Concentration of a Nutrient Suppose compartments A and B shown in Figure 3.3.10 are...

Concentration of a Nutrient Suppose compartments A and B shown in Figure 3.3.10 are filled with fluids and

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

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 same set of coordinate axes, possible solution curves of the system. Explain your reasoning. Discuss the behavior of the solutions over a long period of time.