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Architecture of attractor determines dynamics on mutualistic complex networks. (English) Zbl 1370.34109

The paper presents a full mathematical study of the system \[ \begin{aligned} \frac{dS_{p_i}}{dt}&=S_{p_i}\left(\alpha_{p_i}-\sum\limits_{j=1}^P\beta_{p_{ij}}S_{p_j}+ \sum\limits_{k=1}^A\gamma_{p_{ik}}S_{a_k}\right),\\ \frac{dS_{a_i}}{dt}&=S_{a_i}\left(\alpha_{a_i}-\sum\limits_{j=1}^A\beta_{a_{ij}}S_{a_j}+ \sum\limits_{k=1}^P\gamma_{a_{ik}}S_{p_k}\right),\\ S_{p_i}(0)&=S_{p_{i0}},\;\;\;S_{a_i}(0)=S_{a_{i0}}. \end{aligned} \] The system describes a relationship of plants and animals. Here, \(S_{p_i}\) and \(S_{a_i}\) are the species density populations for the \(i\)-th species of plant and of animal respectively; numbers \(\alpha_{p_i}\) and \(\alpha_{a_i}\) represent the intrinsic growth rates in the absence of competition and cooperation; \(\beta_{p_{ij}}\) and \(\beta_{a_{ij}}\) denote the competitive interactions and \(\gamma_{p_{ij}}\) and \(\gamma_{a_{ij}}\) denote the mutualistic strenghts.
In fact, the authors study the following essentially simplified model (M): \[ \begin{aligned} \frac{du_i}{dt}&=u_{i}\left(\alpha_{p_i}-u_i-\sum\limits_{j\neq i}^P\beta u_j+ \sum\limits_{k=1}^A\gamma_1u_k\right), \;i=1,\dots, P,\\ \frac{dv_i}{dt}&=v_{i}\left(\alpha_{a_i}-v_i-\sum\limits_{j\neq i}^A\beta v_j+ \sum\limits_{k=1}^P\gamma_2u_k\right), \;i=1,\dots, A,\\ u_i(0)&=u_{i0},\;\;\;, \;i=1,\dots, P,\\ v_i(0)&=v_{i0},\;\;\;, \;i=1,\dots, A, \end{aligned} \] where \(u_i\) and \(v_i\) represent plants and animals, respectively, \(\alpha_{p_i},\alpha_{a_i}\in \mathbb{R}\), \(\beta\geq0\) and \(\gamma_1, \gamma_2\geq0\).
Let \(n = P + A\) be the total number of species, \(w=(u, v) = (u_1,\dots, u_P, v_1,\dots,v_A)\) and so \(w_0=(u_{10},\dots,u_{P0},v_{10},\dots,v_{A0})\). The natural phase space of the system (M) is the invariant positive cone \(\mathbb{R}^n_+=\{w\in \mathbb{R}^n|w_i\geq0, i =1,\dots,n\}\). The authors obtain sufficient conditions for the continuation of positive solutions on \((0,+\infty)\): \[ \beta<1, \;\;\gamma_1\gamma_2<G=\frac{(1+\beta(P-1))(1+\beta(A-1))}{PA}, \] and show that any positive solution blows up in finite time if \(\alpha_{p_i}=\alpha_{a_i}=\alpha>0\) for all \(i,j\) and \(\gamma_1\gamma_2>G\).
The authors also show that system (M) has a global attractor if \(\gamma_1\gamma_2<G\). The geometrical structure of the global attractor for system (M) is described as well.
As application a 3D-model from [G. Guerrero et al., Discrete Contin. Dyn. Syst. 34, No. 10, 4107–4126 (2014; Zbl 1327.92041)] is discussed.

MSC:

34D45 Attractors of solutions to ordinary differential equations
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations

Citations:

Zbl 1327.92041
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References:

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