On a property of a generalized Kolmogorov population model
Consider the Kolmogorov system \[ {dx_k\over dt}= x_k g_k(x),\quad 1\leq k\leq N,\tag{\(*\)} \] where \(x^T= (x_1,\dotsc, x_N)\). System \((*)\) is called competitive if \(\partial g_k/\partial x_j\geq 0\) for \(j\neq k\), it is called competitive in case \(\partial g_k/\partial x_j\leq 0\) for \(j\neq k\). In the latter case, system \((*)\) has no non-trivial periodic solution. In the paper under consideration, the authors assume that the positive orthant (cone) \(\mathbb{R}^+_N\) is positively invariant under the flow of \((*)\) and that the condition \[ \sum^N_{j=1} {\partial g_j(x)\over\partial x_j} x_j= -\alpha(x)> 0\quad\text{for }x\in \mathbb{R}^+_N \] is valid, where \(\alpha\) is continuous. Thus, \((*)\) is neither cooperative nor competitive. It is shown that under these assumptions any compact invariant set \(A\) has measure zero and that there is no compact orbitally stable set in \(\mathbb{R}^+\). These results imply that in case \(N= 2\) system \((*)\) has no limit cycle and no homoclinic orbit, in the general case system \((*)\) has no nontrivial closed orbit which is orbitally stable.
scientific article; zbMATH DE number 6146670