Abstract
The topic of this paper is concerned with an investigation of the qualitative properties of solutions to the following problem.
Let Ξ©βR^n^ be a bounded domain with boundary βΞ©.
We seek solutions P,ΟβR^m+1^ of the system
subject to the βnoβfluxβ boundary condition
where n denotes the inward pointing normal to βΞ©. To close the system we prescribe the initial conditions
In this system D is a constant diffusion coefficient, P is a population density and Ο is a vector of nutrients or chemicals whose dynamics influences the movement of P.
Notice here that the substances Ο do not diffuse. If they do then the second equation of (1) is modified to
where d is a positive semiβdefinite diagonal matrix. This more general system includes the soβcalled KellerβSegel model of Biology ([1] Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology 1970; 26: 339β415). To motivate our study of system (1)β(3) we begin by outlining two themes. One basic to developmental biology and the other from angiogenesis. Copyright Β© 2001 John Wiley & Sons, Ltd.