Input-output analysis of mathematical models of ecosystems

Necessary and sufficient conditions for the convergence of the solutions of linear and nonlinear time varying compartmental models described by systems of differential equations are reviewed. Similar conditions for discrete models described by systems of difference equations are derived.

For continuous and discrete models, the concept of environ analysis is extended to advanced linear systems and for the first time to systems with time varying coefficient matrices A̎(t) and [A̍(t)]^T . Generalizations to nonlinear models satisfying certain regularity assumptions and to linear systems with time delay are introduced. Output and input environ partitioning flow and storage matrices for a two trophic level aquatic system are derived in the form of integral equations. Also, for the same system, estimates for the deviation of the environ partitions at any time from their asymptotic values are found.

As a step towards the important goal of controlling the eutrophication phenomenon, two phytoplankton population models in natural waters are presented. In the first model, a nonlinear function general enough to include the effects of feeding saturation intraspecific consumer interference, and eutrophication phenomenon is used to present the transfer of material or energy from phytoplankton to zooplankton populations. The model using this grazing rate function is subjected to equilibrium and stability analysis to ascertain its mathematical implications. It is shown that, for a certain range of one of the parameters in this function all equilibrium points of the system become stable even with nutrient enrichment. In the second model, dynamics of both nitrogen and phosphorus cycles are combined. Persistence results for both models are proved and compared.

The influence of direct human control added to different aquatic models is studied in detail. Optimal control theory is used to obtain optimal strategies for the control of these models with several cost functions. It is found that the control program in each problem depends on the model considered and on the function to be optimized. Explicit expressions of singular control in each case are given as functions of the state and costate variables.