Dissipativity and Boundedness of Positive Solutions of a Class of Systems of Nonlinear Differential Equations

Abstract

Consider the system of nonlinear ordinary differential equations

\(\begin{array}{*{20}{c}}
{x' = c(t)(Y(t) - y)x - {k_1}(t)x,}&{y' = a(t)(Y(t) - y)x - {k_2}(t)y,}
\end{array}\)

where the coefficients c(t), a(t), Y(t) k₁(t), and k₂(t) are continuous, positive, bounded, and bounded away from zero on the entire real line, the coefficient Y(t) is uniformly continuous, and the function Y(t) - k₁(t)/c(t) is bounded below by a positive constant. For this system, we prove (i) the dissipativity of positive solutions on the positive half-line (0, +∞); (ii) the existence of solutions that are positive and bounded on the entire real line (-∞, +∞); (iii) the existence of positive periodic or recurrent solutions under the condition that the coefficients of the equations are periodic or jointly recurrent, respectively. This system is a dynamic model of production and sales of goods under time-varying conditions.