A note on interacting populations that disperse to avoid crowding

Abstract

Abstract. In this note we derive partial differential equations for populations that disperse to avoid crowding, paying particular attention to situations in which the ease of dispersal is not uniform among individuals. We develop equations for the dispersal of a finite number of interacting biological groups and for a single age-structured group, and we give conditions under which the latter equations reduce to the former. In all cases the equations generalize the classical porous flow equation-a degenerate parabolic equation that exhibits a myriad of interesting effects. For the special case of two groups we deduce a simple solution in which the species remain segregated for all time.

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

“…In the context of epidemic models, we can mention as one of the first nonlinear cross-diffusion model introduced by Busenberg and Travis in [20] (see also Gurtin and Pipkin [21] for a related model). Analysis of a simplified model given in [20] particularized to a tumor model is studied in [22][23][24].…”

Traveling waves for some biological systems with density dependent diffusion

“…Mimura and Kawasaki [26] and Mimura [25] studied the spatial patterns of solutions for this system when self-population pressures do not exist, that 298 Y. HosoNo is fl11= fl22 = 0. Recently, Gurtin and Pipkin [19] proposed a model for the dispersal of a finite number of interacting biological groups: (1.3) (ui),=kidiv(uiVU)+Ji (ul, ..., uN) , i= 1, ..., N, where U= ~Ns=l uj is the total population density and k i ate nonnegative constants.…”

A system of nonlinear degenerate parabolic equations.

“…This case is related to the general age-dependent diffusion problems studied by McCamy [10] and by Busenberg and lannelli [5]. 6) with the k t not equal is discussed by Gurtin and Pipkin [8] and is analyzed in the special case where N = 2, k t >0 and k 2 = 0 (only one diffusing specie u^. Equation (1.…”