Locally Precise Response Surface Models for the Generalization of Controlled Dynamic Systems and Associated Performance Measures

Abstract

This treatment demonstrates the utility of response surface models (RSMs) as predictive, companion tools which aid in the development of harvesting (control) strategies applicable to predator-prey dynamic systems. To this end a control algorithm is derived that considers the regulation of a predator-prey natural resource while considering revenue for commercial ventures and regulatory agencies. Numerical simulations provide the mechanism to quantify performance measures associated with control algorithms, yet complicated problems require that all “tools” available be considered. Complex problems may be more tractable when simulation results are combined with alternate, continuous models exhibiting predictive capacities. For this reason, RSMs are appealing; analytic evaluation of the state, the gradient, and the Hessian matrix is possible. From these models we may glean valuable information linked to the gathered data revealing information about the “true nature” of the ecological system. Therefore, we propose to create RSMs based on scattered data obtained from the ordinary differential equation (ODE) dynamic system model. These response surface models are constructed using radial basis functions (RBFs); RSMs so created have the desirable property of matching the objective function value exactly at each sampled data point. Furthermore, they have the ability to interpolate to any desired point throughout the parameter space. This is powerful as the “objective function” may be any function of critical importance to the analyst which in this treatment is the predator biomass time rate of change (ODE) itself. This has the immediate implication of providing a single ODE model, with a “locally” or even perhaps a “globally” precise nature. Since such models are constructed from scattered data, which is consistent with what would be collected from field measurements, a further connection of theory to practice is realized. It will be shown that these RSMs provide greater insight into ecological systems, with special emphasis on parameter estimation.