## Abstract and Applied Analysis

### Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal

#### Abstract

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 135843, 17 pages.

Dates
First available in Project Euclid: 16 March 2010

https://projecteuclid.org/euclid.aaa/1268745581

Digital Object Identifier
doi:10.1155/2009/135843

Mathematical Reviews number (MathSciNet)
MR2521130

Zentralblatt MATH identifier
1181.34088

#### Citation

Huo, Hai-Feng; Ma, Zhan-Ping; Liu, Chun-Ying. Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal. Abstr. Appl. Anal. 2009 (2009), Article ID 135843, 17 pages. doi:10.1155/2009/135843. https://projecteuclid.org/euclid.aaa/1268745581

#### References

• J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.
• J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Heidelberg, Germany, 2nd edition, 1993.
• H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
• Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, River Edge, NJ, USA, 1996.
• F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2001.
• P. H. Leslie, Some further notes on the use of matrices in population mathematics,'' Biometrika, vol. 35, pp. 213--245, 1948.
• M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes,'' Applied Mathematics Letters, vol. 16, no. 7, pp. 1069--1075, 2003.
• A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,'' Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1104--1118, 2006.
• A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models,'' Applied Mathematics Letters, vol. 14, no. 6, pp. 697--699, 2001.
• H. Guo and X. Song, An impulsive predator-prey system with modified Leslie-Gower and Holling type II schemes,'' Chaos, Solitons & Fractals, vol. 36, no. 5, pp. 1320--1331, 2008.
• X. Song and Y. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect,'' Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 64--79, 2008.
• 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.
• Y. Takeuchi, Diffusion-mediated persistence in two-species competition Lotka-Volterra model,'' Mathematical Biosciences, vol. 95, no. 1, pp. 65--83, 1989.
• S. A. Levin, Dispersion and population interactions,'' The American Naturalist, vol. 108, pp. 207--228, 1974.
• S. A. Levin and L. A. Segel, Hypothesis to explain the origion of planktonic patchness,'' Nature, vol. 259, p. 659, 1976.
• R. Xu, M. A. J. Chaplain, and F. A. Davidson, Global stability of a stage-structured predator-prey model with prey dispersal,'' Applied Mathematics and Computation, vol. 171, no. 1, pp. 293--314, 2005.
• W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure,'' Mathematical Biosciences, vol. 101, no. 2, pp. 139--153, 1990.
• R. Xu, M. A. J. Chaplain, and F. A. Davidson, Persistence and global stability of a ratio-dependent predator-prey model with stage structure,'' Applied Mathematics and Computation, vol. 158, no. 3, pp. 729--744, 2004.
• R. Xu, M. A. J. Chaplain, and F. A. Davidson, Persistence and periodicity of a delayed ratio-dependent predator-prey model with stage structure and prey dispersal,'' Applied Mathematics and Computation, vol. 159, no. 3, pp. 823--846, 2004.
• R. Xu, M. A. J. Chaplain, and F. A. Davidson, Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay,'' Applied Mathematics and Computation, vol. 159, no. 3, pp. 863--880, 2004.
• W. Wang, G. Mulone, F. Salemi, and V. Salone, Permanence and stability of a stage-structured predator-prey model,'' Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 499--528, 2001.
• X. Song, L. Cai, and A. U. Neumann, Ratio-dependent predator-prey system with stage structure for prey,'' Discrete and Continuous Dynamical Systems. Series B, vol. 4, no. 3, pp. 747--758, 2004.
• X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure,'' Mathematical Biosciences, vol. 170, no. 2, pp. 173--186, 2001.
• R. Xu and L. Chen, Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment,'' Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 577--588, 2000.
• Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993.
• Y. Song, M. Han, and J. Wei, Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,'' Physica D, vol. 200, no. 3-4, pp. 185--204, 2005.