Suppose compartments A and B shown in the following figure are fi­lled with fluidsand are separated by a permeable membrane. The fi­gure is a compartmental representation of the exterior and interior of a cell. Suppose, too, that a nutrient necessary for cell growth passes through the membrane. A model for the concentrations x(t) and y(t) of the nutrient in compartments A and B, respectively, at time t is given by the linear system of differential equations

dx/dt = k/V_A (y - x)

dy/dt = k/V_B (x - y) ,

where V_A and V_B are the volumes of the compartments, and k > 0 is a permeability factor. Let x(0) = x₀ and y(0) = y₀ denote the initial concentrations of the nutrient. Solely on the basis of the equations in the system and the assumption x₀ > y₀ > 0, sketch, on the sameset of coordinate axes, possible solution curves of the system. Explain your reasoning. Discuss the behavior of the solutions over a long period of time.

Transcribed Image Text:

fluid at fluid at concentration У) concentration x(t) A membrane