Exclusion principles for density-dependent discrete pioneer-climax models
In ecological literature, the most widely used model in describing the interactions of \(n\) species with discretely reproducing populations is the system of difference equations \[ x_i(k+1)= x_i(k) \lambda_i \Biggl( \sum_{j=1}^n a_{ij} x_j (k) \Biggr), \qquad i=1,2, \dots,n, \tag{1} \] where \(x_i(k)\) denotes the population of species \(i\) at generation \(k\). The density effects in system (1) are modeled by assuming that each growth function, \(\lambda_i: \mathbb{R}_+\to \mathbb{R}_+\), is a continuous function whose argument is a linear combination of the densities of the individual species (where \(\mathbb{R}_+= [0,\infty))\). Therefore, the composition \(\lambda_i (\sum^n_{j=1} a_{ij} x_j (k))\) maps the nonnegative cone \(\mathbb{R}^n_+\) to \(\mathbb{R}_+\). We will always require that each constant \(a_{ij}\) be strictly positive, making any pair of two species interact directly. It is well known that, even if all the species being modeled by (1) are missing except one, the long-run behavior of the reduced system may be extremely complicated. We are concerned with the exclusion of species, a central problem in population ecology. Not withstanding the complex dynamics associated with (1), we establish sufficient conditions to allow populations in the long run to remain near the boundary \(\partial \mathbb{R}^n_+\) of the nonnegative cone \(\mathbb{R}^n_+\) (corresponding to extinction of at least one species).
scientific article; zbMATH DE number 703375