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.

scholarly journals Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Jia-Fang Zhang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Positive Constant ◽  
Functional Differential Equations ◽  
Neumann Boundary ◽  
Steady State Solution ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model

This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

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 Convective instability in a diffusive predator–prey system

2021 ◽  
pp. 1-9
Author(s):  
Hui Chen ◽  
Xuelian Xu
Keyword(s):  
Steady State ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Steady States ◽  
Small Diffusion ◽  
Predator Prey ◽  
Positive Steady States ◽  
Existence And Stability ◽  
Biological Pattern Formation ◽  
Turing Mechanism

It is well known that biological pattern formation is the Turing mechanism, in which a homogeneous steady state is destabilized by the addition of diffusion, though it is stable in the kinetic ODEs. However, steady states that are unstable in the kinetic ODEs are rarely mentioned. This paper concerns a reaction diffusion advection system under Neumann boundary conditions, where steady states that are unstable in the kinetic ODEs. Our results provide a stabilization strategy for the same steady state, the combination of large advection rate and small diffusion rate can stabilize the homogeneous equilibrium. Moreover, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis.

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.

scholarly journals Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

2016 ◽  
Vol 26 (02) ◽  
pp. 1650033 ◽  
Author(s):  
Ling Jin ◽  
Qi Wang ◽  
Zengyan Zhang
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Neumann Boundary ◽  
Steady States ◽  
Local Theory ◽  
Logistic Growth ◽  
Homogeneous Steady State ◽  
Cellular Aggregation ◽  
Chemotaxis Models ◽  
Steady State Stability

In this paper, we investigate pattern formation in Keller–Segel chemotaxis models over a multidimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate [Formula: see text] increases. Then using Crandall–Rabinowitz local theory with [Formula: see text] being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally, we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.

Effects of Delay and Diffusion on the Dynamics of a Leslie–Gower Type Predator–Prey Model

2014 ◽  
Vol 24 (04) ◽  
pp. 1450043
Author(s):  
Jia-Fang Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Periodic Solutions ◽  
Positive Constant ◽  
Neumann Boundary ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Delayed System ◽  
The Stability ◽  
And Diffusion

In this paper, we consider the effects of time delay and space diffusion on the dynamics of a Leslie–Gower type predator–prey system. It is shown that under homogeneous Neumann boundary condition the occurrence of space diffusion does not affect the stability of the positive constant equilibrium of the system. However, we find that the incorporation of a discrete delay representing the gestation of prey species can not only destabilize the positive constant equilibrium of the system but can also cause a Hopf bifurcation at the positive constant equilibrium as it crosses some critical values. In particular, we prove that these Hopf bifurcations' periodic solutions are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as periodic solutions of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system will generate spatially nonhomogeneous periodic solutions. The results in this work demonstrate that diffusion plays an important role in deriving complex spatiotemporal dynamics.

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