A Simple Stochastic Process Model for River Environmental Assessment Under Uncertainty

We consider a new simple stochastic single-species population dynamics model for understanding the flow-regulated benthic algae bloom in uncertain river environment: an engineering problem. The population dynamics are subject to regime-switching flow conditions such that the population is effectively removed in a high-flow regime while it is not removed at all in a low-flow regime. A focus in this paper is robust and mathematically rigorous statistical evaluation of the disutility by the algae bloom under model uncertainty. We show that the evaluation is achieved if the optimality equation derived from a dynamic programming principle is solved, which is a coupled system of non-linear and non-local degenerate elliptic equations having a possibly discontinuous coefficient. We show that the system is solvable in continuous viscosity and asymptotic senses. We also show that its solutions can be approximated numerically by a convergent finite difference scheme with a demonstrative example.