Global stability of the predator-prey model with a sigmoid functional response

Abstract

A predator-prey model with Sigmoid functional response is studied. The main purpose is to investigate the global stability of a positive (coexistence) equilibrium, whenever it exists. A recent developed approach shows that, associated with the model, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing an analytic and geometrical analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium. With this crucial property we are able to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional response.

Dynamics of the predator–prey model with the Sigmoid functional response

“…As mentioned in Ref. 2, page 1160, little has been done in the literature on the analysis of the global dynamics of the system (1) with the Sigmoid functional response (2). When + + 2 can be factorized as a product of two linear factors, there obtained some sufficient conditions in Ref.…”

“…As shown in Ref. 2, the parameters and both can be taken 1 after a rescaling if necessary. So, in what follows we always take in (1) = = 1, and set ( ) = (1 − )∕ ( ) for ≠ 0.…”

“…Theorem 1. For the system (1) with the Sigmoid functional response (2) and having a positive equilibrium, the following statements hold.…”

“…For proving Theorem 1, we first present some basic facts on the system (1) with the functional response (2), where parts of them had appeared in Ref. 2.…”

Global Stability of a Predator-Prey Model with Ivlev-Type Functional Response

“…The systems that share the above properties include the Lotka-Volterra models, the model with the general Holling Type functional response [3], the model with the sigmoid functional response [15]. Even the model which contains a complicated toxin-determined functional response can share the above properties [1].…”