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August  2016, 21(6): 2011-2037. doi: 10.3934/dcdsb.2016034

Intermittent dispersal population model with almost period parameters and dispersal delays

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China, China

Received  April 2015 Revised  April 2016 Published  June 2016

We establish a class of intermittent bidirectional dispersal population models with almost periodic parameters and dispersal delays between two patches. The form of dispersal discussed in this paper is different from both continuous and impulsive dispersals, in which the dispersal behavior occurs either in a sustained manner or instantaneously; instead, it is a synthesis of these types. Dynamical properties such as permanence, existence, uniqueness, and globally asymptotic stability of almost periodic solutions are investigated by using Liapunov-Razumikhin type technique, using the comparison theorem, constructing a suitable Lyapunov functional, using almost periodic functional hull theory and analysis approach, etc. Finally, numerical simulations are presented and discussed to illustrate our analytic results, by which we find that intermittent dispersal systems are more complicated than continuous or impulsive dispersal systems.
Citation: Long Zhang, Gao Xu, Zhidong Teng. Intermittent dispersal population model with almost period parameters and dispersal delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2011-2037. doi: 10.3934/dcdsb.2016034
References:
[1]

L. Allen, Persistence, extinction and critical patch number for island populations,, J. Math. Biol., 24 (1987), 617.  doi: 10.1007/BF00275506.  Google Scholar

[2]

S. Ahmad and I. M. Stamova, Asymptotic stability of competitive systems with delay and impulsive perturbations,, J. Math. Anal. Appl., 334 (2007), 686.  doi: 10.1016/j.jmaa.2006.12.068.  Google Scholar

[3]

S. Ahmad and G. Stamov, Almost periodic solutions of N-dimensional impulsive competitive systems,, Nonlinear Anal: RWA., 10 (2009), 1846.  doi: 10.1016/j.nonrwa.2008.02.020.  Google Scholar

[4]

D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications,, Longman, (1993).   Google Scholar

[5]

D. D. Bainov, A. D. Myshkis and G. T. Stamov, Dichotomies and almost periodicity of the solutions of systems of impulsive differential equations,, Dyna. Syst. Appl., 5 (1996), 145.   Google Scholar

[6]

G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations,, Dyna. Cont. Disc. Impul. Syst., 5 (1999), 579.  doi: 10.1080/00036810008840804.  Google Scholar

[7]

E. Beretta, P. Fergola and C. Tenneriello, Ultimate boundedness of nonautonomous diffusive Lotka-Volterra patches,, Math. Biosci., 92 (1988), 29.  doi: 10.1016/0025-5564(88)90004-1.  Google Scholar

[8]

E. Beretta and Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays,, Bull. Math. Biol., 49 (1987), 431.  doi: 10.1007/BF02458861.  Google Scholar

[9]

J. Cui and L. Chen, The effect of diffusion on the time varying logistic population growth,, Comput. Math. Appl., 36 (1998), 1.  doi: 10.1016/S0898-1221(98)00124-2.  Google Scholar

[10]

J. Cui, Y. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems,, J. Math. Anal. Appl., 298 (2004), 73.  doi: 10.1016/j.jmaa.2004.02.059.  Google Scholar

[11]

H. Dingle, Migration: The Biology of Life on the Move,, Oxford University Press, (1996).  doi: 10.1093/acprof:oso/9780199640386.001.0001.  Google Scholar

[12]

C. H. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations,, Appl. Math. Comput., 136 (2003), 487.  doi: 10.1016/S0096-3003(02)00072-3.  Google Scholar

[13]

H. I. Freedman, J. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Math. Biosci., 95 (1989), 111.  doi: 10.1016/0025-5564(89)90055-2.  Google Scholar

[14]

H. I. Freedman and Q. Peng, Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment,, Nonlinear Anal., 36 (1999), 981.  doi: 10.1016/S0362-546X(97)00712-8.  Google Scholar

[15]

X. Fu, B. Yan and Y. Liu, Introduction to Impulsive Differential Systems,, Science Press, (2005).   Google Scholar

[16]

K. Gopalsamy, Global asymptotic stability in an almost periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1986), 346.  doi: 10.1017/S0334270000004975.  Google Scholar

[17]

K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses,, J. Math. Anal. Appl., 139 (1989), 110.  doi: 10.1016/0022-247X(89)90232-1.  Google Scholar

[18]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluwer Academic Publishers, (1992).  doi: 10.1007/978-94-015-7920-9.  Google Scholar

[19]

