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