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Semilinear Caputo time-fractional pseudo-parabolic equations
Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model
Laboratoire de Mathématiques Appliquées du Havre, Normandie University, FR CNRS 3335, ISCN, 76600 Le Havre, France |
The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements.
References:
[1] |
R. Adams and J. Fournier, Sobolev spaces, Academic press, 2003.
![]() |
[2] |
B. Ambrosio, M. Aziz-Alaoui and V. Phan,
Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type, IMA J. Appl. Math., 84 (2019), 416-443.
doi: 10.1093/imamat/hxy064. |
[3] |
M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226. Google Scholar |
[4] |
I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,
Synchronization and graph topology, Int. J. Bifurcat. Chaos, 15 (2005), 3423-3433.
doi: 10.1142/S0218127405014143. |
[5] |
G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, Int. J. Bifurcat. Chaos, 27 (2017), 1750213.
doi: 10.1142/S0218127417502133. |
[6] |
G. Cantin and A. Thorel, Approximation of a fourth order parabolic problem by a complex network of reaction-diffusion systems, Submitted. Google Scholar |
[7] |
G. Cantin, N. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA J. Appl. Math., 2019.
doi: 10.1093/imamat/hxz022. |
[8] |
G. Cantin, N. Verdière, V. Lanza, M. Aziz-Alaoui, R. Charrier, C. Bertelle, D. Provitolo and E. Dubos-Paillard, Control of panic behavior in a non identical network coupled with a geographical model, In PhysCon 2017, University, Firenze, 2017. Google Scholar |
[9] |
C. Carrère,
Spreading speeds for a two-species competition-diffusion system, J. Differ. Equ., 264 (2018), 2133-2156.
doi: 10.1016/j.jde.2017.10.017. |
[10] |
G. Chen, X. Wang and X. Li, Fundamentals of Complex Networks: Models, Structures and Dynamics, John Wiley & Sons, 2014. Google Scholar |
[11] |
S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comput., 70(236), 2001.
doi: 10.1090/S0025-5718-00-01277-1. |
[12] |
M. Di. Francesco, K. Fellner and P. A. Markowich,
The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464 (2008), 3273-3300.
doi: 10.1098/rspa.2008.0214. |
[13] |
A. Ducrot, M. Langlais and P. Magal,
Qualitative analysis and travelling wave solutions for the si model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.
doi: 10.3934/cpaa.2012.11.97. |
[14] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 1994. |
[15] |
M. Efendiev, A. Miranville and S. Zelik,
Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. A, 460 (2004), 1107-1129.
doi: 10.1098/rspa.2003.1182. |
[16] |
M. Efendiev, E. Nakaguchi and K. Osaki,
Dimension estimate of the exponential attractor for the chemotaxis–growth system, Glasgow Math. J., 50 (2008), 483-497.
doi: 10.1017/S0017089508004357. |
[17] |
M. Golubitsky and I. Stewart,
Nonlinear dynamics of networks: the groupoid formalism, B. Am. Math. Soc., 43 (2006), 305-364.
doi: 10.1090/S0273-0979-06-01108-6. |
[18] |
M. Haase, The functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006.
doi: 10.1007/3-7643-7698-8. |
[19] |
N. M. Haddad, L. A. Brudvig, J. Clobert, K. F. Davies, A. Gonzalez, R. D. Holt, T. E. Lovejoy, J. O. Sexton, M. P. Austin and C. D. Collins, Habitat fragmentation and its lasting impact on Earth's ecosystems, Sci. Adv., 1 (2015), 9 pp. Google Scholar |
[20] |
I. Hanski, M. E. Gilpin and D. E. McCauley, Metapopulation biology, Elsevier, 1997. Google Scholar |
[21] |
S. B. Hsu, J. Jiang and F. B. Wang,
Reaction–diffusion equations of two species competing for two complementary resources with internal storage, J. Differ. Equ., 251 (2011), 918-940.
doi: 10.1016/j.jde.2011.05.003. |
[22] |
S. B. Hsu and P. Waltman,
On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[23] |
S. Iwasaki, Asymptotic convergence of solutions of keller–segel equations in network shaped domains, Nonlinear Anal., 197 (2020), 111839.
doi: 10.1016/j.na.2020.111839. |
[24] |
O. V. Kapustyan, P. O. Kasyanov and J. Valero,
Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906.
doi: 10.3934/cpaa.2014.13.1891. |
[25] |
P. E. Kloeden and J. Simsen,
Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.
doi: 10.3934/cpaa.2014.13.2543. |
[26] |
O. Ladyzhenskaya, Attractors for semi-groups and evolution equations, CUP Archive, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
A. Leung,
Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.
doi: 10.1016/0022-247X(80)90028-1. |
[28] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Springer Science & Business Media, 2012. |
[29] |
J. Mallet-Paret and G. Sell,
Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 805-866.
doi: 10.2307/1990993. |
[30] |
M. Marion,
Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.
