A Single Species Model with Symmetric Bidirectional Impulsive Diffusion and Dispersal Delay

ABSTRACT

In the natural ecosystem, impulsive diffusion provides a more natural description for population dynamics. In addition, dispersal processes often involve with time delay. In view of these facts, a single species model with impulsive diffusion and dispersal delay is formulated. By the stroboscopic map of the discrete dynamical system and other analysis methods, the permanence of the system is investigated. Moreover, sufficient conditions on the existence and uniqueness of a positive periodic solution for the system are derived from the intermediate value theorem. We also demonstrate the global stability of the positive periodic solution by the theory of discrete dynamical system. Finally, numerical simulations and discussion are presented to validate our theoretical results.

In the natural ecosystem, impulsive diffusion provides a more natural description for population dynamics. In addition, dispersal processes often involve with time delay. In view of these facts, a single species model with impulsive diffusion and dispersal delay is formulated. By the stroboscopic map of the discrete dynamical system and other analysis methods, the permanence of the system is investigated. Moreover, sufficient conditions on the existence and uniqueness of a positive periodic solution for the system are derived from the intermediate value theorem. We also demonstrate the global stability of the positive periodic solution by the theory of discrete dynamical system. Finally, numerical simulations and discussion are presented to validate our theoretical results.

Cite this paper

H. Wan, L. Zhang and H. Li, "A Single Species Model with Symmetric Bidirectional Impulsive Diffusion and Dispersal Delay,"*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 1079-1088. doi: 10.4236/am.2012.39159.

H. Wan, L. Zhang and H. Li, "A Single Species Model with Symmetric Bidirectional Impulsive Diffusion and Dispersal Delay,"

References

[1] Z. Teng and Z. Lu, “The Effect of Dispersal on SingleSpecies Nonautonomous Dispersal Models with Delays,” Journal of Mathematical Biology, Vol. 42, No. 5, 2001, pp. 439-454. doi:10.1007/s002850000076

[2] Z. Teng and L. Chen, “Permanence and Extinction of Periodic Predator-Prey Systems in a Patchy Environment with Delay,” Nonlinear Analysis: Real World Applications, Vol. 4, No. 2, 2003, pp. 335-364. doi:10.1016/S1468-1218(02)00026-3

[3] L. Buttel, R. Durrett and S. Levin, “Competition and Species Packing in Patchy Environments,” Theoretical Population Biology, Vol. 61, No. 3, 2002, pp. 265-276. doi:10.1006/tpbi.2001.1569

[4] J. Cui, Y. Takeuchi and Z. Lin, “Permanence and Extinction for Dispersal Population Systems,” Journal of Mathematical Analysis and Applications, Vol. 298, No. 1, 2004, pp. 73-93. doi:10.1016/j.jmaa.2004.02.059

[5] Y. Takeuchi, “Diffusion Effect on Stability of LotkaVolterra Models,” Bulletin of Mathematical Biology, Vol. 48, No. 5-6, 1986, pp. 585-601.

[6] Y. Takeuchi, J. Cui, R. Miyazak and Y. Saito, “Permanence of Delayed Population Model with Dispersal Loss,” Mathematical Biosciences, Vol. 201, No. 1-2, 2006, pp. 143-156. doi:10.1016/j.mbs.2005.12.012

[7] E. Beretta and Y. Takeuchi, “Global Stability of SingleSpecies Diffusion Volterra Models with Continuous Time Delays,” Bulletin of Mathematical Biology, Vol. 49, 1987, pp. 431-448.

