# Understanding the model of a fishery's effect on a fish population

The following differential equation is given to model the population of fish which is affected by a fishery:

$$\frac{\text{d}{U}}{\text{d}T}=rU\left(1-\frac{U}{K}\right)-\alpha\frac{U}{H_2+U}$$

The first half of this equation describes the logistic growth of the population and is pretty easy to understand. The second half though seems way less obvious. No further information is given on what $$\alpha$$ or $$H_2$$ are supposed to be. $$U$$ is the concentration of fish present and $$K$$ is the capacity.

The term seems to always give us a value between $$0$$ and $$1$$, which sounds like a ratio. But we subtract that value, rather than multiplying, which seems to show that it is not a ratio. Furthermore, I am not even sure what $$H_2$$ is supposed to represent. If it is a representation of the effort the fishery puts into fishing, then it is placed somewhat counterproductively. It seems like the higher $$H_2$$ is, the less fish is actually fished.

How is the second part supposed to be interpreted so that this is a sensible model? What are $$H_2$$ and $$\alpha$$?

• I'm afraid you forgot to ask a question. – user436658 May 15 '20 at 12:16
• @ProfessorVector Thanks for the headsup, definitely forgot to make my question clear. Fixed now! – Vladis Becker May 15 '20 at 12:20
• @Moo , Could you maybe elaborate on that? I am not sure how a uniform distribution is relevant in this case. – Vladis Becker May 15 '20 at 12:33
• @Moo How is this expressed in the function? I am not very experienced in this area, so I need a more extensive explanation, if possible. – Vladis Becker May 15 '20 at 12:37

That last term is from Michaelis-Menten kinetics. The parameter $$\alpha$$ is the maximum fishing rate, and the parameter $$H_2$$ is such that the fishing rate equals $$\alpha/2$$ if the population of fish satisfies $$U = H_2$$. No fishing occurs if there is no fish, and fishing increases with the quantity of fish (increase with slope $$\alpha/H_2$$ at the origin). The fishing rate is bounded by $$\alpha$$, the maximum fishing rate. In a certain way, $$H_2$$ characterizes the fishery's lack of responsiveness, as it gives an idea of how fast the fishing rate is adapted to the quantity of fish. The fishing term with coefficient $$\alpha$$ is subtracted from the logistic term, since it is responsible for a diminution of the fish population's growth rate. This effect is illustrated below: