Architecture of attractor determines dynamics on mutualistic complex networks (Q332351)

The paper presents a full mathematical study of the system NEWLINE\[NEWLINE \begin{aligned} \frac{dS_{p_i}}{dt}&=S_{p_i}\left(\alpha_{p_i}-\sum\limits_{j=1}^P\beta_{p_{ij}}S_{p_j}+ \sum\limits_{k=1}^A\gamma_{p_{ik}}S_{a_k}\right),\\ \frac{dS_{a_i}}{dt}&=S_{a_i}\left(\alpha_{a_i}-\sum\limits_{j=1}^A\beta_{a_{ij}}S_{a_j}+ \sum\limits_{k=1}^P\gamma_{a_{ik}}S_{p_k}\right),\\ S_{p_i}(0)&=S_{p_{i0}},\;\;\;S_{a_i}(0)=S_{a_{i0}}. \end{aligned} NEWLINE\]NEWLINE 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.NEWLINENEWLINEIn fact, the authors study the following essentially simplified model (M): NEWLINE\[NEWLINE \begin{aligned} \frac{du_i}{dt}&=u_{i}\left(\alpha_{p_i}-u_i-\sum\limits_{j\neq i}^P\beta u_j+ \sum\limits_{k=1}^A\gamma_1u_k\right), \;i=1,\dots, P,\\ \frac{dv_i}{dt}&=v_{i}\left(\alpha_{a_i}-v_i-\sum\limits_{j\neq i}^A\beta v_j+ \sum\limits_{k=1}^P\gamma_2u_k\right), \;i=1,\dots, A,\\ u_i(0)&=u_{i0},\;\;\;, \;i=1,\dots, P,\\ v_i(0)&=v_{i0},\;\;\;, \;i=1,\dots, A, \end{aligned} NEWLINE\]NEWLINE where \(u_i\) and \(v_i\) represent plants and animals, respectively, \(\alpha_{p_i},\alpha_{a_i}\in \mathbb{R}\), \(\beta\geq0\) and \(\gamma_1, \gamma_2\geq0\).NEWLINENEWLINELet \(n = P + A\) 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 \((0,+\infty)\): NEWLINE\[NEWLINE \beta<1, \;\;\gamma_1\gamma_2<G=\frac{(1+\beta(P-1))(1+\beta(A-1))}{PA}, NEWLINE\]NEWLINE 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 \(\gamma_1\gamma_2>G\).NEWLINENEWLINEThe 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.NEWLINENEWLINEAs application a 3D-model from [\textit{G. Guerrero} et al., Discrete Contin. Dyn. Syst. 34, No. 10, 4107--4126 (2014; Zbl 1327.92041)] is discussed.