Mechanism of reaction-diffusion systems modeling pattern formation phenomena in biology

在由单个反应扩散方程和常微分方程组描述的过程中，有一些图案形成的数学模型。这类系统源于对细胞过程和扩散生长因子之间相互作用的建模，并且表现出扩散驱动的不稳定性。首先，研究了非齐次定常解的不稳定性。结果表明，系统的某种自然(自催化)性质导致了所有非均相固定溶液的不稳定性。接下来，我考虑了解的一个可能的大时间行为。结果表明，即使空间齐次解在时间上一致有界，系统的空间非齐次解在有限或无限时间内都是无界的。

There are some mathematical models of a pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with an ordinary differential equation. Such systems arise from modeling of interactions between cellular processes and diffusing growth factors, and exhibit the diffusion-driven instability.In this study, a general reaction-diffusion-ODE system with a single diffusion operator is considered. First, I studied the instability of inhomogeneous stationary solutions. It was shown that a certain natural (autocatalysis) property of a system led to instability of all inhomogeneous stationary solutions. Next, I considered a possible large time behavior of solutions. It was seen that space inhomogeneous solutions of the system became unbounded in either finite or infinite time, even if space homogeneous solutions were bounded uniformly in time.