# Turing instability in two-patch predator-prey population dynamics

Volume 18, Issue 3, pp 255--261 Publication Date: February 22, 2018       Article History
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### Authors

Ali Al-Qahtani - Department of Mathematics, Faculty of Science, King Khalid University, Saudi Arabia
Aesha Almoeed - Department of Mathematics, Faculty of Science, King Khalid University, Saudi Arabia
Bayan Najmi - Department of Mathematics, Faculty of Science, King Khalid University, Saudi Arabia
Shaban Aly - Department of Mathematics, Faculty of Science, King Khalid University, Saudi Arabia $\&$ Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

### Abstract

In this paper, a spatio-temporal model as systems of ODE which describe two-species Beddington-DeAngelis type predator-prey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one's density, i.e., there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

### Keywords

• Self-diffusion
• cross-diffusion
• diffusive instability
• pattern formation

•  35K57
•  92B25
•  93D20

### References

• [1] U. Dieckmann, R. Law, J. A. Metz, The Geometry of Ecological Interaction: Simplifying Spatial Complexity, Cambridge University Press, Cambridge (2005)

• [2] M. Farkas, Two ways of modeling cross diffusion, Nonlinear Anal., 30 (1997), 1225–1233

• [3] M. Farkas, Dynamical Models in Biology, Academic Press, San Diego (2001)

• [4] Y. Huang, O. Diekmann, Interspecific influence on mobility and Turing instability, Bull. Math. Biol., 65 (2003), 143–156

• [5] J. D. Murray , Mathematical Biology, Springer-Verlag, Berlin (1989)

• [6] Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra system, World Scientific Publishing Co., Singapore (1996)

• [7] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37–72