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

Consider the system of nonlinear ordinary differential equations x '=c(t)(Y(t)-y)x-k1(t)x,y '=a(t)(Y(t)-y)x-k2(t)y\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}$$\end{document} where the coefficients c(t), a(t), Y(t) k(1)(t), and k(2)(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) - k(1)(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, +infinity); (ii) the existence of solutions that are positive and bounded on the entire real line (-infinity, +infinity); (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.