Analysis Of The Instability Of A Multi-Species Keller-Segel Chemotaxis Model And The Stability Of A Two Species Predator-Prey Model

In this paper we study the instability of a generalized Keller-Segel chemotaxis model and stability of a two species predator-prey model with a prey refuge. we discuss Turing instability of a generalized Keller-Segel chemotaxis model involving n-species and m-chemoattractants. By using combinatorial matrix theory we ex-tend the result on the instability of a single-species and n-chemicals Keller-Segel chemotaxis system(P. Leenheer, J. Gopalakrishnan, E. Zuhr, Instability in a gen-eralized Keller-Segel model, Journal of Biological Dynamics,6 (2012) 974-991) to a general (n+m) × (n+m) system. A sufficient condition for chemotaxis driven Turing instability of the multi-species model is obtained. Moreover, two examples of the theoretical results are presented. Then we study the stability of a two species predator-prey model with a prey refuge by linearization technique, especially the influence of the prey refuge on stability.