On the number of positive periodic solutions for planar competing Lotka-Volterra systems
The author considers the competing Lotka-Volterra system: \[ u''=u[a(t)-b(t)u-c(t)v],\qquad v''=v[d(t)-e(t)u-f(t)v], \tag{\(*\)} \] where \(a,b,c,d,e,f\) are continuous \(T\)-periodic functions for some \(T > 0\); \(b,c,e,f\) are positive and \(a,d\) have positive averages. Let \(S\) be the set of all nonnegative and nontrivial \(T\)-periodic solutions to \((*)\). It is known that \(S\) contains the semitrivial solutions \((U,0)\) and \((0,V)\), where \(U\) (resp. \(V)\) is the uniquely positive solution of the logistic equation: \(x''=x[a(t)-b(t)x]\) (resp. \(x''=x[d(t)-f(t)x])\). The author gives an accurate description of the set \(S\), which is stated as the following theorem: Let \(S(0)=\{(u(0),v(0)):(u,v)\in S\}\). Then, either \( S\) is finite or \(S(0)\) is the graph of the function \(\phi\) by \textit{P. de Mottoni} and \textit{A. Schiaffino} [J. Math. Biol. 11, 319-335 (1991; Zbl 0474.92015)].
scientific article; zbMATH DE number 811215