Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model. (English) Zbl 1177.34103

Authors’ abstract: A delayed Holling-Tanner predator-prey model with ratio-dependent functional response is considered. It is proved that the model system is permanent under certain conditions. The local asymptotic stability and the Hopf-bifurcation results are discussed. Qualitative behaviour of the singularity \((0,0)\) is explored by using a blow up transformation. Global asymptotic stability analysis of the positive equilibrium is carried out. Numerical simulations are presented for the support of our analytical findings.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
34K25 Asymptotic theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Kot, M., Elements of Mathematical Ecology (2001), Cambridge University Press: Cambridge University Press Cambridge
[2] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0682.92001
[3] Bazykin, A. D., Nonlinear Dynamics of Interacting Populations (1998), World Scientific: World Scientific Singapore · Zbl 0605.92015
[4] May, R. M., Stability and Complexity in Ecosystems (2001), Princeton University Press: Princeton University Press Princeton
[5] Thieme, H. R., Mathematics in Population Biology (2003), Princeton University Press: Princeton University Press Princeton · Zbl 1054.92042
[6] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[7] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[8] Rosenzweig, M. L.; McArthur, R. H., Graphical representation and stability conditions of predator-prey interactions, Amer. Natur., 47, 209-223 (1963)
[9] Holling, C. S., The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 46, 1-60 (1965)
[10] Tanner, J. T., The stability and intrinsic growth rates of prey and predator populations, Ecology, 56, 313-347 (1975)
[11] Hsu, S. B.; Hwang, T. W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 763-783 (1995) · Zbl 0832.34035
[12] Gasull, A.; Kooij, R. E.; Torregrosa, J., Limit cycles in the Holling-Tanner model, Publ. Mat., 41, 149-167 (1997) · Zbl 0880.34028
[13] Saez, E.; Gonzalez-Olivares, E., Dynamics of predator-prey model, SIAM J. Appl. Math., 59, 1867-1878 (1999) · Zbl 0934.92027
[14] Aziz-Alaoui, M. A.; Daher-Okiye, M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Hollingtype-II schemes, Appl. Math. Lett., 16, 1069-1075 (2003) · Zbl 1063.34044
[15] Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551 (1992)
[16] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R., Variation in Plankton densities among lakes: A case for ratio-dependent models, Amer. Natur., 138, 1287-1296 (1991)
[17] Gutierrez, A. P., The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholoson’s blowfies as an example, Ecology, 73, 1552-1563 (1992)
[18] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics, ratio-dependence, J. Theoret. Biol., 139, 311-326 (1989)
[19] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: An experiment test with cladocerans, OIKOS, 60, 69-75 (1991)
[20] Rosenzweig, M. L., Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171, 385-387 (1971)
[21] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten type ratio-dependent predator-prey systems, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[22] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey model, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[23] Kuang, Y., Rich dynamics of Gause-type ratio-dependent predator-prey systems, Fields Inst. Commun., 21, 325-337 (1999) · Zbl 0920.92032
[24] Kuang, Y.; Beretta, E., Global qualitative analysis of ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[25] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal., 32, 3, 381-408 (1998) · Zbl 0946.34061
[26] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 268-290 (2001) · Zbl 1007.34031
[27] Liang, Z.; Pan, H., Qualitative analysis of a ratio-dependent Holling-Tanner model, J. Math. Anal. Appl., 334, 954-964 (2007) · Zbl 1124.34030
[28] Zhang, Z.; Ding, T.; Huang, W.; Dong, Z., Qualitative Theory of Differential Equations, Transl. Math. Monogr., vol. 101 (1991), Amer. Math. Soc.: Amer. Math. Soc. Providence
[29] Perko, L., Differential Equations and Dynamical Systems, Texts Appl. Math., vol. 7 (1996), Springer-Verlag · Zbl 0854.34001
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.