Lotka-Volterra and Related Systems (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

Z.Hou, London Met. Univ.; B.Lisena, UniBa, Bari; Z.Teng, Xinjiang Univ., Urumqi; F.Zanolin, UniUd, Udine.

Preface 5
Permanence, global attraction and stability 9
1 Introduction 9
2 Existence of a compact uniform attractor 11
3 Proof of Theorems 2.1, 2.2 and 2.3 16
4 Partial permanence and permanence 23
5 Necessary conditions for permanence of Lotka-Volterra systems 34
6 Sufficient condition for permanence of Lotka-Volterra systems 39
7 Further notes 47
8 Global attraction and stability of Lotka-Volterra systems 47
9 Global stability by Lyapunov functions 48
10 Global stability by split Lyapunov functions 50
10.1 Checking the conditions (10.2) and (10.8) 54
10.2 Examples 55
11 Global stability of competitive Lotka-Volterra systems 56
12 Global attraction of competitive Lotka-Volterra systems 63
13 Some notes 68
Bibliography 68
Competitive Lotka-Volterra systems with periodic coefficients 71
1 Introduction 71
2 The autonomous model. The logistic equation 72
3 Two species periodic models 76
4 Competitive exclusion 84
5 One species extinction in three-dimensional models 90
6 The impulsive logistic equation 99
7 Two species systems with impulsive effects. A look at the N-dimensional case 103
8 The influence of impulsive perturbations on extinction in three-species models 117
Bibliography 129
Fixed points, periodic points and chaotic dynamics for continuous maps with applications to population dynamics 131
1 Introduction 131
2 Notation 133
3 Search of fixed points for maps expansive along one direction 135
4 The planar case 136
4.1 Stretching along the paths and variants 136
4.2 The Crossing Lemma 151
5 The N-dimensional setting: Intersection Lemma 160
5.1 Zero-sets of maps depending on parameters 165
5.2 Stretching along the paths in the N-dimensional case 171
6 Chaotic dynamics for continuous maps 176
7 Definitions and main results 180
8 Symbolic dynamics 189
9 On various notions of chaos 198
10 Linked twist maps 206
11 Examples from the ODEs 214
12 Predator-prey model 215
12.1 The effects of a periodic harvesting 215
12.2 Technical details and proofs 223
Bibliography 233
Index 243