Asymptotic behaviour of a logistic lattice system

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October 2014, 34(10): 4019-4037. doi: 10.3934/dcds.2014.34.4019

Asymptotic behaviour of a logistic lattice system

Tomás Caraballo ^1, , Francisco Morillas ^2, and José Valero ^3,

1.	Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla
2.	Department d'Economia Aplicada, Facultat d'Economia, Universitat de València, Campus del Tarongers s/n, 46022-València
3.	Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received December 2012 Revised February 2013 Published April 2014

Abstract
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In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of infinite ordinary differential equations which can be obtained after the spatial discretization of a logistic equation with diffusion. We prove that a global attractor exists in suitable weighted spaces of sequences.

Keywords: logistic equation, Lattice dynamical systems, population models., global attractor, ordinary differential equations.

Mathematics Subject Classification: 35B40, 35B41, 35K55, 34G20, 37L30, 37L6.

Citation: Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019

References:

[1]	V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos, 4 (1994), 631-637. doi: 10.1142/S0218127494000459.
[2]	J. M. Amigó, A. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modelling suspensions, Internat. J. Bifur. Chaos, 20 (2010), 2681-2700. doi: 10.1142/S0218127410027295.
[3]	P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.
[4]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics & Dynamics, 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
[5]	P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.
[6]	J. Bell, Some threshhold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.
[7]	J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl.Math., 42 (1984), 1-14.
[8]	W. J. Beyn and S. Yu. Pilyugin, Attractors of Reaction Diffusion Systems on Infinite Lattices, J. Dynam. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.
[9]	T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.
[10]	T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Equat. App., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.
[11]	T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.
[12]	S.-N. Chow and J. Mallet-Paret, Pattern formulation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.
[13]	S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations., 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.
[14]	S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996), 109-178.
[15]	S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781. doi: 10.1137/S0036139994261757.
[16]	L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.
[17]	L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.
[18]	L. O. Chua and L. Yang, Cellular neural neetworks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.
[19]	R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation, Internat. J. Bifur. Chaos, 8 (1988), 211-257. doi: 10.1142/S0218127498000152.
[20]	T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244. doi: 10.1016/0167-2789(93)90208-I.
[21]	M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166.
[22]	X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.
[23]	X. Han, W. Shen and Sh. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.
[24]	R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.
[25]	J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.
[26]	J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. doi: 10.1016/S0022-5193(05)80465-5.
[27]	O. A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian), Zap. Nauchn. Sem. LOMI, 182 (1990), 102-112; English translation in J. Soviet Math., 62 (1992), 1789-1794. doi: 10.1007/BF01671002.
[28]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[29]	J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934. doi: 10.1021/j100191a038.
[30]	J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127. doi: 10.1023/A:1021841618074.
[31]	F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578. doi: 10.1142/S0218127409023196.
[32]	F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems, J. Diff. Equat. App., 18 (2012), 675-692. doi: 10.1080/10236198.2011.574621.
[33]	A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-d array of nonlinear circuits, IEEE Trans. Circuits Syst., 40 (1993), 872-877.
[34]	N. Rashevsky, Mathematical Biophysics, 3rd revised edition, Dover Publications, Inc., New York, 1960.
[35]	A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254. doi: 10.1007/BF02479046.
[36]	W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399. doi: 10.1137/S0036139995282670.
[37]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
[38]	B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.
[39]	B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.
[40]	E. Zeidler, Nonlinear Functional Analysis and Its Applciations, Springer, New-York, 1986. doi: 10.1007/978-1-4612-4838-5.
[41]	B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.
[42]	S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2.
[43]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.
[44]	S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

show all references

References:

