Logistic Growth Models (interpreting r value)

In summary, the conversation discusses the relationship between birth rates, death rates, and the intrinsic growth rate (r) in the context of exponential and logistic growth models. The discussion also touches on the practical application of these models and the necessary information needed to determine r and the carrying capacity (K).
  • #1
thelema418
132
4
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.
 
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  • #2
thelema418 said:
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 [itex]P(t)[/itex] is governed by the ODE
[tex]
\dot P = aP - bP^2
[/tex]
where [itex]a[/itex] is (birth rate - death rate), which is assumed to be constant, and [itex]b \geq 0[/itex], which is assumed to be constant, is a parameter representing the effects of competition for resources. In effect [itex]bP[/itex] is the death rate from competition, which is not constant but is proportional to the size of the population, whereas [itex]a[/itex] is birth rate less death rate from all other causes. When [itex]b = 0[/itex] we recover exponential growth and there are no competition-related deaths.

For [itex]b \neq 0[/itex] the ODE can also be written in the form
[tex]
\dot P = rP(K - P)
[/tex]
where [itex]r = b[/itex] and [itex]K = a/b[/itex], or in the form
[tex]
\dot P = sP\left(1 - \frac{P}{K}\right)
[/tex]
where [itex]s = a[/itex] and again [itex]K = a/b[/itex].
 
  • #3
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?
 
  • #4
thelema418 said:
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 [itex]r[/itex] and [itex]K[/itex] you need to know [itex]P[/itex] and [itex]dP/dt[/itex] at two different times. Knowledge of [itex]P^{-1} dP/dt[/itex], which is really all that the birth and death rates give you, at just a single time is not sufficient.
 
  • #5


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.
 

FAQ: Logistic Growth Models (interpreting r value)

1. What is a logistic growth model?

A logistic growth model is a mathematical function that describes the growth of a population over time. It takes into account the carrying capacity of the environment, which is the maximum number of individuals that can be sustained by the available resources.

2. How is the r value interpreted in a logistic growth model?

The r value, also known as the growth rate, represents the rate at which the population is increasing. A positive r value indicates that the population is growing, while a negative r value indicates that the population is declining. The magnitude of the r value can also provide information about the speed of the population growth or decline.

3. What is the significance of the carrying capacity in a logistic growth model?

The carrying capacity is an important factor in a logistic growth model as it represents the maximum number of individuals that can be sustained by the available resources in a given environment. When the population reaches the carrying capacity, the growth rate will slow down and eventually level off.

4. How is a logistic growth model different from an exponential growth model?

A logistic growth model differs from an exponential growth model in that it takes into account the carrying capacity of the environment. In an exponential growth model, the population continues to grow at an increasing rate with no limit, while in a logistic growth model, the population growth slows down and reaches a plateau once the carrying capacity is reached.

5. What are some real-life applications of logistic growth models?

Logistic growth models are commonly used in ecology to study the growth of populations in different environments. They are also applied in economics to analyze the growth of markets and consumer demand. Additionally, logistic growth models are used in public health to study the spread of diseases and the effectiveness of interventions.

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