Stationary and time periodic patterns of two-predator and one-prey systems with prey-taxis

摘要

This paper concerns pattern formation in a class ofreaction-advection-diffusion systems modeling the population dynamics of twopredators and one prey. We consider the biological situation that bothpredators forage along the population density gradient of the preys which candefend themselves as a group. We prove the global existence and uniformboundedness of positive classical solutions for the fully parabolic system overa bounded domain with space dimension $N=1,2$ and for the parabolic--parabolic-elliptic system over higher space dimensions. Linearized stabilityanalysis shows that prey-taxis stabilizes the positive constant equilibrium ifthere is no group defense while it destabilizes the equilibrium otherwise. Thenwe obtain stationary and time-periodic nontrivial solutions of the system thatbifurcate from the positive constant equilibrium. Moreover, the stability ofthese solutions is also analyzed in detail which provides a wave mode selectionmechanism of nontrivial patterns for this strongly coupled system. Finally, weperform numerical simulations to illustrate and support our theoreticalresults.