Abstract
The present chapter is devoted to the mathematical modeling and the analysis of the dynamics of predator prey systems on a circular domain. We first give some reminders on the Laplace operator and spectral theory on a disc. Then, we analyze the dynamics of two mathematical models with two or three reaction diffusion equations, defined on a circular domain. The results are given in terms of local/global stability and of emergence of spatio-temporal patterns due to symmetry-breaking bifurcations. One basic type of such a phenomenon is Turing bifurcation which gives rise to pattern formation, a process by which a spatially uniform state loses stability to a non-uniform state. We derive, theoretically, the conditions for Turing diffusion driven instability to occur, and perform numerical simulations to illustrate how biological processes can affect spatiotemporal pattern formation in a spatial domain.
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Yafia, R., Aziz-Alaoui, M.A., El Yacoubi, S. (2016). Modeling and Dynamics of Predator Prey Systems on a Circular Domain. In: Cushing, J., Saleem, M., Srivastava, H., Khan, M., Merajuddin, M. (eds) Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics & Statistics, vol 186. Springer, New Delhi. https://doi.org/10.1007/978-81-322-3640-5_1
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