# American Institute of Mathematical Sciences

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 & Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283
##### References:

show all references

##### References:
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
Three topologies for a complex network of competing species. (a) Oriented chain. (b) Star oriented from center toward periphery. (c) Bi-directed complete graph
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 $(4)$
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
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
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
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 & 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 & 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 & Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [4] Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393 [5] Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989 [6] Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841 [7] Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure & 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 & 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 & 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 & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 [11] Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 [12] Atsushi Yagi. Exponential attractors for competing species model with cross-diffusions. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1091-1120. doi: 10.3934/dcds.2008.22.1091 [13] Linfang Liu, Xianlong Fu, Yuncheng You. Pullback attractor in $H^{1}$ for nonautonomous stochastic reaction-diffusion equations on $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3629-3651. doi: 10.3934/dcdsb.2017143 [14] Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343 [15] 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 & Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155 [16] 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 & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 [17] Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087 [18] Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169 [19] Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 [20] Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227

2020 Impact Factor: 1.916