Stability analysis of a discrete competitive system with nonlinear interinhibition terms
Tóm tắt
We propose and study a discrete competitive system of the following form:
$$\begin{aligned} &x_{1}(n+1)=x_{1}(n)\exp{\biggl[r_{1}-a_{1}x_{1}(n)- \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)}\biggr]}, \\ &x_{2}(n+1)=x_{2}(n)\exp{\biggl[r_{2}-a_{2}x_{2}(n)- \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]}. \end{aligned}$$
We obtain some conditions for the local stability of the equilibria. Using the iterative method and the comparison principle of a difference equation, we also obtain a set of sufficient conditions that ensure the global stability of the interior equilibrium. Numeric simulations show the feasibility of the main results. Our results supplement and complement some known results.
Tài liệu tham khảo
Chen, GY, Teng, ZD: On the stability in a discrete two-species competition system. J. Appl. Math. Comput. 38(1), 25-39 (2012)
Qin, WJ, Liu, ZJ, Chen, YP: Permanence and global stability of positive periodic solutions of a discrete competitive system. Discrete Dyn. Nat. Soc. 2009, Article ID 830537 (2009)
Chen, BG: Global attractivity of a two-species competitive system with nonlinear inter-inhibition terms. J. Math. Comput. Sci. 16, 481-494 (2016)
Zhuo, XL, Zhang, FX: Stability for a new discrete ratio-dependent predator-prey system. Qual. Theory Dyn. Syst. (2017). doi:10.1007/s12346-017-0228-1
Chen, FD, Wang, HN: Dynamic behaviors of a Lotka-Volterra competitive system with infinite delays and single feedback control. J. Funct. Anal. 2016, Article ID 43 (2016)
Chen, FD, Xie, XD, Wang, HN: Global stability in a competition model of plankton allelopathy with infinite delay. J. Syst. Sci. Complex. 28(5), 1070-1079 (2015)
Wang, K, Zhu, YL: Global attractivity of positive periodic solution for a Volterra model. Appl. Math. Comput. 203(2), 493-501 (2008)
Li, Z, Chen, FD: Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances. J. Comput. Appl. Math. 231(1), 143-153 (2009)
Wang, QL, Liu, ZJ, Li, ZX: Existence and global asymptotic stability of positive almost periodic solutions of a two-species competitive system. Int. J. Biomath. 7(04), 1450040 (2014)
Pu, LQ, Xie, XD, Chen, FD, Miao, ZS: Extinction in two-species nonlinear discrete competitive system. Discrete Dyn. Nat. Soc. 2016, Article ID 2806405 (2016)
Wang, QL, Liu, ZJ: Uniformly asymptotic stability of positive almost periodic solutions for a discrete competitive system. J. Appl. Math. 2013, Article ID 182158 (2013)
Xie, XD, Chen, FD, Xue, YL: Note on the stability property of a cooperative system incorporating harvesting. Discrete Dyn. Nat. Soc. 2014, Article ID 327823 (2014)
Yang, K, Xie, XD, Chen, FD: Global stability of a discrete mutualism model. Abstr. Appl. Anal. 2014, Article ID 709124 (2014)
Xie, XD, Xue, YL, Wu, RX, Zhao, L: Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Adv. Differ. Equ. 2016, Article ID 268 (2016)
Wang, L, Wang, MQ: Ordinary Difference Equations. XinJiang University Press, Urmuqi (1989)
Wang, XL, Du, ZJ, Liang, J: Existence and global attractivity of positive periodic solution to a Lotka-Volterra model. Nonlinear Anal., Real World Appl. 11(5), 4054-4061 (2010)
Hu, ZY, Teng, ZD, Zhang, L: Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Anal., Real World Appl. 12(4), 2356-2377 (2011)
Chen, FD, Xie, XD, Miao, ZS, Pu, LQ: Extinction in two species nonautonomous nonlinear competitive system. Appl. Math. Comput. 274(1), 119-124 (2016)
Yu, SB: Permanence for a discrete competitive system with feedback controls. Commun. Math. Biol. Neurosci. 2015, Article ID 16 (2015)
Chen, FD, Chen, XY, Huang, SY: Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances. Open Math. 14(1), 1157-1173 (2016)
Chen, LJ, Chen, FD: Extinction in a discrete Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. Int. J. Biomath. 8(01), 1550012 (2015)
Yang, K, Miao, ZS, Chen, FD, Xie, XD: Influence of single feedback control variable on an autonomous Holling-II type cooperative system. J. Math. Anal. Appl. 435(1), 874-888 (2016)
Han, RY, Xie, XD, Chen, FD: Permanence and global attractivity of a discrete pollination mutualism in plant-pollinator system with feedback controls. Adv. Differ. Equ. 2016, Article ID 199 (2016)
Chen, FD, Wu, HL, Xie, XD: Global attractivity of a discrete cooperative system incorporating harvesting. Adv. Differ. Equ. 2016, Article ID 268 (2016)