# Natural Growth and Logistic Models

• Beez
In summary, the logistic model and the exponential model are nearly identical AT SMALL TIMES, but when time has gone a bit the logistic growth restriction kicks in and flattens the population level to K. Problem remains when trying to do the same thing for P(t) = 400e^1.0986t and P(t) = 10000/(1 + 24e^-1.0986t).
Beez
Hi, I have a question regarding natural growth and logistic models. According to my textbook, any exponential equations, P(t) = e^kt can be expressed logistically as P(t) = K/(1 + Ae^-kt) where A = (K-P0)/P0. When I applied this rule to P(t) = e ^0.0015t, and P(t) = 6000 / (1 + 5e^-0.0015t) it worked fine. The results were so close. But when I tried to do the same thing for P(t) = 400e^1.0986t and P(t) = 10000/(1 + 24e^-1.0986t), the results were so different from each other; 1199.98 and 1111.098 for P(1). Did I do something wrong here or they don't work if k>1?

I would appreciate it if I can get any help with this problem. Thank you in advance.

Naoko

No, you have misuderstood the issue here:

The logistic model and the exponential model are nearly identical AT SMALL TIMES!
That is, it is only when time has gone a bit that the logistic growth restriction kicks in, and flattens the population level to K.

Problem remains

I understood there is nothing to do with the value of k. Thank you.
But then how come the second logistic equation does not work for only t=1?

N0=400
N1=1200 (after a year the population tripled)
The lake can hold up to 10,000 fish
P(1)= 400e^(1.09861*1)=1199.99... =N1

P(1) = 10,000/(1+24e^(-1.09861*1))=1111.10... where 24 = (10000-400)/400

Am I doing something wrong here?

Thanks.

Do you understand that the natural growth model is a DIFFERENT model than the logistic model; that is, there are some effects included in the logistic model that the natural growth model does not take into account?

Yes I do

Yes I do. The logistic model takes the maximum population, in this case 10,000, and A, in this case (K-P0)/Po, into account. But what I understood was while t is very small, they really don't have much effect and the exponential model and logistic model are almost identical. I used t=1 for the problem and the results were already so different. I used t=50 for the other problems that I mentioned earlier, but they still showed identical answers. That is why I thought the value of k or A might be the factors which determine if a size of the population satisfies both the exponential and logistic models or the only exponential model. But you told that it was nothing to do with them. Thus my question remained.

I hope I described my questions more clearly this time.

So, according to your explanation, the size of the population in question does not satisfy the logistic model. Am I right?

Thanks.

First, wherever have I stated that some of the parameters don't matter?
I've never said anything to that effect!

Let's take it in full:
Let's call the exponential work model $$p_{E}(t)=P_{0}e^{kt}$$
And the logistic model $$p_{L}(t)=\frac{K}{1+\frac{K-P_{0}}{P_{0}}e^{-kt}}$$
Rewrite the logistic model as $$P_{L}(t)=\frac{P_{0}e^{kt}}{1+\frac{P_{0}(e^{kt}-1)}{K}}=\frac{P_{E}(t)}{1+\frac{P_{0}(e^{kt}-1)}{K}}$$
Thus, if we have: $$\frac{P_{0}(e^{kt}-1)}{K}<<1$$, then it follows $$P_{L}(t)\approx{P}_{E}(t)$$
Let us make this into a condition on "t":
We rewrite, and find:
$$t<<\frac{1}{k}ln(1+\frac{K}{P_{0}})$$
This condition determines what a "small time" is, in this particular context.

If you plug into your values of $$k,P_{0},K$$ here, you'll see why the approximation holds in one case but not in the other..
(that is, the "small times"-regions are different)

Note that a concept like "small time" is not really meaningful if you haven't specified a REFERENCE time.
That is, "small times" means :the time is small compared with the reference time for the problem.

I think it was this usage of mine of "small times" which confused you.
I hope that I have made it more explicit now.

Last edited:
Thank you

Thank you so much for providing me the detailed explanation.
Yes, I was taking "small amount" literally.
Now my sky is as blue as it could be.
Thank you for your help again.

## 1. What is the difference between natural growth and logistic growth models?

Both natural growth and logistic growth models are used to describe the growth of a population over time. However, natural growth models assume that the population grows at a constant rate, while logistic growth models take into account limiting factors such as resources and competition that can slow down growth.

## 2. How do we determine the carrying capacity in a logistic growth model?

The carrying capacity, or the maximum population size that an environment can sustain, is determined by the limiting factors in the environment. These factors can include resources, competition, and predation. In a logistic growth model, the carrying capacity is represented by the point at which the growth curve levels off.

## 3. Can natural growth and logistic growth models be applied to all populations?

No, these models are most applicable to populations that have a stable environment and are not experiencing significant changes in resources or competition. In rapidly changing environments, these models may not accurately predict population growth.

## 4. How do natural growth and logistic growth models compare to other growth models?

Natural growth and logistic growth models are two of the most commonly used models to describe population growth. Other models include exponential growth models, which assume that populations grow at an unlimited rate, and density-dependent models, which take into account the effects of population density on growth.

## 5. What are some real-world applications of natural growth and logistic growth models?

Natural growth and logistic growth models are commonly used in ecology, biology, and environmental science to study and predict population dynamics. These models can also be applied in fields such as economics and marketing to predict the growth of consumer demand over time.

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