Abstract
In this paper, we propose and investigate a predator–prey model with harvesting and reserve area for prey in the presence of toxicity. We research the boundedness of the solutions and the existence of the equilibria of this system. By analyzing the characteristic equations, the local asymptotically stability of feasible equilibria is discussed. By use of Lyapunov function method, we also obtain some sufficient conditions for globally stability of these equilibria. The optimal harvesting policy are discussed by using the Pantryagin’s maximum principle. Finally, numerical simulations are carried to verify the theoretical conclusions.
Similar content being viewed by others
References
Chakraborty, K., Das, K.: Modeling and analysis of a two-zooplankton one-phytoplankton system in the presence of toxicity. Appl. Math. Model. 39, 1241–1265 (2015)
Gao, S., Chen, L., Teng, Z.: Hopf bifurcation and global stability for a delayed predator prey system with stage structure for predator. Appl. Math. Comput. 202, 721–729 (2008)
Jana, D., Agrawal, R., Upadhyay, R.K.: Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain. Chaos, Solitons Fractals 69, 50–63 (2014)
Jana, S., Ghorai, A., Guria, S., Kar, T.K.: Global dynamics of a predator,weaker prey and stronger prey system. Appl. Math. Comput. 250, 235–248 (2015)
Kar, T.K., Chaudhury, K.S.: Harvesting in a two-prey one-predator fishery. ANZIAM 45, 443–456 (2004)
Lv, Y., Pei, Y., Gao, S., Li, C.: Harvesting of a phytoplankton-zooplankton model. Nonlinear Anal.: Real World Appl. 11, 3608–3619 (2010)
Zeng, Z.: Asymptotically periodic solution and optimal harvesting policy for gompertz system. Nonlinear Anal. RWA 12, 1401–1409 (2011)
Beverton, R., Holt, S.: On the Dynamics of Exploited Fish Populations. Chapman and Hall, London (1957)
Gunette, S., Lauck, T., Clark, C.: Marine reserves: from beverton and holt to the present. Rev. Fish Biol. Fish. 8, 251–272 (1998)
Dubey, B.: Peeyush Chandra, Praeal Sinha, A model for fishery resource with reserve area. Nonlinear Anal.: Real World Appl. 4, 625–637 (2003)
Mukherjee, D.: Bifurcation and stability analysis of a prey-predator system with a reserved area. World J. Model. Simul. 8, 285–292 (2012)
Murthy, M.V.R., Bahlool, D.K.: Modeling and analysis of a prey-predator dependent predator system with reserved area. J. Math. Comput. Sci. 4, 1114–1138 (2014)
Kar, T.K.: Stablity and optimal harvesting of a prey-predator model with stage structure for predator. Appl. Math. 32(3), 279–291 (2005)
Lv, Y., Yuan, R., Pei, Y.: A prey-predator model with harvesting for fishery resource with reserve area. Appl. Math. Model. 37, 3048–3062 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by Natural Science Foundation of Shanxi province (2013011002-2).
Rights and permissions
About this article
Cite this article
Yang, H., Jia, J. Harvesting of a predator–prey model with reserve area for prey and in the presence of toxicity. J. Appl. Math. Comput. 53, 693–708 (2017). https://doi.org/10.1007/s12190-016-0989-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-016-0989-8