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The diffusion-driven instability and complexity for a single-handed discrete Fisher equation
Yan, Yubin Zhang, Guang Zhang, Ruixuan
Yan, Yubin
Zhang, Guang
Zhang, Ruixuan
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2019-12-19
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Abstract
For a reaction diffusion system, it is well known that the diffusion coefficient of the
inhibitor must be bigger than that of the activator when the Turing
instability is considered. However, the diffusion-driven instability/Turing
instability for a single-handed discrete Fisher equation with the Neumann
boundary conditions may occur and a series of 2-periodic patterns have been
observed. Motivated by these pattern formations, the existence of 2-periodic
solutions is established. Naturally, the periodic double and the chaos
phenomenon should be considered. To this end, a simplest two elements system
will be further discussed, the flip bifurcation theorem will be obtained by
computing the center manifold, and the bifurcation diagrams will be
simulated by using the shooting method. It proves that the Turing
instability and the complexity of dynamical behaviors can be completely
driven by the diffusion term. Additionally, those effective methods of
numerical simulations are valid for experiments of other patterns, thus, are
also beneficial for some application scientists.
Citation
Zhang, G., Zhang, R. and Yan, Y. (2020). The diffusion-driven instability and complexity for a single-handed discrete Fisher equation, Applied Mathematics and Computation, 371, 124946.
Publisher
Elsevier
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Applied Mathematics and Computation
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Article