Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator–prey pursuit and evasion example

We have studied properties of non-linear waves in a mathematical model of a predator–prey system with pursuit and evasion. We demonstrate a new type of propagating wave in this system. The mechanism of propagation of these waves essentially depends on the “taxis”, represented by non-linear “ cross-diffusion ” terms in the mathematical formulation. We have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms, “parabolic” and “linear”. Transition from one form to the other correlates with changes in the shape of the wave profile. Dependence of the propagation velocity on diffusion in this system differs from the square root dependence typical of reaction–diffusion waves. We also demonstrate that, for systems with negative and positive taxis, for example, pursuit and evasion, there typically exists a large region in the parameter space, where the waves demonstrate quasi-soliton interaction; colliding waves can penetrate through each other, and waves can also reflect from impermeable boundaries.