[8] E. Beretta and Y. Takeuchi, “Global Asymptotic Stability of Lotka-Volterra Diffusion Models with Continuous Time Delays,” SIAM Journal on Applied Mathematics, Vol. 48, No. 3, 1998, pp. 627-651. doi:10.1137/0148035

[9] E. Beterra, P. Fergola and C. Tenneriello, “Ultimate Boundedness of Nonautonomous Diffusive Lotka-Volterra Patches,” Mathematical Biosciences, Vol. 92, No. 1, 1988, pp. 29-53. doi:10.1016/0025-5564(88)90004-1

[10] H. I. Freedman, J. Shukla and Y. Takeuchi, “Population Diffusion in a Two-Patch Environment,” Mathematical Biosciences, Vol. 95, No. 1, 1989, pp. 111-123. doi:10.1016/0025-5564(89)90055-2

[11] A. Hastings, “Dynamics of a Single Species in a Spatially Varying Environment: The Stability Role of High Dispersal Rates,” Journal of Mathematical Biology, Vol. 16, No. 1, 1982, pp. 49-55. doi:10.1007/BF00275160

[12] W. Wang and L. Chen, “Global Stability of a Population Dispersal in a Two-Patch Environment,” Dynamic Systems & Applications, Vol. 6, 1997, pp. 207-216.

[13] L. Zhang and Z. Teng, “Permanence for a Class of Periodic Time-Dependent Competitive System with Delays and Dispersal in a Patchy-Environment,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 855-864. doi:10.1016/j.amc.2006.10.037

[14] L. Zhang and Z. Teng, “Permanence for a Delayed Periodic Predator-Prey Model with Prey Dispersal in MultiPatches and Predator Density-Independent,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 175-193. doi:10.1016/j.jmaa.2007.05.016

[15] E. Beretta, F. Solimano and Y. Takeuchi, “Global Stability and Periodic Orbits for Two Patch Predator-Prey Diffusion-Delay Models,” Mathematical Biosciences, Vol. 85, No. 2, 1987, pp. 153-183. doi:10.1016/0025-5564(87)90051-4

[16] R. Mahbuba and L. Chen, “On the Nonautonomous Lotka-Volterra Competion System with Diffusion,” Differential Equations and Dynamical Systems, Vol. 2, 1994, pp. 243-253.

[17] J. G. Skellam, “Random Dispersal in Theoretical Population,” Biometrika, Vol. 38, 1951, pp. 196-218.

[18] J. Hui and L. Chen, “A Single Species Model with Impulsive Diffusion,” Acta Mathematicae Applicatae Sinica. English Series, Vol. 21, No. 1, 2005, pp. 43-48. doi:10.1007/s10255-005-0213-3

[19] L. Wang and L. Chen, “Impulsive Diffusion in Single Species Model,” Chaos Solitons, Fractals, Vol. 33, No. 4, 2007, pp. 1213-1219. doi:10.1016/j.chaos.2006.01.102

[20] A. Lakmeche and O. Arino, “Bifurcation of Nontrivial Periodic Solution of Impulsive Differential Equations Arising Chemotherapeutic Treatment,” Dynamics of Continuous, Discrete and Impulsive Systems, Vol. 7, 2000, pp. 265-287.

[21] L. Zhang and Z. Teng, “N-Species Non-Autonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 6, 2011, pp. 3152-3169. doi:10.1016/j.nonrwa.2011.05.015

[22] J. Vandermeer, L. Stone and B. Blasius, “Categories of Chaos and Fractal Basin Boundaries in Forced Predator-Preymodels,” Chaos Solitons, Fractals, Vol. 12, No. 2, 2001, pp. 265-276. doi:10.1016/S0960-0779(00)00111-9

[23] L. Dong, L. Chen and L. Sun, “Optimal Harvesting Policy for Inshore-Offshore Fishery Model with Impulsive Diffusion,” Acta Mathematica Scientia, Vol. 27, No. 2, 2007, pp. 405-412. doi:10.1016/S0252-9602(07)60040-X

[24] Z. Zhao, X. Zhang and L. Chen, “The Effect of Pulsed Harvesting Policy on the Inshore-Offshore Fishery Model with the Impusive Diffusion,” Nonlinear Dynamic, Vol. 63, No. 4, 2011, pp. 537-545. doi:10.1007/s11071-009-9527-7

[25] Y. Kuang, “Delay Differential Equations with Applications in Population Dynamics,” Academic Press, New York, 1993.

[26] X. Zhao, “Dynamical Systems in Population Biology,” Springer-Verlag, New York, 2003.

