American Institute of Mathematical Sciences

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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

Guillaume Cantin , and  M. A. Aziz-Alaoui

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  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.

Keywords: Dynamical system, reaction-diffusion, complex network, exponential attractor, fractal dimension, synchronization, competing species.
Mathematics Subject Classification: Primary: 35A01, 35B40; Secondary: 35B41, 35K57.
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 & Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283
References:
[1] R. Adams and J. Fournier, Sobolev spaces, Academic press, 2003.   Google Scholar
[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.  Google Scholar

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M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. Haase, The functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

O. Ladyzhenskaya, Attractors for semi-groups and evolution equations, CUP Archive, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Springer Science & Business Media, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics., Springer, New York, NY, USA, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

G. Strang, Accurate partial difference methods, Numer. Math., 6 (1964), 37-46.  doi: 10.1007/BF01386051.  Google Scholar

[38]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[39]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Science & Business Media, 2009. doi: 10.1007/978-3-642-04631-5.  Google Scholar

show all references

References:
[1] R. Adams and J. Fournier, Sobolev spaces, Academic press, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Haase, The functional Calculus for Sectorial Operators, Birkhäuser Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

O. Ladyzhenskaya, Attractors for semi-groups and evolution equations, CUP Archive, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Springer Science & Business Media, 2012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Murray, Mathematical Biology I: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics., Springer, New York, NY, USA, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

G. Strang, Accurate partial difference methods, Numer. Math., 6 (1964), 37-46.  doi: 10.1007/BF01386051.  Google Scholar

[38]

R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[39]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Science & Business Media, 2009. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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
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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
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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) $
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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
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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
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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
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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
$\alpha_{1,3}$ $0.5$ $\alpha_{1,4}$ $10.0$
$\alpha_{2,3}$ $0.5$ $\alpha_{2,4}$ $10.0$
$\beta_{1,3}$ $0.1$ $\beta_{1,4}$ $5.0$
$\beta_{2,3}$ $0.1$ $\beta_{2,4}$ $5.0$
$\gamma_{1,3}$ $0.5$ $\gamma_{1,4}$ $4.0$
$\gamma_{2,3}$ $0.5$ $\gamma_{2,4}$ $4.0$
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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
$\alpha_{1,3}$ $0.5$ $\alpha_{1,4}$ $10.0$
$\alpha_{2,3}$ $0.5$ $\alpha_{2,4}$ $10.0$
$\beta_{1,3}$ $0.1$ $\beta_{1,4}$ $5.0$
$\beta_{2,3}$ $0.1$ $\beta_{2,4}$ $5.0$
$\gamma_{1,3}$ $0.5$ $\gamma_{1,4}$ $4.0$
$\gamma_{2,3}$ $0.5$ $\gamma_{2,4}$ $4.0$
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