2. Consider the model chemical reaction

A+ X 2X,

in which one molecule of a chemical X combines with another molecule of A to form two

molecules of X. This implies that the chemical X stimulates its own production, a process called

autocatalysis. This positive feedback process results in a chain reaction that eventually is limited

by a "back reaction" in which 2X returns to A+X.

According to the law of mass action, the rate of an elementary reaction is proportional to

the product of the concentrations of the reactants. Let x and a denote the concentrations of X

and A, respectively. Then, the equation for the kinetics of x is

x = k₁ax-k_₁x²

where k₁ and k_ are positive rate constant parameters.

a) Find all the equilibrium points and determine their stability by linearization.

b) Sketch the solution curves in (t, x) plane for various initial conditions x(0) = xo.

(Strogatz: Nonlinear Dynamics and Chaos; pp. 39)