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

有一些数学模型的图案形成过程中所描述的一个系统的一个单一的反应扩散方程耦合一个常微分方程。这类系统是由细胞过程和扩散生长因子之间的相互作用所引起的，并且表现出扩散驱动的不稳定性。首先，我研究了非齐次平稳解的不稳定性。结果表明，一个系统的某种自然（自催化）性质导致所有非均匀定态溶液的不稳定性。接下来，我考虑了解的可能的大时间行为。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上也会变为无界。

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.