Variations on Lyapunov's stability criterion and periodic prey-predator systems
The author recovers a stability criterion applied to the linear equation \(\ddot{x} + \alpha(t) x =0\) where \(\alpha(t)\) is \(T\)-periodic and proved by Lyapunov. The author obtains related stability criteria for several equations. First for the same equation, then for the dissipative Hill's equation \[ \ddot{x} + c \dot{x} + \alpha(t) x =0, \] where \(c>0\) is a constant. Then, he considers the linear prey-predator system \[ \dot{x}_1 = -a_{11}(t) x_1 -a_{12}(t) x_2, \ \dot{x}_2 = a_{21}(t) x_1 -a_{22}(t) x_2, \] where all the coefficients \(a_{ij}(t)\) are \(T\)-periodic and positive. The author looks for applications of these results to Lotka--Volterra prey-predator systems of the type \[ \begin{array}{lll} \dot{u} & = & \displaystyle u\left( a(t)- b(t) u - c(t) v\right), \vspace{0.2cm} \\ \dot{v} & = & \displaystyle v \left(d(t)+e(t)u-f(t)v \right), \end{array} \] where \(u>0\), \(v>0\), all the coefficients are \(T\)-periodic and \(b(t)\), \(c(t)\), \(e(t)\) and \(f(t)\) are positive.
scientific article; zbMATH DE number 7438963