scholarly journals Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-(n+1) Functional Response

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xiao-zhou Feng ◽  
Zhi-guo Wang
Keyword(s):  
Functional Response ◽  
Neumann Boundary ◽  
Steady States ◽  
Diffusion Term ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Prey System ◽  
Homogeneous Neumann Boundary Condition ◽  
Constant Solution ◽  
Positive Steady States

This paper discusses a predator-prey system with Holling-(n+1) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution(u~,v~)is asymptotically stable when the parameterksatisfies some conditions.

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 Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

Stability and Bifurcation in a Predator–Prey System with Prey-Taxis

2020 ◽  
Vol 30 (02) ◽  
pp. 2050022 ◽  
Author(s):  
Huanhuan Qiu ◽  
Shangjiang Guo ◽  
Shangzhi Li
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Steady States ◽  
Mode Interaction ◽  
Small Range ◽  
Steady State Mode ◽  
Predator Prey ◽  
Prey System

In this paper, we consider a generalized predator–prey system with prey-taxis under Neumann boundary condition, that is, the predators can survive even in the absence of the prey species. It is proved that for an arbitrary spatial dimension, the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range. Moreover, the local stabilities of constant steady states (including trivial, semi-trivial and positive constant steady states) are investigated. A further study on the coexistence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influences the steady-state/Hopf bifurcations (if they exist). Analyses of steady-state bifurcation, Hopf bifurcation, and even Hopf/steady-state mode interaction are carried out in detail by means of the Lyapunov–Schmidt procedure. In particular, we obtain stable or unstable steady states, time-periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found in numerical simulations.

Positive steady states of the Holling–Tanner prey–predator model with diffusion

2005 ◽  
Vol 135 (1) ◽  
pp. 149-164 ◽  
Author(s):  
Rui Peng ◽  
Mingxin Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Steady States ◽  
Homogeneous Neumann Boundary Condition ◽  
Positive Steady States ◽  
Predator Model

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

scholarly journals Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

2019 ◽  
Vol 16 (6) ◽  
pp. 6753-6768
Author(s):  
Xiaoling Li ◽  
◽  
Guangping Hu ◽  
Xianpei Li ◽  
Zhaosheng Feng ◽  
...  
Keyword(s):  
Steady States ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Positive Steady States ◽  
Ratio Dependent
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