Global Dynamical Properties of Lotka-Volterra SystemsWorld Scientific, 1996 - 302 Mathematical ecology is a subject which recently attracts attentions of many mathematicians and biologists. One of the most important and fundamental mathematical models in ecology is of Lotka-Volterra type. This book gives global dynamical properties of L-V systems. The properties analyzed are global stability of the equilibria, persistence or permanence of the systems (which ensures the survival of all the biological-species composed of the systems for the long term) and the existence of periodic or chaotic solutions. The special subject of this book is to consider the effects of the systems structure, diffusion of the biological species and time delay on the global dynamical properties of the systems. |
Spis treści
SingleSpecies Growth | 5 |
Global Stability | 21 |
Periodic and Chaotic Behavior | 57 |
Global Stability and Persistence | 95 |
Periodic and Chaotic Behavior | 181 |
Global Stability and Permanence | 211 |
References | 265 |
Appendix | 283 |
B Stability of Equilibria | 289 |
Linear Complementarity Problem | 295 |
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Kluczowe wyrazy i wyrażenia
assume b₁ b₁/b2 and ẞ b₂/b₁ Beretta bistable competitive patch completes the proof condition Consider system D₁ defined delay denote density diagonal matrix diffusion disperse dynamics E+++ E++++ eigenvalues eigenvector ensures equation Exercise exists a positive extinction Further globally stable equilibrium globally stable positive Hence heteroclinic cycle Hint homotopy Hopf bifurcation implies initial value interaction matrix Jacobian matrix K₁ L-V systems Lemma let us consider Liapunov function limit cycle linear linear complementarity problem Lotka-Volterra systems Math negative definite nonnegative and globally nonnegative equilibrium point Note obtained one-predator P-matrix parameter periodic orbit persistent population positive constant positive equilibrium point positively invariant predator predator-mediated coexistence prey proof of Theorem refuge satisfying Section stable equilibrium point stable iff stable positive equilibrium stable with respect strongly monotone Suppose two-prey unique unstable vector xi(t y₁ αβ μ₁