We consider a reaction-diffusion equation ut = uxx + f(u), where f has exactly three zeros 0, α and 1 (0 α having two interfaces at a large distance, u(x, t) approaches a pair of travelling wave solutions φ(x − p1(t)) + φ(−x + p2(t)) for a long time, and then the travelling fronts eventually disappear by colliding with each other. While our results establish this process, they show that there is a (backward) global solution ψ(x, t) and that the annihilation process is approximated by a solution ψ(x−x0, t−t0).

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Backward global solutions characterizing annihilation dynamics of travelling fronts (Nonlinear Diffusive Systems and Related Topics)

Traveling waves for a generalized Holling-Tanner predator-prey model

We study a predator–prey model with two alien predators and one aborigine prey in which the net growth rates of both predators are negative. We characterize the invading speed of these two predators by the minimal wave speed of traveling wave solutions connecting the predator-free state to the co-existence state. The proof of the existence of traveling waves is based on a standard method by constructing (generalized) upper-lower-solutions with the help of Schauder’s fixed point theorem. However, in this three species model, we are able to construct some suitable pairs of upper-lower-solutions not only for the super-critical speeds but also for the critical speed. Moreover, a new form of shrinking rectangles is introduced to derive the right-hand tail limit of wave profile.

Traveling wave solutions for a predator–prey system with two predators and one prey

This paper is concerned with the propagation dynamics of a predator–prey system with nonlocal dispersal. We obtain a threshold phenomenon for the invasion of the predator into the habitat of the aborigine prey. It turns out that this threshold is the so-called spreading speed of the predator as well as the minimal wave speed of traveling wave solutions connecting the predator-free state to a nontrivial state.

Chih-Hong Yao

Papers

Traveling wave solutions for a continuous and discrete diffusive predator–prey model

Entire solutions originating from monotone fronts to the Allen–Cahn equation