The authors consider the following system of nonlinear partial differential equations \[ \partial u_ i/ \partial t = \Delta_{\alpha_ i} u_ i + V_ i \sum^ k_{j = 1} (m_{ij} - \delta_{ij}) u_ j - V_ i \sum^ k_{j = 1} c_{ij} u_ j^{1 + \beta_{ij}}, \quad u_ i (x,0) = f_ i(x), \quad i = 1, \dots, k, \] where \(f = (f_ 1, \dots, f_ k)\), \(f_ i \in C_ c (\mathbb{R}^ d)_ +\) (the nonnegative continuous functions on \(\mathbb{R}^ d\) with compact support), \(\Delta_{\alpha_ i} \equiv - ( - \Delta)^{\alpha_ i}/2\), \(\alpha_ i \in (0,2]\), \((\Delta = \) Laplacian on \(\mathbb{R}^ d)\), \(V_ i>0\), \(\beta_{ij} \in (0,1]\), \(c_{ij} \in [0,m_{ij}/(1+\beta_{ij})]\) for each \(i,j\), and \((m_{ij})\) is an irreducible nonnegative matrix with maximal eigenvalue 1. The main result can be formulated as follows:
If at least one \(f_ i\) is not 0, then for each \(i=1,\dots,k\), \[ \lim_{t \to \infty} \bigl \| u_ i (f;t) \bigr \|_{L^ 1}>0 \quad \text{ if } \quad d>d_ c, \qquad \text{and} \quad = 0 \quad \text{ if } \quad d \leq d_ c, \] where \(d_ c = \min \alpha_ i/ \min \beta_{ij}\).