Abstract
Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.
Highlights
Several models in mathematical biology take the form of a reactiondiffusion system ut = ε2∆u + f (u, v)τ vt = D∆v + g(u, v) in Ω × (0, T ) (1) with ∂u ∂v= =0 on ∂Ω × (0, T ) (2)∂ν ∂ν where ε, τ, and D are positive constants, Ω is a bounded domain in RN with smooth boundary ∂Ω, and ν is the outer unit normal vector
Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits
∗Key Words: reaction-diffusion equation, Gierer-Meinhardt system, Turing pattern, Hamilton structure, asymptotic behavior of the solution, AMS Mathematical Subject Classification 2010: 35K57, 35Q92 of them is the Gierer-Meinhardt system in morphogenesis [2] which is the case of up ur f (u, v) = −u + vq, g(u, v) = −v + vs with p > 1, q, r > 0, s > −1. It is concerned with pattern formations of spatial tissue structures of hydra, where u = u(x, t) > 0 and v = v(x, t) > 0 stand for the activator and inhibitor, respectively. Fundamental ideas of this model are from Turing [17], that is, instability of constant stationary solutions is driven by diffusion terms
Summary
Condition (6) implies 0 < γ < 1 and global-in-time existence of the solution of this reduced system. In spite of such a stable profile of the stationary solution (u, v) = (1, 1) in ODE, it becomes unstable as a steady state of ut ε2∆u. This Lyapunov function is valid only in the case of (20) and (21) In spite of these additional restrictions, parameters (p, q, r, s) satisfying all the requirements of Theorem 1 exist. ∂ν ∂ν where a, b, c, d > 0 are constants (see [1, 10]), that is, the ODE part takes always time-periodic orbits and the PDE solution is absorbed into one of them. The proof of Theorem 1 is given in the final section
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