doi: 10.1137/0520057. |
[31] |
J. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics., Springer, New York, NY, USA, 2002. |
[32] |
A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications mathématiques de l'I.H.É.S., 19 (1964), 5–68. Google Scholar |
[33] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[34] |
M. Pierre,
Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78 (2010), 417-455.
doi: 10.1007/s00032-010-0133-4. |
[35] |
B. Rink and J. Sanders,
Coupled cell networks: semigroups, Lie algebras and normal forms, T. Am. Math. Soc., 367 (2015), 3509-3548.
doi: 10.1090/S0002-9947-2014-06221-1. |
[36] |
P. Souplet,
Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth, J. Evol. Equ., 18 (2018), 1713-1720.
doi: 10.1007/s00028-018-0458-y. |
[37] |
G. Strang,
Accurate partial difference methods, Numer. Math., 6 (1964), 37-46.
doi: 10.1007/BF01386051. |
[38] |
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4684-0313-8. |
[39] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978. |
[40] |
J. Wang, H. Wu, T. Huang and S. Ren, Analysis and Control of Coupled Neural Networks with Reaction-Diffusion Terms, Springer, 2018.
doi: 10.1007/978-981-10-4907-1. |
[41] |
X. S. Wang, H. Wang and J. Wu,
Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. A, 32 (2012), 3303-3324.
doi: 10.3934/dcds.2012.32.3303. |
[42] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Science & Business Media, 2009.
doi: 10.1007/978-3-642-04631-5. |
show all references
References:
[1] |
R. Adams and J. Fournier, Sobolev spaces, Academic press, 2003.
![]() |
[2] |
B. Ambrosio, M. Aziz-Alaoui and V. Phan,
Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type, IMA J. Appl. Math., 84 (2019), 416-443.
doi: 10.1093/imamat/hxy064. |
[3] |
M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226. Google Scholar |
[4] |
I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,
Synchronization and graph topology, Int. J. Bifurcat. Chaos, 15 (2005), 3423-3433.
doi: 10.1142/S0218127405014143. |
[5] |
G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, Int. J. Bifurcat. Chaos, 27 (2017), 1750213.
doi: 10.1142/S0218127417502133. |
[6] |
G. Cantin and A. Thorel, Approximation of a fourth order parabolic problem by a complex network of reaction-diffusion systems, Submitted. Google Scholar |
[7] |
G. Cantin, N. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA J. Appl. Math., 2019.
doi: 10.1093/imamat/hxz022. |
[8] |
G. Cantin, N. Verdière, V. Lanza, M. Aziz-Alaoui, R. Charrier, C. Bertelle, D. Provitolo and E. Dubos-Paillard, Control of panic behavior in a non identical network coupled with a geographical model, In PhysCon 2017, University, Firenze, 2017. Google Scholar |
[9] |
C. Carrère,
Spreading speeds for a two-species competition-diffusion system, J. Differ. Equ., 264 (2018), 2133-2156.
doi: 10.1016/j.jde.2017.10.017. |
[10] |
G. Chen, X. Wang and X. Li, Fundamentals of Complex Networks: Models, Structures and Dynamics, John Wiley & Sons, 2014. Google Scholar |
[11] |
S. Descombes, Convergence of a splitting method of high order for reaction-diffusion systems, Math. Comput., 70(236), 2001.
doi: 10.1090/S0025-5718-00-01277-1. |
[12] |
M. Di. Francesco, K. Fellner and P. A. Markowich,
The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 464 (2008), 3273-3300.
doi: 10.1098/rspa.2008.0214. |
[13] |
A. Ducrot, M. Langlais and P. Magal,
Qualitative analysis and travelling wave solutions for the si model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.
doi: 10.3934/cpaa.2012.11.97. |
[14] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 1994. |
[15] |
M. Efendiev, A. Miranville and S. Zelik,
Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. A, 460 (2004), 1107-1129.
doi: 10.1098/rspa.2003.1182. |
[16] |
M. Efendiev, E. Nakaguchi and K. Osaki,
Dimension estimate of the exponential attractor for the chemotaxis–growth system, Glasgow Math. J., 50 (2008), 483-497.
doi: 10.1017/S0017089508004357. |
[17] |
M. Golubitsky and I. Stewart,
Nonlinear dynamics of networks: the groupoid formalism, B. Am. Math. Soc., 43 (2006), 305-364.
doi: 10.1090/S0273-0979-06-01108-6. |
[18] |
M. Haase, The functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006.
doi: 10.1007/3-7643-7698-8. |
[19] |
N. M. Haddad, L. A. Brudvig, J. Clobert, K. F. Davies, A. Gonzalez, R. D. Holt, T. E. Lovejoy, J. O. Sexton, M. P. Austin and C. D. Collins, Habitat fragmentation and its lasting impact on Earth's ecosystems, Sci. Adv., 1 (2015), 9 pp. Google Scholar |
[20] |
I. Hanski, M. E. Gilpin and D. E. McCauley, Metapopulation biology, Elsevier, 1997. Google Scholar |
[21] |
S. B. Hsu, J. Jiang and F. B. Wang,
Reaction–diffusion equations of two species competing for two complementary resources with internal storage, J. Differ. Equ., 251 (2011), 918-940.
