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1. | College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China, China |
References:
[1] |
L. Allen, Persistence, extinction and critical patch number for island populations, J. Math. Biol., 24 (1987), 617-625.
doi: 10.1007/BF00275506. |
[2] |
S. Ahmad and I. M. Stamova, Asymptotic stability of competitive systems with delay and impulsive perturbations, J. Math. Anal. Appl., 334 (2007), 686-700.
doi: 10.1016/j.jmaa.2006.12.068. |
[3] |
S. Ahmad and G. Stamov, Almost periodic solutions of N-dimensional impulsive competitive systems, Nonlinear Anal: RWA., 10 (2009), 1846-1853.
doi: 10.1016/j.nonrwa.2008.02.020. |
[4] |
D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. |
[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), 145-152. |
[6] |
G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations, Dyna. Cont. Disc. Impul. Syst., 5 (1999), 579-591.
doi: 10.1080/00036810008840804. |
[7] |
E. Beretta, P. Fergola and C. Tenneriello, Ultimate boundedness of nonautonomous diffusive Lotka-Volterra patches, Math. Biosci., 92 (1988), 29-53.
doi: 10.1016/0025-5564(88)90004-1. |
[8] |
E. Beretta and Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays, Bull. Math. Biol., 49 (1987), 431-448.
doi: 10.1007/BF02458861. |
[9] |
J. Cui and L. Chen, The effect of diffusion on the time varying logistic population growth, Comput. Math. Appl., 36 (1998), 1-9.
doi: 10.1016/S0898-1221(98)00124-2. |
[10] |
J. Cui, Y. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems, J. Math. Anal. Appl., 298 (2004), 73-93.
doi: 10.1016/j.jmaa.2004.02.059. |
[11] |
H. Dingle, Migration: The Biology of Life on the Move, Oxford University Press, New York, 1996.
doi: 10.1093/acprof:oso/9780199640386.001.0001. |
[12] |
C. H. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations, Appl. Math. Comput., 136 (2003), 487-494.
doi: 10.1016/S0096-3003(02)00072-3. |
[13] |
H. I. Freedman, J. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci., 95 (1989), 111-123.
doi: 10.1016/0025-5564(89)90055-2. |
[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), 981-996.
doi: 10.1016/S0362-546X(97)00712-8. |
[15] |
X. Fu, B. Yan and Y. Liu, Introduction to Impulsive Differential Systems, Science Press, Beijing, 2005. |
[16] |
K. Gopalsamy, Global asymptotic stability in an almost periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1986), 346-360.
doi: 10.1017/S0334270000004975. |
[17] |
K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110-122.
doi: 10.1016/0022-247X(89)90232-1. |
[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. |
[19] |
W. Hamilton and R. May, Dispersal in stable habitats, Nature (London), 269 (1977), 578-581.
doi: 10.1038/269578a0. |
[20] |
C. Y. He, Almost Periodic Differential Equations, Higher Education Publishing House, Beijing, 1992 (In Chinese). |
[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), 145-160.
doi: 10.1016/j.jmaa.2010.10.031. |
[22] |
H. Hu, Permanence for nonautonomous predator-prey Kolmogorov systems with impulses and its applications, Appl. Math. Comput., 223 (2013), 54-75.
doi: 10.1016/j.amc.2013.07.093. |
[23] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego,CA, 1993. |
[24] |
V. Lakshmikantham, D. D. Banov and P. S. Simeonov, Theory of Impulsive Differential Equations, 6 World Scientific Press, Singapore, 1989.
doi: 10.1142/0906. |
[25] |
S. A. Levin, Dispersion and population interactions, Amer. Nat., 108 (1974), 207-228.
doi: 10.1086/282900. |
[26] |
X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41 (2001), 903-915.
doi: 10.1016/S0898-1221(00)00328-X. |
[27] |
X. Liu and L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. Math. Anal. Appl., 289 (2004), 279-291.
doi: 10.1016/j.jmaa.2003.09.058. |
[28] |
Z. Liu, Z. Teng and L. Zhang, Two patches impulsive diffusion periodic single-species logistic model, Int. J. Biomath., 3 (2010), 127-141.
