A note on interacting populations that disperse to avoid crowding

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. 1. Basic equations. We consider the dispersal1 of N biological groups in Rw. The groups may correspond, for example, to different biological species or to different age classes of the same species. We assume that the dispersal of each group is described by three functions: un(t,x), the population density, v„( t, x) the dispersal velocity, o„(f,x), the population supply. The field un(t,\) gives the number of individuals of group n, per unit "volume", at position x at time t\ its integral over any region R gives the population of that group in R. The flow of population from point to point is described by the dispersal velocity v„(f,x), which represents the average velocity of individuals of group n. The field <Jn(t, x) gives the rate at which individuals are supplied at x, for example, by births and deaths. These fields are assumed to be consistent with the balance law2, du„/dt = -div(M„v„) + an. •Received March 4, 1983. The authors would like to thank M. Bertsch, D. Hilhorst, and L. A. Peletier for valuable discussions. This work was supported by the National Science Foundation. 'General discussions of dispersal are contained in the review articles of Levin [1976] and McMurtrie [1978] and in the books of Okubo [1980] and Nisbet and Gurney [1982]. 2Cf„ e.g., [1977, Eq. (2.3)]. We use standard notation: lightface letters are scalars, boldface letters are vectors in R w; v, div, and A, respectivelu, denote the gradient, divergence, and laplacian in R1' ©1984 Brown University 88 MORTON E. GURTIN AND A. C. PIPKIN We limit our attention to groups which disperse to avoid crowding.3 With this in mind, we suppose that y„ = -knvU, (1.1) with kn a constant called the dispersivity and c/=2«» (1-2) n the total population, so that each group disperses (locally) toward lower values of total population,4 Thus assuming °n = °n(u)> " = ("i, u2,...,uN), the underlying partial differential equation takes the form du„/dt = kndi\{unvU) + o„(u). (1.3) Since we are concerned with dispersal in all of Rw, we need only adjoin to (1.3) initial conditions of the form K„(0,x) = W„(x), with un prescribed. Note that for kn — k the same for all n and 2<t„(«) a function a(U) of U alone, (1.3) yields ^ = (t/2) + a(t/), (1.4) the porous flow equation with kinetics.5 The most interesting and unusual behavior occurs, however, when the dispersivity kn varies from group to group, for then the groups disperse with different speeds. Remark 1. In the relations (1.1), (1.2) all groups are assigned equal weight. This assumption is not crucial. Indeed, in place of (1.2) we could take U = 2Pnun, n with each /?„ a strictly-positive constant. Then, defining Wn = A,«n. we once again recover the basic equations (1.2) and (1.3), but with w„ replaced by wn. 3 The effects of population on dispersal are demonstrated in the field studies and experiments of Morisita [1950,1954] (water striders and ant lions), Ito [1952] (aphids), Kono [1952] (rice weevils). These studies are discussed by Okubo [1980, §6] and Shigesada [1980], See also the remarks of Carl [1971] concerning the dispersal of arctic ground squirrels. 4Cf. the discussion of Gurtin and MacCamy [1977, §6]. See also Busenberg and Travis [1983], who utilize an assumption of the form (1.1) with kn independent of n. We saw this paper after completing our study. 5 Theories of population dynamics based on (1.4) have been given by Gurney and Nisbet [1975, 1982] and Gurtin and MacCamy [1977] (see also Shigesada [1980]). For general studies concerning the porous flow equation see, for example, Oleinik [1965], Aronson [1969], Peletier [1981]. POPULATIONS THAT DISPERSE TO AVOID CROWDING 89 Remark 2. In some instances it might be appropriate to allow the dispersivities kn to be functions of un and U. In this instance (1.3) is replaced by 0 = di\[unkn{un, £/)vt/] + on{u). Remark 3. Theories of the type presented here are often based on the balance law