American Institute of Mathematical Sciences

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

Chaos in delay differential equations with applications in population dynamics

Alfonso Ruiz-Herrera 1,

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.
Keywords: stretching along paths, Topological horseshoes, global continuation of horseshoes, delay, predator-prey systems..
Mathematics Subject Classification: 92D40, 37D50, 34K23, 34K6.
Citation: Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

J. M. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.2307/2695795.  Google Scholar

[12]

J. Kennedy and J. A. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513.  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.  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.  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.  doi: 10.1016/0022-5193(74)90168-4.  Google Scholar

[16]

Y. Kuang, "Delay-differential Equations with Applications in Population Dynamics,", Academic, (1993).   Google Scholar

[17]

Y. Kuang, Global stability in delay differential systems without dominanting instantaneous negative feedbacks,, J. Differential Equations, 119 (1995), 503.  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.  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.  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.   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.  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.  doi: 10.1137/0136023.  Google Scholar

[23]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.  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.  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.  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.  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.  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.   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.   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.   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.  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.  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.  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.   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).   Google Scholar

[36]

K. Wójcik and P. Zgliczyński, Topological horseshoes and delay differential equations,, Discrete Contin. Dyn. Syst., 12 (2005), 827.   Google Scholar

[37]

P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32.   Google Scholar

show all references

References:
[1]

B. Aulbach and B. Kieninger, On three definitions of chaos,, Nonlinear Dyn. Syst. Theory, 1 (2001), 23.   Google Scholar

[2]

J. M. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.2307/2695795.  Google Scholar

[12]

J. Kennedy and J. A. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513.  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.  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.  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.  doi: 10.1016/0022-5193(74)90168-4.  Google Scholar

[16]

Y. Kuang, "Delay-differential Equations with Applications in Population Dynamics,", Academic, (1993).   Google Scholar

[17]

Y. Kuang, Global stability in delay differential systems without dominanting instantaneous negative feedbacks,, J. Differential Equations, 119 (1995), 503.  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.  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.  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.   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.  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.  doi: 10.1137/0136023.  Google Scholar

[23]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.  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.  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.  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.  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.  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.   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.   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.   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.  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.  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.  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.   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).   Google Scholar

[36]

K. Wójcik and P. Zgliczyński, Topological horseshoes and delay differential equations,, Discrete Contin. Dyn. Syst., 12 (2005), 827.   Google Scholar

[37]

P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32.   Google Scholar

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