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Stochastic model for fish stock management

Xn : Biomass in a year n
r : Intrinsic growth rate
M : Maximum biomass that can live in a given site
C : Amount of biomass harvested by fishers
C : can be defined as C=pXn

Python program:

The Schaefer model :

Vn : Environmental variability for year n

How can we choose a fishing rate so as to be the maximum sustainable yield?

→ Not causing the extinction of the fish species.
→ Not deprecate our economy.

Introduction :

we've used a python and matplotlib to make diagram

The stochastic model for fish stock management is a model theorized and developped by M. Schaefer to optimize fishing rate without killing the fish population.

I. The population growth

(Assuming the sequence is convergent )


we tried to solve (E) so we calculate Δ with Δ=b²-4ac

Then we know that the
maximum value
of C is rM/4

From the previous calcul, we've deduced the best yield :

the maximum percentage than we can fish each year is :

For example: this is a graphic
of the fish biomass with
a decaying exponential function:

With this model, there are three possible ways for the population to grow:
1. decrease of the fish population
2. fish population growth
3. stationary fish population

for each of those 3 populations growth, there are differents ways to reach these differents populations growth cases

We've worked on two of these cases; the stationnary fish population and the fish population growth

Conclusion :

From our previous work: we noticed three population growth and we've tried to transform the sequence in function to find the different formula for each of these population growth, we've calculated for the most interesting population growth, the growing population one (graph.1) what value we needed to get the best yield without killing the whole fish population, it's the best case we've found, because the population will still grow, even with a really high fishing rate.

But even with a model to help us, in real life, fish population growth has too many variables to be "so easily" calculated .

II. The Stationnary population

for a stationnary population, we've only calculated a constant , without trying to have the best yield:

With this formula, we've found one way to get a stationnary population, but we can also get one with a sinusoide