W. Hamilton and R. May, Dispersal in stable habitats,, Nature (London), 269 (1977), 578.  doi: 10.1038/269578a0.  Google Scholar

[20]

C. Y. He, Almost Periodic Differential Equations,, Higher Education Publishing House, (1992).   Google Scholar

[21]

H. Hu, K. Wang and D. Wu, Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays,, J. Math. Anal. Appl., 377 (2011), 145.  doi: 10.1016/j.jmaa.2010.10.031.  Google Scholar

[22]

H. Hu, Permanence for nonautonomous predator-prey Kolmogorov systems with impulses and its applications,, Appl. Math. Comput., 223 (2013), 54.  doi: 10.1016/j.amc.2013.07.093.  Google Scholar

[23]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).   Google Scholar

[24]

V. Lakshmikantham, D. D. Banov and P. S. Simeonov, Theory of Impulsive Differential Equations,, 6 World Scientific Press, 6 (1989).  doi: 10.1142/0906.  Google Scholar

[25]

S. A. Levin, Dispersion and population interactions,, Amer. Nat., 108 (1974), 207.  doi: 10.1086/282900.  Google Scholar

[26]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations,, Comput. Math. Appl., 41 (2001), 903.  doi: 10.1016/S0898-1221(00)00328-X.  Google Scholar

[27]

X. Liu and L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations,, J. Math. Anal. Appl., 289 (2004), 279.  doi: 10.1016/j.jmaa.2003.09.058.  Google Scholar

[28]

Z. Liu, Z. Teng and L. Zhang, Two patches impulsive diffusion periodic single-species logistic model,, Int. J. Biomath., 3 (2010), 127.  doi: 10.1142/S1793524510000842.  Google Scholar

[29]

Z. Luo and J. Shen, New Razumikhin type theorems for impulsive functional differential equations,, Appl. Math. Comput., 125 (2002), 375.  doi: 10.1016/S0096-3003(00)00139-9.  Google Scholar

[30]

X. Meng, J. Jiao and L. Chen, Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays,, Nonlinear Anal., 68 (2008), 3633.  doi: 10.1016/j.na.2007.04.006.  Google Scholar

[31]

A. Muchnik, A. Semenov and M. Ushakov, Almost periodic sequences,, Theoret. Comput. Sci., 304 (2003), 1.  doi: 10.1016/S0304-3975(02)00847-2.  Google Scholar

[32]

Y. Muroya, Persistence and global stability in Lotka-Volterra delay differential systems,, Appl. Math. Lett., 17 (2004), 795.  doi: 10.1016/j.aml.2004.06.009.  Google Scholar

[33]

K. Nislow, M. Hudy, B. Letcher and E. Smith, Variation in local abundanceand species richness of stream fishes in relation to dispersal barriers: Implicationsfor management and conservation,, Freshwater Biol., 56 (2011), 2135.  doi: 10.1111/j.1365-2427.2011.02634.x.  Google Scholar

[34]

A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect,, World Scientific, (1995).  doi: 10.1142/9789812798664.  Google Scholar

[35]

J. H. Shen, Razumikhin techniques in impulsive functional differential equations,, Nonlinear Anal., 36 (1999), 119.  doi: 10.1016/S0362-546X(98)00018-2.  Google Scholar

[36]

G. Stamov, On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model,, Appl. Math. Lett., 22 (2009), 516.  doi: 10.1016/j.aml.2008.07.002.  Google Scholar

[37]

G. Stamov, Separated and almost periodic solutions for impulsive differential equations,, Note Math., 20 (): 105.   Google Scholar

[38]

G. Stamov, On the existence of almost periodic Lyapunov functions for impulsive differential equations,, Z. Anal. Anwendungen, 19 (2000), 561.  doi: 10.4171/ZAA/968.  Google Scholar

[39]

I. Stamova and G. Stamov, Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics,, J. Comput. Appl. Math., 130 (2001), 163.  doi: 10.1016/S0377-0427(99)00385-4.  Google Scholar

[40]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of dispersal population model with time delays,, J. Comp. Appl. Math., 192 (2006), 417.  doi: 10.1016/j.cam.2005.06.002.  Google Scholar

[41]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss,, Math. Biosci., 201 (2006), 143.  doi: 10.1016/j.mbs.2005.12.012.  Google Scholar

[42]

B. Tang and Y. Kuang, Permanence in Kolmogorov type systems of nonautonomous functional differential equations,, J. Math. Anal. Appl., 197 (1996), 427.  doi: 10.1006/jmaa.1996.0030.  Google Scholar

[43]

