Consider a human population, with $N=120$ being the maximum possible age in years. Use the following parametric models for mortality and fecundity:\\
(a) Mortality: Write a model

$$
\mu_{j}=\frac{e^{j / w}}{M+e^{j / w}}, \quad[j]=\text { years }
$$

and adjust the parameters $M$ and $w$ based on the following information:\\
(i) The infant mortality is $\mu_{0}=0.005$;\\
(ii) The mortality reaches the value $\mu_{j}=0.5$ at the age $j=85$.\\
(b) Fecundity: Write a model

$$
\beta_{j}=B\left(\frac{j}{j_{0}}\right)^{n} e^{-j / j_{0}}, \quad n=40
$$

Adjust the parameters $j_{0}$ and $B$ based on the following guidelines:\\
(i) The age of maximum fecundity is $j=30$ years.\\
(ii) Choose the scaling $B$ such that the condition
\begin{equation*}
\sum_{j=1}^{N} \beta_{j}\left(\prod_{k\end{equation*}
is satisfied.

Plot the mortality and fecundity values. Run a long simulation, e.g., 500 years, starting with a population of size $P=1$, units e.g., in millions, with a random age distribution. Use the Matlab command rand to generate the initial population, and normalize it so that the sum of the components of the population vector gives a unity. Plot the population distribution with $t=1,20,40,60,120,240,500$. At the end, the population should have settled close to the stable equilibrium. Save the total population size of each simulation and plot the evolution of the population size as a function of time.\\
Comment on the results. Do they look realistic?