We study a system of three partial differential equations modelling the spatiotemporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. More precisely, we consider the initial-boundary value problem forx e £2, t > 0, under homogeneous Neumann boundary conditions in a bounded domain £2 cl",n > 1, with smooth boundary.When 0 < a\ < 1 and 0 < a 2 < 1, this system possesses a uniquely determined spatially homogeneous positive equilibrium (w*, v*). We show that given any such a\ and a 2 and any positive diffusivities d\ and d 2 and crossdiffusivities xi and X2, this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients JX\ and /x 2 .
We consider a mathematical model for the spatio-temporal evolution of two biological species in a competitive situation. Besides diffusing, both species move toward higher concentrations of a chemical substance which is produced by themselves. The resulting system consists of two parabolic equations with Lotka-Volterra-type kinetic terms and chemotactic cross-diffusion, along with an elliptic equation describing the behavior of the chemical. We study the question in how far the phenomenon of competitive exclusion occurs in such a context. We identify parameter regimes for which indeed one of the species dies out asymptotically, whereas the other reaches its carrying capacity in the large time limit.
In this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation. The system arises in chemotaxis, a process whereby living organisms respond to chemical substance by moving toward higher, or lower, concentrations of the chemical substance, or by aggregating or dispersing. We prove that stationary solutions of the system are asymptotically stable.
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