Z. Teng, On the Positive almost periodic solutions of a class of Lotka-Volterra type systems with delays,, J. Math. Anal. Appl., 249 (2000), 433.  doi: 10.1006/jmaa.2000.6891.  Google Scholar

[44]

Z. Teng, Nonautonomous Lotka-Volterra systems with delays,, J. Diff. Equa., 179 (2002), 538.  doi: 10.1006/jdeq.2001.4044.  Google Scholar

[45]

Z. Teng and Z. Lu, The effect of dispersal on single-species nonautonomous dispersal models with delays,, J. Math. Biol., 42 (2001), 439.  doi: 10.1007/s002850000076.  Google Scholar

[46]

W. Wang and L. Chen, Global stability of a population dispersal in a two-patch environment,, Dyna. Syst. Appl., 6 (1997), 207.   Google Scholar

[47]

K. Winemiller and D. B. Jepsen, Effects of seasonality and fish movement on tropical river food webs,, J. Fish Ecol., 53 (1990), 267.  doi: 10.1111/j.1095-8649.1998.tb01032.x.  Google Scholar

[48]

Y. Xiong and K. Wang, Almost periodic solution for a class of ecological system with time delay,, Appl. Math. J. Chin. Univ. Ser. A, 18 (2003), 163.   Google Scholar

[49]

X. Yang, L. Chen and J. Chen, Permanence and positive periodic solution for the single species nonautonomous delay diffusive models,, Comput. Math. Appl., 32 (1996), 109.  doi: 10.1016/0898-1221(96)00129-0.  Google Scholar

[50]

J. Zhang and L. Chen, Periodic solutions of single species nonautonomous diffusion models with continuous time delays,, Math. Comput. Model., 23 (1996), 17.  doi: 10.1016/0895-7177(96)00026-X.  Google Scholar

[51]

L. Zhang, Z. Teng, D. L. DeAngelis and S. Ruan, Single species models with logistic growth and dissymmetric impulse dispersal,, Math. Biosci., 241 (2013), 188.  doi: 10.1016/j.mbs.2012.11.005.  Google Scholar

[52]

L. Zhang, Z. Teng and Z. Liu, Survival analysis for a periodic predator-prey model with prey impulsively unilateral diffusion in two patches,, Appl. Math. Model., 35 (2011), 4243.  doi: 10.1016/j.apm.2011.02.041.  Google Scholar

[53]

L. Zhang and Z. Teng, N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations,, Nonlinear Anal: RWA., 12 (2011), 3152.  doi: 10.1016/j.nonrwa.2011.05.015.  Google Scholar

[54]

L. Zhang, Z. Teng and H. Jiang, Permanence for general nonautonomous impulsive population systems of functional differential equations and its applications,, Acta. Appl. Math., 110 (2010), 1169.  doi: 10.1007/s10440-009-9500-y.  Google Scholar

show all references

References:
[1]

L. Allen, Persistence, extinction and critical patch number for island populations,, J. Math. Biol., 24 (1987), 617.  doi: 10.1007/BF00275506.  Google Scholar

[2]

S. Ahmad and I. M. Stamova, Asymptotic stability of competitive systems with delay and impulsive perturbations,, J. Math. Anal. Appl., 334 (2007), 686.  doi: 10.1016/j.jmaa.2006.12.068.  Google Scholar

[3]

S. Ahmad and G. Stamov, Almost periodic solutions of N-dimensional impulsive competitive systems,, Nonlinear Anal: RWA., 10 (2009), 1846.  doi: 10.1016/j.nonrwa.2008.02.020.  Google Scholar

[4]

D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications,, Longman, (1993).   Google Scholar

[5]

D. D. Bainov, A. D. Myshkis and G. T. Stamov, Dichotomies and almost periodicity of the solutions of systems of impulsive differential equations,, Dyna. Syst. Appl., 5 (1996), 145.   Google Scholar

[6]

G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations,, Dyna. Cont. Disc. Impul. Syst., 5 (1999), 579.  doi: 10.1080/00036810008840804.  Google Scholar

[7]

E. Beretta, P. Fergola and C. Tenneriello, Ultimate boundedness of nonautonomous diffusive Lotka-Volterra patches,, Math. Biosci., 92 (1988), 29.  doi: 10.1016/0025-5564(88)90004-1.  Google Scholar

[8]

E. Beretta and Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays,, Bull. Math. Biol., 49 (1987), 431.  doi: 10.1007/BF02458861.  Google Scholar

[9]

J. Cui and L. Chen, The effect of diffusion on the time varying logistic population growth,, Comput. Math. Appl., 36 (1998), 1.  doi: 10.1016/S0898-1221(98)00124-2.  Google Scholar

