Temporal and spatial patterns in a diffusive ratio-dependent predator-prey system with linear stocking rate of prey species

Li, Wanjun and Gao, Xiaoyan and Fu, Shengmao: Temporal and spatial patterns in a diffusive ratio-dependent predator-prey system with linear stocking rate of prey species. Electronic journal of qualitative theory of differential equations 80. pp. 1-26. (2019)

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Abstract

The ratio-dependent predator–prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in ratio-dependent predator–prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator–prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.

Item Type: Article
Journal or Publication Title: Electronic journal of qualitative theory of differential equations
Date: 2019
Number: 80
Page Range: pp. 1-26
ISSN: 1417-3875
DOI: https://doi.org/10.14232/ejqtde.2019.1.80
Uncontrolled Keywords: Hopf bifurkáció, Matematika
Additional Information: Bibliogr.: p. 23-26. ; összefoglalás angol nyelven
Date Deposited: 2020. Jan. 27. 12:32
Last Modified: 2020. Jan. 27. 12:33
URI: http://acta.bibl.u-szeged.hu/id/eprint/64724

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