July  2009, 12(1): 187-203. doi: 10.3934/dcdsb.2009.12.187

Lyapunov exponents and persistence in discrete dynamical systems

1. 

Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, United States, United States

Received  October 2008 Revised  January 2009 Published  May 2009

The theory of Lyapunov exponents and methods from ergodic theory have been employed by several authors in order to study persistence properties of dynamical systems generated by ODEs or by maps. Here we derive sufficient conditions for uniform persistence, formulated in the language of Lyapunov exponents, for a large class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, having the property that a nontrivial compact invariant set exists on a bounding hyperplane. We require that all so-called normal Lyapunov exponents be positive on such invariant sets. We apply the results to a plant-herbivore model, showing that both plant and herbivore persist, and to a model of a fungal disease in a stage-structured host, showing that the host persists and the disease is endemic.
Citation: Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
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