Analysis In Predator-prey Models And Crime Models With Directed Movement

Taxis phenomena describes the behavior that the organism tends to get close to or keep away from some materials as the environment changes.In this paper,we consider a three-species predator-prey system with predator-taxis and a mathemat-ical model of burglary with fatigue effect.For the predator-prey system,the global existence and uniform-in-time boundedness of classical solutions in any dimension-al bounded domain with the Neumann boundary conditions are obtained.Using Routh-Hurwitz criterion,we know that if the attractive predator-taxis coefficient is over a critical value,the system may generate Hopf bifurcation solutions.For the crime model,choosing the diffusion rate of attractiveness as the parameter,we prove that the local bifurcation can be generated at constant positive steady state of the system.Furthermore,we discuss the type and the stability of bifurcating solutions.