A remark on the disorienting of species due to the fluctuating environment

In this article we study the stabilizing of a primitive pattern of behaviour for the two-species community with chemotaxis due to the short-wavelength external signal. We use a system of Patlak-Keller-Segel type as a model of the community. It is well-known that such systems can produce complex unsteady patterns of behaviour which are usually explained mathematically by bifurcations of some basic solutions that describe simpler patterns. As far as we aware, all such bifurcations in the models of the Patlak-Keller-Segel type had been found for homogeneous (i.e. translationally invariant) systems where the basic solutions are equilibria with homogeneous distributions of all species. The model considered in the present paper does not possess the translational invariance: one of species (the predators) is assumed to be capable of moving in response to a signal produced externally in addition to the signal emitted by another species (the prey). For instance, the external signal may arise from the inhomogeneity of the distribution of an environmental characteristic such as temperature, salinity, terrain relief, etc. Our goal is to examine the effect of short-wavelength inhomogeneity. To do this, we employ a certain homogenization procedure. We separate the short-wavelength and smooth components of the system response and derive a slow system governing the latter one. Analysing the slow system and comparing it with the case of homogeneous environment shows that, generically, a short-wavelength inhomogeneity results in an exponential decrease in the motility of the predators. The loss of motility prevents, to a great extent, the occurrence of complex unsteady patterns and dramatically stabilizes the primitive basic solution. In some sense, the necessity of dealing with intensive small-scale changes of the environment makes the system unable to respond to other challenges.