Consider the modification to the Malthusian equation $$\frac{dN}{dt}= rS(N)N ,$$ where $r > 0 $ is the per capita growth rate, and $S(N)$ is a survival fraction. For some organisms, finding a mate at low population densities may be difficult. In such cases, the survival fraction can take the form $S(N) = \frac{N}{A + N}$, where $A > 0$ is a constant.
(a)
i. By examining what happens to $S(N)$ for $N \gg A$ and $N \ll A$ explain why this fraction models the situation outlined above.
Attempt:
When $N \gg A$ (my assumption is that the sign '$\gg$'means significantly greater than ) so $A$ is much smaller than $N$ then the constant $A$.This would effect the survival fraction in a way which it becomes roughly equal to $1$ since $A$ is really small.
When $N\ll A$ (my assumption is that the sign '$\ll$' means significantly smaller than) so $N$ is much smaller than $A$ will impact the survival fraction in a way which will make it become small.
The survival fraction models matches with the situation above since:
For a low population finding a mate is difficult (a.k.a $N\ll A$ )
For a large population finding a mate is more likely (a.k.a $N\gg A$ )
ii. By examining the form of the equation, determine the long-term behaviour of a population for an initial condition $N(0) > 0$.
Attempt: For the initial conditions stated above the Malthusian equation has the form: $$\frac{dN}{dt}= r\, \frac{N}{A + N}\, N,$$ which has the solution.
$$N(t)=N_{0}e^{\lambda t}$$ If $\lambda>0$ then exponential growth
If $\lambda <0$ then exponential decay
Please could you check if this is correct. If it is please can you suggest any improvement I could add to my attempts.
You don't need to find the exact solution in the second part. The right-hand side is positive for all $N>0$, so there are no equilibria except $N=0$. Hence the solution will increase indefinitely, and for $N \gg A$ you have $\dot N \approx kN$ (as you showed in the first part), so you will end up with approximately exponential growth. It's just that the population grows slower to begin with (if $N(0) \ll A$), since you have $\dot N \approx \frac{r}{A} N^2$ for small $N$ (as you also, more or less, showed in the first part).