Title: A remark on the disorienting of species due to the fluctuating environment,
Abstract: 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.