[10]

J. Cui, Y. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems,, J. Math. Anal. Appl., 298 (2004), 73.  doi: 10.1016/j.jmaa.2004.02.059.  Google Scholar

[11]

H. Dingle, Migration: The Biology of Life on the Move,, Oxford University Press, (1996).  doi: 10.1093/acprof:oso/9780199640386.001.0001.  Google Scholar

[12]

C. H. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations,, Appl. Math. Comput., 136 (2003), 487.  doi: 10.1016/S0096-3003(02)00072-3.  Google Scholar

[13]

H. I. Freedman, J. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment,, Math. Biosci., 95 (1989), 111.  doi: 10.1016/0025-5564(89)90055-2.  Google Scholar

[14]

H. I. Freedman and Q. Peng, Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment,, Nonlinear Anal., 36 (1999), 981.  doi: 10.1016/S0362-546X(97)00712-8.  Google Scholar

[15]

X. Fu, B. Yan and Y. Liu, Introduction to Impulsive Differential Systems,, Science Press, (2005).   Google Scholar

[16]

K. Gopalsamy, Global asymptotic stability in an almost periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1986), 346.  doi: 10.1017/S0334270000004975.  Google Scholar

[17]

K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses,, J. Math. Anal. Appl., 139 (1989), 110.  doi: 10.1016/0022-247X(89)90232-1.  Google Scholar

[18]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Kluwer Academic Publishers, (1992).  doi: 10.1007/978-94-015-7920-9.  Google Scholar

[19]

W. Hamilton and R. May, Dispersal in stable habitats,, Nature (London), 269 (1977), 578.  doi: 10.1038/269578a0.  Google Scholar

[20]

C. Y. He, Almost Periodic Differential Equations,, Higher Education Publishing House, (1992).   Google Scholar

[21]

H. Hu, K. Wang and D. Wu, Permanence and global stability for nonautonomous N-species Lotka-Volterra competitive system with impulses and infinite delays,, J. Math. Anal. Appl., 377 (2011), 145.  doi: 10.1016/j.jmaa.2010.10.031.  Google Scholar

[22]

H. Hu, Permanence for nonautonomous predator-prey Kolmogorov systems with impulses and its applications,, Appl. Math. Comput., 223 (2013), 54.  doi: 10.1016/j.amc.2013.07.093.  Google Scholar

[23]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).   Google Scholar

[24]

V. Lakshmikantham, D. D. Banov and P. S. Simeonov, Theory of Impulsive Differential Equations,, 6 World Scientific Press, 6 (1989).  doi: 10.1142/0906.  Google Scholar

[25]

S. A. Levin, Dispersion and population interactions,, Amer. Nat., 108 (1974), 207.  doi: 10.1086/282900.  Google Scholar

[26]

X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations,, Comput. Math. Appl., 41 (2001), 903.  doi: 10.1016/S0898-1221(00)00328-X.  Google Scholar

[27]

X. Liu and L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations,, J. Math. Anal. Appl., 289 (2004), 279.  doi: 10.1016/j.jmaa.2003.09.058.  Google Scholar

[28]

Z. Liu, Z. Teng and L. Zhang, Two patches impulsive diffusion periodic single-species logistic model,, Int. J. Biomath., 3 (2010), 127.  doi: 10.1142/S1793524510000842.  Google Scholar

[29]

Z. Luo and J. Shen, New Razumikhin type theorems for impulsive functional differential equations,, Appl. Math. Comput., 125 (2002), 375.  doi: 10.1016/S0096-3003(00)00139-9.  Google Scholar

[30]

X. Meng, J. Jiao and L. Chen, Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays,, Nonlinear Anal., 68 (2008), 3633.  doi: 10.1016/j.na.2007.04.006.  Google Scholar

[31]

A. Muchnik, A. Semenov and M. Ushakov, Almost periodic sequences,, Theoret. Comput. Sci., 304 (2003), 1.  doi: 10.1016/S0304-3975(02)00847-2.  Google Scholar

[32]

Y. Muroya, Persistence and global stability in Lotka-Volterra delay differential systems,, Appl. Math. Lett., 17 (2004), 795.  doi: 10.1016/j.aml.2004.06.009.  Google Scholar

[33]

K. Nislow, M. Hudy, B. Letcher and E. Smith, Variation in local abundanceand species richness of stream fishes in relation to dispersal barriers: Implicationsfor management and conservation,, Freshwater Biol., 56 (2011), 2135.  doi: 10.1111/j.1365-2427.2011.02634.x.  Google Scholar

