Mathematical Biosciences and Engineering, 2011, 8(3): 807-825. doi: 10.3934/mbe.2011.8.807

Primary: 58F15, 58F17; Secondary: 53C35.

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Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents

1. Mathematics Department, University of Louisiana at Lafayette, Lafayette, LA 70504

This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence in a class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, generated by maps. Here a unified approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of $\mathbb{R}^m_+$ to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.
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