Nonexistence of periodic solutions in delayed Lotka--Volterra systems.
The authors consider Lotka-Volterra systems with distributed delays \[ {dx_i\over dt}= x_i\Biggl(e_i+ \sum^n_{j=1} a_{i_j} x_j+ \sum^n_{j=1} \gamma_{ij} \int^t_{-\infty} f_{ij}(t- \tau) x_j(\tau)\,d\tau\Biggr),i=1,2,\dots, n, \tag{*} \] where the delay kernels \(f_{ij}\) are such continuous nonnegative functions that \((*)\) can be transformed in an extended system of \(n+p\) autonomous ordinary differential equations. Exploiting the fact that if the extended system has no nonconstant periodic solution, then also system \((*)\) has no nonconstant periodic solution, the authors formulate conditions that the extended autonomous system has no nonconstant periodic solution. Hereby, extended negative criteria of Bendixson by \textit{M. Y. Li} and \textit{J. S. Muldowney} [J. Differ. Equations 106, No.~1, 27--39 (1993; Zbl 0786.34033)] play a crucial role. The case \(n= 2\), \(p\leq 2\) is investigated in detail.
scientific article; zbMATH DE number 1915792
scientific article