Abstract
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.
Highlights
Mathematical modeling of the pattern formation by the active media widely has employed the Patlak–Keller–Segel (PKS) systems for several decades
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
We start with revisiting the onset of instability of the homogeneous equilibria, which is known mainly due to the works by Govorukhin et al [11] and by Arditi et al [12]
Summary
Mathematical modeling of the pattern formation by the active media widely has employed the Patlak–Keller–Segel (PKS) systems for several decades. It is natural to ask to what extent the inhomogeneities influence these instabilities compared to the case of homogeneity These questions become quite tractable provided that the scale of the environmental fluctuations is relatively small as we can address them by borrowing the ideas of Kapitza’s theory of the upside-down pendulum [21] and combining them with the homogenization [22,23]. Examining the stability of the quasi-equilibria shows that, compared to the case of the homogeneous environment, increasing the magnitude of the fluctuations can destabilize the quasi-equilibria and stabilize them Which of these effects occurs depends only on the speed at which the signal-making wave propagates, but they both are exponential in the amplitude of the wave. Appendixes A–C contain the details of proofs of the assertions formulated in the article body
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