[27] H. L. Smith, “Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,” Mathematical Surveys and Monographs, Vol. 41, 1995.

[28] X. Zhao, “Global Attractivity in a Class of Non-Monotone Reaction? Diffusion Equations with Time Delay,” Canadian Applied Mathematics Quarterly, Vol. 17, 2009, pp. 271-281.

[29] X. Meng and L. Chen, “Permanence and Global Stability in an Impulsive Lotka-Volterra N-Species Competitive System with Both Discrete Delays and Continuous Delays,” International Journal of Biomathematics, Vol. 1, No. 2, 2008, pp. 179-196. doi:10.1142/S1793524508000151

[30] H. L. Smith, “Cooperative Systems of Differential Equations with Concave Nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 10, 1986, pp. 1037-1052.

[31] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

[32] D. Bainov and P. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Applications,” Longman, England, 2003.

[1] Z. Teng and Z. Lu, “The Effect of Dispersal on SingleSpecies Nonautonomous Dispersal Models with Delays,” Journal of Mathematical Biology, Vol. 42, No. 5, 2001, pp. 439-454. doi:10.1007/s002850000076

[2] Z. Teng and L. Chen, “Permanence and Extinction of Periodic Predator-Prey Systems in a Patchy Environment with Delay,” Nonlinear Analysis: Real World Applications, Vol. 4, No. 2, 2003, pp. 335-364. doi:10.1016/S1468-1218(02)00026-3

[3] L. Buttel, R. Durrett and S. Levin, “Competition and Species Packing in Patchy Environments,” Theoretical Population Biology, Vol. 61, No. 3, 2002, pp. 265-276. doi:10.1006/tpbi.2001.1569

[4] J. Cui, Y. Takeuchi and Z. Lin, “Permanence and Extinction for Dispersal Population Systems,” Journal of Mathematical Analysis and Applications, Vol. 298, No. 1, 2004, pp. 73-93. doi:10.1016/j.jmaa.2004.02.059

[5] Y. Takeuchi, “Diffusion Effect on Stability of LotkaVolterra Models,” Bulletin of Mathematical Biology, Vol. 48, No. 5-6, 1986, pp. 585-601.

[6] Y. Takeuchi, J. Cui, R. Miyazak and Y. Saito, “Permanence of Delayed Population Model with Dispersal Loss,” Mathematical Biosciences, Vol. 201, No. 1-2, 2006, pp. 143-156. doi:10.1016/j.mbs.2005.12.012

[7] E. Beretta and Y. Takeuchi, “Global Stability of SingleSpecies Diffusion Volterra Models with Continuous Time Delays,” Bulletin of Mathematical Biology, Vol. 49, 1987, pp. 431-448.

[8] E. Beretta and Y. Takeuchi, “Global Asymptotic Stability of Lotka-Volterra Diffusion Models with Continuous Time Delays,” SIAM Journal on Applied Mathematics, Vol. 48, No. 3, 1998, pp. 627-651. doi:10.1137/0148035

[9] E. Beterra, P. Fergola and C. Tenneriello, “Ultimate Boundedness of Nonautonomous Diffusive Lotka-Volterra Patches,” Mathematical Biosciences, Vol. 92, No. 1, 1988, pp. 29-53. doi:10.1016/0025-5564(88)90004-1

[10] H. I. Freedman, J. Shukla and Y. Takeuchi, “Population Diffusion in a Two-Patch Environment,” Mathematical Biosciences, Vol. 95, No. 1, 1989, pp. 111-123. doi:10.1016/0025-5564(89)90055-2

[11] A. Hastings, “Dynamics of a Single Species in a Spatially Varying Environment: The Stability Role of High Dispersal Rates,” Journal of Mathematical Biology, Vol. 16, No. 1, 1982, pp. 49-55. doi:10.1007/BF00275160

[12] W. Wang and L. Chen, “Global Stability of a Population Dispersal in a Two-Patch Environment,” Dynamic Systems & Applications, Vol. 6, 1997, pp. 207-216.

