Abstract
This paper proposes application of nonlocal operators to represent the biological pattern formation mechanism of self-activation and lateral inhibition. The blue-green algae Anabaena is discussed as a model example. The patterns are determined by the kernels of the integrals representing the nonlocal operators. The emergence of patters when varying the size of the support of the kernels is numerically investigated.
Highlights
The theory of pattern formation through local self-activation and long range inhibition accounts for much of the observed pattern forming regulatory phenomena [5]. This mechanism is captured mathematically by considering two species, activator and inhibitor, with different spatial diffusivity, so that the resulting model is a system of reaction diffusion equations
Of Turing’s work, Gierer and Meinhardt derived in 1972 their Theory of Biological Pattern Formation showing that patterns occur only if local self-enhancing reaction is coupled with an antagonistic reaction of long range [5, 8]
The control of the pattern formation is attributed to a peptide, which is produced and released by cells differentiating as heterocysts
Summary
The theory of pattern formation through local self-activation and long range inhibition accounts for much of the observed pattern forming regulatory phenomena [5]. This mechanism is captured mathematically by considering two species, activator and inhibitor, with different spatial diffusivity, so that the resulting model is a system of reaction diffusion equations. The theory was embedded in a model comprising a system of reaction diffusion equations satisfying the Turing conditions This model is known as the Gierer-Meinhardt model. We propose modelling of the activation-inhibition mechanism of pattern formation by using nonlocal integral operators. An advantage of using the nonlocal operator model from the point of view of its theoretical and numerical analysis is that it does not require smoothness of the solution with respect to the spatial variable
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