Steady-state Analysis in a Model for Population Diffusion in a Multi-patch Environment

ONE PROBLEM of recent interest in the theory of mathematical ecology is the attempt to describe the effect of the spatiotemporal structure of a heterogeneous environment on the growth and diffusion of a population. Most single-species models that have been analysed in the literature (cf. [5,6]) have a constant diffusion coefficient and a constant carrying capacity of the environment. However, except for a controlled laboratory environment, the carrying capacity is usually varying in space and time. For details, we refer to [7-9, 1 l-141 and references therein, Several methods can be utilized to describe this environmental heterogeneity and its effect on the growth and diffusion of the population by a reaction-diffusion equation. One method is to allow the diffusion coefficient and reaction term to depend on spatial position. For example, this method has recently been used by Cantrell and Cosner [l] to show the existence of a positive steady-state solution under appropriate assumptions in the case of logistic growth, where the diffusion is held constant, but the carrying capacity continually varies in the reaction term. A second method, which is the one adopted in this paper, is to approximate the environment by a sequence of patches, in each of which the diffusion rate and carrying capacity are constants, not necessarily the same from patch to patch. This technique was utilized in [7] for the case of two patches, where the existence of a positive, monotonic, asymptotically stable steady state solution was proved constructively. However, there are many situations arising in nature where two patches are inadequate for approximating the environmental variation. In [16] it is noted that since the opening of the Suez Canal, there has been a steady diffusion of certain plankton species into the Red Sea, which has gradually spread down its length. Since these plankton form a food basis for many fish and other populations, this has led to a spatially monotonically decreasing carrying capacity from very high to very low levels. In [lo], a study of forest degradation in the Doon Valley, India, due to limestone quarries and other activities, was reported. The degradation was more severe for forest biomass closer to the quarries than for the biomass farther away due to dust