Global Existence and Uniqueness of Periodic Waves in a Population Model with Density-Dependent Migrations and Allee Effect

We report a new result on the traveling wave solutions of a biological invasion model with density-dependent migrations and Allee effect. It has been shown in the literature that such a model can exhibit one periodic wave solution by using Hopf bifurcation theory. In this paper, global bifurcation theory is applied to prove that there exists maximal one periodic solution which can be reached in a large feasible parameter regime. The basic idea used in our technique is to examine the monotonicity of the ratio of related Abelian integrals. Especially, the existence condition for the solution near a homoclinic loop is obtained.