Permanence for the discrete mutualism model with time delays

Sufficient conditions are obtained for the permanence of the following discrete model of mutualism x"1(k+1)=x"1(k)exp{r"1(k)[K"1(k)+@a"1(k)x"2(k-@t"2(k))1+x"2(k-@t"2(k))-x"1(k-@s"1(k))]},x"2(k+1)=x"2(k)exp{r"2(k)[K"2(k)+@a"2(k)x"1(k-@t"1(k))1+x"1(k-@t"1(k))-x"2(k-@s"2(k))]}, where r"i,K"i,@a"i, @t"i and @s"i, with i=1,2, are non-negative sequences bounded above and below by positive constants, and @a"i>K"i, with i=1,2.

[1]  Yongkun Li,et al.  Positive periodic solutions for an integrodifferential model of mutualism , 2001, Appl. Math. Lett..

[2]  Ke Wang,et al.  Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system , 2002 .

[3]  Xitao Yang,et al.  Uniform persistence and periodic solutions for a discrete predator–prey system with delays☆ , 2006 .

[4]  Zhan Zhou,et al.  Stable periodic solutions in a discrete periodic logistic equation , 2003, Appl. Math. Lett..

[5]  Ke Wang,et al.  Asymptotically periodic solution of N-species cooperation system with time delay , 2006 .

[6]  Fengde Chen,et al.  Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems , 2006, Appl. Math. Comput..

[7]  Fengde Chen,et al.  Average conditions for permanence and extinction in nonautonomous Gilpin–Ayala competition model , 2006 .

[8]  Jinde Cao,et al.  Positive Periodic Solutions of a Class of Non–autonomous Single Species Population Model with Delays and Feedback Control , 2005 .

[9]  Cui Jing,et al.  GLOBAL ASYMPTOTIC STABILITY IN N-SPECIES COOPERATIVE SYSTEM WITH TIME DELAYS , 1994 .

[10]  B. Dai,et al.  Periodic solutions of a discrete-time diffusive system governed by backward difference equations , 2005 .

[11]  Wan-Tong Li,et al.  Permanence and global stability for nonautonomous discrete model of plankton allelopathy , 2004, Appl. Math. Lett..

[12]  Fengde Chen,et al.  Some new results on the permanence and extinction of nonautonomous Gilpin–Ayala type competition model with delays , 2006 .

[13]  Wan-Tong Li,et al.  Permanence for a delayed discrete ratio-dependent predator–prey system with Holling type functional response☆ , 2004 .

[14]  Fengde Chen,et al.  The dynamic behavior of N-species cooperation system with continuous time delays and feedback controls , 2006, Appl. Math. Comput..

[15]  Xiaoxin Chen,et al.  Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control , 2004, Appl. Math. Comput..

[16]  Fengde Chen.  Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control , 2005, Appl. Math. Comput..

[17]  Yongkun Li,et al.  Positive periodic solutions of a discrete mutualism model with time delays , 2005, Int. J. Math. Math. Sci..

[18]  Zhan Zhou,et al.  Stable periodic solution of a discrete periodic Lotka–Volterra competition system , 2003 .

[19]  Fengde Chen,et al.  Permanence of a discrete N-species cooperation system with time delays and feedback controls , 2007, Appl. Math. Comput..

[20]  Fengde Chen,et al.  The permanence and global attractivity of Lotka-Volterra competition system with feedback controls , 2006 .

[21]  Samir H. Saker,et al.  Existence of positive periodic solutions of nonlinear discrete model exhibiting the Allee effect , 2005, Appl. Math. Comput..

[22]  Fengde Chen.  On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay , 2005 .

[23]  Fengde Chen.  Positive periodic solutions of neutral Lotka-Volterra system with feedback control , 2005, Appl. Math. Comput..

[24]  Jinlin Shi,et al.  Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays , 2006 .

[25]  Fengde Chen,et al.  Permanence for an integrodifferential model of mutualism , 2007, Appl. Math. Comput..

[26]  Lansun Chen,et al.  Global asymptotic stability in n-species nonautonomous Lotka-Volterra competitive systems with infinite delays , 2002, Appl. Math. Comput..