GLOBAL EXISTENCE AND CONVERGENCE TO STEADY STATES FOR A PREDATOR-PREY MODEL WITH BOTH PREDATOR- AND PREY-TAXIS

In this work we consider a two-species predator-prey chemotaxis model {u(t) = d(1)Delta u + chi(1)del.(u del v) + u(a(1) - b(11)u -b(12)v), x is an element of Omega, t > 0, v(t) = d(2)Delta v + chi(2)del.(u del v) + v(a(2) - b(21)u -b(22)v-b(23)w), x is an element of Omega, t > 0, (*) w(t) = d(3)Delta w - chi(3)del.(w del v) + w(-a(3) - b(32)v-b(33)w), x is an element of Omega, t > 0 in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model (*) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions (u, v, w) exponentially converges to constant stable steady state (u*, v*, w*). Inspired by [5], we employ the special structure of (*) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.

Automated summary

In this work, we considered a two-species predator-prey chemotaxis model and proved that it admits a global boundeness of classical solutions in any physically meaningful dimension. By carefully balancing the triple cross diffusion, we showed that the global classical solutions exponentially converge to a constant stable steady state.