We will consider a fish population in a lake. N will represent the biomass in kg and we will consider initial values for $N_0$ in the range $020000$ kg$.$ The time unit will be months. K $=10000$ kg$.$ The intrinsic growth rate is $0\mathrm{.}6$ per year or with the chosen time unit: $0\mathrm{.}6/12$ per month. One can start with a timeframe of $15$ years or more for studying how the population evolves in time.

The differential equation was given as:

$\frac{dN}{dt}=r(1-\frac{N}{K})N,N(0)=N_0$

Plot the right$-$hand side of this equation as function of N to study how dN$/$dt changes as function of N$.$ Include this in the report. What happens when N $=0$ and N $=$ K$?$ What does these points represent?

For which N value will dN$/$dt have its largest value and what will this value be$?($Show this analytically and check expressions with numbers and the graph$).$ Also identify from the previous time simulations where we can see the effect of this and what does this represent?