Abstract
Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps. In this paper, we discuss a system of reaction-diffusion equations that can be used with a view to modelling this phenomenon and we solve it numerically by means of the method of lines. For the spatial discretization, we use the finite difference method and Galerkin finite element method. We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used. Further, we present the numerical results obtained by solving the model in a cubic domain.
Highlights
According to microbiological experiments, the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied—and surprisingly regular—patterns while the individual cells grow, reproduce and migrate on the dish in clumps
Microbiological experiments show that the colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied spatial patterns while the individual cells grow, reproduce and migrate on the dish in clumps
We present how the spatial patterns obtained depend on the spatial discretization employed and we measure the experimental order of convergence of the numerical schemes used
Summary
The colonies of the bacterium bacillus subtilis placed on a dish filled with an agar medium and nutrient form varied—and surprisingly regular—patterns while the individual cells grow, reproduce and migrate on the dish in clumps (see, e.g., [1,2]). The general approach to the pattern formation in biology is based on continuous and discrete dynamical models or neural models as described in [3]. In this case, the simple character of the motion of the bacteria resembles the Brownian motion and indicates the analogy with the diffusion limited aggregation (see [4]). T for x and t 0 , where d represents a diffusion coefficient, and the function a a u, v is of the form a u,v a0. This problem can be written in the following weak formulation:. Terns in 2 (the darker points correspond to the higher concentration of the cells) that have been found by the author by solving system (1)-(3)
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