Abstract and Applied Analysis

Effects of Dispersal for a Logistic Growth Population in Random Environments

Xiaoling Zou, Dejun Fan, and Ke Wang

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We study a stochastic logistic model with diffusion between two patches in this paper. Using the definition of stationary distribution, we discuss the effect of dispersal in detail. If the species are able to have nontrivial stationary distributions when the patches are isolated, then they continue to do so for small diffusion rates. In addition, we use some examples and numerical experiments to reflect that diffusions are capable of both stabilizing and destabilizing a given ecosystem.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 912579, 9 pages.

First available in Project Euclid: 27 February 2014

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Zou, Xiaoling; Fan, Dejun; Wang, Ke. Effects of Dispersal for a Logistic Growth Population in Random Environments. Abstr. Appl. Anal. 2013 (2013), Article ID 912579, 9 pages. doi:10.1155/2013/912579. https://projecteuclid.org/euclid.aaa/1393511788

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  • J. G. Skellam, “Random dispersal in theoretical populations,” Biometrika, vol. 38, pp. 196–218, 1951.
  • 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.
  • 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.
  • L. J. S. Allen, “Persistence, extinction, and critical patch number for island populations,” Journal of Mathematical Biology, vol. 24, no. 6, pp. 617–625, 1987.
  • 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.
  • W. Wang and L. Chen, “Global stability of a population dispersal in a two-patch environment,” Dynamic Systems and Applications, vol. 6, no. 2, pp. 207–215, 1997.
  • 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.
  • Y. Takeuchi, “Cooperative systems theory and global stability of diffusion models,” Acta Applicandae Mathematicae, vol. 14, no. 1-2, pp. 49–57, 1989.
  • S. A. Levin, “Dispersion and population interactions,” The American Naturalist, vol. 108, no. 960, pp. 207–228, 1974.
  • S. A. Levin, “Spatial patterning and the structure of ecological communities,” in Some Mathematical Questions in Biology, vol. 8 of Lectures on Mathematics in the Life Sciences, pp. 1–35, American Mathematical Society, Providence, RI, USA, 1976.
  • H. I. Freedman and Y. Takeuchi, “Global stability and predator dynamics in a model of prey dispersal in a patchy environment,” Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 8, pp. 993–1002, 1989.
  • Y. Kuang and Y. Takeuchi, “Predator-prey dynamics in models of prey dispersal in two-patch environments,” Mathematical Biosciences, vol. 120, no. 1, pp. 77–98, 1994.
  • Y. Takeuchi, “Diffusion-mediated persistence in two-species competition Lotka-Volterra model,” Mathematical Biosciences, vol. 95, no. 1, pp. 65–83, 1989.
  • J. Li and J. Yan, “Permanence and extinction for a nonlinear diffusive predator-prey system,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 399–417, 2009.
  • Y. Takeuchi and Z. Y. Lu, “Permanence and global stability for competitive Lotka-Volterra diffusion systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 24, no. 1, pp. 91–104, 1995.
  • T. C. Gard, “Persistence in stochastic food web models,” Bulletin of Mathematical Biology, vol. 46, no. 3, pp. 357–370, 1984.
  • T. C. Gard, “Stability for multispecies population models in random environments,” Nonlinear Analysis. Theory, Methods & Applications, vol. 10, no. 12, pp. 1411–1419, 1986.
  • R. M. May, Stability and Complexity in Model Ecosystems, Princeton University, Princeton, NJ, USA, 1973.
  • M. Fan and K. Wang, “Study on harvested population with diffusional migration,” Journal of Systems Science and Complexity, vol. 14, no. 2, pp. 139–148, 2001.
  • X. Zou and K. Wang, “The protection zone of biological population,” Nonlinear Analysis. Real World Applications, vol. 12, no. 2, pp. 956–964, 2011.
  • X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and Their Applications, vol. 97, no. 1, pp. 95–110, 2002.
  • L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, NY, USA, 1972.
  • A. Friedman, Stochastic Differential Equations and Applications. Vol. 2. Probability and Mathematical Statistics, vol. 28, Academic Press, New York, NY, USA, 1976.
  • R. Z. Hasminskiĭ, Stochastic Stability of Differential Equations, vol. 7 of Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
  • C. Ji, D. Jiang, and N. Shi, “A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 435–440, 2011.
  • C. Ji and D. Jiang, “Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 441–453, 2011.
  • X. Mao, “Stationary distribution of stochastic population systems,” Systems & Control Letters, vol. 60, no. 6, pp. 398–405, 2011.
  • T. C. Gard, Introduction to Stochastic Differential Equations, vol. 270, Madison Avenue, New York, NY, USA, 1988.
  • G. Strang, Linear Algebra and Its Applications, Harcourt Brace, Watkins, Minn, USA, 3rd edition, 1988.
  • C. Zhu and G. Yin, “Asymptotic properties of hybrid diffusion systems,” SIAM Journal on Control and Optimization, vol. 46, no. 4, pp. 1155–1179, 2007.
  • S. Pasquali, “The stochastic logistic equation: stationary solutions and their stability,” Rendiconti del Seminario Matematico della Università di Padova, vol. 106, pp. 165–183, 2001.
  • D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.