A higher degenerated invasive-invaded species interaction

Invasive-invaded species problems are of relevance in mathematics applied to population dynamics. In this paper, the mentioned dynamics is introduced based on a fourth-order parabolic operator, together with coupled non-linear reaction terms. The fourth-order operator allows us to model a heterogeneous diffusion, as introduced by the Landau-Ginzburg free energy approach. The reaction terms are given by a coupled non-linear effect in the invasive species, to account for the action of the invaded species and limited resources, and by a non-Lipschitz term in the invaded species, to account for possible sprouts, once the invasion occurs. The analysis starts by the proof of existence and uniqueness of solutions, making use of the semi-group theory and a fixed point argument. Asymptotic solutions to the invasive species are explored with an exponential scaling. Afterward, the problem is analyzed with traveling wave profiles, for which a region of positive solutions is explored.