Pattern Formation in Consumer-Finite Resource Reaction-Diffusion Systems

In 1952, two paradoxes of diffusion were demonstrated by simple differential equations. The first paradox was shown by a great mathematician, A. Turing, who is well known as a pioneer in the field of computer science. He proposed a simple reaction-diffusion (RD) system for which a spatially constant equilibrium state is possibly destabilized so that non-constant equilibrium states appear [1]. The occurrence of such destabilization indicates that diffusion does not necessarily enhance homogeneity in space. This instability is called “diffusion-induced instability”. Mathematically speaking, it is some bifurcation phenomenon which is interpreted as the destabilization of spatially constant equilibrium solutions, when certain parameters in the system are suitably varied. He claimed that such destabilization plays a role in cell differentiation and morphogenesis arising in biological systems. The second was contributed by two neurophysiologists, A. L. Hodgkin and A. F. Huxley, who investigated the mechanism of impulses propagating along nerve fiber from experimental viewpoints [2]. One of the important neurophysiological problems was to understand the reason why a nerve impulse constantly propagates with fixed shape. In the same year as Turing’ paradox was stated, they proposed a model of nonlinear partial differential equations to describe the propagation of impulses along fibers. The model is given by a coupling of a single RD equation with three ODEs. Since it contains high nonlinearity, its analysis was so hard at that time. But the model could be managed to numerically solve. It is thus