Non-existence of wave solutions for the class of reaction--diffusion equations given by the Volterra interacting-population equations with diffusion.

Abstract The system of equations is reduced to a single nonlinear parabolic equation on which a maximum principle can be used. It is then shown that the effect of uniform diffusion on the Volterra equations for any even number of interacting populations which have non-zero equilibrium values, is to damp out all spatial variations. The inclusion of population saturation terms is shown to enhance the damping process, as would be expected. The main consequence of the results is that such reaction-diffusion equations (given in section 5) cannot have physically realistic wave-like solutions, that is stable solutions, with non-negative values of the concentrations, which evolve from a time dependent solution.