Improved model of a fishery: $\dot N=rN(1-\frac{N}{K})-H\frac{N}{A+N}$

Question

Strogatz exercise $3.7.4.a:$

An improved model of a fishery is:

$$\dot N=rN\left(1-\frac{N}{K}\right)-H\frac{N}{A+N}.$$

a) Give a biological interpretation of the parameter $A$; what does it measure?

Here's what I did:

I non-dimensionalized the system $(\frac{dN}{dt}=N(1-N)-h\frac{N}{a+N})$ and used the 'Manipulate' function of Mathematica to gain a better understanding of what '$a$' is biologically, but still, I couldn't see it.

So here's my question: How could I approach answering such a question and gain a better insight of it?

Thank you in advance.

Answer 1

The term $HN/(A+N)$ is a typical saturation term (see Michaelis–Menten kinetics, or type II functional response).

For small $N$ (in comparison to $A$), the harvesting is roughly proportional to how much fish there is, since $HN/(A+N) \approx (H/A)N$.

But for large $N$ (in comparison to $A$) there is saturation; no matter how much fish there is available, one can't harvest at a higher rate than $H$, since $HN/(A+N) \nearrow H$ as $N \to \infty$.

Answer 2

I don't think you are expected to do any manipulation, you are just supposed to explain what $A$ does in the model. I would answer along the lines of

It reflects that harvesting gets less when fish get scarce. If $N=A$ the harvesting drops in half compared to when $N$ is very large compared with $A$.