Resumen de A Quasilinear Predator-Prey Model with Indirect Prey-Taxis

Jie Xing, Pan Zheng, Xu Pan

• This paper deals with a quasilinear predator-prey model with indirect prey-taxis ⎧ ⎪⎨ ⎪⎩ ut =∇· (D(u)∇u) −∇· (S(u)∇w) + rug(v) − uh(u), (x, t) ∈ × (0,∞), wt = dww − μw + αv, (x, t) ∈ × (0,∞), vt = dvv + f (v) − ug(v), (x, t) ∈ × (0,∞), under homogeneous Neumann boundary conditions in a smooth bounded domain ⊂ Rn, n ≥ 1, where dw, dv, α, μ,r > 0 and the functions g, h, f ∈ C2([0,∞)).

  The nonlinear diffusivity D and chemosensitivity S are supposed to satisfy D(s) ≥ a(s + 1) −γ and 0 ≤ S(s) ≤ bs(s + 1) β−1 for all s ≥ 0, with a, b > 0 and γ,β ∈ R. Suppose that γ + β < 1 + 1 n and γ < 2 n , it is proved that the problem has a unique global classical solution, which is uniformly bounded in time. In addition, we derive the asymptotic behavior of globally bounded solution in this system according to the different predation conditions.