[13] L. Zhang and Z. Teng, “Permanence for a Class of Periodic Time-Dependent Competitive System with Delays and Dispersal in a Patchy-Environment,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 855-864. doi:10.1016/j.amc.2006.10.037

[14] L. Zhang and Z. Teng, “Permanence for a Delayed Periodic Predator-Prey Model with Prey Dispersal in MultiPatches and Predator Density-Independent,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 175-193. doi:10.1016/j.jmaa.2007.05.016

[15] E. Beretta, F. Solimano and Y. Takeuchi, “Global Stability and Periodic Orbits for Two Patch Predator-Prey Diffusion-Delay Models,” Mathematical Biosciences, Vol. 85, No. 2, 1987, pp. 153-183. doi:10.1016/0025-5564(87)90051-4

[16] R. Mahbuba and L. Chen, “On the Nonautonomous Lotka-Volterra Competion System with Diffusion,” Differential Equations and Dynamical Systems, Vol. 2, 1994, pp. 243-253.

[17] J. G. Skellam, “Random Dispersal in Theoretical Population,” Biometrika, Vol. 38, 1951, pp. 196-218.

[18] J. Hui and L. Chen, “A Single Species Model with Impulsive Diffusion,” Acta Mathematicae Applicatae Sinica. English Series, Vol. 21, No. 1, 2005, pp. 43-48. doi:10.1007/s10255-005-0213-3

[19] L. Wang and L. Chen, “Impulsive Diffusion in Single Species Model,” Chaos Solitons, Fractals, Vol. 33, No. 4, 2007, pp. 1213-1219. doi:10.1016/j.chaos.2006.01.102

[20] A. Lakmeche and O. Arino, “Bifurcation of Nontrivial Periodic Solution of Impulsive Differential Equations Arising Chemotherapeutic Treatment,” Dynamics of Continuous, Discrete and Impulsive Systems, Vol. 7, 2000, pp. 265-287.

[21] L. Zhang and Z. Teng, “N-Species Non-Autonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 6, 2011, pp. 3152-3169. doi:10.1016/j.nonrwa.2011.05.015

[22] J. Vandermeer, L. Stone and B. Blasius, “Categories of Chaos and Fractal Basin Boundaries in Forced Predator-Preymodels,” Chaos Solitons, Fractals, Vol. 12, No. 2, 2001, pp. 265-276. doi:10.1016/S0960-0779(00)00111-9

[23] L. Dong, L. Chen and L. Sun, “Optimal Harvesting Policy for Inshore-Offshore Fishery Model with Impulsive Diffusion,” Acta Mathematica Scientia, Vol. 27, No. 2, 2007, pp. 405-412. doi:10.1016/S0252-9602(07)60040-X

[24] Z. Zhao, X. Zhang and L. Chen, “The Effect of Pulsed Harvesting Policy on the Inshore-Offshore Fishery Model with the Impusive Diffusion,” Nonlinear Dynamic, Vol. 63, No. 4, 2011, pp. 537-545. doi:10.1007/s11071-009-9527-7

[25] Y. Kuang, “Delay Differential Equations with Applications in Population Dynamics,” Academic Press, New York, 1993.

[26] X. Zhao, “Dynamical Systems in Population Biology,” Springer-Verlag, New York, 2003.

[27] H. L. Smith, “Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,” Mathematical Surveys and Monographs, Vol. 41, 1995.

[28] X. Zhao, “Global Attractivity in a Class of Non-Monotone Reaction? Diffusion Equations with Time Delay,” Canadian Applied Mathematics Quarterly, Vol. 17, 2009, pp. 271-281.

[29] X. Meng and L. Chen, “Permanence and Global Stability in an Impulsive Lotka-Volterra N-Species Competitive System with Both Discrete Delays and Continuous Delays,” International Journal of Biomathematics, Vol. 1, No. 2, 2008, pp. 179-196. doi:10.1142/S1793524508000151

[30] H. L. Smith, “Cooperative Systems of Differential Equations with Concave Nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 10, 1986, pp. 1037-1052.

[31] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

[32] D. Bainov and P. Simeonov, “Impulsive Differential Equations: Periodic Solutions and Applications,” Longman, England, 2003.