doi: 10.1142/S1793524510000842. |
[29] |
Z. Luo and J. Shen, New Razumikhin type theorems for impulsive functional differential equations, Appl. Math. Comput., 125 (2002), 375-386.
doi: 10.1016/S0096-3003(00)00139-9. |
[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), 3633-3645.
doi: 10.1016/j.na.2007.04.006. |
[31] |
A. Muchnik, A. Semenov and M. Ushakov, Almost periodic sequences, Theoret. Comput. Sci., 304 (2003), 1-33.
doi: 10.1016/S0304-3975(02)00847-2. |
[32] |
Y. Muroya, Persistence and global stability in Lotka-Volterra delay differential systems, Appl. Math. Lett., 17 (2004), 795-800.
doi: 10.1016/j.aml.2004.06.009. |
[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), 2135-2144.
doi: 10.1111/j.1365-2427.2011.02634.x. |
[34] |
A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
[35] |
J. H. Shen, Razumikhin techniques in impulsive functional differential equations, Nonlinear Anal., 36 (1999), 119-130.
doi: 10.1016/S0362-546X(98)00018-2. |
[36] |
G. Stamov, On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model, Appl. Math. Lett., 22 (2009), 516-520.
doi: 10.1016/j.aml.2008.07.002. |
[37] |
G. Stamov, Separated and almost periodic solutions for impulsive differential equations,, Note Math., 20 (): 105.
|
[38] |
G. Stamov, On the existence of almost periodic Lyapunov functions for impulsive differential equations, Z. Anal. Anwendungen, 19 (2000), 561-573.
doi: 10.4171/ZAA/968. |
[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), 163-171.
doi: 10.1016/S0377-0427(99)00385-4. |
[40] |
Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of dispersal population model with time delays, J. Comp. Appl. Math., 192 (2006), 417-430.
doi: 10.1016/j.cam.2005.06.002. |
[41] |
Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss, Math. Biosci., 201 (2006), 143-156.
doi: 10.1016/j.mbs.2005.12.012. |
[42] |
B. Tang and Y. Kuang, Permanence in Kolmogorov type systems of nonautonomous functional differential equations, J. Math. Anal. Appl., 197 (1996), 427-447.
doi: 10.1006/jmaa.1996.0030. |
[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), 433-444.
doi: 10.1006/jmaa.2000.6891. |
[44] |
Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Diff. Equa., 179 (2002), 538-561.
doi: 10.1006/jdeq.2001.4044. |
[45] |
Z. Teng and Z. Lu, The effect of dispersal on single-species nonautonomous dispersal models with delays, J. Math. Biol., 42 (2001), 439-454.
doi: 10.1007/s002850000076. |
[46] |
W. Wang and L. Chen, Global stability of a population dispersal in a two-patch environment, Dyna. Syst. Appl., 6 (1997), 207-216. |
[47] |
K. Winemiller and D. B. Jepsen, Effects of seasonality and fish movement on tropical river food webs, J. Fish Ecol., 53 (1990), 267-296.
doi: 10.1111/j.1095-8649.1998.tb01032.x. |
[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), 163-170. |
[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), 109-116.
doi: 10.1016/0898-1221(96)00129-0. |
[50] |
J. Zhang and L. Chen, Periodic solutions of single species nonautonomous diffusion models with continuous time delays, Math. Comput. Model., 23 (1996), 17-27.
doi: 10.1016/0895-7177(96)00026-X. |
[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), 188-198.
doi: 10.1016/j.mbs.2012.11.005. |
[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), 4243-4256.
doi: 10.1016/j.apm.2011.02.041. |
[53] |
L. Zhang and Z. Teng, N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations, Nonlinear Anal: RWA., 12 (2011), 3152-3169.
doi: 10.1016/j.nonrwa.2011.05.015. |
[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), 1169-1197.
doi: 10.1007/s10440-009-9500-y. |
show all references
References:
[1] |
L. Allen, Persistence, extinction and critical patch number for island populations, J. Math. Biol., 24 (1987), 617-625.
doi: 10.1007/BF00275506. |
[2] |
S. Ahmad and I. M. Stamova, Asymptotic stability of competitive systems with delay and impulsive perturbations, J. Math. Anal. Appl., 334 (2007), 686-700.
