Optimal control for a nonlinear stochastic parabolic model of population competition

Abstract

In this investigation, an optimal control problem for a stochastic mathematical model of population competition is studied. We have considered the stochastic model of population competition by adding the stochastic terms to the deterministic model to take into account the random perturbations and uncertainties caused by the environment to have more reliable model. The model has formulated the population densities competing against each other to be saved from destruction, attract more members and so on, using two nonlinear stochastic parabolic equations. Four factors including the status and the growth rates of the populations are considered as the control variables (which can be controlled by the members of the populations or policy makers who make decisions for the populations for particular purposes) to control the evolution of the population densities. Then, the optimal control problem for the stochastic model of population competition is studied. Employing the tangent-normal cone techniques, the Ekeland variational principle and other theorems proved throughout the paper, we have shown there exists unique stochastic optimal control. We have also presented the exact form of the optimal control in terms of stochastic adjoint states.

Keywords:

• Optimal control
• nonlinear stochastic parabolic equation
• existence and uniqueness of stochastic optimal control
• population competition

2010 Mathematics Subject Classifications:

• 49K20
• 49J20
• 35R60

Acknowledgements

The authors are very grateful to reviewers for carefully reading the paper and for their valuable comments and suggestions which improved the original submission of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).