The Prospective Role of Mathematical Theory in Plant Ecology

Despite the recent flowering of mathematical theory in animal ecology, comparable developments in plant population biology are few. Most theory in animal ecology presumes uniform mixing of individuals and, consequently, spatially homogeneous populations. This assumption, which some animal ecologists can scarcely tolerate, is wholly inappropriate for plants. The simplest relaxations of the assumption of uniform mixing lead either to ad hoc theories that are not easily generalized or to second-order partial differential equations that are susceptible neither to mathematical analysis nor to computer simulation. It is suggested that developing methods to deal with spatial heterogeneity, which is the inevitable consequence of imperfect mixing, is crucial to the useful application of mathematics to plant ecology. From time to time the more theoretically minded sort of zoologist may wonder why there has been no eruption of mathematical theory among plant ecologists to match that which made Hutchinson, MacArthur, and Levins household names in animal ecology. Quite likely, our zoologist is unaware of just how intricate (and how different from the algebra of population growth) the mathematics of plant ecology can be-for example, the methods used to detect environmental gradients and associations or "natural groupings" of species. Nor is he likely to know the literature on spiral leaf arrangement (Thompson, 1917; Leigh, 1972), unless his bent runs to D'Arcy Thompson and functional anatomy. Even the paper in which Skellam (1951) showed how an inferior competitor might coexist with a superior one that would invariably replace it whenever the two came into contact, is likely to be unfamiliar, and so also the extensions of Leslie's (1945) stable age matrix in which age classes of animals are replaced by size classes of trees (Usher, 1969). No doubt there are other more compelling evidences of botanical expertise in mathematics. But this only sharpens the point of our zoologist's question: Why, despite the high level of mathematical sophistication among certain sorts of botanists and plant ecologists, has there been no pandemic of theory in plant population biology? Is this an accident of history, or is there a message here for the animal ecologist as well? To answer this question, let us consider the way in which animal ecologists typically construct a theoretical basis for their science. We begin with 1 We thank H. R. Pulliam, M. L. Rosenzweig, and Patricia A. Werner for useful discussion. 2 Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721. 3 Smithsonian Tropical Research Institute, Balboa, Canal Zone. This content downloaded from 207.46.13.113 on Thu, 06 Oct 2016 04:33:49 UTC All use subject to http://about.jstor.org/terms 210 SYSTEMATIC BOTANY [Volume 1 the growth of a single species and hence with the "logistic equation" of Pearl and Verhulst. Suppose that N(t) is the density of animals at time t. Let r be the per capita rate of increase, i.e. births minus deaths, when N(t) is near zero, and let a measure the reduction in r occasioned per unit increase in density. Then, [l/N(t)] [dN(t)/dt] (d log N)/dt r aN. (1) Integration of this expression yields the familiar sigmoid growth curve, with N(t) asymptotically increasing to the equilibrium density,