# American Institute of Mathematical Sciences

April  2013, 33(4): 1633-1644. doi: 10.3934/dcds.2013.33.1633

## Chaos in delay differential equations with applications in population dynamics

 1 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Received  October 2011 Revised  January 2012 Published  October 2012

We develop a geometrical method to detect the presence of chaotic dynamics in delay differential equations. An application to the classical Lotka-Volterra model with delay is given.
Citation: Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
##### References:
 [1] B. Aulbach and B. Kieninger, On three definitions of chaos, Nonlinear Dyn. Syst. Theory, 1 (2001), 23-37.  Google Scholar [2] J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95. doi: 10.1137/0132006.  Google Scholar [3] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.  Google Scholar [4] J. Grasman and E. Veling, An asymptotic formula for the period of a Volterra-Lotka system, Mathematical Biosciences, 18 (1973), 185-189. doi: 10.1016/0025-5564(73)90029-1.  Google Scholar [5] J. K. Hale and S. M. Tanaka, Square and pulse waves with two delays, J. Dynam. Differential Equations, 12 (2000), 1-30. doi: 10.1023/A:1009052718531.  Google Scholar [6] U. an der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, J. Differential Equations, 47 (1983), 273-295. doi: 10.1016/0022-0396(83)90037-2.  Google Scholar [7] S.-B. Hsu, A remark on the period of the periodic solution in the Lotka-Volterra system, J. Math. Anal. Appl.,95 (1983), 428-436. doi: 10.1016/0022-247X(83)90117-8.  Google Scholar [8] S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition with seasonal sucession,, J. Math. Biol., ().   Google Scholar [9] A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol.,236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar [10] M. Keeling, P. Rohani and B. T. Grenfell, Seasonally forced disease dynamics explored as switching between attractors, Physica D, 148 (2001), 317-335. doi: 10.1016/S0167-2789(00)00187-1.  Google Scholar [11] J. Kennedy, S. Koçcak and J. A. Yorke, A chaos lemma, Amer. Math. Monthly, 108 (2001), 411-423. doi: 10.2307/2695795.  Google Scholar [12] J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7.  Google Scholar [13] U. Kirchgraber and D. Stoffer , On the definition of chaos, Z. Angew. Math. Mech., 69 (1989), 175-185. doi: 10.1002/zamm.19890690703.  Google Scholar [14] C. A. Klausmeier , Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics, J. Theor. Biol., 262 (2010), 584-595. doi: 10.1016/j.jtbi.2009.10.018.  Google Scholar [15] A. L. Koch , Coexistence resulting from an alternation of density dependent and density independent growth, J. Theor. Biol., 44 (1974), 373-386. doi: 10.1016/0022-5193(74)90168-4.  Google Scholar [16] Y. Kuang, "Delay-differential Equations with Applications in Population Dynamics," Academic, Boston, MA, 1993.  Google Scholar [17] Y. Kuang, Global stability in delay differential systems without dominanting instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532. doi: 10.1006/jdeq.1995.1100.  Google Scholar [18] B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999), 901-945. doi: 10.1090/S0002-9947-99-02351-X.  Google Scholar [19] B. Lani-Wayda and R. Srzednicki, A generalized Lefschetz fixed point theorem and symbolic dynamics in delay equations, Ergodic Theory Dynam. Systems, 22 (2002), 1215-1232. doi: 10.1017/S0143385702000639.  Google Scholar [20] B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback. I: A transversality criterion, Differential Integral Equations, 8 (1995), 1407-1452.  Google Scholar [21] B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback. II: Construction of nonlinearities, Math. Nachr., 180 (1996), 141-211. doi: 10.1002/mana.3211800109.  Google Scholar [22] A. Leung, Conditions for global stability concerning a prey-predator model with delay effect, SIAM J. Appl. Math., 36 (1979), 3602-3608. doi: 10.1137/0136023.  Google Scholar [23] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.  Google Scholar [24] T. Malik and H. L. Smith, Does dormancy increase fitness of bacterial populations in time-varying environments?, Bull. Math. Biol., 70 (2008), 1140-1162. doi: 10.1007/s11538-008-9294-5.  Google Scholar [25] R. May, Time-Delay Versus Stability in Population Models with Two and Three Trophic, Ecology, 54 (1973), 315-325. doi: 10.2307/1934339.  Google Scholar [26] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions: A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3283-3309. doi: 10.1142/S0218127409024761.  Google Scholar [27] T. Namba and S. Takahashi, Competitive coexistence in a seasonally fluctuating environment II: Multiple stable states and invasion succession, Theor. Popul. Biol., 44 (1995), 374-402. doi: 10.1006/tpbi.1993.1033.  Google Scholar [28] D. Papini and F. Zanolin, Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equations, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 115-157.  Google Scholar [29] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics, Topol. Methods Nonlinear Anal., 30 (2007), 279-319.  Google Scholar [30] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on $N$-dimensional cells, Adv. Nonlinear Stud., 5 (2005), 411-440.  Google Scholar [31] C. E. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food chain dynamics: model predictions and experimental validation, Ecology, 90 (2009), 3099-3107. doi: 10.1890/08-2377.1.  Google Scholar [32] X. H. Tang and X. Zou, Global attractivity in a predator prey system with pure delay, Proc. Edinb. Math. Soc., 51 (2008), 495-508. doi: 10.1017/S0013091506000988.  Google Scholar [33] J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl., 114 (1986), 178-184. doi: 10.1016/0022-247X(86)90076-4.  Google Scholar [34] H.-O. Walther, Homoclinic solution and chaos in $\dot x(t)=f(x(t-1))$, Nonlinear Anal., 5 (1981), 775-788.  Google Scholar [35] H.-O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations, Mem. Amer. Math. Soc., 79 (1989) iv+104 pp.  Google Scholar [36] K. Wójcik and P. Zgliczyński, Topological horseshoes and delay differential equations, Discrete Contin. Dyn. Syst., 12 (2005), 827-852.  Google Scholar [37] P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.  Google Scholar

