Reduction of slow-fast asymptotically autonomous systems with applications to gradostat models

Coupling of Bio-Reactors to Increase Maximum Sustainable Yield

In the field of fisheries management, the objective is to obtain an optimal catch while maintaining the fishery resource at a sufficiently high level to avoid the extinction of the exploited species. In mathematical fishery models, the fishing effort that must be implemented to have a sustainable fishery with a maximum harvest rate in the long term is sought. This goal is called the “Maximum Sustainable Yield” (MSY). In the chemostat, the substrate can be seen as prey of which the predator is the product. MSY search is thus extended to the classical chemostat model with a Monod function. There exists a dilution rate that maximizes the product synthesis. The study is extended to the case of the gradostat with fast substrate and product exchanges between two coupled bioreactors. The existence of two time scales makes it possible to apply methods of aggregation of variables to derive a reduced model governing a few global variables describing the dynamics of the complete system at the slow time scale. The analysis of the mathematical aggregated model is performed. Existence of equilibria as well as local and global stability are studied. The overall product yield in the system of coupled bioreactors may be greater than the sum of the yields of the two uncoupled bioreactors, i.e., if they functioned without connection between them. The increase in product yield is all the more important as the distribution of the substrate and of the product is asymmetrical between the two coupled bioreactors. The model is applied to fish farming by considering the coupling of two breeding sites. Here again, the model makes it possible to find the fast fish exchanges that must be established between the two breeding basins to optimize the overall yield of the farm.

Reduction of nonautonomous population dynamics models with two time scales

The purpose of this work is reviewing some reduction results to deal with systems of nonautonomous ordinary differential equations with two time scales. They could be included among the so-called approximate aggregation methods. The existence of different time scales in a system, together with some long-term features, are used to build up a simpler system governed by a lesser number of state variables. The asymptotic behavior of the latter system is then used to describe the asymptotic behaviour of the former one. The reduction results are stated in two particular but important cases: periodic systems and asymptotically autonomous systems. The reduction results are illustrated with the help of simple spatial SIS epidemic models including either periodic or asymptotically autonomous terms.