×

zbMATH — the first resource for mathematics

On a Leslie-Gower predator-prey model incorporating a prey refuge. (English) Zbl 1167.92032
The authors propose a Leslie-Gower predator-prey model incorporating a prey refuge, which is an extension of the model first introduced by Leslie where the carrying capacity of the predator’s environment is proportional to the number of prey. By constructing a Korobeinikov-type Lyapunov function, it is shown that for this ecosystem prey refuge has no influence on the persistence property of the system as the unique positive equilibrium of the system is globally stable. Also, increasing the amount of refuge can increase prey density. As far as the predator species is concerned, under certain conditions, increasing the amount of prey refuge can decrease the predator density. There exists a threshold such that for the prey refuge smaller than this threshold, increasing the amount of prey refuge can increase the predator density, and if the prey refuge is larger than the threshold, increasing the amount of prey refuge can decrease the predator density.

MSC:
92D40 Ecology
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Leslie, P.H., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303
[2] Leslie, P.H., A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45, 16-31, (1958) · Zbl 0089.15803
[3] Pielou, E.C., Mathematical ecology, (1977), John Wiley & Sons New York · Zbl 0259.92001
[4] Korobeinikov, A., A Lyapunov function for leslie – gower predator – prey models, Applied mathematics letters, 14, 697-699, (2001) · Zbl 0999.92036
[5] Kumar Kar, T., Stability analysis of a prey-predator model incorporating a prey refuge, Communications in nonlinear science and numeric simulation, 10, 681-691, (2005) · Zbl 1064.92045
[6] Srinivasu, P.D.N.; Gayatri, I.L., Influence of prey reserve capacity on predator – prey dynamics, Ecological modelling, 181, 191-202, (2005)
[7] P. Aguirre, E. González-Olivares, E. Sáez, Two limit cycles in a Leslie-Gower predator – prey model with additive Allee effect, Nonlinear Analysis: Real World Applications (in press)
[8] Ko, W.; Ryu, K., Qualitative analysis of a predator – prey model with Holling type II functional response incorporating a prey refuge, Journal of differential equations, 231, 534-550, (2006) · Zbl 1387.35588
[9] Huang, Y.; Chen, F.; Li, Z., Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Applied mathematics and computation, 182, 672-683, (2006) · Zbl 1102.92056
[10] Kumar Kar, T., Modelling and analysis of a harvested prey-predator system incorporating a prey refuge, Journal of computational and applied mathematics, 185, 19-33, (2006) · Zbl 1071.92041
[11] González-Olivares, E.; Ramos-Jiliberto, R., Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological modelling, 166, 135-146, (2003)
[12] Hoy, M.A., (), 229-310
[13] Aziz-Alaoui, M.A.; Okiye, M.Daher, Boundedness and global stability for a predator – prey model with modified leslie – gower and Holling-type II schemes, Applied mathematics letters, 16, 7, 1069-1075, (2003) · Zbl 1063.34044
[14] Nindjin, A.F.; Aziz-Alaoui, M.A.; Cadivel, M., Analysis of a predator – prey model with modified leslie – gower and Holling-type II schemes with time delay, Nonlinear analysis: real world applications, 7, 5, 1104-1118, (2007) · Zbl 1104.92065
[15] Chen, F.D.; You, M.S., Permanence, extinction and periodic solution of the predator – prey system with beddington-deangelis functional response and stage structure for prey, Nonlinear analysis: real world applications, 9, 2, 207-221, (2008) · Zbl 1142.34051
[16] Chen, F.D.; Shi, J.L., On a delayed nonautonomous ratio-dependent predator – prey model with Holling type functional response and diffusion, Applied mathematics and computation, 192, 2, 358-369, (2007) · Zbl 1193.34140
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.