doi: 10.1016/j.jmaa.2006.12.068. |
[3] |
S. Ahmad and G. Stamov, Almost periodic solutions of N-dimensional impulsive competitive systems, Nonlinear Anal: RWA., 10 (2009), 1846-1853.
doi: 10.1016/j.nonrwa.2008.02.020. |
[4] |
D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993. |
[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), 145-152. |
[6] |
G. Ballinger and X. Liu, Existence and uniqueness results for impulsive delay differential equations, Dyna. Cont. Disc. Impul. Syst., 5 (1999), 579-591.
doi: 10.1080/00036810008840804. |
[7] |
E. Beretta, P. Fergola and C. Tenneriello, Ultimate boundedness of nonautonomous diffusive Lotka-Volterra patches, Math. Biosci., 92 (1988), 29-53.
doi: 10.1016/0025-5564(88)90004-1. |
[8] |
E. Beretta and Y. Takeuchi, Global stability of single-species diffusion Volterra models with continuous time delays, Bull. Math. Biol., 49 (1987), 431-448.
doi: 10.1007/BF02458861. |
[9] |
J. Cui and L. Chen, The effect of diffusion on the time varying logistic population growth, Comput. Math. Appl., 36 (1998), 1-9.
doi: 10.1016/S0898-1221(98)00124-2. |
[10] |
J. Cui, Y. Takeuchi and Z. Lin, Permanence and extinction for dispersal population systems, J. Math. Anal. Appl., 298 (2004), 73-93.
doi: 10.1016/j.jmaa.2004.02.059. |
[11] |
H. Dingle, Migration: The Biology of Life on the Move, Oxford University Press, New York, 1996.
doi: 10.1093/acprof:oso/9780199640386.001.0001. |
[12] |
C. H. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations, Appl. Math. Comput., 136 (2003), 487-494.
doi: 10.1016/S0096-3003(02)00072-3. |
[13] |
H. I. Freedman, J. Shukla and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci., 95 (1989), 111-123.
doi: 10.1016/0025-5564(89)90055-2. |
[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), 981-996.
doi: 10.1016/S0362-546X(97)00712-8. |
[15] |
X. Fu, B. Yan and Y. Liu, Introduction to Impulsive Differential Systems, Science Press, Beijing, 2005. |
[16] |
K. Gopalsamy, Global asymptotic stability in an almost periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1986), 346-360.
doi: 10.1017/S0334270000004975. |
[17] |
K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110-122.
doi: 10.1016/0022-247X(89)90232-1. |
[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. |
[19] |
W. Hamilton and R. May, Dispersal in stable habitats, Nature (London), 269 (1977), 578-581.
doi: 10.1038/269578a0. |
[20] |
C. Y. He, Almost Periodic Differential Equations, Higher Education Publishing House, Beijing, 1992 (In Chinese). |
[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), 145-160.
doi: 10.1016/j.jmaa.2010.10.031. |
[22] |
H. Hu, Permanence for nonautonomous predator-prey Kolmogorov systems with impulses and its applications, Appl. Math. Comput., 223 (2013), 54-75.
doi: 10.1016/j.amc.2013.07.093. |
[23] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego,CA, 1993. |
[24] |
V. Lakshmikantham, D. D. Banov and P. S. Simeonov, Theory of Impulsive Differential Equations, 6 World Scientific Press, Singapore, 1989.
doi: 10.1142/0906. |
[25] |
S. A. Levin, Dispersion and population interactions, Amer. Nat., 108 (1974), 207-228.
doi: 10.1086/282900. |
[26] |
X. Liu and G. Ballinger, Uniform asymptotic stability of impulsive delay differential equations, Comput. Math. Appl., 41 (2001), 903-915.
doi: 10.1016/S0898-1221(00)00328-X. |
[27] |
X. Liu and L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. Math. Anal. Appl., 289 (2004), 279-291.
doi: 10.1016/j.jmaa.2003.09.058. |
[28] |
Z. Liu, Z. Teng and L. Zhang, Two patches impulsive diffusion periodic single-species logistic model, Int. J. Biomath., 3 (2010), 127-141.
