Asymptotic properties and optimal control of rainfall models involving forest-lake effects

This study focuses on the dynamical properties and optimal control problems for rainfall models involving effects of forests and lakes. First a deterministic model is established and the existence and asymptotic stability of equilibria as well as the optimal control problem are investigated. Then white noise is incorporated into the deterministic model and it is transformed to a stochastic differential system. The existence of solutions and the extent of white noise's effect on the steady state are considered for the resulting stochastic model. The optimal control problem for this stochastic model is studied as well. Additionally, L'evy noise is also introduced to the rainfall model to consider the influence of extreme climate conditions on rainfall. For this model existence of solutions and the specific disturbance caused by L'evy noise to the steady state are explored. Finally, some numerical simulations, discussion and an application are provided to illustrate the achieved results.