Effects of Diffusivity and Chemotaxis on Cell Growth
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چکیده
We are interested in the effects of diffusivity and chemotaxis on the competition of several species for a limited resource. Diffusivity of cells is also called motility in some engineering literature. Chemotaxis is the oriented movement of cells in response to the concentration gradient of chemical substances in their environment. It is ”anti-diffusion”. It was observed experimentally that motility and chemotaxis of cells play a dominant role in the cell growth: when several species of cells compete for a limited resource, the species with smaller diffusion rate and larger chemotaxis rate enjoys better growth, even when the other species have superior growth kinetics. The mathematical modeling on chemotaxis by using PDE’s started only in 1970 by Keller and Segal (see [KS]). To illustrate the general idea involved, consider a species of cells which responds chemotactically to a chemical which attracts the cells. Let u(x, t) and v(x, t) be the concentration of the chemical and density of cells, respectively. The chemical is assumed to be diffusive with diffusion coefficient = D1; The cells are assumed both diffusive and chemotactic with diffusion coefficient = D2; The chemotaxis flux of cells is assumed to have the form −→ J chemo = χvφ ′(u)∇u, (1.1) where χ is a nonnegative constant, called chemotaxis coefficient; φ′(u) > 0, and ∇u is the gradient of u with respect to the spatial variable x.
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