Stabilization to a positive equilibrium for some reaction–diffusion systems

Spreading dynamics for a predator-prey system with two predators and one prey in a shifting habitat

“…for some large enough positive constant B, a simple comparison gives V ≤ v. Indeed, this can be done by checking that v is a supersolution of (15) (14), this completes the proof of (13). Therefore, the theorem is proved.…”

“…Proof of Theorem 1.4. We apply a contradiction argument used in [15]. Suppose that there exist δ > 0 and a sequence of points {(x n , t n )} with t n → ∞ and x n ∈ [(s + ε)t n , (s * − ε)t n ] such that…”

Spreading properties of a three-component reaction-diffusion model for the population of farmers and hunter-gatherers

“…Then, by parabolic estimates, possibly along a subsequence, one may assume that However, by applying the Lyapunov function Ψ(C, H) which satisfying (4.42), (4.47) contradicts to the convergence result of Theorem 1.1 in [13]. Therefore, the proof of (2) of Theorem 1.7 is complete.…”