Determine the maximum sustainable yield (the maximum harvesting and still keep the population stable) of population of logistic growth is Kr/4.
I know the logistic equation with constant harvest rate, h, is dx/dt=rx((1-x/k)-h)
I do not know how to apply that to the problem...
To determine the maximum sustainable yield (MSY) of a population governed by the logistic growth model, you need to find the value of the harvest rate (h) that allows the population to stay stable at its carrying capacity (K).
The logistic equation with a constant harvest rate can be written as:
dx/dt = rx((1 - x/K) - h)
To find the MSY, we want to determine the value of h that keeps the population stable.
1. Set dx/dt = 0 to find the equilibrium point. This occurs when the population is neither increasing nor decreasing.
0 = rx((1 - x/K) - h)
2. Simplify the equation by distributing the rx term.
0 = r(1 - x/K - hx)
3. Expand the equation.
0 = r(1 - x/K) - rhx
4. Move the rhx term to the other side of the equation.
rhx = r(1 - x/K)
5. Divide both sides of the equation by rh.
x = (1 - x/K)
6. Multiply both sides by K.
Kx = K - x
7. Move the x term to the left side of the equation.
Kx + x = K
8. Combine like terms.
(K + 1)x = K
9. Divide both sides by (K + 1) to solve for x.
x = K/(K + 1)
Now that you have found the equilibrium population size, x, you can substitute it back into the logistic equation to find the MSY.
MSY = K * (1 - K/(K + 1))
Simplifying further:
MSY = K * (1 - 1/(1 + 1/K))
10. Multiply the 1/K term by K/K for simpler calculation.
MSY = K * (1 - K/(K + 1))
Now, you can use this equation to calculate the maximum sustainable yield (MSY) of the population, given the value of K and r.