H emodialysis is a process by which a machine is used to filter urea and other waste products from a patient’s blood if their kidneys fail. The concentration of a patient’s urea during dialysis is sometimes modeled by supposing there are two compartments within the patient—the blood, which is directly filtered by the dialysis machine, and another compartment that cannot be directly filtered but that is connected to the blood. A system of two differential equations describing this is

where c and p are the urea concentrations in the blood (in mgymL) and in the inaccessible pool, respectively, and all constants are positive.

(a) Explain each term of the system of differential equations.

(b) Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. What happens to the urea concentration as t l `?

dc K dp c+ap-bc -ap + bc dt V dt