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.

 

[CellCoree]

I can provide some insights into the interpretation of r values in logistic growth models. In the context of this model, the intrinsic growth rate (r) represents the maximum rate at which a population can grow under ideal conditions, without any limiting factors. This is often referred to as the "biotic potential" of a population.

The birth rates and death rates are related to the intrinsic growth rate in the sense that they determine the net growth rate of a population. If the birth rate is higher than the death rate, the population will grow, and if the death rate is higher than the birth rate, the population will decline. However, in the logistic growth model, the intrinsic growth rate (r) is not a constant value, but it changes as the population approaches its carrying capacity (K). As the population size approaches K, the intrinsic growth rate decreases, and eventually, the birth rate and death rate will become equal, resulting in a stable population size.

In terms of the specific equation you mentioned (P(t) = K/2), this represents the point where the population reaches half of its carrying capacity. At this point, the growth rate is at its maximum, and both birth and death rates are occurring at the same time. However, this does not mean that they only occur at this specific time, as birth and death rates are ongoing processes in a population.

I hope this clarifies the relationship between r, birth rates, and death rates in the logistic growth model. It is important to keep in mind that these values are theoretical and may not always accurately reflect real-life populations, as there are many other factors that can affect population growth.