doi: 10.1142/S1793524510000842. |
[29] |
Z. Luo and J. Shen, New Razumikhin type theorems for impulsive functional differential equations, Appl. Math. Comput., 125 (2002), 375-386.
doi: 10.1016/S0096-3003(00)00139-9. |
[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), 3633-3645.
doi: 10.1016/j.na.2007.04.006. |
[31] |
A. Muchnik, A. Semenov and M. Ushakov, Almost periodic sequences, Theoret. Comput. Sci., 304 (2003), 1-33.
doi: 10.1016/S0304-3975(02)00847-2. |
[32] |
Y. Muroya, Persistence and global stability in Lotka-Volterra delay differential systems, Appl. Math. Lett., 17 (2004), 795-800.
doi: 10.1016/j.aml.2004.06.009. |
[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), 2135-2144.
doi: 10.1111/j.1365-2427.2011.02634.x. |
[34] |
A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
[35] |
J. H. Shen, Razumikhin techniques in impulsive functional differential equations, Nonlinear Anal., 36 (1999), 119-130.
doi: 10.1016/S0362-546X(98)00018-2. |
[36] |
G. Stamov, On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model, Appl. Math. Lett., 22 (2009), 516-520.
doi: 10.1016/j.aml.2008.07.002. |
[37] |
G. Stamov, Separated and almost periodic solutions for impulsive differential equations,, Note Math., 20 (): 105.
|
[38] |
G. Stamov, On the existence of almost periodic Lyapunov functions for impulsive differential equations, Z. Anal. Anwendungen, 19 (2000), 561-573.
doi: 10.4171/ZAA/968. |
[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), 163-171.
doi: 10.1016/S0377-0427(99)00385-4. |
[40] |
Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of dispersal population model with time delays, J. Comp. Appl. Math., 192 (2006), 417-430.
doi: 10.1016/j.cam.2005.06.002. |
[41] |
Y. Takeuchi, J. Cui, R. Miyazaki and Y. Saito, Permanence of delayed population model with dispersal loss, Math. Biosci., 201 (2006), 143-156.
doi: 10.1016/j.mbs.2005.12.012. |
[42] |
B. Tang and Y. Kuang, Permanence in Kolmogorov type systems of nonautonomous functional differential equations, J. Math. Anal. Appl., 197 (1996), 427-447.
doi: 10.1006/jmaa.1996.0030. |
[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), 433-444.
doi: 10.1006/jmaa.2000.6891. |
[44] |
Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Diff. Equa., 179 (2002), 538-561.
doi: 10.1006/jdeq.2001.4044. |
[45] |
Z. Teng and Z. Lu, The effect of dispersal on single-species nonautonomous dispersal models with delays, J. Math. Biol., 42 (2001), 439-454.
doi: 10.1007/s002850000076. |
[46] |
W. Wang and L. Chen, Global stability of a population dispersal in a two-patch environment, Dyna. Syst. Appl., 6 (1997), 207-216. |
[47] |
K. Winemiller and D. B. Jepsen, Effects of seasonality and fish movement on tropical river food webs, J. Fish Ecol., 53 (1990), 267-296.
doi: 10.1111/j.1095-8649.1998.tb01032.x. |
[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), 163-170. |
[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), 109-116.
doi: 10.1016/0898-1221(96)00129-0. |
[50] |
J. Zhang and L. Chen, Periodic solutions of single species nonautonomous diffusion models with continuous time delays, Math. Comput. Model., 23 (1996), 17-27.
doi: 10.1016/0895-7177(96)00026-X. |
[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), 188-198.
doi: 10.1016/j.mbs.2012.11.005. |
[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), 4243-4256.
doi: 10.1016/j.apm.2011.02.041. |
[53] |
L. Zhang and Z. Teng, N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbations, Nonlinear Anal: RWA., 12 (2011), 3152-3169.
doi: 10.1016/j.nonrwa.2011.05.015. |
[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), 1169-1197.
doi: 10.1007/s10440-009-9500-y. |
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