Lotka-Volterra and Related Systems: Recent Developments in Population Dynamics (De Gruyter Series in Mathematics and Life Sciences)

Main description: This book facilitates research in the general area of population dynamics by presenting some of the recent developments involving theories, methods and application in this important area of research. The underlying common feature of the studies included in the book is that they are related, either directly or indirectly, to the well-known Lotka-Volterra systems which offer a variety of mathematical concepts from both theoretical and application points of view Cover 1 Title 4 Copyright 5 Preface 6 Contents 8 Permanence, global attraction and stability 10 1 Introduction 10 2 Existence of a compact uniform attractor 12 3 Proof of Theorems 2.1, 2.2 and 2.3 17 4 Partial permanence and permanence 24 5 Necessary conditions for permanence of Lotka-Volterra systems 35 6 Sufficient condition for permanence of Lotka-Volterra systems 40 7 Further notes 48 8 Global attraction and stability of Lotka-Volterra systems 48 9 Global stability by Lyapunov functions 49 10 Global stability by split Lyapunov functions 51 10.1 Checking the conditions (10.2) and (10.8) 55 10.2 Examples 56 11 Global stability of competitive Lotka-Volterra systems 57 12 Global attraction of competitive Lotka-Volterra systems 64 13 Some notes 69 Bibliography 69 Competitive Lotka-Volterra systems with periodic coefficients 72 1 Introduction 72 2 The autonomous model. The logistic equation 73 3 Two species periodic models 77 4 Competitive exclusion 85 5 One species extinction in three-dimensional models 91 6 The impulsive logistic equation 100 7 Two species systems with impulsive effects. A look at the N-dimensional case 104 8 The influence of impulsive perturbations on extinction in three-species models 118 Bibliography 130 Fixed points, periodic points and chaotic dynamics for continuous maps with applications to population dynamics 132 1 Introduction 132 2 Notation 134 3 Search of fixed points for maps expansive along one direction 136 4 The planar case 137 4.1 Stretching along the paths and variants 137 4.2 The Crossing Lemma 152 5 The N-dimensional setting: Intersection Lemma 161 5.1 Zero-sets of maps depending on parameters 166 5.2 Stretching along the paths in the N-dimensional case 172 6 Chaotic dynamics for continuous maps 177 7 Definitions and main results 181 8 Symbolic dynamics 190 9 On various notions of chaos 199 10 Linked twist maps 207 11 Examples from the ODEs 215 12 Predator-prey model 216 12.1 The effects of a periodic harvesting 216 12.2 Technical details and proofs 224 Bibliography 234 Index 244

In recent years, there has been a tremendous amount of research activity in the general area of population dynamics, particularly the Lotka-Volterra system, which has been a rich source of mathematical ideas from both theoretical and application points of view.

In spite of the technological advances, many authors seem to be unaware of the bulk of the work that has been done in this area recently. This often leads to duplication of work and frustration to the authors as well as to the editors of various journals. This book is built out of lecture notes and consists of three chapters written by four mathematicians with overlapping expertise that cover a broad sector of the research in this area. Each chapter consists of carefully written introductory exposition, main breakthroughs, open questions and bibliographies.

The chapters present recent developments on topics involving the dynamic behavior of solutions and topics such as stability theory, permanence, persistence, extinction, existence of positive solutions for the Lotka-Volterra and related systems. This fills a void in the literature, by making available a source book of relevant information on the theory, methods and applications of an important area of research.