For a fish population modeled by a depensation growth model with harvesting, we have
$\dfrac {dN} {dt} = F (N) – H(N)$
where
$F(N) = rN \left (\dfrac N {N_c} – 1 \right)\left( 1 – \dfrac N K \right)$
models the growth rate of the fish population without harvesting and
$H(N) = qEN$
is the rate at which fish are harvested.
I'm trying to find the sustained yield $H(N^*_3)$ and the unsustainable yield $H(N^*_2)$
as functions of effort ($E$). ($N^*_3$ is the nontrivial stable equilibrium of $N$ and $N^*_2$ is the unstable equilibrium.) I'm asking you for help.
I'm having a little trouble getting those functions. I have to determine the maximum effort and look into trends (can you recover from going above $E_\max$, etc) as well, but I don't think that'll be hard if I can figure out the equations.
Thanks for your help!
(Note TeX code: E_max changed to E_\max)
Best Answer
The equilibrium values are simply the solutions of $F(N) - H(N) = 0$, which (after dividing by $N$) is a quadratic equation.