The role of convection in the existence of wavefronts for biased movements

Abstract

We investigate a model, inspired by Johnston et al. (2017) to describe the movement of a biological population which consists of isolated and grouped organisms. We introduce biases in the movements and then obtain a scalar reaction–diffusion equation that includes a convective term as a consequence of the biases. We focus on the case the diffusivity makes the parabolic equation of forward–backward–forward type and the reaction term models a strong Allee effect, with the Allee parameter lying between the two internal zeros of the diffusion. In such a case, the unbiased equation (i.e., without convection) possesses no smooth traveling‐wave solutions; on the contrary, in the presence of convection, we show that traveling‐wave solutions do exist for some significant choices of the parameters. We also study the sign of their speeds, which provides information on the long term behavior of the population, namely, its survival or extinction.