Article Contents

2013, Volume 18, Issue 8: 2123-2142. Doi: 10.3934/dcdsb.2013.18.2123

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Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates

Yun Kang^1,

1.
Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212

Received: November 2011

Revised: January 2013

Published: October 2013

Abstract / Introduction Related Papers Cited by

Abstract

The per-capita growth rate of a species is influenced by density-independent, positive and negative density-dependent factors. These factors can lead to nonlinearity with a consequence that species may process multiple nontrivial equilibria in its single state (e.g., Allee effects). This makes the study of permanence of discrete-time multi-species population models very challenging due to the complex boundary dynamics. In this paper, we explore the permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates for the first time. We find a simple sufficient condition for guaranteeing the permanence of the system by applying and extending the ecological concept of the relative nonlinearity to estimate systems' external Lyapunov exponents. Our method allows us to fully characterize the effects of nonlinearities in the per-capita growth functions and implies that the fluctuated populations may devastate the permanence of systems and lead to multiple attractors. These results are illustrated with specific two species competition and predator-prey models with generic nonlinear per-capita growth functions. Finally, we discuss the potential biological implications of our results.

Keywords:

Allee effects,
nonlinear per-capita growth rates,
permanence,
relative nonlinearity,
two-species-interaction population models.

Mathematics Subject Classification: Primary: 39A22, 39A33; Secondary: 92B05, 92D25, 92D40.

Citation:

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