Chaos in predator-prey systems with/without impulsive effect. (English) Zbl 1238.34086
Summary: We prove analytically that the seasonal effect can cause chaos in predator-prey systems. Our method of proof is based on some recent results on topological horseshoes. Some applications in systems with impulsive effect are given.
MSC:
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
| 37N25 | Dynamical systems in biology |
| 92D40 | Ecology |
| 34A37 | Ordinary differential equations with impulses |
Keywords:
stretching along paths; predator; prey systems; complex dynamics; topological horseshoes; impulsive effectReferences:
| [1] | Medio, A.; Pireddu, M.; Zanolin, F., Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19, 3283-3309 (2009) · Zbl 1182.37027 |
| [2] | Kirchgraber, U.; Stoffer, D., On the definition of chaos, Z. Angew. Math. Mech., 69, 175-185 (1989) · Zbl 0713.58035 |
| [3] | Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021 |
| [4] | Awrejcewicz, J., Bifurcation and Chaos in Coupled Oscillators (1991), World Scientific: World Scientific Singapore · Zbl 0824.58034 |
| [5] | Awrejcewicz, J.; Lamarque, C.-H., Bifurcation and Chaos in Nonsmooth Mechanical Systems (2003), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 1067.70001 |
| [6] | Guckenheimer, J.; Holmes, P., (Nonlinear Oscillations, Dynamical Ssystems, and Bifurcations of Vector Fields. Nonlinear Oscillations, Dynamical Ssystems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42 (1990), Springer-Verlag: Springer-Verlag New York), Revised and corrected reprint of the 1983 original |
| [7] | Robinson, C., (Dynamical Systems. Stability, Symbolic Dynamics, and Chaos. Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics (1999), CRC Press: CRC Press Boca Raton, FL) · Zbl 0914.58021 |
| [8] | Srzednicki, R.; Wójcik, K.; Zgliczyński, P., Fixed point results based on the Ważewski method, (Handbook of Topological Fixed Point Theory (2005), Springer: Springer Dordrecht), 905-943 · Zbl 1079.37012 |
| [9] | de Mottoni, P.; Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. Math. Biol., 11, 319-335 (1981) · Zbl 0474.92015 |
| [10] | Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman Scientific and Technical: Longman Scientific and Technical New York · Zbl 0815.34001 |
| [11] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |
| [12] | Pei, Y.; Liu, S.; Li, C., Complex dynamics of an impulsive control system in which predator species share a common prey, J. Nonlinear Sci., 19, 249-266 (2009) · Zbl 1172.92039 |
| [13] | Ahmad, S.; Stamova, I., Asymptotic stability of an \(N\)-dimensional impulsive competitive system, Nonlinear Anal. RWA, 8, 654-663 (2007) · Zbl 1152.34342 |
| [14] | Qian, L.; Lu, Q.; Meng, Q.; Feng, Z., Dynamical behaviors of a prey-predator system with impulsive control, J. Math. Anal. Appl., 363, 345-356 (2010) · Zbl 1187.34060 |
| [15] | Wanga, W.; Shenb, J.; Luoc, Z., Partial survival and extinction in two competing species with impulses, Nonlinear Anal. RWA, 10, 1243-1254 (2009) · Zbl 1162.34308 |
| [16] | He, M.; Li, Z.; Chen, F., Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses, Nonlinear Anal. RWA, 11, 1537-1551 (2010) · Zbl 1255.34056 |
| [17] | Wang, L.; Chen, L.; Nieto, J. J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Anal. RWA, 11, 1374-1386 (2010) · Zbl 1188.93038 |
| [18] | Kennedy, J.; Yorke, J. A., Topological horseshoes, Trans. Amer. Math. Soc., 353, 2513-2530 (2001) · Zbl 0972.37011 |
| [19] | Zanini, C.; Zanolin, F., Complex dynamics in a nerve fiber model with periodic coefficients, Nonlinear Anal. RWA, 10, 1381-1400 (2009) · Zbl 1165.34318 |
| [20] | Burra, L.; Zanolin, F., Chaotic dynamics in a vertically driven planar pendulum, Nonlinear Anal., 72, 1462-1476 (2010) · Zbl 1191.34057 |
| [21] | Pascoletti, A.; Zanolin, F., Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach, J. Math. Anal. Appl., 339, 1179-1198 (2008) · Zbl 1139.34037 |
| [22] | Papini, D.; Zanolin, F., Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells, Fixed Point Theory Appl., 2, 113-134 (2004) · Zbl 1086.37012 |
| [23] | Pireddu, M.; Zanolin, F., Chaotic dynamics in the Volterra predator-prey model via linked twist maps, Opuscula Math., 28, 567-592 (2008) · Zbl 1167.34341 |
| [24] | Waldvogel, J., The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl., 114, 178-184 (1986) · Zbl 0588.92018 |
| [25] | Amine, Z.; Ortega, R., A periodic prey-predator system, J. Math. Anal. Appl., 185, 477-489 (1994) · Zbl 0808.34043 |
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.