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##### References:
 [1] B. Aulbach and B. Kieninger, On three definitions of chaos, Nonlinear Dyn. Syst. Theory, 1 (2001), 23-37.  Google Scholar [2] J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32 (1977), 82-95. doi: 10.1137/0132006.  Google Scholar [3] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.  Google Scholar [4] J. Grasman and E. Veling, An asymptotic formula for the period of a Volterra-Lotka system, Mathematical Biosciences, 18 (1973), 185-189. doi: 10.1016/0025-5564(73)90029-1.  Google Scholar [5] J. K. Hale and S. M. Tanaka, Square and pulse waves with two delays, J. Dynam. Differential Equations, 12 (2000), 1-30. doi: 10.1023/A:1009052718531.  Google Scholar [6] U. an der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, J. Differential Equations, 47 (1983), 273-295. doi: 10.1016/0022-0396(83)90037-2.  Google Scholar [7] S.-B. Hsu, A remark on the period of the periodic solution in the Lotka-Volterra system, J. Math. Anal. Appl.,95 (1983), 428-436. doi: 10.1016/0022-247X(83)90117-8.  Google Scholar [8] S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition with seasonal sucession,, J. Math. Biol., ().   Google Scholar [9] A. Huppert, B. Blasius, R. Olinky and L. Stone, A model for seasonal phytoplankton blooms, J. Theoret. Biol.,236 (2005), 276-290. doi: 10.1016/j.jtbi.2005.03.012.  Google Scholar [10] M. Keeling, P. Rohani and B. T. Grenfell, Seasonally forced disease dynamics explored as switching between attractors, Physica D, 148 (2001), 317-335. doi: 10.1016/S0167-2789(00)00187-1.  Google Scholar [11] J. Kennedy, S. Koçcak and J. A. Yorke, A chaos lemma, Amer. Math. Monthly, 108 (2001), 411-423. doi: 10.2307/2695795.  Google Scholar [12] J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513-2530. doi: 10.1090/S0002-9947-01-02586-7.  Google Scholar [13] U. Kirchgraber and D. Stoffer , On the definition of chaos, Z. Angew. Math. Mech., 69 (1989), 175-185. doi: 10.1002/zamm.19890690703.  Google Scholar [14] C. A. Klausmeier , Successional state dynamics: a novel approach to modeling nonequilibrium foodweb dynamics, J. Theor. Biol., 262 (2010), 584-595. doi: 10.1016/j.jtbi.2009.10.018.  Google Scholar [15] A. L. Koch , Coexistence resulting from an alternation of density dependent and density independent growth, J. Theor. Biol., 44 (1974), 373-386. doi: 10.1016/0022-5193(74)90168-4.  Google Scholar [16] Y. Kuang, "Delay-differential Equations with Applications in Population Dynamics," Academic, Boston, MA, 1993.  Google Scholar [17] Y. Kuang, Global stability in delay differential systems without dominanting instantaneous negative feedbacks, J. Differential Equations, 119 (1995), 503-532. doi: 10.1006/jdeq.1995.1100.  Google Scholar [18] B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999), 901-945. doi: 10.1090/S0002-9947-99-02351-X.  Google Scholar [19] B. Lani-Wayda and R. Srzednicki, A generalized Lefschetz fixed point theorem and symbolic dynamics in delay equations, Ergodic Theory Dynam. Systems, 22 (2002), 1215-1232. doi: 10.1017/S0143385702000639.  