×

Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator-prey model with linear harvesting rate: analysis and computation. (English) Zbl 1185.34047

Math. Comput. Modelling 50, No. 3-4, 360-379 (2009); corrigendum ibid. 51, No. 5-6, 852-853 (2010).
Summary: We propose a ratio-dependent predator-prey model with linear harvesting rate, which can describe the real predator-prey system better than the previously published model with constant harvesting in [D. Xiao, L.S. Jennings, SIAM J. Appl. Math. 65, No. 3, 737–753 (2005; Zbl 1094.34024)]. We show the different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of “conditional coexistence” depending on the initial values. By choosing the modulus of linear harvesting rate as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator-prey models, is that the origin is a non-hyperbolic equilibrium, whose characteristics crucially determine the main properties of the model. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. Experimental motivation is made to summarize the effect of harvesting on coexistence and extinction.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology
92D25 Population dynamics (general)

Citations:

Zbl 1094.34024
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139, 311-326 (1989)
[2] Huffaker, C. B., Experimental studies on predation: Dispersion factors and predator-prey oscillations, Hilgardia, 27, 343-383 (1958)
[3] Luckinbill, L. S., Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum, Ecology, 54, 1320-1327 (1973)
[4] Luck, R. F., Evaluation of natural enemies for biological control: A behavior approach, Trends Ecol. Evol., 5, 196-199 (1990)
[5] Arditi, R.; Berryman, A. A., The biological control paradox, Trends Ecol. Evol., 6, 32 (1991)
[6] Hassell, M. P.; Varley, C. C., New inductive population model for insect parasites and its bearing on biological control, Nature, 223, 1133-1137 (1969)
[7] DeAngelis, D. L.; Goldstein, R. A.; ONeill, R. V., A model for trophic interactions, Ecology, 56, 881-892 (1975)
[8] Beddington, J. R., Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44, 331-340 (1975)
[9] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R., Variation in plankton densities among lakes: A case for ratio-dependent predation models, Am. Nat., 138, 1287-1296 (1991)
[10] Hanski, I., The functional response of predators: Worries about scale, Trends Ecol. Evol., 6, 141-142 (1991)
[11] Michalski, J.; Arditi, R., Food web structure at equilibrium and far from it: Is it the same?, Proc. Roy. Soc. London B, 259, 217-222 (1995)
[12] Arditi, R.; Michalski, J., Nonlinear food web models and their responses to increased basal productivity, (Polis, G. A.; Winemiller, K. O., FoodWebs: Integration of Patterns and Dynamics (1996), Chapman and Hall: Chapman and Hall New York), 122-133
[13] Berezovskaya, F.; Karev, G.; Arditi, R., Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol., 43, 221-246 (2001) · Zbl 0995.92043
[14] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten type ratiodependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[15] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[16] Kuang, Y., Rich dynamics of Gause-type ratio-dependent predator-prey system, (Ruan, S.; Wolkowicz, G. S.K.; Wu, J., Differential Equations with Applications to Biology. Differential Equations with Applications to Biology, Fields Inst. Commun., vol. 21 (1999), AMS: AMS Providence, RI), 325-337 · Zbl 0920.92032
[17] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[18] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 268-290 (2001) · Zbl 1007.34031
[19] Xiao, D.; Jennings, L. S., Bifurcation of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65, 3, 737-753 (2005) · Zbl 1094.34024
[20] Tang, Y.; Zhang, W., Heteroclinic bifurcation in a ratio-dependent predator-prey system, J. Math. Biol., 50, 699-712 (2005) · Zbl 1067.92050
[21] Li, B.; Kuang, Y., Heteroclinic bifurcation in the Michaelis-Menten type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67, 1453-1464 (2007) · Zbl 1132.34320
[22] Ruan, S.; Tang, Y.; Zhang, W., Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response, J. Math. Biol., 57, 223-241 (2008) · Zbl 1160.34048
[23] Brauer, F.; Soudack, A. C., Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol., 8, 55-71 (1979) · Zbl 0406.92020
[24] Brauer, F.; Soudack, A. C., Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12, 101-114 (1981) · Zbl 0482.92015
[25] Clark, C. W., Mathematical Bioeconomics, The Optimal Management of Renewable Resources (1990), Wiley: Wiley New York · Zbl 0712.90018
[26] Yodzis, P., Predator-prey theory and management of multispecies fisheries, Ecol. Appl., 4, 51-58 (2004)
[27] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[28] Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G., Qualitative Theory of Second-order Dynamic Systems (1973), John Wiley & Sons, Translation · Zbl 0282.34022
[29] Berezovskaya, F. S.; Medvedeva, N. B., A complicated singular point of “center-focus” type and the Newton diagram, Sel. Math. Formely Sov., 13, 1-15 (1994) · Zbl 0792.58032
[30] Berezovskaya, F. S.; Novozhilov, A. S.; Karev, G. P., Topological normal form for a system of two differential equations, Math. Biosci., 208, 270-299 (2007)
[31] Dumortier, F., Singularities of vector fields in the plane, J. Differential Equations, 23, 53 (1977) · Zbl 0346.58002
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.