Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

Abstract

We have investigated the pattern formation in systems described by the nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation for the cases where the dimension of the pattern concentration area is less than that of independent variables space. We have obtained a system of integro-differential equations which describe the dynamics of the concentration area and the semiclassically limited distribution of a pattern in the class of trajectory concentrated functions. Also, asymptotic large-time solutions have been obtained that describe the semiclassically limited distribution of a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the prediction of a numerical simulation.

Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model

“…The Fisher-KPP equation has been introduced in [68,69]. For its nonlocal generalizations, see, e.g., [67,[70][71][72][73].…”

“…In this section, we briefly review a theoretical approach developed in [73] for the nonlocal Fisher-KPP equation:…”

“…Nonlocal Fisher-KPP models like (54) take into account that competition for the resource can be long range. Compared to the (local) Fisher-KPP equation, in nonlocal models, pattern formation can arise due to nonlocal competitive losses and diffusion (e.g., [70][71][72][73] under a certain choice of parameters. This is quite different from the well-known Turing morphogenesis where the mechanism of pattern formation relies on the competition between the activator and the inhibitor [77,78].…”

“…The approach developed in [73] considers special patterns described by Equation (54). These patterns are concentrated on k-dimensional manifolds Λ k in the space R n of independent variables x in Equation (54), k < n. For simplicity, we will consider the manifolds defined as:…”

“…For a relevant theory, we may state that the solution u( x, t) of Equation (54) generates a distribution ρ(t, s) on the manifold Λ k t , which is defined in [73] as a semiclassically limited distribution (SLD), as D → 0, on the space R k . The SLD is governed by more simple evolution equations compared to the original Equation (54).…”