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The permanence and global attractivity of Lotka--Volterra competition system with feedback controls. (English) Zbl 1103.34038
The author studies the following Lotka-Volterra type competitive system with feedback control \aligned \dot x_i(t)=x_i(t)[b_i(t)-\sum_{j=1}^n a_{ij}(t)x_j(t)-d_i(t)u_i(t)],\\ \dot u_i(t)=r_i(t)-e_i(t)u_i(t)+f_i(t)x_i(t),\quad i=1,2,\dots,n, \endaligned \tag1 where $x_i(t)$ represents the density of the $i$th species at time $t$, respectively, $i=1,2,\dots,n$, and $u_i(t)$ is the control variable, $i=1,2,\dots,n$. $a_{ij}(t), b_i(t), d_i(t),$ $r_i(t), e_i(t), f_i(t)$, $i,j=1,2,\dots, n$, are continuous functions defined on $[c,+\infty)$. Given a function $g(t)$ defined on $[c,+\infty)$, let $g_M=\sup\{g(t)\vert c\leq t<+\infty\}, g_L=\inf\{g(t)\vert c\leq t<+\infty\}.$ It is assumed in (1) that $a_{ij}(t)\geq 0, a_{ijM}<+\infty, a_{ijL}>0, b_i(t)>0, b_{iM}<+\infty, b_{iL}>0, d_i(t)\geq 0, d_{iM}<+\infty, d_{iL}\geq 0, r_i(t)\geq 0, r_{iM}<+\infty, r_{iL}\geq 0, e_i(t)>0, e_{iM}<+\infty, e_{iL}>0, f_i(t)>0, f_{iM}<+\infty$ and $f_{iL}>0$. Some average conditions for the permanence of system (1) and sufficient conditions for the global attractivity of positive solutions of system (1) are derived, respectively. The results developed by {\it J. Zhao, J. Jiang} and {\it A. C. Lazer} [Nonlinear Anal., Real World Appl. 5, 265--276 (2004; Zbl 1085.34040)] are generalized.

##### MSC:
 34D05 Asymptotic stability of ODE 34C25 Periodic solutions of ODE 92D25 Population dynamics (general) 34D20 Stability of ODE 34D40 Ultimate boundedness (MSC2000)
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##### References:
 [1] Ahmad, S.: On the nonautonomous Volterra -- Lotka competition equations. Proc. amer. Math. soc. 117, 199-204 (1993) · Zbl 0848.34033 [2] Ahmad, S.; Lazer, A. C.: Average conditions for global asymptotic stability in a nonautonomous Lotka -- Volterra system. Nonlinear anal. 40, 37-49 (2000) · Zbl 0955.34041 [3] Chen, F. D.: Periodicity in a nonlinear predator -- prey system with state dependent delays. Acta math. Appl. sinica engl. Ser. 21, No. 1, 1-10 (2005) [4] Chen, F. D.: Positive periodic solutions of neutral Lotka -- Volterra system with feedback control. Appl. math. Comput. 162, No. 3, 1279-1302 (2005) · Zbl 1125.93031 [5] Chen, F. D.; Lin, S. J.: Periodicity in a logistic type system with several delays. Comput. math. Appl. 48, No. 1 -- 2, 35-44 (2004) · Zbl 1061.34050 [6] Chen, F. D.; Lin, F. X.; Chen, X. X.: Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control. Appl. math. Comput. 158, No. 1, 45-68 (2004) · Zbl 1096.93017 [7] Chen, F. D.; Sun, D. X.; Lin, F. X.: Periodicity in a food-limited population model with toxicants and state dependent delays. J. math. Anal. appl. 288, No. 1, 132-142 (2003) [8] Fan, M.; Wong, P. J. Y.; Agarwal, R. P.: Periodicity and stability in periodic n-species Lotka -- Volterra competition system with feedback controls and deviating arguments. Acta math. Sinica 19, No. 4, 801-822 (2003) · Zbl 1047.34080 [9] Gopalsamy, K.: Global asymptotic stability in a periodic Lotka -- Volterra system. J. austral. Math. soc. Ser. B 27, 66-72 (1985) · Zbl 0588.92019 [10] Gopalsamy, K.; Weng, P. X.: Feedback regulation of logistic growth. Internat. J. Math. sci. 16, No. 1, 177-192 (1993) · Zbl 0765.34058 [11] He, C.: On almost periodic solutions of Lotka -- Volterra almost periodic competition systems. Ann. differential equations 9, No. 1, 26-36 (1993) · Zbl 0781.34037 [12] Huang, Z. K.; Chen, F. D.: Almost periodic solution of two species model with feedback regulation and infinite delay. J. eng. Math. 20, No. 5, 33-40 (2004) · Zbl 1138.34344 [13] Huo, H.; Li, W.: Positive periodic solutions of a class of delay differential system with feedback control. Appl. math. Comput. 148, No. 1, 35-46 (2004) · Zbl 1057.34093 [14] Li, X. Y.; Fan, M.; Wang, K.: Positive periodic solution of single species model with feedback regulation and infinite delay. Appl. math. J. chin. Univ. ser. A 17, No. 1, 13-21 (2002) · Zbl 1005.34039 [15] Teng, Z.: The almost periodic komogorov competitive systems. Nonlinear analysis 42, 1221-1230 (2000) · Zbl 1135.34319 [16] Tineo, A.: An iterative scheme for the N-competing species problem. J. differential equations 116, 1-15 (1995) · Zbl 0823.34048 [17] Tineo, A.; Alvarz, C.: A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem. J. math. Anal. appl. 159, 44-55 (1991) · Zbl 0729.92025 [18] Weng, P. X.: Global attractivity in a periodic competition system with feedback controls. Acta appl. Math. 12, 11-21 (1996) · Zbl 0859.34061 [19] Weng, P. X.: Existence and global stability of positive periodic solution of periodic integro-differential systems with feedback controls. Comput. math. Appl. 40, 747-759 (2000) · Zbl 0962.45003 [20] Xia, Y. H.; Cao, J. D.; Zhang, H. Y.; Chen, F. D.: Almost periodic solutions in n-species competitive system with feedback controls. J. math. Anal. appl. 294, 504-522 (2004) · Zbl 1053.34040 [21] Xiao, Y. N.; Tang, S. Y.; Chen, J. F.: Permanence and periodic solution in competition system with feedback controls. Math. comput. Model. 27, No. 6, 33-37 (1998) · Zbl 0896.92032 [22] Yang, F.; Jiang, D. Q.: Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments. Ann. differential equations 17, No. 4, 337-384 (2001) · Zbl 1004.34030 [23] Yin, F. Q.; Li, Y. K.: Positive periodic solutions of a single species model with feedback regulation and distributed time delay. Appl. math. Comput. 153, 475-484 (2004) · Zbl 1087.34051 [24] Zhao, X. Q.: The qualitative analysis of n-species Lotka -- Volterra periodic competition systems. Math. comput. Model. 15, No. 11, 3-8 (1991) · Zbl 0756.34048 [25] Zhao, J. D.; Jiang, J. F.; Lazer, A. C.: The permanence and global attractivity in a nonautonomous Lotka -- Volterra system. Nonlinear anal.real world appl. 5, 265-276 (2004) · Zbl 1085.34040