Let's say you want to develop a strategy for controlling the vector by which malaria is transmitted. You have

developed a model of a sudden outbreak of the mosquito population with the following equation:

$\frac{dN}{dt}=RN(1-\frac{N}{C})-\frac{r{N}^2}{N_C^2+N^2}$

where $N$ is the number of mosquitos, $R$ is an intrinsic growth rate, and $C$ is the carrying capacity of the local

environment. The second term represents the effects of a predator $($e$.$g$.,$ fish that eat the mosquitos$).$ Its effect

becomes significant when the population reaches a critical size $N_C.r$ is the maximum value that the second

term can reach at large values of $N.$

$($a$)$ Write a code to numerically solve this equation for the period of $50$ days. Use a starting set of following

parameters: growth rate $R=0\mathrm{.}55$ days ${?}^{-1},N(0)=10,000,C={10}^4,N_C={10}^4,$ and $r={10}^4\frac{1}{?}$ day.

$($b$)$ Change the growth rate value to $R=0\mathrm{.}005$ days ${?}^{-1}$ and illustrate the difference in the dynamics of this

system compared to $($a$).$ Make one plot comparing the two solutions and interpret how the change in

results reflects the respective change in parameters.