Abstract
A predator-prey model with Ivlev-Type functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recently developed approach shows that for certain classes of models, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing a detailed analytic analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium, which enable us 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 responses.
Highlights
The predator-prey systems described by differential equations have served as important models in the studies of dynamical interaction of predator-prey species in ecological systems
The functional response considered here is assumed to depend on the prey population
Even the model which contains a complicated toxin-determined functional response can share the above properties [1]. Some models, such as the model with t Holling Types II and III functional response, share the following property stronger than Property (P2) [2, 4, 9]: (P3) The system (1.1) has a unique positive equilibrium E∗, and all positive, non-constant solutions converge to a unique limit cycle, a global attract, whenever E∗ is unstable
Summary
The predator-prey systems described by differential equations have served as important models in the studies of dynamical interaction of predator-prey species in ecological systems. Ivlev-Type functional response; Positive equilibrium; Local and Global stability. It has been confirmed that, for a large class of functional responses, the global dynamics of the predator-prey system (1.1) can be classified as (P1) The system (1.1) has a unique positive equilibrium E∗, and E∗ is globally stable whenever it is locally stable.
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