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 - 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.

[2]

J. M. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82. doi: 10.1137/0132006.

[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.

[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.

[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.

[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.

[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.

[8]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition with seasonal sucession,, J. Math. Biol., ().

[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.

[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.

[11]

J. Kennedy, S. Koçcak and J. A. Yorke, A chaos lemma,, Amer. Math. Monthly, 108 (2001), 411. doi: 10.2307/2695795.

[12]

J. Kennedy and J. A. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513. doi: 10.1090/S0002-9947-01-02586-7.

[13]

U. Kirchgraber and D. Stoffer, On the definition of chaos,, Z. Angew. Math. Mech., 69 (1989), 175. doi: 10.1002/zamm.19890690703.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254.

[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.

[25]

R. May, Time-Delay Versus Stability in Population Models with Two and Three Trophic,, Ecology, 54 (1973), 315. doi: 10.2307/1934339.

[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.

[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.

[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.

[29]

M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics,, Topol. Methods Nonlinear Anal., 30 (2007), 279.

[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.

[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.

[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.

[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.

[34]

H.-O. Walther, Homoclinic solution and chaos in $\dot x(t)=f(x(t-1))$,, Nonlinear Anal., 5 (1981), 775.

[35]

H.-O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations,, Mem. Amer. Math. Soc., 79 (1989).

[36]

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

[37]

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

show all references

References:
[1]

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

[2]

J. M. Cushing, Periodic time-dependent predator-prey systems,, SIAM J. Appl. Math., 32 (1977), 82. doi: 10.1137/0132006.

[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.

[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.

[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.

[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.

[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.

[8]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition with seasonal sucession,, J. Math. Biol., ().

[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.

[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.

[11]

J. Kennedy, S. Koçcak and J. A. Yorke, A chaos lemma,, Amer. Math. Monthly, 108 (2001), 411. doi: 10.2307/2695795.

[12]

J. Kennedy and J. A. Yorke, Topological horseshoes,, Trans. Amer. Math. Soc., 353 (2001), 2513. doi: 10.1090/S0002-9947-01-02586-7.

[13]

U. Kirchgraber and D. Stoffer, On the definition of chaos,, Z. Angew. Math. Mech., 69 (1989), 175. doi: 10.1002/zamm.19890690703.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[23]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254.

[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.

[25]

R. May, Time-Delay Versus Stability in Population Models with Two and Three Trophic,, Ecology, 54 (1973), 315. doi: 10.2307/1934339.

[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.

[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.

[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.

[29]

M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics,, Topol. Methods Nonlinear Anal., 30 (2007), 279.

[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.

[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.

[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.

[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.

[34]

H.-O. Walther, Homoclinic solution and chaos in $\dot x(t)=f(x(t-1))$,, Nonlinear Anal., 5 (1981), 775.

[35]

H.-O. Walther, Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations,, Mem. Amer. Math. Soc., 79 (1989).

[36]

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

[37]

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

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