Intermittent dispersal population model with almost period parameters and dispersal delays

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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

Long Zhang ^1, , Gao Xu ^1, and Zhidong Teng ^1,

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

Received April 2015 Revised April 2016 Published June 2016

Abstract
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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.

Keywords: intermittent dispersal, global stability, Almost period, permanence., delays.

Mathematics Subject Classification: Primary: 92D25, 34D20; Secondary: 34D1.

Citation: Long Zhang, Gao Xu, Zhidong Teng. Intermittent dispersal population model with almost period parameters and dispersal delays. Discrete and 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), 617-625. 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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