Consider the model

$\frac{dN}{dt}=N(K-N{)}^2,K>0$

$($a$)$ Calculate the steady state solutions.

$($b$)$ Determine the stability of each steady state solution.

$($c$)$ Draw a phase line diagram representing the steady states and the direction of solution curves.

$2.$ Consider the chemical reaction

$A^{k-1}{?}^{k_1}B.$

$($a$)$ Using the Law of Mass Action, write a pair of differential equations to track the changes to a and $b.$

$($b$)$ Determine the nullclines and steady states of the resulting system.

$($c$)$ We expect that the total concentration of chemicals $C$ does not change. Use the equation $b=C-a$ to obtain a single differential equation for $a.$

$($d$)$ Solve the resulting equation to obtain an equation for $a(t).$