Prey-Taxis vs. An External Signal: Short-Wave Asymptotic and Stability Analysis

We consider two models of the predator–prey community with prey-taxis. Both models take into account the capability of the predators to respond to prey density gradients and also to one more signal, the production of which occurs independently of the community state (such a signal can be due to the spatiotemporal inhomogeneity of the environment arising for natural or artificial reasons). We call such a signal external. The models differ to one another through the description of their responses: the first one employs the Patlak–Keller–Segel law for both responses, and the second one employs Cattaneo’s model of heat transfer for both responses following to Dolak and Hillen. Assuming a short-wave external signal, we construct the complete asymptotic expansions of the short-wave solutions to both models. We use them to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria forced by the external signal. Such an approach refers back to Kapitza’s theory for an upside-down pendulum. The overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal.