Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion

Abstract

In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domaininduced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reactionkinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

Bifurcation analysis of domain-dependent nonlinear dynamics for a cross-diffusive reaction-diffusion model

“…Solving (9) for λ requires to find the roots of the corresponding characteristic polynomial, which in terms of the trace and determinant of the stability matrix on the left hand-side of ( 9) is a quadratic equation in λ of the form…”

“…Both Theorem 1 and 2 are limited to serve only as necessary conditions for the respected dynamical behaviors in the presence of cross-diffusion. This means, to ensure the expected dynamical behaviors we need further conditions involving relationships between L, d u , d v and d. Such conditions are derived by requiring the positivity of the determinant D(α, β) of the stability matrix (9). The determinant of the stability matrix ( 9) is expanded and factorized in the form of a product of a strictly positive quantity 1 α+β with a cubic polynomial in β given by…”

“…One of these forms of diffusion is cross diffusion, which serves to explain a process in which a gradient in concentration of one chemical or biological species induces the fluxes of another species [5]. It has also been shown that the presence of cross-diffusion in RDSs has a significant and positive effect on the ability of reaction-diffusion systems to form patterns in the presence of cross-diffusion [5,7,8,9]. Cross-diffusive induced pattern formation has been extensively studied in several biological applications [12,13,14,15,16].…”

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

“…Reaction-diffusion partial differential equations have been widely employed to provide mathematical models across many physical, chemical and biological processes [1-3], with classic applications including the spread of bushfires, the development of animal coat patterns, and the spread of epidemics. In recent decades the fundamental development of reaction-diffusion equations has focussed on extensions to incorporate physical complexities in two key areas; anomalous subdiffusion [4][5][6][7][8][9][10][11], and domain growth [3,[12][13][14][15][16][17][18][19][20][21]. Anomalous subdiffusion, which has been reported in numerous experimental observations [22][23][24][25][26][27][28][29][30], refers to diffusion processes in which the mean square displacement scales as a sublinear power law in time.…”

Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

“…In the modeling of chemical reactions or population dynamics, several authors have considered generalizations of the diffusion terms, choosing a positive value of the coefficient d in (1.1) to indicate a random diffusivity for the inhibitor [35,54], or introducing cross-diffusion terms to account for evasion-pursuit mechanisms [31,32,38,58,61,62,63].…”