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The discrete Rosenzweig model. (English) Zbl 0694.92014
Summary: Discrete time versions of the Rosenzweig predator-prey model [see {\it M. L. Rosenzweig}, Science 171, 385-387 (1971)] are studied by analytic and numerical methods. The interaction of the Hopf bifurcation leading to periodic orbits and the period-doubling bifurcation are investigated. It is shown that for certain choices of the parameters there is stable coexistence of both species together with a local attractor at which the prey is absent.

92D25Population dynamics (general)
39A12Discrete version of topics in analysis
39A11Stability of difference equations (MSC2000)
65C20Models (numerical methods)
Full Text: DOI
[1] Aronson, D. G.; Chory, M. A.; Hall, G. R.; Mcgehee, R. P.: Bifurcations from an invariant circle for two-parameter families of maps on the plane: A computer assisted study. Commun. math. Phys. 83, 303-354 (1982) · Zbl 0499.70034
[2] Aronson, D. G.; Chory, M. A.; Hall, G. R.; Mcgehee, R. P.: Resonance phenomena for two-parameter families of maps of the plane: uniqueness and nonuniqueness of rotation numbers. Nonlinear dynamics and turbulence, 34-47 (1983)
[3] Devaney, R. L.: An introduction to chaotic dynamical systems. (1989) · Zbl 0695.58002
[4] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. (1983) · Zbl 0515.34001
[5] Gumovski, I.; Mira, C.: Dynamique chaotique. (1980)
[6] Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. entomol. Soc. can. 45, 1-60 (1965)
[7] Kolmogorov, A. N.: Sulla teoria di Volterra Della lotta per l’esistenza. Giorn. isti. Ital. attuar 7, 74-80 (1936)
[8] Kolmogorov, A. N.; Scudo, F. M.; Zeigler, J. R.: On Volterra’s theory of the struggle for existence. The golden age of theoretical ecology 22, 78 (1923--1940)
[9] Kuang, Y.; Freedman, H. I.: Uniqueness of limit cycles in gause-type models of predator-prey systems. Math. biosci. 88, 67-84 (1988) · Zbl 0642.92016
[10] Liou, L. P.; Cheng, K. S.: On the uniqueness of a limit cycle for a predator-prey system. SIAM J. Math. anal. 19, 867-878 (1988) · Zbl 0655.34022
[11] Lu, Y. -C.: Singularity theory and an introduction to catastrophe theory. (1976) · Zbl 0354.58008
[12] May, R. M.: Stability and complexity in model ecosystems. (1973)
[13] May, R. M.; Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am. nat. 110, 573 (1976)
[14] Pounder, J. R.; Rogers, T. D.: The geometry of chaos: dynamics of a nonlinear second order difference equation. Bull. math. Biol. 42, 551-597 (1980) · Zbl 0439.39001
[15] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171, 385-387 (1971)
[16] Rotenberg, M.: Mappings of the plane that simulate prey-predator systems. J. math. Biol. 26, 169-192 (1988) · Zbl 0713.92026
[17] Ruelle, D.; Takens, F.: On the nature of turbulence. Commun. math. Phys. 23, 343-344 (1971) · Zbl 0227.76084
[18] Smale, S.; Williams, R. F.: The qualitative analysis of a difference equation of population growth. J. math. Biol. 3, 1-4 (1976) · Zbl 0342.92014
[19] Schmidt, J. W.; Hess, W.: Positivity of cubic polynomials on intervals and positive spline interpolation. Bit 28, 340-352 (1988) · Zbl 0642.41007
[20] Thompson, J. M. T.; Stewart, H. B.: Nonlinear dynamics and chaos. (1986) · Zbl 0601.58001