Modeling Population Growth [dP/dt = k P - A P2 - h]

[NZBRU]

Does anyone know how to solve dP/dt = k P - A P² - h for P. I understand partial fractions are needed and I have already solved dP/dt = k P - A P². Is anyone able to solve it, Cheers NZBRU.

 

[sweet springs]

Hi.

dt = dP/(k P - A P＾2 - h) = -1/A dP/[ (P- k/A)^2-{(k/A)^2+h} ] = -1/A dp/[p^2 - {(k/A)^2+h}) ] , p= P- k/A

= -1/A dp/[p - sqrt{(k/A)^2+h} ][p +sqrt {(k/A)^2+h} ]

now you can integrate.

 

[NZBRU]

If I typed that in correctly the line would be [-1/A [dp]/[[p - sqrt{(k/A)^2+h} ][p +sqrt {(k/A)^2+h} ]]]=dt or would it be: [-1/A [dp] [p +sqrt {(k/A)^2+h} ]/[p - sqrt{(k/A)^2+h} ]]=dt? (I have not used ASCIIMath extensively). Thank you for the fast response.

 

[JJacquelin]

Hi !
integration gives t(P)
The inverse fonction is P(t)

 

[blue_raver22]

Hello NZBRU,

Thank you for your question. The equation dP/dt = k P - A P2 - h is known as the logistic growth model, which is commonly used in population biology to model the growth of a population over time. In this equation, P represents the population size, t represents time, k is the intrinsic growth rate, A is the carrying capacity, and h is the environmental resistance.

To solve this equation for P, we can use the method of separation of variables. This involves isolating the variables on either side of the equation and then integrating both sides. In this case, we can rewrite the equation as:

dP/(k P - A P2 - h) = dt

Next, we can use partial fractions to simplify the left-hand side of the equation. This involves breaking down the denominator into simpler fractions. In this case, we can rewrite the equation as:

dP/[(A/k)P - P2 - h/k] = dt

Using partial fractions, we can express the right-hand side as:

dP/[(A/k)P - P2 - h/k] = [1/(A/k) - 1/(P + h/k)]dP

Now, we can integrate both sides of the equation, which will give us:

ln[(A/k)P - P2 - h/k] = (1/A)P - (1/k)ln(P + h/k) + C

Where C is the constant of integration. We can then rearrange this equation to solve for P:

[(A/k)P - P2 - h/k] = e^(1/A)P - C

Solving for P, we get:

P = (A/k) - (1/k)e^(1/A)P + C/(1/k)e^(1/A)

I hope this helps you solve the equation. Please let me know if you have any further questions.

Best,