Abstract This paper is concerned with a coupled system of first order nonlinear partial differential equations. This system is, but not limited in, the extended case of the general blood–tissue exchange model (BTEX). We use the solutions of a coupled system of first order ordinary differential equations to construct a pair of ordered lower and upper solutions for the nonlinear partial differential system. By monotone iterative methods we show the existence and uniqueness of the solution of the coupled system of nonlinear partial differential equations. The asymptotic behavior of the solutions to the coupled nonlinear partial differential system can be obtained by investing the asymptotic behavior of the solution to the coupled system of first order ordinary differential equations. Finally we apply these results to the mathematical models of general blood–tissue exchange and the gas–solid inter-phase heat transfer for the fast igniting catalytic converter problems.