spring of 1977. Their main purpose is to inves­ tigate the effect that delays (of Volterra integral type) have when placed in the differential models of mathematical ecology, as far as stability of equilibria and the nature of oscillations of species densities are concerned. A secondary pur­ pose of the course out of which they evolved was to give students an (at least elementary) introduction to some mathematical modeling in ecology as well as to some purely mathematical subjects, such as stability theory for integrodifferentia1 systems, bifurcation theory, and some simple topics in perturbation theory. The choice of topics of course reflects my personal interests; and while these notes were not meant to exhaust the topics covered, I think they and the list of refer­ ences come close to covering the literature to date, as far as integrodifferentia1 models in ecology are concerned. I would like to thank the students who took the course and consequently gave me the opportunity and stimulus to organize these notes. Special thanks go to Professor Paul Fife and Dr. George Swan who also sat in the course and were quite helpful with their comments and observations. Also deserving thanks are Professor Robert O'Malley and Ms. Louise C. Fields of the Applied Mathematics Program here at the University of Arizona. Ms. Fields did an outstandingly efficient and accu­ rate typing of the manuscript.

Table of Contents:
1: Introductory Remarks.- 2: Some Preliminary Remarks on Stability.- 2.1 Linearization.- 2.2 Autonomous Linear Systems.- 3: Stability and Delay Models for a Single Species.- 3.1 Delay Logistic Equations.- 3.2 The Logistic Equation with a Constant Time Lag.- 3. 3 Some Other Models.- 3.4 Some General Results.- 3.5 A General Instability Result.- 3.6 The Stabilizing Effect of Delays.- 4: Stability and Multi-Species Interactions with Delays.- 4.1 Volterra’s Predator-Prey Model with Delays.- 4. 2 Predator-Prey Models with Density Terms.- 4.3 Predator-Prey Models with Response Delays to Resource Limitation.- 4.4 Stability and Vegetation-Herbivore-Carnivore Systems.- 4.5 Some Other Delay Predator-Prey Models.- 4.6 The Stabilization of Predator-Prey Interactions.- 4.7 A General Predator-Prey Model.- 4.8 Competition and Mutualism.- 4.9 Stability and Instability of n-Species Models.- 4.10 Delays Can Stabilize an Otherwise Unstable Equilibrium.- 5: Oscillations and Single Species Models with Delays.- 5.1 Single Species Models and Large Delays.- 5.2 Bifurcation of Periodic Solutions of the Delay Logistic.- 5.3 Other Results on Nonconstant Periodic Solutions.- 5.4 Periodically Fluctuating Environments.- 6: Oscillations and Multi-Species Interactions with Delays.- 6.1 A General Bifurcation Theoren.- 6.2 Periodic Oscillations Due to Delays in Predator-Prey Interactions..- 6.3 Numerically Integrated Examples of Predator-Prey Models with Delays.- 6.4 Oscillations and Predator-Prey Models with Delays.- 6.5 Two Species Competition Models with Linear Response Functionals.- 6.6 Two Species Mutualism Models with Linear Response Functionals.- 6.7 Delays in Systems with More than Two Interacting Species.- 6.8 Periodically Fluctuating Environments.- 7: Some Miscellaneous Topics.- References.