Pattern formation in parabolic equations containing hysteresis with diffusive thresholds

Abstract

We consider a reaction-diffusion system with discontinuous reaction terms modeled by non-ideal relays. The system is motivated by an epigenetic population model of evolution of two-phenotype bacteria which switch phenotype in response to variations of environment. We prove the formation of patterns in the phenotype space. The mechanism responsible for pattern formation is based on memory (hysteresis) of the non-ideal relays.

Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds

“…Below we are interested in the limiting behavior of the configuration function r(x, t) given by (3.3), or, in other words, in the limiting distribution of phenotypes over the population of thresholds. In [23], we have proved that r(x, t) converges to a step like profile as t → ∞, see Fig. 5.1.…”

“…This integral can be interpreted as the Preisach operator [8,13,31,32,34,35,43,44,46,50] with a time dependent density (the density is a component of the solution describing the varying distribution of bacteria). In [23], we have shown that fitness, competition and diffusion can act together to select a nontrivial distribution of phenotypes (states) over the population of thresholds.…”

“…The model exhibits formation of patterns in the space of distributions of the two phenotypes over an admissible range of the switching thresholds. The convergence of solutions to a stationary pattern observed numerically in [17] was proved in [23]. The objective of this work is to obtain asymptotic formulas for the pattern and the timing of its formation under the assumption of slow diffusion.…”