Indirect Taxis on a Fluctuating Environment

In this article, we study a Patlak-Keller-Siegel (PKS) model of a community of two species placed in the inhomogeneous environment. We employ PKS law for modeling tactic movement due to interspecific taxis and in response to the environmental fluctuations. These fluctuations can arise for natural reasons, e.g., the terrain relief, the sea currents and the food resource distribution, and there are artificial ones. The main result in the article elucidates the effect of the small-scale environmental fluctuations on the large-scale pattern formation in PKS systems. This issue remains uncharted, although numerous studies have addressed the pattern formation while assuming an homogeneous environment. Meanwhile, exploring the role of the fluctuating environment is substantial in many respects, for instance, for predicting the side effects of human activity or for designing the control of biological systems. As well, it is necessary for understanding the roles played in the dynamics of trophic communities by the natural environmental inhomogeneities-those mentioned above, for example. We examined the small-scale environmental inhomogeneities in the spirit of Kapitza's theory of the upside-down pendulum, but we used the homogenization instead of classical averaging. This approach is novel for the dynamics of PKS systems (though used commonly for other areas). Employing it has unveiled a novel mechanism of exerting the effect from the fluctuating environment on the pattern formation by the drift of species arising upon the homogenization of the fluctuations.