In this paper, we study in detail the existence and stability of localized pulses in a Gierer-Meinhardt equation with an additional "slow" nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray-Scott and Gierer-Meinhardt. We investigate the existence of these pulses using the methods of geometric singular perturbation theory. The additional nonlinearity has a profound impact on both the stability analysis of the pulse-compared to Gray-Scott/Gierer-Meinhardt-type models a distinct extension of the Evans function approach has to be developed-and the stability properties of the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray-Scott/Gierer-Meinhardt-type models, the pulse solutions of the model exhibit a rich and complex behavior near the Hopf bifurcations.
The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.
Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator–inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives
vis-à-vis
the classical Turing analysis, and give an overview of recent results and future challenges for both approaches.
This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.
We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast-3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast-slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.
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