Competitive exclusion is proved for a discrete-time, size-structured, nonlinear matrix model of m-species competition in the chemostat. The winner is the population able to grow at the lowest nutrient concentration. This extends the results of earlier work of the rst author 11] where the case m = 2 was treated.
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"Here, x i is the number of individuals of species i; r i is the maximal rate of increase of species i; K i is the carrying capacity for species i in the absence of competing species; and a ij ≥ 0 represents interaction among species i and j, more specifically the decrease in the growth rate of species i due to the competitive pressure of j. Subsequently, much work on competitive dynamics has been reported (see, e.g.,              and the references cited therein). "
[Show abstract][Hide abstract] ABSTRACT: In this paper, a class of non-autonomous N-species Lotka–Volterra-type competitive system with time delays and impulsive perturbations is investigated. New criteria on coexistence and global attractivity for all species other than some inclining to extinction are established. From our results, we find that the extinction of some species due to competition exclusion and weak adaption to a stochastically variable environment characterized by impulsive perturbations is necessary for the coexistence and global attractivity of other survivors in a limited resource ecosystem. Meanwhile, the dynamic behaviors of competition models with impulse perturbations are more complicated and different from prior research results obtained from continuous competition models. The impulse perturbations impose either negative or positive influences on the survival of species, which results in evidently regulative effects, and even make the inferior competitors exclude or coexist with the superior competitors. The theoretical results are confirmed by a special example and numerical simulations, by which we find some interesting phenomena.
Full-text · Article · May 2011 · Nonlinear Analysis Real World Applications
"For structured populations considerably less work has been done due to the complexity of these models. In  competitive exclusion is proved for a discrete-time, size-structured, non-linear matrix model of m competing species in a chemostat. The winner is the population that is able to grow at the lowest nutrient concentration. "
[Show abstract][Hide abstract] ABSTRACT: We present a quasilinear size-structured model which describes the dynamics of a population with n competing ecotypes. We assume that the vital rates of each subpopulation depend on the total population due to competition. We provide conditions on the individual rates which guarantee competitive exclusion in the case of closed reproduction (offspring always belongs to the same ecotype as the parent). In particular, our results suggest that the ratio of the reproduction and mortality rates is a good measure to determine the winning ecotype. Meanwhile, we show that in the case of open reproduction all ecotypes coexist.
Full-text · Article · Jan 2005 · Mathematical Biosciences