Logistic Growth Models (interpreting r value)

[thelema418]

I originally posted this on the Biology message boards. But I have not received any responses.

In models of exponential growth, we have an intrinsic growth rate (r) that is calculated as the difference of birth rates to death rates.

With the logistic growth model, we also have an intrinsic growth rate (r). How then do birth rates and death rates relate to the intrinsic growth rate in the context of this model? Specifically, if you have a model where you have been given values for r and K, does the birth rate and death rate associated with r occur at a particular time? I'm wondering if this specifically relates to P(t) = K/2 since this is where the maximum growth occurs.

Thanks.

 

[pasmith]

I originally posted this on the Biology message boards. But I have not received any responses.

In models of exponential growth, we have an intrinsic growth rate (r) that is calculated as the difference of birth rates to death rates.

With the logistic growth model, we also have an intrinsic growth rate (r). How then do birth rates and death rates relate to the intrinsic growth rate in the context of this model? Specifically, if you have a model where you have been given values for r and K, does the birth rate and death rate associated with r occur at a particular time? I'm wondering if this specifically relates to P(t) = K/2 since this is where the maximum growth occurs.

Thanks.

Logistic growth of a population $P(t)$ is governed by the ODE
$$\dot P = aP - bP^2$$
where $a$ is (birth rate - death rate), which is assumed to be constant, and $b \geq 0$, which is assumed to be constant, is a parameter representing the effects of competition for resources. In effect $bP$ is the death rate from competition, which is not constant but is proportional to the size of the population, whereas $a$ is birth rate less death rate from all other causes. When $b = 0$ we recover exponential growth and there are no competition-related deaths.

For $b \neq 0$ the ODE can also be written in the form
$$\dot P = rP(K - P)$$
where $r = b$ and $K = a/b$, or in the form
$$\dot P = sP\left(1 - \frac{P}{K}\right)$$
where $s = a$ and again $K = a/b$.

 

[thelema418]

Yes, those are the models I'm speaking about.

But my question concerns, I guess, "practical guidance" of the model. Consider a model where the population reaches capacity at t = 500. If a researcher measures the birth rate and death rate after t = 500, the number of births would be the same as the number of deaths.

This is again why I'm wondering if the inflection point is significant to the concept of r. If I have a birth rate and death rate relative to a specific time and I know the model is logistic, is this enough information to find r for the logistic equation?

 

[pasmith]

Yes, those are the models I'm speaking about.

But my question concerns, I guess, "practical guidance" of the model. Consider a model where the population reaches capacity at t = 500. If a researcher measures the birth rate and death rate after t = 500, the number of births would be the same as the number of deaths.

This is again why I'm wondering if the inflection point is significant to the concept of r. If I have a birth rate and death rate relative to a specific time and I know the model is logistic, is this enough information to find r for the logistic equation?

I don't think so. To determine $r$ and $K$ you need to know $P$ and $dP/dt$ at two different times. Knowledge of $P^{-1} dP/dt$, which is really all that the birth and death rates give you, at just a single time is not sufficient.