Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat

Abstract

We explore an approach based on the theory of cosymmetry to model interaction of predators and prey in a two-dimensional habitat. The model under consideration is formulated as a system of nonlinear parabolic equations with spatial heterogeneity of resources and species. Firstly, we analytically determine system parameters, for which the problem has a nontrivial cosymmetry. To this end, we formulate cosymmetry relations. Next, we employ numerical computations to reveal that under said cosymmetry relations, a one-parameter family of steady states is formed, which may be characterized by different proportions of predators and prey. The numerical analysis is based on the finite difference method (FDM) and staggered grids. It allows to follow the transformation of spatial patterns with time. Eventually, the destruction of the continuous family of equilibria due to mistuned parameters is analyzed. To this end, we derive the so-called cosymmetric selective equation. Investigation of the selective equation gives an insight into scenarios of local competition and coexistence of species, together with their connection to the cosymmetry relations. When the cosymmetry relation is only slightly violated, an effect we call 'memory on the lost family' may be observed. Indeed, in this case, a slow evolution takes place in the vicinity of the lost states of equilibrium.

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