Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference

Hong Bo Shi, Shigui Ruan

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

Original languageEnglish (US)
Pages (from-to)1534-1568
Number of pages35
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Volume80
Issue number5
DOIs
StatePublished - Oct 28 2014

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Spatio-temporal Patterns
Predator-prey Model
Hopf Bifurcation
Hopf bifurcation
Interference
Reaction-diffusion Model
Turing
Asymptotic stability
Uniform Persistence
Local Asymptotic Stability
Turing Instability
Bifurcation Curve
Functional Response
Predator-prey System
Predator prey systems
Global Asymptotic Stability
Neumann Boundary Conditions
Reaction-diffusion System
Intersect
Coexistence

Keywords

  • diffusive predator-prey model
  • Hopf bifurcation
  • mutual interference
  • positive non-constant steady states
  • Turing instability
  • Turing-Hopf bifurcation

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

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title = "Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference",
abstract = "In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.",
keywords = "diffusive predator-prey model, Hopf bifurcation, mutual interference, positive non-constant steady states, Turing instability, Turing-Hopf bifurcation",
author = "Shi, {Hong Bo} and Shigui Ruan",
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T1 - Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference

AU - Shi, Hong Bo

AU - Ruan, Shigui

PY - 2014/10/28

Y1 - 2014/10/28

N2 - In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

AB - In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction-diffusion predator-prey system with mutual interference described by the Crowley-Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction-diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing-Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction-diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.

KW - diffusive predator-prey model

KW - Hopf bifurcation

KW - mutual interference

KW - positive non-constant steady states

KW - Turing instability

KW - Turing-Hopf bifurcation

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