doi: 10.1016/j.jde.2011.05.003. |
[22] |
S. B. Hsu and P. Waltman,
On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[23] |
S. Iwasaki, Asymptotic convergence of solutions of keller–segel equations in network shaped domains, Nonlinear Anal., 197 (2020), 111839.
doi: 10.1016/j.na.2020.111839. |
[24] |
O. V. Kapustyan, P. O. Kasyanov and J. Valero,
Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906.
doi: 10.3934/cpaa.2014.13.1891. |
[25] |
P. E. Kloeden and J. Simsen,
Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.
doi: 10.3934/cpaa.2014.13.2543. |
[26] |
O. Ladyzhenskaya, Attractors for semi-groups and evolution equations, CUP Archive, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
A. Leung,
Equilibria and stabilities for competing-species reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl., 73 (1980), 204-218.
doi: 10.1016/0022-247X(80)90028-1. |
[28] |
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Springer Science & Business Media, 2012. |
[29] |
J. Mallet-Paret and G. Sell,
Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 805-866.
doi: 10.2307/1990993. |
[30] |
M. Marion,
Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.
doi: 10.1137/0520057. |
[31] |
J. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics., Springer, New York, NY, USA, 2002. |
[32] |
A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications mathématiques de l'I.H.É.S., 19 (1964), 5–68. Google Scholar |
[33] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[34] |
M. Pierre,
Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78 (2010), 417-455.
doi: 10.1007/s00032-010-0133-4. |
[35] |
B. Rink and J. Sanders,
Coupled cell networks: semigroups, Lie algebras and normal forms, T. Am. Math. Soc., 367 (2015), 3509-3548.
doi: 10.1090/S0002-9947-2014-06221-1. |
[36] |
P. Souplet,
Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth, J. Evol. Equ., 18 (2018), 1713-1720.
doi: 10.1007/s00028-018-0458-y. |
[37] |
G. Strang,
Accurate partial difference methods, Numer. Math., 6 (1964), 37-46.
doi: 10.1007/BF01386051. |
[38] |
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4684-0313-8. |
[39] |
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978. |
[40] |
J. Wang, H. Wu, T. Huang and S. Ren, Analysis and Control of Coupled Neural Networks with Reaction-Diffusion Terms, Springer, 2018.
doi: 10.1007/978-981-10-4907-1. |
[41] |
X. S. Wang, H. Wang and J. Wu,
Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Syst. A, 32 (2012), 3303-3324.
doi: 10.3934/dcds.2012.32.3303. |
[42] |
A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Science & Business Media, 2009.
doi: 10.1007/978-3-642-04631-5. |






Vertex 1 | Vertex 2 | ||
Parameter | Value | Parameter | Value |
$\alpha_{1,2}$ | $1.0$ | ||
$\alpha_{2,2}$ | $1.0$ | ||
$\beta_{1,2}$ | $1.0$ | ||
$\beta_{2,2}$ | $0.1$ | ||
$\gamma_{1,2}$ | $1.0$ | ||
$\gamma_{2,2}$ | $0.1$ | ||
$\textbf{Vertex 3} $ | $\textbf{Vertex 4} $ | ||
Parameter | Value | Parameter | Value |
$\alpha_{1,3}$ | $0.5$ | $\alpha_{1,4}$ | $10.0$ |
$\alpha_{2,3}$ | $0.5$ | $\alpha_{2,4}$ | $10.0$ |
$\beta_{1,3}$ | $0.1$ | $\beta_{1,4}$ | $5.0$ |
$\beta_{2,3}$ | $0.1$ | $\beta_{2,4}$ | $5.0$ |
$\gamma_{1,3}$ | $0.5$ | $\gamma_{1,4}$ | $4.0$ |
$\gamma_{2,3}$ | $0.5$ | $\gamma_{2,4}$ | $4.0$ |
Vertex 1 | Vertex 2 | ||
Parameter | Value | Parameter | Value |
$\alpha_{1,2}$ | $1.0$ | ||
$\alpha_{2,2}$ | $1.0$ | ||
$\beta_{1,2}$ | $1.0$ | ||
$\beta_{2,2}$ | $0.1$ | ||
$\gamma_{1,2}$ | $1.0$ | ||
$\gamma_{2,2}$ | $0.1$ | ||
$\textbf{Vertex 3} $ | $\textbf{Vertex 4} $ | ||
Parameter | Value | Parameter | Value |
$\alpha_{1,3}$ | $0.5$ | $\alpha_{1,4}$ | $10.0$ |
$\alpha_{2,3}$ | $0.5$ | $\alpha_{2,4}$ | $10.0$ |
$\beta_{1,3}$ | $0.1$ | $\beta_{1,4}$ | $5.0$ |
$\beta_{2,3}$ | $0.1$ | $\beta_{2,4}$ | $5.0$ |
$\gamma_{1,3}$ | $0.5$ | $\gamma_{1,4}$ | $4.0$ |
$\gamma_{2,3}$ | $0.5$ | $\gamma_{2,4}$ | $4.0$ |
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