Wavefronts in Forward-Backward Parabolic Equations and Applications to Biased Movements

We consider a discrete biological model concerning the movements of organisms, whose population is formed by isolated and grouped individuals. The movement occurs in a random way in one spatial dimension and the transition probabilities per unit time for a one-step jump are assigned. Differently from other papers on the same subject, we assume that the random walk is biased and so, by passing to the limit, we obtain a parabolic equation which includes a convective term. The noteworthy feature of the equation is that the diffusivity changes sign. We investigate the existence of wavefront solutions for this equation, their qualitative properties and we estimate their admissible speeds; in this way we generalize some recent results concerning the case of unbiased movements. Our discussion makes use of some results obtained by the authors on the existence of wavefront solutions in backward-forward parabolic equations.