Due to over$-$harvesting, the population of western Atlantic cod collapsod in the carly $1990$s$.$ The government has since declared a moratorium on the Northern Cod fishery, which drastically changed the fishing industry in Atlantic Canada.

Suppose that the cod population off the eastern coast of Canada recovers enough to support fishing. The Department of Fisheries and Occans will want to decide on a safe fixed proportion of fish, $h,$ to authorize for harvest cach ycar, following the reproduction scason. This means that $h$ is a non$-$negative real number and the size of the catch is determined by multiplying $h$ times the cod population at the end of the previous year.

The modified discrete$-$time dynamical system will be:

$N_{t+1}=(\frac{r}{1+0\mathrm{.}01N_t})N_t-h{N}_t$

Use this for problem $3$ and onward.

For the problems below, $N$ will still represent fish in millions and $t$ will still be in years. We will work to find the maximal number of fish that can be harvested, while maintaining a stable population.

$3.$ Again suppose that for Atlantic codr$=2\mathrm{.}3,$ and find the equilibria points for this new system algebraically $($the expression for one of the equilibrium points will contain the parameter $h).$

$4.$ To prevent another collapse in the population of Atlantic cod, we need to harvest from a stable cquilibrium population. Assume for the moment that the positive equilibrium point that you found in problem $3$ is stable.

The annual yicld of the cod population at equilibrium will be the product of two terms:

the positive equilibrium value and

the safe proportion value $h.$

Write down the formula for the annual yicld, $Y(h),$ a function of $h.$