Qualitative analysis of a harvested predator-prey system with Holling type III functional response
Tóm tắt
A harvested predator-prey system with Holling type III functional response is considered. By applying qualitative theory of differential equations, we show the instability and global stability properties of the equilibria and the existence and uniqueness of limit cycles for the model. The possibility of existence of a bionomic equilibrium is discussed. The optimal harvesting policy is studied from the view point of control theory. Numerical simulations are carried out to illustrate the validity of our results.
Tài liệu tham khảo
Berryman AA: The origins and evolutions of predator-prey theory. Ecology 1992, 73: 1530–1535. 10.2307/1940005
Gause GF: La Theorie mathematique de la lutte pour la vie. Hermann, Paris; 1935.
Clark CW: Mathematical Bioeconomics: The Optimal Management of Renewable Resources. 2nd edition. Wiley, New York; 1990.
Chaudhuri KS: A bioeconomic model of harvesting a multispecies fishery. Ecol. Model. 1986, 32: 267–279. 10.1016/0304-3800(86)90091-8
Chaudhuri KS: Dynamic optimization of combined harvesting of a two species fishery. Ecol. Model. 1988, 41: 17–25. 10.1016/0304-3800(88)90041-5
Matsuda H, Katsukawa T: Fisheries management based on ecosystem dynamics and feedback control. Fish. Oceanogr. 2002, 11(6):366–370. 10.1046/j.1365-2419.2002.00214.x
Matsuda H, Abrams PA: Maximal yields from multi-species fisheries systems: rules for systems with multiple trophic. Ecol. Appl. 2006, 16(1):225–237. 10.1890/05-0346
Dai G, Tang M: Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 1998, 58(1):193–210. 10.1137/S0036139994275799
Xiao D, Ruan S: Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. Fields Inst. Commun. 1999, 21: 493–506.
Kar TK, Matsuda H: Global dynamics and controllability of a harvested prey-predator system with Holling type III function response. Nonlinear Anal. Hybrid Syst. 2007, 1: 59–67. 10.1016/j.nahs.2006.03.002
Chaudhuri KS: Dynamic optimization of combined harvesting of a two species fishery. Ecol. Model. 1988, 41: 17–25. 10.1016/0304-3800(88)90041-5
Hassard BD, Kazarinoff ND, Wan YH: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge; 1981.
Kuang Y, Freedman HI: Uniqueness of limit cycles in Gauss-type models of predator-prey systems. Math. Biosci. 1988, 88: 67–84. 10.1016/0025-5564(88)90049-1
Kar TK, Chaudhuri KS: On non-selective harvesting of a two competing fish species in the presence of toxicity. Ecol. Model. 2003, 161(1–2):125–137. 10.1016/S0304-3800(02)00323-X