The stability of harmonic solutions of Lotka-Volterra systems
Consider the periodic Lotka-Volterra system (1) \(\dot p = h_1 (t,E) p - h_3 (t,E) pq\), \(\dot q = - h_2 (t,E) q + h_4 (t,E) pq\), where \(E\) is a parameter in \(J \subset \mathbb{R}\), and the coefficient functions \(h_i (t,E) > 0(i = 1, \dots, 4)\) are smooth in \((t,E)\), and \(2 \pi\)-periodic in \(t \in \mathbb{R}\). In recent years, existence theorems for \(2 \pi\)-periodic solutions of (1) are considered in literature under various conditions for the above coefficient functions. In 1993, the reviewer and F. Zanolin proved the existence of at least one \(T\)-periodic solution of (1) with continuous positive coefficient \(T\)-periodic functions, as well as at least \(2N_k\) subharmonic solutions of any order \(k \geq k_0\) (for some integer \(k_0 > 1)\) with \(N_k \to \infty\) as \(k \to \infty\). The purpose of this paper is to study the stability of the \(2 \pi\)-periodic solution of (1), denoted by \(z(t,E) = (p(t,E), q(t,E))\). Its main result is stated as follows: There is a subset \(J_1\) of \(J\), such that for each \(E \in J_1\), the harmonic solution \(z(t,E)\) is stable in the Lyapunov sense. Moreover, there is at least one subharmonic solution of (1) with minimal period \(2k \pi\) for every sufficiently large \(k\) and \(E \in J_1\). Furthermore, \(\text{mes} [J \backslash J_1] = 0\).
scientific article; zbMATH DE number 859231