On systems of reaction-diffusion equations modeling the growth of harmful algae and the spread of dengue fever

The chemostat is a continuous culture device of constant volume for microorganisms into which a nutrient medium is pumped, balanced by an outflow that removes nutrients and organisms. The chemostat has played an important role in ecology. Although the chemostat provides a simple model for many microbial habitats, the assumption of well-mixing is often questionable. Since coexistence of competing species is obvious in the nature, a candidate for an explanation is to remove the ''well-mixing" hypothesis. For the unstirred chemostat, flow enters at one boundary supplying nutrient resource(s), and exits at another, removing nutrients and organisms, while diffusion transports organisms and nutrient across the habitat domain. Motivated by considering habitats such as broad high-order rivers or riverine reservoirs constructed by damming a river, there are various mathematical extensions of the flow reactor model. The complementary resource model is highly relevant, because many ecosystems have two limiting nutrients, such as nitrogen and phosphorus. Furthermore, if we consider the periodic time dependence in the nutrient concentration to account for seasonal or daily changes, then the model will become more realistic. Thus, we consider two species competition for two, growth-limiting, nonreproducing essential resources with seasonalities. It is quite natural to assume that each competitor produces the inhibitor at some cost to its own growth and the inhibitor is detrimental to its competitor. The effect of the inhibitor is to retard growth (and hence, uptake, since one of the basic chemostat assumptions is that these are proportional). For this purpose, we propose a flowing competition model where each organism produces an inhibitor retarding the growth of its competitor. In the following, we further study two basic questions: What is the longitudinal distribution of algal abundance and toxicity in flowing habitats with a hydraulic storage zone ? Secondly, what differences arise between a fringing cove and a main lake? For the first question, advection-dispersion-reaction systems are proposed. For the second question, a simpler, two-compartment model of algal dynamics is constructed, in which one compartment is a small cove connected to a larger lake. Traditional mathematical models in epidemiology, and related areas typically assume that the environment is uniform in time and space. In reality, many environments are heterogeneous in time and/or space. Incorporating heterogeneity into models for biological processes and then analyzing such models leads to significant new mathematical challenges. To understand how the spatial heterogeneity infulence the dynamics of epidemiology, we propose and study a delayed reaction-diffusion system modeling the dengue fever with a fixed latent period.

Project ID:PA10202-0661
External Project ID:NSC101-2115-M182-003-MY2