On existence, multiplicity, uniqueness and stability of positive solutions of a Leslie–Gower type diffusive predator–prey system

2014 ◽  
Vol 19 (4) ◽  
pp. 669-688
Author(s):  
Jun Zhou ◽  
Keyword(s):  
Positive Solutions ◽  
Predator Prey ◽  
Prey System

scholarly journals Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

2021 ◽  
Vol 16 ◽  
pp. 25
Author(s):  
Pan Xue ◽  
Yunfeng Jia ◽  
Cuiping Ren ◽  
Xingjun Li
Keyword(s):  
Positive Solutions ◽  
A Priori ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Stationary Solutions ◽  
Existence Results ◽  
Predator Prey ◽  
Prey System ◽  
Homogeneous Neumann Boundary Condition ◽  
Constant Solution

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establish a priori estimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

scholarly journals Asymptotic Behaviors of a Delayed Nonautonomous Predator-Prey System Governed by Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lili Liu ◽  
Zhijun Liu
Keyword(s):  
Numerical Simulations ◽  
Positive Solutions ◽  
Differential System ◽  
Difference Equations ◽  
Global Attractivity ◽  
Asymptotic Behaviors ◽  
Predator Prey ◽  
Prey System ◽  
Discretization Technique ◽  
Theoretical Results

Based on a predator-prey differential system with continuously distributed delays, we derive a corresponding difference version by using the method of a discretization technique. A good understanding of permanence of system and global attractivity of positive solutions of system is gained. An example and its numerical simulations are presented to substantiate our theoretical results.

Dynamical behaviors of a diffusive predator–prey system with Beddington–DeAngelis functional response

2014 ◽  
Vol 07 (03) ◽  
pp. 1450033
Author(s):  
Er-Dong Han ◽  
Peng Guo
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Positive Solutions ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Lower Solution ◽  
Comparison Theory ◽  
Predator Prey ◽  
Prey System

In this paper, we present a diffusive predator–prey system with Beddington–DeAngelis functional response, where the prey species can disperse between the two patches, and there is competition between the two predators. Sufficient conditions for the permanence and extinction of system are established based on the upper and lower solution methods and comparison theory of differential equation. Furthermore, the global asymptotic stability of positive solutions is obtained by constructing a suitable Lyapunov function. By using the continuation theorem in coincidence degree theory, we show the periodicity of positive solutions. Finally, we illustrate global asymptotic stability of the model by a simulation figure.

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