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 language | English (US) |
|---|---|
| Pages (from-to) | 1534-1568 |
| Number of pages | 35 |
| Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |
| Volume | 80 |
| Issue number | 5 |
| DOIs | |
| State | Published - Oct 28 2014 |
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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
Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference. / Shi, Hong Bo; Ruan, Shigui.
In: IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications), Vol. 80, No. 5, 28.10.2014, p. 1534-1568.Research output: Contribution to journal › Article
}
TY - JOUR
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
UR - http://www.scopus.com/inward/record.url?scp=84943381915&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84943381915&partnerID=8YFLogxK
U2 - 10.1093/imamat/hxv006
DO - 10.1093/imamat/hxv006
M3 - Article
AN - SCOPUS:84943381915
VL - 80
SP - 1534
EP - 1568
JO - IMA Journal of Applied Mathematics
JF - IMA Journal of Applied Mathematics
SN - 0272-4960
IS - 5
ER -