Species Survival versus Eigenvalues

The purpose of this work is to analyze mathematical models describing the behavior of individuals inhabiting an environment with spatial heterogeneity. The latter means that the environment influences the reproductivity and mobility of the species. This effect is obtained if we model the density function of these organisms by evolution partial differential equations with nonconstant coefficients. More precisely, we introduce a function which corrects the reproductive capabilities of these species and characterizes the most and least favorable regions for its development. With this at hand, we wish to know if such an environment constitutes a refuge, that is, if this environment has good enough conditions for the studied species persistence, which in mathematical terms implies the existence of a stable nontrivial stationary solution—the limit density of these individuals in the steady state. Such a stationary solution is obtained by making use of the fibering method [7, 13, 14, 15, 16], introduced and developed by Pohozaev. This method implies that the fundamental eigenvalue of a certain spectral problem must be of a determined magnitude. This paper is organized as follows. In the next section, we present the details of the considered mathematical model. In Section 3, we apply the fibering method to a problem studied by Cantrell and Cosner [3, 4, 5, 6] and discuss the relationship between their approach and the one used here. Section 4 contains concluding remarks and comments. In the appendix, we adapt the method to a particular case considered by Ludwig et al. [10].