Critical threshold: One equation of change that arises in the study of epidemics is Here r and T are positive constants, and T is called the threshold level. In this problem assume that r = 0.5 and T...

Critical threshold: One equation of change that arises in the study of epidemics is

Here r and T are positive constants, and T is called the threshold level. In this problem assume that r = 0.5 and T = 10.

a. Write the equation of change for N under these assumptions.

b. Make a graph of dN dt versus N and use it to find the equilibrium solutions. (Include a span of N = 0 to N = 12.) Does one of the equilibrium solutions correspond to the threshold level?

c. For what values of N is the graph of N versus t increasing, and for what values is it decreasing?

 d. What happens if initially N is close to 10 but smaller than 10? What if initially N is close to 10 but larger than 10? (Note: Such an equilibrium solution is said to be unstable.)