Mathematical modelling - Fishery - Harvest equation

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Main Question or Discussion Point

A logistic function has formula U(n+1)=rUn
This models the growth in the fish population from year n to year n+1. If you now decided to harvest H fish, your equation looks like this: U(n+1)=rUn-H
Now, they want me to find the maximum H for which the population stays constant (growth factor r=1). Ie., if H is too large the population dies out!


Homework Equations



The exact equation is U(n+1)=[(-1x10^-5)(Un^2)+1.6Un]-H


The Attempt at a Solution


Somehow I need to solve for H and then look at the formula as a quadratic, look at the discriminant and then solve for H....
I think H needs to replace some variable in the equation...
 

Answers and Replies

  • #2
HallsofIvy
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A logistic function has formula U(n+1)=rUn
This models the growth in the fish population from year n to year n+1. If you now decided to harvest H fish, your equation looks like this: U(n+1)=rUn-H
Now, they want me to find the maximum H for which the population stays constant (growth factor r=1). Ie., if H is too large the population dies out!


Homework Equations



The exact equation is U(n+1)=[(-1x10^-5)(Un^2)+1.6Un]-H
Why is that the "exact" equation? You just said "Un+1= rUn[/sup]-H". Also, please tell us where the "-1x10-5" and 1.6 came from. Finally, "the maximum H for which the population stays constant" does NOT mean "r= 1". r should be a given "growth rate" without harvesting. From what you say, it looks like the correct equation for constant population should be Un+1= Un= rUn- H or just U= rU- H so that H= (r+1)U where U is the original population.
 

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