A series of one-predator one-prey models are studied using two parameter Hopf bifurcation techniques which allow the determination of two periodic orbits. The biological implications of the results, in terms of domains of attraction and multiple stable states, are discussed.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Andronov, A. A., Leontovich, E. A., Gordon, I. I., Maier, A. G.: Theory of bifurcations — Dynamic systems on a plane. New York: Halsted Press 1973
Arnol'd, V.I.: Lectures on bifurcations in versal families. Russian Math. Surveys 27, 54–123 (1972)
Beddington, J. R., Free, C. A., Lawton, J. H.: Concepts of stability and resilience in predator prey models. J. Anim. Ecol. 45, 791–816 (1976)
Fitzhugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological engineering (H. P. Schwann, ed.), pp. 1–85. New York: McGraw Hill 1968
Goh, B. S.: Global stability in many-species systems. Amer. Nat. 111, 135–143 (1977)
Hale, J.: Generic bifurcations with applications. In: Nonlinear analysis and mechanics: HeriotWatt symposium, Vol. 1 (R. J. Knops, ed.), pp. 59–157. San Francisco: Pitman 1977
Hastings, A.: Global stability of two species systems, J. Math. Biol. 5, 399–403 (1978)
Hastings, A.: Spatial heterogeneity and the stability of predator-prey systems: Predator-mediated coexistence. Theor. Pop. Biol. 143, 380–395 (1978)
Hastings, A.: Spatial heterogeneity and the stability of predator-prey systems: Population cycles. In: Applied nonlinear analysis (V. Lakshmikantham, ed.), New York: Academic Press 1979
Hastings, A.: Population dynamics in patchy environments. In: Proc. Conf. Ord. Diff. Eq. and Models in Biology (T. A. Burton, ed.) 1979
Holling, C. S.: The components of predation as revealed by a study of small-mammal predation of the European Pine sawfly. Can. Entom. 91, 293–320 (1959)
Holling, C. S.: Resilience and stability of ecosystems. Ann. Rev. Ecol. Syst. 4, 1–24 (1974)
Hsu, I. D., Kazarinoff, N.: An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J. Math. Anal. Appl. 60, 47–57 (1977)
Hsu, S. B.: The application of the Poincaré-transform to the Lotka-Volterra model. J. Math. Biol. 6, 67–73 (1978)
Huffaker, C. B.: Experimental studies on predation: Dispersion factors and predator-prey osculations. Hilgardia 27, 343–383 (1958)
Kazarinoff, N., van den Driessche, P.: A model predator-prey system with functional response. Math. Biosci. 39, 125–134 (1978)
Levin, S.: Population dynamics in heterogeneous environments. Ann. Rev. Ecol. Syst. 7, 287–310 (1976)
Luckinbill, L. S.: The effect of space and enrichment on a predator-prey system. Ecol. 55, 1142–1147 (1974)
MacDonald, N.: Time delay in prey-predator models. Math. Biosci. 28, 321–330 (1976)
MacDonald, N.: Time delay in prey-predator models. II. Bifurcation theory. Math. Biosci. 33, 227–234 (1977)
Marsden, J. E., McCracken, M.: The Hopf bifurcation and its applications. New York: Springer-Verlag 1976
Negrini, P., Salvadori, L.: Attractivity and Hopf bifurcation. Nonlin. Anal. 3, 87–100 (1979)
Takens, F.: Unfolding of certain singularities of vector fields: Generalized Hopf bifurcations. J. Diff. Eq. 14, 476–493 (1973)
Troy, W.: Bifurcation phenomena in Fitzhugh's nerve condition equations. J. Math. Anal. Appl. 54, 678–690 (1976)
Zeigler, B. P.: Persistence and patchiness of predator-prey systems induced by discrete event population exchange mechanisms. J. Theor. Biol. 67, 687–713 (1977)
About this article
Cite this article
Hastings, A. Multiple limit cycles in predator-prey models. J. Math. Biology 11, 51–63 (1981). https://doi.org/10.1007/BF00275824
- Predator-prey models
- Periodic orbits
- Hopf bifurcation