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

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. * Three chapters consisting of carefully written introductory exposition, main breakthroughs, open questions and bibliographies * Presents recent developments * Fills a void in the literature by making available a source book Preface Permanence, global attraction and stability 1 Introduction 2 Existence of a compact uniform attractor 3 Proof of Theorems 2.1, 2.2 and 2.3 4 Partial permanence and permanence 5 Necessary conditions for permanence of Lotka-Volterra systems 6 Sufficient condition for permanence of Lotka-Volterra systems 7 Further notes 8 Global attraction and stability of Lotka-Volterra systems 9 Global stability by Lyapunov functions 10 Global stability by split Lyapunov functions 10.1 Checking the conditions (10.2) and (10.8) 10.2 Examples 11 Global stability of competitive Lotka-Volterra systems 12 Global attraction of competitive Lotka-Volterra systems 13 Some notes Bibliography Competitive Lotka-Volterra systems with periodic coefficients 1 Introduction 2 The autonomous model. The logistic equation 3 Two species periodic models 4 Competitive exclusion 5 One species extinction in three-dimensional models 6 The impulsive logistic equation 7 Two species systems with impulsive effects. A look at the N-dimensional case 8 The influence of impulsive perturbations on extinction in three-species models Bibliography Fixed points, periodic points and chaotic dynamics for continuous maps with applications to population dynamics 1 Introduction 2 Notation 3 Search of fixed points for maps expansive along one direction 4 The planar case 4.1 Stretching along the paths and variants 4.2 The Crossing Lemma 5 The N-dimensional setting: Intersection Lemma 5.1 Zero-sets of maps depending on parameters 5.2 Stretching along the paths in the N-dimensional case 6 Chaotic dynamics for continuous maps 7 Definitions and main results 8 Symbolic dynamics 9 On various notions of chaos 10 Linked twist maps 11 Examples from the ODEs 12 Predator-prey model 12.1 The effects of a periodic harvesting 12.2 Technical details and proofs Bibliography Index

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.

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