[34]

A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect,, World Scientific, (1995).  doi: 10.1142/9789812798664.  Google Scholar

[35]

J. H. Shen, Razumikhin techniques in impulsive functional differential equations,, Nonlinear Anal., 36 (1999), 119.  doi: 10.1016/S0362-546X(98)00018-2.  Google Scholar

[36]

G. Stamov, On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model,, Appl. Math. Lett., 22 (2009), 516.  doi: 10.1016/j.aml.2008.07.002.  Google Scholar

[37]

G. Stamov, Separated and almost periodic solutions for impulsive differential equations,, Note Math., 20 (): 105.   Google Scholar

[38]

G. Stamov, On the existence of almost periodic Lyapunov functions for impulsive differential equations,, Z. Anal. Anwendungen, 19 (2000), 561.  doi: 10.4171/ZAA/968.  Google Scholar

[39]

I. Stamova and G. Stamov, Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics,, J. Comput. Appl. Math., 130 (2001), 163.  doi: 10.1016/S0377-0427(99)00385-4.  Google Scholar

[40]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of dispersal population model with time delays,, J. Comp. Appl. Math., 192 (2006), 417.  doi: 10.1016/j.cam.2005.06.002.  Google Scholar

[41]

Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss,, Math. Biosci., 201 (2006), 143.  doi: 10.1016/j.mbs.2005.12.012.  Google Scholar

[42]

B. Tang and Y. Kuang, Permanence in Kolmogorov type systems of nonautonomous functional differential equations,, J. Math. Anal. Appl., 197 (1996), 427.  doi: 10.1006/jmaa.1996.0030.  Google Scholar

[43]

Z. Teng, On the Positive almost periodic solutions of a class of Lotka-Volterra type systems with delays,, J. Math. Anal. Appl., 249 (2000), 433.  doi: 10.1006/jmaa.2000.6891.  Google Scholar

[44]

Z. Teng, Nonautonomous Lotka-Volterra systems with delays,, J. Diff. Equa., 179 (2002), 538.  doi: 10.1006/jdeq.2001.4044.  Google Scholar

[45]

Z. Teng and Z. Lu, The effect of dispersal on single-species nonautonomous dispersal models with delays,, J. Math. Biol., 42 (2001), 439.  doi: 10.1007/s002850000076.  Google Scholar

[46]

W. Wang and L. Chen, Global stability of a population dispersal in a two-patch environment,, Dyna. Syst. Appl., 6 (1997), 207.   Google Scholar

[47]

K. Winemiller and D. B. Jepsen, Effects of seasonality and fish movement on tropical river food webs,, J. Fish Ecol., 53 (1990), 267.  doi: 10.1111/j.1095-8649.1998.tb01032.x.  Google Scholar

[48]

Y. Xiong and K. Wang, Almost periodic solution for a class of ecological system with time delay,, Appl. Math. J. Chin. Univ. Ser. A, 18 (2003), 163.   Google Scholar

[49]

X. Yang, L. Chen and J. Chen, Permanence and positive periodic solution for the single species nonautonomous delay diffusive models,, Comput. Math. Appl., 32 (1996), 109.  doi: 10.1016/0898-1221(96)00129-0.  Google Scholar

[50]

J. Zhang and L. Chen, Periodic solutions of single species nonautonomous diffusion models with continuous time delays,, Math. Comput. Model., 23 (1996), 17.  doi: 10.1016/0895-7177(96)00026-X.  Google Scholar

[51]

L. Zhang, Z. Teng, D. L. DeAngelis and S. Ruan, Single species models with logistic growth and dissymmetric impulse dispersal,, Math. Biosci., 241 (2013), 188.  doi: 10.1016/j.mbs.2012.11.005.  Google Scholar

[52]

L. Zhang, Z. Teng and Z. Liu, Survival analysis for a periodic predator-prey model with prey impulsively unilateral diffusion in two patches,, Appl. Math. Model., 35 (2011), 4243.  doi: 10.1016/j.apm.2011.02.041.  Google Scholar

[53]

L. Zhang and Z. Teng, N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations,, Nonlinear Anal: RWA., 12 (2011), 3152.  doi: 10.1016/j.nonrwa.2011.05.015.  Google Scholar

[54]

L. Zhang, Z. Teng and H. Jiang, Permanence for general nonautonomous impulsive population systems of functional differential equations and its applications,, Acta. Appl. Math., 110 (2010), 1169.  doi: 10.1007/s10440-009-9500-y.  Google Scholar

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