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Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model

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  • 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.

    Mathematics Subject Classification: Primary: 35A01, 35B40; Secondary: 35B41, 35K57.


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