A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations
Volterra introduced systems of the form \[ \dot x_j = \varepsilon_j x_j + \sum_{k=1}^n a_{jk} x_j x_k, j = 1, \dots, n, \] in the context of the interaction of \(n\) biological species. Attention is confined to the restriction to \({\mathbb R}_+^n\). Such a system is ``stably dissipative'' if for it and any nearby system there exists a diagonal matrix \(D > 0\) such that \(A D \leqslant 0\), where \(A\) is the \(n \times n\) interaction matrix \(A = (a_{jk})\). This survey is organized into two parts. In the first one, the author discusses such systems and proves that if there is a singularity then the system has a global attractor on which the dynamics are Hamiltonian. In the second part systems with delays, also introduced by Volterra, are discussed. Examples are included that illustrate the complexity of the dynamics possible in such systems.
scientific article; zbMATH DE number 6381941