Google Scholar [20] B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback. I: A transversality criterion, Differential Integral Equations, 8 (1995), 1407-1452.  Google Scholar [21] B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback. II: Construction of nonlinearities, Math. Nachr., 180 (1996), 141-211. doi: 10.1002/mana.3211800109.  Google Scholar [22] A. Leung, Conditions for global stability concerning a prey-predator model with delay effect, SIAM J. Appl. Math., 36 (1979), 3602-3608. doi: 10.1137/0136023.  Google Scholar [23] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254.  Google Scholar [24] T. Malik and H. L. Smith, Does dormancy increase fitness of bacterial populations in time-varying environments?, Bull. Math. Biol., 70 (2008), 1140-1162. doi: 10.1007/s11538-008-9294-5.  Google Scholar [25] R. May, Time-Delay Versus Stability in Population Models with Two and Three Trophic, Ecology, 54 (1973), 315-325. doi: 10.2307/1934339.  Google Scholar [26] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions: A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 3283-3309. doi: 10.1142/S0218127409024761.  Google Scholar [27] T. Namba and S. Takahashi, Competitive coexistence in a seasonally fluctuating environment II: Multiple stable states and invasion succession, Theor. Popul. Biol., 44 (1995), 374-402. doi: 10.1006/tpbi.1993.1033.  Google Scholar [28] D. Papini and F. Zanolin, Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equations, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 115-157.  Google Scholar [29] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics, Topol. Methods Nonlinear Anal., 30 (2007), 279-319.  Google Scholar [30] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on $N$-dimensional cells, Adv. Nonlinear Stud., 5 (2005), 411-440.  Google Scholar [31] C. E. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food chain dynamics: model predictions and experimental validation, Ecology, 90 (2009), 3099-3107. doi: 10.1890/08-2377.1.  Google Scholar [32] X. H. Tang and X. Zou, Global attractivity in a predator prey system with pure delay, Proc. Edinb. Math. Soc., 51 (2008), 495-508. doi: 10.1017/S0013091506000988.  Google Scholar [33] J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl., 114 (1986), 178-184. doi: 10.1016/0022-247X(86)90076-4.  Google Scholar [34] H.-O. Walther, Homoclinic solution and chaos in $\dot x(t)=f(x(t-1))$, Nonlinear Anal., 5 (1981), 775-788.  Google Scholar [35] H.-O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations, Mem. Amer. Math. Soc., 79 (1989) iv+104 pp.  Google Scholar [36] K. Wójcik and P. Zgliczyński, Topological horseshoes and delay differential equations, Discrete Contin. Dyn. Syst., 12 (2005), 827-852.  Google Scholar [37] P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.  Google Scholar
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