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 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 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.

A diffusive predator–prey system with additional food and intra-specific competition among predators

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Ruizhi Yang ◽  
Ming Liu ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delay System ◽  
Additional Food ◽  
Predator Prey ◽  
Prey System ◽  
Boundary Equilibrium ◽  
The Stability

In this paper, a diffusive predator–prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For delay system, instability and Hopf bifurcation induced by time delay and global stability of boundary equilibrium are discussed. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived.

Qualitative analysis of a ratio-dependent predator–prey system with diffusion

2003 ◽  
Vol 133 (4) ◽  
pp. 919-942 ◽  
Author(s):  
Peter Y. H. Pang ◽  
Mingxin Wang
Keyword(s):  
Differential Equation ◽  
Differential System ◽  
Neumann Boundary ◽  
Qualitative Properties ◽  
Partial Differential ◽  
Predator Prey ◽  
Prey System ◽  
Homogeneous Neumann Boundary Condition ◽  
Ratio Dependent ◽  
Predator Prey Models

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.

Turing Pattern Induced by Cross-Diffusion in a Predator–Prey Model with Pack Predation-Herd Behavior

2020 ◽  
Vol 30 (07) ◽  
pp. 2050103
Author(s):  
Jingen Yang ◽  
Tonghua Zhang ◽  
Sanling Yuan
Keyword(s):  
Neumann Boundary ◽  
Multiple Scale ◽  
Turing Instability ◽  
Average Density ◽  
Field Analysis ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition

In this paper, we propose a diffusive predator–prey model with herd behavior in prey and pack predation behavior in predator under the homogeneous Neumann boundary condition. Due to the pack predation-herd behavior, the predator–prey interaction occurs only at the outer edge of each group. First, we analyze the existence and local stability of the equilibria of temporal model using vector field analysis and characteristic equations. Second, we deduce the conditions under which Turing instability occurs with the help of linear stability analysis. Then, using standard multiple-scale analysis, we derive the amplitude equations for the excited modes. By numerical simulations, we find that the model exhibits complex pattern replication by varying the value of parameters [Formula: see text] and [Formula: see text], and study the relationships between the average density of both prey and predator and these two parameters.

scholarly journals Dynamics and Pattern Formation of a Diffusive Predator-prey Model in the Subdiffusive Regime in Presence of Toxicity

2020 ◽  
Author(s):  
Daniel Cassidi Bitang A Ziem ◽  
Carlos Lawrence Gninzanlong ◽  
Conrad Bertrand Tabi ◽  
Timoléon Crépin Kofane
Keyword(s):  
Neumann Boundary ◽  
Turing Instability ◽  
Stability Domain ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Diffusive Regime ◽  
Ecological Literature ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Motivated by the fact that the restrictive conditions for a Turing instability are relaxed in sub- diffusive regime, we investigate the effects of subdiffu- sion in the predator−prey model with toxins under the homogeneous Neumann boundary condition. First, the stability analysis of the corresponding ordinary differ- ential equation is carried out. From this analysis, it fol- lows that stability is closely related to the coefficient of toxicity. In addition, the temporal fractional derivative does not systematically widen the range of parameters to maintain a point in the stability domain. Further- more, we derive the condition which links the Turing instability to the coefficient of toxicity in the subdif- fusive regime. System parameters are varied in order to test our mathematical predictions while comparing them to ecological literature. It turns out that the mem- ory effects, linked to the transport process can, depend- ing on the parameters, either stabilize an ecosystem or make a completely different configuration.

Spatiotemporal Dynamics of a Diffusive Leslie–Gower Predator–Prey Model with Ratio-Dependent Functional Response

2015 ◽  
Vol 25 (05) ◽  
pp. 1530014 ◽  
Author(s):  
Hong-Bo Shi ◽  
Shigui Ruan ◽  
Ying Su ◽  
Jia-Fang Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Ratio Dependent ◽  
Theoretical Results

This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. It is shown that the model exhibits spatial patterns via Turing (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, the existence of spatiotemporal patterns is established via Turing–Hopf bifurcation at the degenerate points where the Turing instability curve and the Hopf bifurcation curve intersect. Various numerical simulations are also presented to illustrate the theoretical results.

scholarly journals Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Wenjie Zuo
Keyword(s):  
Global Stability ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Prey System ◽  
Existence And Stability ◽  
Ratio Dependent ◽  
Stability And Bifurcations

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.

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