[1]	V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Internat. J. Bifur. Chaos, 4 (1994), 631-637. doi: 10.1142/S0218127494000459.
[2]	J. M. Amigó, A. Giménez, F. Morillas and J. Valero, Attractors for a lattice dynamical system generated by non-newtonian fluids modelling suspensions, Internat. J. Bifur. Chaos, 20 (2010), 2681-2700. doi: 10.1142/S0218127410027295.
[3]	P. W. Bates and A. Chmaj, On a discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.
[4]	P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stochastics & Dynamics, 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
[5]	P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.
[6]	J. Bell, Some threshhold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190. doi: 10.1016/0025-5564(81)90085-7.
[7]	J. Bell and C. Cosner, Threshold behaviour and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly Appl.Math., 42 (1984), 1-14.
[8]	W. J. Beyn and S. Yu. Pilyugin, Attractors of Reaction Diffusion Systems on Infinite Lattices, J. Dynam. Differential Equations, 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.
[9]	T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.
[10]	T. Caraballo, F. Morillas and J. Valero, Random Attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Equat. App., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.
[11]	T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.
[12]	S.-N. Chow and J. Mallet-Paret, Pattern formulation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.
[13]	S.-N. Chow, J. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations., 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.
[14]	S.-N. Chow, J. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Computational Dynamics, 4 (1996), 109-178.
[15]	S.-N. Chow and W. Shen, Dynamics in a discrete Nagumo equation: Spatial topological chaos, SIAM J. Appl. Math., 55 (1995), 1764-1781. doi: 10.1137/S0036139994261757.
[16]	L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.
[17]	L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.
[18]	L. O. Chua and L. Yang, Cellular neural neetworks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273-1290. doi: 10.1109/31.7601.
[19]	R. Dogaru and L. O. Chua, Edge of chaos and local activity domain of Fitz-Hugh-Nagumo equation, Internat. J. Bifur. Chaos, 8 (1988), 211-257. doi: 10.1142/S0218127498000152.
[20]	T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244. doi: 10.1016/0167-2789(93)90208-I.
[21]	M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166.
[22]	X. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493. doi: 10.1016/j.jmaa.2010.11.032.
[23]	X. Han, W. Shen and Sh. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266. doi: 10.1016/j.jde.2010.10.018.
[24]	R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.
[25]	J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.
[26]	J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. doi: 10.1016/S0022-5193(05)80465-5.
[27]	O. A. Ladyzhenskaya, Some comments to my papers on the theory of attractors for abstract semigroups (in russian), Zap. Nauchn. Sem. LOMI, 182 (1990), 102-112; English translation in J. Soviet Math., 62 (1992), 1789-1794. doi: 10.1007/BF01671002.
[28]	O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[29]	J. P. Laplante and T. Erneux, Propagating failure in arrays of coupled bistable chemical reactors, J. Phys. Chem., 96 (1992), 4931-4934. doi: 10.1021/j100191a038.
[30]	J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations, 11 (1999), 49-127. doi: 10.1023/A:1021841618074.
[31]	F. Morillas and J. Valero, A Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578. doi: 10.1142/S0218127409023196.
[32]	F. Morillas and J. Valero, On the connectedness of the attainability set for lattice dynamical systems, J. Diff. Equat. App., 18 (2012), 675-692. doi: 10.1080/10236198.2011.574621.
[33]	A. Pérez-Muñuzuri, V. Pérez-Muñuzuri, V. Pérez-Villar and L. O. Chua, Spiral waves on a 2-d array of nonlinear circuits, IEEE Trans. Circuits Syst., 40 (1993), 872-877.
[34]	N. Rashevsky, Mathematical Biophysics, 3rd revised edition, Dover Publications, Inc., New York, 1960.
[35]	A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254. doi: 10.1007/BF02479046.
[36]	W. Shen, Lifted lattices, hyperbolic structures, and topological disorders in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399. doi: 10.1137/S0036139995282670.
[37]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
[38]	B. Wang, Dynamics of systems of infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.
[39]	B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.
[40]	E. Zeidler, Nonlinear Functional Analysis and Its Applciations, Springer, New-York, 1986. doi: 10.1007/978-1-4612-4838-5.
[41]	B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.
[42]	S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61. doi: 10.1016/S0167-2789(02)00807-2.
[43]	S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.
[44]	S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.

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