## Prey-Predators systems (*)

### The problem

Umberto d'Ancona, an Italian biologist, was surprised with the abnormal growth of the percentage of sharks in the total catch of fish, during the period of the World War I (1914 - 1918).

Mean percentage of sharks in the fish catch
(Fiume port in the Adriatic Sea)

Indeed, during the war, the level of fishing was reduced, and consequently D'Ancona asked himself:

How does the variation in the level of fishing affect the size of the populations of the different species of fish?

Can the reduction of the level of fishing favor the population of sharks in relation to the population of pouts?

The sharks eat the pouts and so, when they meet they are affected mutually. Qualitatively, we expect that:

• the number of pouts grows when there are few sharks
• the number of sharks grows when there are lots of pouts

and that will result in cycles of variation which are repeated along the time.

(Click on the figure to display an animation of the joint variation of the number of elements of each specie.)

(Click on the figure to display the relation between the species along the time.)

Analyse the mean quantity of pouts and sharks in one of these cycles and, taking into account how fishing affects those values, try to solve the problem.

(Click on the figure to display the effect of the decrease of the mean values of the level of fishing.)

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

(*) This work was carried out under the guidance of Professor Maria Carvalho from the Universidade of Porto, under a grant by the Calouste Gulbenkian Foundation to develop a project for the promotion of Mathematics in Atractor.

Difficulty level: University