February  2021, 20(2): 623-650. doi: 10.3934/cpaa.2020283

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

* Corresponding author

Received  April 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

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.

Citation: Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure and Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283
References:
[1] R. Adams and J. Fournier, Sobolev spaces, Academic press, 2003. 
[2]

B. AmbrosioM. 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. 

[4]

I. BelykhM. HaslerM. 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.

[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.

[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.

[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. FrancescoK. 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. DucrotM. 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. EfendievA. 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. EfendievE. 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.

[20]

I. Hanski, M. E. Gilpin and D. E. McCauley, Metapopulation biology, Elsevier, 1997.

[21]

S. B. HsuJ. 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. KapustyanP. 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.

[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. WangH. 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. AmbrosioM. 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. 

[4]

I. BelykhM. HaslerM. 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.

[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.

[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.

[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. FrancescoK. 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. DucrotM. 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. EfendievA. 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. EfendievE. 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.

[20]

I. Hanski, M. E. Gilpin and D. E. McCauley, Metapopulation biology, Elsevier, 1997.

[21]

S. B. HsuJ. 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. KapustyanP. 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.

[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. WangH. 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.

Figure 1.  Several graph topologies. (a) Star oriented from interior toward exterior. (b) Oriented chain. (c) Oriented cycle. (d) Oriented complete topology. (e) Star oriented from exterior toward interior. (f) Bi-directed chain. (g) Bi-directed cycle. (h) Bi-directed complete topology
Figure 2.  Three topologies for a complex network of competing species. (a) Oriented chain. (b) Star oriented from center toward periphery. (c) Bi-directed complete graph
Figure 3.  Numerical simulation of a complex network of competing species models in absence of coupling, showing the densities $ u_1 $, $ u_2 $, $ u_3 $ and $ u_4 $ for three different times (similar computations would show the densities $ v_1 $, $ v_2 $, $ v_3 $ and $ v_4 $): $ u_1 $ persists on vertex $ (1) $, whereas $ u_2 $ vanishes on vertex $ (2) $; in parallel, $ u_3 $ and $ v_3 $ coexist on vertex $ (3) $, and similarly, $ u_4 $ and $ v_4 $ coexist on vertex $ (4) $
Figure 4.  Numerical simulation of a complex network of competing species models, built on an oriented chain: the domination of $ u_1 $ on vertex $ (1) $ is attenuated; $ u_2 $ seems to persist on vertex $ (2) $, whereas $ u_2 $ vanishes in absence of coupling; $ u_3 $ dominates on vertex $ (3) $, whereas $ u_3 $ and $ v_3 $ coexist in absence of coupling; $ u_4 $ and $ v_4 $ still coexist
Figure 5.  Numerical simulation of a complex network of competing species models, built on a star oriented from center toward periphery: the asymptotic dynamics are modified; in particular, $ u_2 $ persists on vertex $ (2) $, whereas $ u_2 $ vanishes in absence of coupling
Figure 6.  Numerical simulation of a complex network of competing species models, built on a bi-directed complete graph topology. After a brief transitional phase, synchronization of the four vertices occurs rapidly, which illustrates Theorem 4.2
Table 1.  Values of the parameters for a complex network of $ 4 $ non-identical competing species models
Vertex 1 Vertex 2
Parameter Value Parameter Value
$ \alpha_{1, 1} $ $ 1.0 $ $\alpha_{1,2}$ $1.0$
$ \alpha_{2, 1} $ $ 1.0 $ $\alpha_{2,2}$ $1.0$
$ \beta_{1, 1} $ $ 0.1 $ $\beta_{1,2}$ $1.0$
$ \beta_{2, 1} $ $ 1.0 $ $\beta_{2,2}$ $0.1$
$ \gamma_{1, 1} $ $ 0.1 $ $\gamma_{1,2}$ $1.0$
$ \gamma_{2, 1} $ $ 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, 1} $ $ 1.0 $ $\alpha_{1,2}$ $1.0$
$ \alpha_{2, 1} $ $ 1.0 $ $\alpha_{2,2}$ $1.0$
$ \beta_{1, 1} $ $ 0.1 $ $\beta_{1,2}$ $1.0$
$ \beta_{2, 1} $ $ 1.0 $ $\beta_{2,2}$ $0.1$
$ \gamma_{1, 1} $ $ 0.1 $ $\gamma_{1,2}$ $1.0$
$ \gamma_{2, 1} $ $ 1.0 $ $\gamma_{2,2}$ $0.1$
$\textbf{Vertex 3} $ $\textbf{Vertex 4} $
Parameter Value Parameter Value
$\alpha_{1,3}$ $0.5$ $\alpha_{1,4}$ $10.0$
$\alpha_{2,3}$ $0.5$ $\alpha_{2,4}$ $10.0$
$\beta_{1,3}$ $0.1$ $\beta_{1,4}$ $5.0$
$\beta_{2,3}$ $0.1$ $\beta_{2,4}$ $5.0$
$\gamma_{1,3}$ $0.5$ $\gamma_{1,4}$ $4.0$
$\gamma_{2,3}$ $0.5$ $\gamma_{2,4}$ $4.0$
[1]

M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395

[2]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[3]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

[4]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[5]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[6]

Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393

[7]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[8]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407

[9]

Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843

[10]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[11]

Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103

[12]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[13]

Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003

[14]

Atsushi Yagi. Exponential attractors for competing species model with cross-diffusions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 1091-1120. doi: 10.3934/dcds.2008.22.1091

[15]

Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143

[16]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[17]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[18]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[19]

Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087

[20]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (338)
  • HTML views (97)
  • Cited by (1)

Other articles
by authors

[Back to Top]