# American Institute of Mathematical Sciences

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, 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] Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164 [2] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [3] Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049 [4] Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 [5] Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024 [6] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [7] Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126 [8] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [9] Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [11] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [12] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [13] Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329 [14] Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004 [15] El Haj Laamri, Michel Pierre. Stationary reaction-diffusion systems in $L^1$ revisited. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 455-464. doi: 10.3934/dcdss.2020355 [16] Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334 [17] Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019 [18] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [19] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376 [20] Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

2019 Impact Factor: 1.105

## Tools

Article outline

Figures and Tables