Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

2021 ◽  
Author(s):  
Haixia Li ◽  
Wenbin Yang ◽  
Meihua Wei ◽  
Aili Wang
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
A Priori Bound ◽  
Steady State Solutions

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

scholarly journals Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Changtong Li ◽  
Hao Sun ◽  
Yuzhen Wang
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Bifurcation Theory ◽  
A Priori Estimates ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
Steady State Solutions

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

2019 ◽  
Vol 29 (11) ◽  
pp. 1950145 ◽  
Author(s):  
Yu-Xia Wang ◽  
Wan-Tong Li
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Spatial Patterns ◽  
Bifurcation Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Prey System

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

Bifurcation of steady-state solutions in predator-prey and competition systems

1984 ◽  
Vol 97 ◽  
pp. 21-34 ◽  
Author(s):  
J. Blat ◽  
K. J. Brown
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Predator Prey ◽  
Interacting Species ◽  
Global Bifurcation Theory ◽  
Steady State Solutions

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

scholarly journals Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Meng Lin ◽  
Yanyou Chai ◽  
Xuguang Yang ◽  
Yufeng Wang
Keyword(s):  
Neumann Boundary ◽  
Spatiotemporal Pattern ◽  
Hopf Bifurcations ◽  
Herd Behavior ◽  
Predator Prey ◽  
Infinite Dimensional ◽  
Prey System ◽  
Dimensional Dynamical System ◽  
Pattern Formations ◽  
Steady State Solutions

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

Spatiotemporal Patterns of a Predator–Prey System with an Allee Effect and Holling Type III Functional Response

2016 ◽  
Vol 26 (05) ◽  
pp. 1650088 ◽  
Author(s):  
Yuanyuan Li ◽  
Jinfeng Wang
Keyword(s):  
Allee Effect ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Rigorous Results ◽  
Predator Prey ◽  
Type Iii ◽  
Prey System ◽  
Response Subject ◽  
Steady State Solutions

A diffusive Gause type predator–prey system with Allee effect in prey growth and Holling type III response subject to Neumann boundary conditions is investigated. Existence of nonconstant positive steady state solutions is proved by Leray–Schauder degree theory and bifurcation theory. Global stability of the positive equilibrium of the system is also investigated. Moreover, bifurcations of spatially homogeneous and nonhomogeneous periodic solutions are analyzed. Our rigorous results justify some recent ecological observations.

scholarly journals Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 432 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Yuzhen Bai
Keyword(s):  
Allee Effect ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Stability Switches ◽  
Predator Prey ◽  
Prey System ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Bifurcation And Stability

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.

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