Cross-Diffusion-Driven Instability in a Reaction-Diffusion Harrison Predator-Prey Model

and Applied Analysis 3 If b > h, from the second equation of model (6), one has: . v ≤ v (b − h − hmv) . (9) A standard comparison argument shows that lim sup t→∞ v (t) ≤ b − h hm ≜ α. (10) If b ≤ h, we have the following differential inequality: . v ≤ −hmv2 mv + 1 , (11) and the same argument above yields lim sup t→∞ v (t) ≤ 0. (12) In either case, the second inequality of (7) holds. 2.2. Boundedness Theorem 2. All the solutions of model (6)which initiate inR+2 are uniformly bounded within the region Γ, where Γ = {(u, v) : 0 ≤ u + c bv ≤ 1 + 1 4h} . (13) Proof. Let us define the function: w (t) = u (t) + c bv (t) . (14) Calculating the time derivative of w(t) along the trajectories of model (6), we get . w (t) = . u + . v = u (1 − u) − ch b v. (15)