Consider the the model chemical reaction A + X k1 ⇄ k−1 2X in which one molecule of X combines with one molecule of A to form two molecules of X. This means that the chemical X stimulates its own production, a process called autocatalysis. Eventually, this leads to a limitation based on the number of molecules of A. Thus there is a “back reaction” in which 2X returns to A + X. According to the law of mass action of chemical kinetics, the rate of an elementary reaction is proportional to the product of the concentrations of the reactants. We denote the concentrations by lowercase letters x = [X] and a = [A]. Assume that there’s an enormous surplus of chemical A, so that its concentration a can be regarded as constant. Then the equation for the kinetics of x is ̇x = k1ax − k−1x2 . (a) Explain in words what each term means in the context of the model of auto- catalysis. (Note: All the information you need is in the problem, you do not have to know more than that about autocatalysis.) (b) Find all the fixed points of this equation and classify their stability. (c) Sketch the graph of x(t) for various initial values x0. Interpret each trajectory by describing the behavior in the context of autocatalysis. Make sure to graph solutions!