Hyperincursive Simulation of Ecosystems Chaos and Patchiness by Diffusive Chaos

In Pearl-Verhulst's finite difference equation, R. May showed that fractal chaos appears for large values of the command parameter. In this paper, it is shown that, surprisingly, chaos emerges for small values of the command parameter when Laplacian spatial diffusion is taken into account. For small diffusion the space pattern is uniform and stable and for large diffusion, discrete space-time structures emerge and then a chaotic patchiness. A mathematical demonstration by incursion shows that the emergence of such structures is due to the space diffusion parameter which gives rise to a bifurcation cascade and chaos. This is a new type of emergence of space-time structures what I suggest to call "diffusive chaos" different from the Turing "morphogenesis by diffusive instability". A gradient spatial transport by advection can also give rise to bifurcations and chaos, what I call "advective chaos" depending of the velocity intensity. A simulation with negative diffusion shows stable fractal periodic patterns. In Lotka-Volterra's discrete model, numerical instabilities occur. D. Dubois had found a new method for stabilising such instabilities by the concept and method of incursion, an inclusive recursion, where the equations are sequentially computed. With space diffusion such incursive equations show the emergence of a chaotic space-time patchiness which is followed by continuous space patchiness represented by travelling waves. Diffusive chaos could explain space-time structures called patchiness in marine plankton.