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Fractal bodies invisible in 2 and 3 directions
Chaos in delay differential equations with applications in population dynamics
1. | Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain |
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., (). 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. |
[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., (). 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. |
[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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