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

Wanjun Li¹  Shengmao Fu²  Xiaoyan Gao³ 

[1] Longdong University, Qingyang, Gansu, P.R. China;School of Mathematics and Statistics, Northwest Normal University,Lanzhou, P.R. China;School of Mathematics and Statistics, Northwest Normal University, Lanzhou, P.R. China;

关键词: ratio-dependent;    stocking rate;    hopf bifurcation;    turing instability;    steady-state;    patter;   

DOI  :  10.14232/ejqtde.2019.1.80

来源: DOAJ

【 摘 要 】

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 a 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.

【 授权许可】

Unknown