The mechanism of pattern transitions between formation and dispersion.

On the surface of living organisms, a wide variety of patterns can be observed, some of which change during their growth process. For instance, Pelodiscus sinensis exhibits distinct black patterns on its vivid orange plastron during the embryonic and juvenile stages, but as it matures, the black patterns gradually disappear, resulting in a whitened plastron. This pattern transition is a mysterious phenomenon that forms and vanishes on the plastron, a ventral part with low visibility to both predators and peers. Our research aims to focus on understanding the mechanisms behind such pattern transitions and proposes a model capable of representing pattern formation and dispersion. To understand the changing patterns, we propose a hypothesis based on a reaction-diffusion system with a time-dependent growing spatial domain. This mathematical framework suggests the occurrence of the dispersion phenomenon. Specifically, we focus on the dilution term within the system under the growing-domain condition. While previous studies have investigated the effects of growth domains, this study specifically addresses the role of the time-dependently growing domain effects - change of diffusion coefficient and dilution - in reaction-diffusion systems. Our research sheds light on the intricate phenomenon of pattern formation and dispersion on the surface of living organisms, proposing a natural system based on the effects of growing domain, namely, a model of reaction-dilution-diffusion systems.