Global boundedness in a chemotaxis quasilinear parabolic predator–prey system with pursuit-evasion

Abstract

We analyze some configurations of the general chemotaxis predator–prey model with pursuit-evasion dynamics ∂tu−∇⋅(Fu(u)∇u)+∇⋅(Fp(u)∇p)=uF1(w)−F2(u)∂tw−∇⋅(Fw(w)∇w)−∇⋅(Fq(w)∇q)=wF3(w)−δuwin Ω×(0,T) with Neumann boundary condition and non-negative initial data, where p and q are the predator’s and the prey’s pheromone, respectively, modeled by parabolic or elliptic equations, and Ω⊂Rd, with d≥1, is a smooth bounded domain. We assume Fu and Fw to be smooth positive functions satisfying kusp1≤Fu(s) and kwsp2≤Fw(s) when s≥s0>1, Fp,Fq smooth non-negative functions such that kp1sp0≤Fp(s)≤kp1sp0 when s≥s0 and Fq(s)≡kqsq0 for all s≥0, with q0=1 or q0≥2. We also assume F1,F2 and F3 to be smooth with F1(0)=F2(0)=F3(0)=0, F2≥0, F1(s)≤k1sθ, F2(s)≥k2s1+b, b≥0, F3(s)≤k3−k4sa, for s≥s0, a>0, k3≥0, k4>0. We prove that for θ,a,b,q0 and p0 satisfying some relation there exists a unique classical solution to the system which is global in time and bounded. The result in independent on p1,p2∈R.