## Journal of Applied Mathematics

### Analysis of a Single Species Model with Dissymmetric Bidirectional Impulsive Diffusion and Dispersal Delay

#### Abstract

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in the real natural ecosystem, impulsive diffusion provides a more suitable manner to model the actual dispersal (or migration) behavior for many ecological species. In addition, the species not only requires some time to disperse or migrate among the patches but also has some possibility of loss during dispersal. In view of these facts, a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay is formulated. Criteria on the permanence and extinction of species are established. Furthermore, the realistic conditions for the existence, uniqueness, and the global stability of the positive periodic solution are obtained. Finally, numerical simulations and discussion are presented to illustrate our theoretical results.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 701545, 11 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305673

Digital Object Identifier
doi:10.1155/2014/701545

Mathematical Reviews number (MathSciNet)
MR3198396

Zentralblatt MATH identifier
07010717

#### Citation

Wan, Haiyun; Zhang, Long; Teng, Zhidong. Analysis of a Single Species Model with Dissymmetric Bidirectional Impulsive Diffusion and Dispersal Delay. J. Appl. Math. 2014 (2014), Article ID 701545, 11 pages. doi:10.1155/2014/701545. https://projecteuclid.org/euclid.jam/1425305673

#### References

• S. A. Levin, “Dispersion and population interactions,” The American Naturalist, vol. 108, pp. 207–228, 1974.
• Z. Teng and Z. Lu, “The effect of dispersal on single-species nonautonomous dispersal models with delays,” Journal of Mathematical Biology, vol. 42, no. 5, pp. 439–454, 2001.
• J. Cui, Y. Takeuchi, and Z. Lin, “Permanence and extinction for dispersal population systems,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 73–93, 2004.
• J. Cui and L. Chen, “Permanence and extinction in logistic and Lotka-Volterra systems with diffusion,” Journal of Mathematical Analysis and Applications, vol. 258, no. 2, pp. 512–535, 2001.
• R. Xu and Z. Ma, “The effect of dispersal on the permanence of a predator-prey system with time delay,” Nonlinear Analysis. Real World Applications, vol. 9, no. 2, pp. 354–369, 2008.
• Y. Takeuchi, J. Cui, R. Miyazaki, and Y. Saito, “Permanence of delayed population model with dispersal loss,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 143–156, 2006.
• E. Beretta and Y. Takeuchi, “Global stability of single-species diffusion Volterra models with continuous time delays,” Bulletin of Mathematical Biology, vol. 49, no. 4, pp. 431–448, 1987.
• E. Beretta and Y. Takeuchi, “Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delay,” SIAM Journal on Applied Mathematics, vol. 48, no. 3, pp. 627–651, 1988.
• E. Beretta, P. Fergola, and C. Tenneriello, “Ultimate boundedness for nonautonomous diffusive Lotka-Volterra patches,” Mathematical Biosciences, vol. 92, no. 1, pp. 29–53, 1988.
• H. I. Freedman, J. B. Shukla, and Y. Takeuchi, “Population diffusion in a two-patch environment,” Mathematical Biosciences, vol. 95, no. 1, pp. 111–123, 1989.
• A. Hastings, “Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates,” Journal of Mathematical Biology, vol. 16, no. 1, pp. 49–55, 1982/83.
• W. Wang, L. Chen, and Z. Lu, “Global stability of a population dispersal in a two-patch environment,” Dynamic Systems and Applications, vol. 6, no. 2, pp. 207–215, 1997.
• L. Dong, L. Chen, and P. Shi, “Periodic solutions for a two-species nonautonomous competition system with diffusion and impulses,” Chaos, Solitons & Fractals, vol. 32, no. 5, pp. 1916–1926, 2007.
• J. Hui and L.-S. Chen, “A single species model with impulsive diffusion,” Acta Mathematicae Applicatae Sinica, vol. 21, no. 1, pp. 43–48, 2005.
• L. Wang, Z. Liu, Jinghui, and L. Chen, “Impulsive diffusion in single species model,” Chaos, Solitons & Fractals, vol. 33, no. 4, pp. 1213–1219, 2007.
• G. Ballinger and X. Liu, “Permanence of population growth models with impulsive effects,” Mathematical and Computer Modelling, vol. 26, no. 12, pp. 59–72, 1997.
• 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, pp. 3152–3169, 2011.
• J. Vandermeer, L. Stone, and B. Blasius, “Categories of chaos and fractal basin boundaries in forced predator-prey models,” Chaos, Solitons & Fractals, vol. 12, no. 2, pp. 265–276, 2001.
• L. Zhang, Z. Teng, D. L. DeAngelis, and S. Ruan, “Single species models with logistic growth and dissymmetric impulse dispersal,” Mathematical Biosciences, vol. 241, no. 2, pp. 188–197, 2013.
• Z. Zhao, X. Zhang, and L. Chen, “The effect of pulsed harvesting policy on the inshore-offshore fishery model with the impulsive diffusion,” Nonlinear Dynamics, vol. 63, no. 4, pp. 537–545, 2011.
• X.-P. Yan and W.-T. Li, “Hopf bifurcation and global periodic solutions in a delayed predator-prey system,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 427–445, 2006.
• L. Zhang and Z. Teng, “Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 175–193, 2008.
• H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1995.
• X.-Q. Zhao, “Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay,” Canadian Applied Mathematics Quarterly, vol. 17, no. 1, pp. 271–281, 2009.
• Y. Takeuchi, W. Wang, and Y. Saito, “Global stability of population models with patch structure,” Nonlinear Analysis: Real World Applications, vol. 7, no. 2, pp. 235–247, 2006.
• 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, pp. 179–196, 2008.
• H. I. Freedman, B. Rai, and P. Waltman, “Mathematical models of population interactions with dispersal. II. Differential survival in a change of habitat,” Journal of Mathematical Analysis and Applications, vol. 115, no. 1, pp. 140–154, 1986.
• R. H. MacArthur and E. O. Wilson, The Theory of Island Biogeography, Princeton University Press, 1967.
• H. L. Smith, “Cooperative systems of differential equations with concave nonlinearities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 10, no. 10, pp. 1037–1052, 1986.
• V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
• D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, London, UK, 2003.
• F. Zhang, S. Gao, and Y. Zhang, “Effects of pulse culling on population growth of migratory birds and economical birds,” Nonlinear Dynamics, vol. 67, no. 1, pp. 767–779, 2012. \endinput