zbMATH — the first resource for mathematics

Qualitative analysis of a ratio-dependent Holling-Tanner model. (English) Zbl 1124.34030
The authors consider a so-called ratio-dependent Holling-Tanner predator-prey model, where both components contain coefficients that depend on the ratio of predator and prey densities. By rescaling, the original six positive parameters are reduced to three, which are restricted in such a way that there exists a unique equilibrium \(E\) in the open positive quadrant. Generically, \(E\) is an attractor or repeller. In the first case, an extra condition on the parameters allows the construction of a Lyapunov function showing that \(E\) is globally attractive. If \(E\) is a repeller, the Poincaré-Bendixson theorem yields the existence of a limit cycle. Its uniqueness is, moreover, proved by transforming the given system into a Liénard system and using known results on the uniqueness of limit cycles for such systems.

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
92D25 Population dynamics (general)
Full Text: DOI
[1] Korobeinikov, A., A Lyapunov function for leslie – gower predator – prey model, Appl. math. lett., 14, 697-699, (2001) · Zbl 0999.92036
[2] 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
[3] Saez, E.; Gonzalez-Olivares, E., Dynamics of predator – prey model, SIAM J. appl. math., 59, 1867-1878, (1999) · Zbl 0934.92027
[4] Gasull, A.; Kooij, R.E.; Torregrosa, J., Limit cycles in the holling – tanner model, Publ. mat., 41, 149-167, (1997) · Zbl 0880.34028
[5] Aziz-Alaoui, M.A.; Daher-Okiye, M., Boundedness and global stability for a predator – prey model with modified leslie – gower and Holling-type II schemes, Appl. math. lett., 16, 1069-1075, (2003) · Zbl 1063.34044
[6] Leslie, P.H., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303
[7] Leslie, P.H.; Gower, J.C., The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrika, 47, 219-234, (1960) · Zbl 0103.12502
[8] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley-Interscience New York · Zbl 0259.92001
[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] Hassell, M.P.; May, R.M., Stability in insect host-parasite models, J. anim. ecol., 42, 693-726, (1973)
[11] Arditi, R.; Saiah, H., Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73, 1544-1551, (1992)
[12] Arditi, R.; Ginzburg, L.R.; Akcakaya, H.R., Variation in plankton densities among lakes: A case for ratio-dependent models, Amer. natural, 138, 1287-1296, (1991)
[13] Gutierrez, A.P., The physiological basis of ratio-dependent predator – prey theory: A metabolic pool model of Nicholson’s blowflies as an example, Ecology, 73, 1552-1563, (1992)
[14] Arditi, R.; Ginzburg, L.R., Coupling in predator – prey dynamics: ratio-dependence, J. theor. biol., 139, 311-326, (1989)
[15] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, Oikos, 60, 69-75, (1991)
[16] Tang, S.Y.; Chen, L.S., Global qualitative analysis for a ratio-dependent predator – prey model with delay, J. math. anal. appl., 266, 401-419, (2002) · Zbl 1069.34122
[17] Hsu, S.B.; Hwang, T.W.; Kuang, Y., Global analysis of the michaelis – menten type ratio-dependence predator – prey system, J. math. biol., 42, 489-506, (2001) · Zbl 0984.92035
[18] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator – prey systems, Nonlinear anal., 32, 381-408, (1998) · Zbl 0946.34061
[19] Rui, X.; Chen, L.S., Persistence and global stability for n-species ratio-dependent predator – prey system with time delays, J. math. anal. appl., 275, 27-43, (2002) · Zbl 1039.34069
[20] Hale, J., Ordinary differential equation, (1980), Krieger Publ. Co. Malabar
[21] Coppel, W.A., A new class of quadratic system, J. differential equations, 92, 360-372, (1991) · Zbl 0733.58037
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.