The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics.
In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts… (More)
We derive models of the effects of periodic, discrete dosing or constant dosing of antibiotics on a bacterial population whose growth is checked by nutrient-limitation and possibly by host defenses. Mathematically rigorous results providing sufficient conditions for treatment success, i.e., the elimination of the bacteria, as well as for treatment failure,… (More)
We develop a simple mathematical model of a bacterial colonization of host tissue which takes account of nutrient availability and innate immune response. The model features an infection-free state which is locally but not globally attracting implying that a super-threshold bacterial inoculum is required for successful colonization and tissue infection. A… (More)
We analyzed the local dynamics of a three-dimensional Ricker type discrete-time competition model that is analogous to the May-Leonard (M-L) differential equation model. The symmetric discrete M-L model is mentioned by Hofbauer et al. [7, J. Math. Biol., 25:553–570,1987] as " perhaps one of the most difficult three species problems ". Both of the discrete… (More)
Biological systematics studies suggest that species are discretized in niche space. That is, rather than seeing a continuum of organism types with respect to continuous environmental variations, observers instead find discrete species or clumps of species, with one clump separated from another in niche space by a gap. Here, using a simple one dimensional… (More)
A mathematical model of a mixed culture of bacteria in a fully three dimensional flow reactor which accounts for the colonization of the reactor wall surface by the microbes is studied both analytically and by simulation. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow. The primary focus is on… (More)
We construct a Lyapunov function for a tridiagonal competitive-cooperative systems. The same function is a Lyapunov function for Kolmogorov tridiagonal systems, which are deened on a closed positive orthant in R n. We show that all bounded orbits converge to the set of equilibria. Moreover, we show that there can be no heteroclinic cycles on the boundary of… (More)