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

Abstract

We study traveling wave solutions for Holling–Tanner type predator–prey models, where the predator equation has a singularity at zero prey population. The traveling wave solutions here connect the prey only equilibrium (1,0) with the unique constant coexistence equilibrium (u⁎,v⁎). First, we give a sharp existence result on weak traveling wave solutions for a rather general class of predator–prey systems, with minimal speed explicitly determined. Such a weak traveling wave (u(ξ),v(ξ)) connects (1,0) at ξ=−∞ but needs not connect (u⁎,v⁎) at ξ=∞. Next we modify the Holling–Tanner model to remove its singularity and apply the general result to obtain a weak traveling wave solution for the modified model, and show that the prey component in this weak traveling wave solution has a positive lower bound, and thus is a weak traveling wave solution of the original model. These results for weak traveling wave solutions hold under rather general conditions. Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions, namely they converge to the coexistence equilibrium as ξ→∞.