Bifurcation and patterns induced by flow in a prey-predator system with Beddington-DeAngelis functional response

In this paper, a prey-predator system described by a couple of advection-reaction-diffusion equations is studied theoretically and numerically, where the migrations of both prey and predator are considered and depicted by the unidirectional flow (advection term). To investigate the effect of population migration, especially the relative migration between prey and predator, on the population dynamics and spatial distribution of population, we systematically study the bifurcation and pattern dynamics of a prey-predator system. Theoretically, we derive the conditions for instability induced by flow, where neither Turing instability nor Hopf instability occurs. Most importantly, linear analysis indicates the instability induced by flow depends only on the relative flow velocity. Specifically, when the relative flow velocity is zero, the instability induced by flow does not occur. Moreover, the diffusion-driven patterns at the same flow velocity may not be stationary because of the contribution of flow. Numerical bifurcation analyses are consistent with the analytical results and show that the patterns induced by flow may be traveling waves with different wavelengths, amplitudes, and speeds, which are illustrated by numerical simulations.