Abstract
This paper considers a discrete predator-prey system with Beddington-DeAngelis functional response. Sufficient conditions are obtained for the existence of the almost periodic solution which is uniformly asymptotically stable by constructing a Lyapunov function.
Highlights
In the past decades, the predator-prey competition models have been extensively studied by many authors
Let (x1(n), x2(n))T be any positive solution of system (6); from the first equation of system (6), it follows that x1
Let ε be an arbitrary small positive number. It follows from Theorem 9 that there exists a positive integer N0 such that xi∗ − ε ≤ xi (n) ≤ xi∗ + ε, i = 1, 2, ∀n > N0. (40)
Summary
The predator-prey competition models have been extensively studied by many authors (see [1,2,3,4,5,6,7]). Cai et al [9] studied the positive periodic solution for a multispecies competition-predator system with the Holling III functional and time delays. A well-known model of such systems is the predator-prey model with a Beddington-DeAngelis functional response which was originally proposed by Beddington [10] and DeAngelis et al [11], independently The dynamics of this model is described by the following differential equations: ẋ. With the method of the theory of difference inequality and constructing a suitable Lyapunov function, they obtained the permanence and the almost periodic solution of the system. We give some examples and numerical simulations to verify our results
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