Cancer progression The development of many cancers, such as colorectal cancer, proceeds through a series of precancerous stages. Suppose there are n 2 1 precancerous stages before cancer develops at stage n. A simple system of differential equations modeling this is x90 − 2u0x0 x9i

− ui21xi21 2 uixi x9n − un21xn21 where xi is the fraction of the population in state i, the ui’s are constants, and i − 1, . . . , n 2 1.

(a) Suppose n − 2. What is the system of three differential equations?

(b) Suppose u0 − 1 and u1 − 1. Construct the phase plane for the variables x0 and x1 alone, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane.

(c) From your answer to part (a), obtain a differential equation for dx1ydx0 for the parametric curves describing the trajectories in the x0-x1 plane. Use this equation to plot the vector field. Then sketch several solutions curves for x1 as a function of x0, assuming that x0s0d − a and x1s0d − 1 2

a, where 0 , a , 1.

(d) Your answer to part

(c) will produce a family of functions in the constant

a. Provide a biological interpretation for these parametric curves.