BIFURCATION AND SPATIAL PATTERNS DRIVEN BY PREDATOR-TAXIS IN A PREDATOR-PREY SYSTEM WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

We consider a reaction-diffusion predator-prey model with predatortaxis, in which both the cost and the benefit induced by fear of predators are incorporated. In particular, the benefit of fear is reflected in the Beddington- DeAngelis functional response that can be derived in view of avoidance behaviours of prey. Critical conditions for Turing instability are determined with the help of the first Turing bifurcation curve in the two-parameter plane composed by the random diffusion rate of prey and the predator-taxis rate. For predator-taxis-induced bifurcation from simple eigenvalues, the existence and stability of non-homogeneous positive steady state solutions are established. Especially, we use decomposition in space and apply the implicit function theorem to obtain the bifurcation theorem with double eigenvalues. Theoretical and numerical results show that low predator-taxis sensitivity may cause the occurrence of spatial patterns when the random diffusion rate of prey is slow. However, the presence of predator-taxis can not lead to pattern formation when the random diffusion rate of prey is fast enough. In addition, increasing the level of fear may stabilize the system at the expense of the density of prey or predators.