# zbMATH — the first resource for mathematics

Complex dynamic behaviors of a discrete-time predator-prey system. (English) Zbl 1130.92056
Summary: The dynamics of a discrete-time predator-prey system is investigated in the closed first quadrant $$R^2_+$$. It is shown that the system undergoes flip bifurcation and Hopf bifurcation in the interior of $$R^2_+$$ by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-5, 6, 9, 10, 14, 18, 20, 25 orbits, cascades of period-doubling bifurcations in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.

##### MSC:
 92D40 Ecology 39A11 Stability of difference equations (MSC2000) 37N25 Dynamical systems in biology 39A12 Discrete version of topics in analysis
Full Text:
##### References:
 [1] Verhulst, P.F., Notice sur la loi que la population suit dans son accroissement, Corr math phys, 10, 113-121, (1838) [2] Pearl, R.; Reed, L.J., On the rate of growth of the population of the united states Since 1790 and its mathematical representation, Proc natl acad sci, 6, 275-288, (1920) [3] Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology, (2001), Springer-Verlag New York · Zbl 0967.92015 [4] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), New York Marcel Dekker · Zbl 0448.92023 [5] Murray, J.D., Mathematical biology, (1993), Springer-Verlag Berlin · Zbl 0779.92001 [6] Li, T.Y.; Yorke, J.A., Period three implies chaos, Amer math monthly, 82, 985-992, (1975) · Zbl 0351.92021 [7] Strogatz, S., Nonlinear dynamics and chaos, (1994), Addison-Wesley Reading, Mass [8] Saha, P.; Strogatz, S., The birth of period 3, Math mag, 68, 42-47, (1995) · Zbl 0856.58012 [9] May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974) [10] May, R.M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088 [11] Summers, D.; Cranford, J.G.; Healey, B.P., Chaos in periodically forced discrete-time ecosystem models, Chaos, solitons & fractals, 11, 2331-2342, (2000) · Zbl 0964.92044 [12] Gao, Y.H., Chaos and bifurcation in the space-clamped Fitzhugh-Nagumo system, Chaos, solitons & fractals, 21, 943-956, (2004) · Zbl 1045.37058 [13] Jing, Z.J.; Jia, Z.; Wang, R., Chaos behavior in the discrete BVP oscillator, Int J bifurcat & chaos, 12, 619-627, (2002) · Zbl 1069.65141 [14] Jing, Z.J.; Chang, Y.; Guo, B., Bifurcation and chaos in discrete Fitzhugh-Nagumo system, Chaos, solitons & fractals, 21, 701-720, (2004) · Zbl 1048.37526 [15] Rosenzweig, M.L.; MacArthur, R.H., Graphical representation and stability conditions of predator-prey interactions, Amer naturalist, 97, 209-223, (1963) [16] Beddington, J.R.; Free, C.A.; Lawton, J.H., Dynamic complexity in predator-prey models framed in difference equations, Nature, 255, 58-60, (1975) [17] Lopez-Ruiz, R.; Fournier-Prunaret, R., Indirect allee effect, bistability and chaotic oscillations in a predator-prey discrete model of logistic type, Chaos, solitons & fractals, 24, 85-101, (2005) · Zbl 1066.92053 [18] Xiao, Y.N.; Cheng, D.Z.; Tang, S.Y., Dynamic complexities in predator-prey ecosystem models with age-structure for predator, Chaos, solitons & fractals, 14, 1403-1411, (2002) · Zbl 1032.92033 [19] Fan, M.; Wang, K., Periodic solutions of a discrete time non-autonomous ratio-dependent predator-prey system, Math comput model, 35, 951-961, (2002) · Zbl 1050.39022 [20] Wang, W.D.; Lu, Z.Y., Global stability of discrete models of Lotka-Volterra type, Nonlinear anal, 35, 1019-1030, (1999) · Zbl 0919.92030 [21] Huo, H.F.; Li, W.T., Stable periodic solution of the discrete periodic Leslie-gower predator-prey model, Math comput model, 40, 261-269, (2004) · Zbl 1067.39008 [22] Robinson, C., Dynamical systems, stability, symbolic dynamics and chaos, (1999), Boca Raton London, New York, Washington (DC) · Zbl 0914.58021 [23] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical system and bifurcation of vector fields, (1983), Springer-Verlag New York, p.160-5
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.