Mathematical modelling - Fishery - Harvest equation

[jinx]

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...

 

[HallsofIvy]

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 "U_n+1= rU_{n[/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.}

 

[tacman]

To find the maximum H for which the population stays constant, we need to set the growth factor r=1 in the equation U(n+1) = rUn - H. This will give us the equation U(n+1) = Un - H. We can then solve for H by setting U(n+1) = Un, since the population is staying constant. This gives us H = 0. Therefore, the maximum value for H that will keep the population constant is 0. If H is larger than 0, the population will start to decrease and eventually die out.

We can also use the given equation U(n+1) = [(-1x10^-5)(Un^2)+1.6Un]-H to find the maximum H. Since the growth factor r=1, we can simplify the equation to U(n+1) = [(-1x10^-5)(Un^2)+1.6Un]-H = Un - H. This gives us the quadratic equation (-1x10^-5)(Un^2)+0.6Un - H = 0. We can then use the quadratic formula to solve for H, which will give us two possible values for H. However, since we are looking for the maximum H for which the population stays constant, we only need to consider the positive value of H.

In summary, to find the maximum H for which the population stays constant, we can either set U(n+1) = Un - H and solve for H, or use the simplified quadratic equation and solve for the positive value of H. This will give us the maximum value for H that will keep the population constant and prevent it from dying out.