Abstract
A new control strategy for preventing the extinction of predator–prey ecosystem in a random environment is proposed. First, a controlled stochastic Lotka–Volterra system is introduced and its some properties are presented. The stochastic averaging method is applied to transfer the original controlled system to a partially averaged one. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The optimal control is determined from the maximum condition and solving the forward-backward stochastic differential equation (FBSDE). For infinite time-interval ergodic control, the adjoint process is stationary and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the prey of optimally controlled system is obtained to show that the controlled ecosystem is more stable than the uncontrolled one. The mean transition time of optimally controlled system is also calculated to show that the proposed control strategy is effective in preventing the possible extinction.
Similar content being viewed by others
References
Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins, Baltimore (1925)
Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie d’animani conviventi. Mem. Acad. lincei 2, 31–113 (1926)
May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)
Khasminskii, R.Z., Klebaner, F.C.: Long term behavior of solutions of the Lotka-Volterra system under small random perturbations. Ann. Appl. Probab. 11, 952–963 (2001)
Zhou, J.: Bifurcation analysis of a diffusive predator–prey model with ratio-dependent Holling type iii functional response. Nonlinear Dyn. 81, 1535–1552 (2015)
Jana, S., Guria, S., Das, U., Kar, T.K., Ghorai, A.: Effect of harvesting and infection on predator in a prey–predator system. Nonlinear Dyn. 81, 917–930 (2015)
Cai, G.Q., Lin, Y.K.: Stochastic analysis of Lotka-Volterra model for ecosystems. Phys. Rev. E 70, 041910 (2004)
Cai, G.Q., Lin, Y.K.: Stochastic analysis of predator-prey type ecosystems. Ecol. Complex. 4, 242–249 (2007)
Wu, Y., Zhu, W.Q.: Stochastic analysis of a pulse-type prey-predator model. Phys. Rev. E 77, 041911 (2008)
Qi, L.Y., Xu, W., Gao, W.T.: Stationary response of Lotka-Volterra system with real noises. Commun. Theor. Phys. 59, 503 (2013)
Yuan, L.G., Yang, Q.G.: Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system. Appl. Math. Modell. 39, 2345–2362 (2015)
Shastri, Y., Diwekar, U.: Sustainable ecosystem management using optimal control theory: part 1 (deterministic systems). J. Theor. Biol. 241, 506 (2006)
Shastri, Y., Diwekar, U.: Sustainable ecosystem management using optimal control theory: part 2 (stochastic systems). J. Theor. Biol. 241, 522 (2006)
Kumar, D., Chakrabarty, S.P.: A comparative study of bioeconomic ratio-dependent predator-prey model with and without additional food to predators. Nonlinear Dyn. 80, 23–28 (2015)
Wang, W.K., Ewald, C.O.: A stochastic differential fishery game for a two species fish population with ecological interaction. J. Econ. Dyn. Control 34, 844 (2010)
Wong, E., Zakai, W.: On the relation between ordinary and stochastic differential equations. Int. J. Eng. Sci. 3, 213–229 (1965)
Stranovich, R.L.: Topics in the Theory of Random Noise, vol. 1. Gordon and Breach, New York (1963)
Khasminskii, R.Z.: A limit theorem for solution of differential equations with random right hand sides. Theory Probab. Appl. 12, 144–147 (1966)
Gu, X.D., Zhu, W.Q.: Time-delayed optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle. Int. J. Nonlinear Mech. 58, 199–205 (2014)
Kovaleva, A.S.: Asymptotic solution of the problem of the optimal control of non-linear oscillations in the neighbourhood of a resonance. J. Appl. Math. Mech. 62(6), 843–852 (1998)
Kushner, H.J.: Optimality conditions for the average cost per unit time problem with a diffusion model. SIAM J. Control Optim. 16, 330–346 (1978)
Yong, J.M., Zhou, X.Y.: Stochastic Controls Hamiltonian Systems and HJB Equations. Springer, New York (1999)
Andronov, A., Pontryagin, L., Witt, A.: On the statistical treatment of dynamical systems. Zh. Eksp. Teor. Fiz 3, 172 (1933)
Acknowledgments
This study was supported by the National Nature Science Foundation of China under NSFC Grant Nos. 11321202, 11432010, 11432012, 11502201 and the Basic Research Fund of Northwestern Polytechnical University under Grant No. G2015KY0104.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gu, X.D., Zhu, W.Q. Stochastic optimal control of predator–prey ecosystem by using stochastic maximum principle. Nonlinear Dyn 85, 1177–1184 (2016). https://doi.org/10.1007/s11071-016-2752-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-016-2752-y