Abstract
In this paper, exact wave solutions to the single-species population dynamical model with density-dependent migrations and Allee effect are studied. First, the non-linear evolution equation is reduced to a planar system by transformation of variables, then based on the planar dynamical systems theory, its first integral is determined by computing singular point quantities, and a phase-portrait analysis of its singular points is presented. From this, some explicit expressions of the bounded travelling-wave solutions are obtained for the single-species model, which correspond to the real patterns of spread during biological invasions. In terms of the technique of finding exact travelling-wave solutions of a non-linear partial differential equation, the work is new.
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