Build logistic differential equation for given data

• killer2008
In summary, the equation for US oil production is N=14.9395; P0=0.23349 and k=0.102758. If the above equation is correct, linear least squares while suffice. Just cross multiply: P(t)(N+P_0\cdot \left(e^{kt}-1\right))=N\cdot P_0 \cdot e^{kt} Then rearrange: P(t)P_0\cdot \left(e^{kt}-1\right)=N (P_0 \cdot e^{kt} - P(t)) P_0 is the initial production level and use least squares to solve for N.
killer2008
http://img258.imageshack.us/img258/3518/01ck5.png

I would like to build a logistic differential equation for US oil production from the given data above.
dP/dt = k(1-P/N)P

I tried many different models but nothing came close to the given data.

Any help and suggestion?

Last edited by a moderator:
Why should it be logistic?

John Creighto said:
Why should it be logistic?

This is a project from the textbook, there are many requirements (answering about 12 questions) and one of them is the equation must be logistic.

I think you will have to model the resulting function with the data using a least squares fitting. This equation is:

$$P(t)=\frac{N\cdot P_0 \cdot e^{kt}}{N+P_0\cdot \left(e^{kt}-1\right)}$$

It's very late over here, I will have a look at it tomorrow, it might be that it is non-linear and then I have to adapt my program to do this.

coomast said:
I think you will have to model the resulting function with the data using a least squares fitting. This equation is:

$$P(t)=\frac{N\cdot P_0 \cdot e^{kt}}{N+P_0\cdot \left(e^{kt}-1\right)}$$

It's very late over here, I will have a look at it tomorrow, it might be that it is non-linear and then I have to adapt my program to do this.

If the above equation is correct, linear least squares while suffice. Just cross multiply:$$P(t)(N+P_0\cdot \left(e^{kt}-1\right))=N\cdot P_0 \cdot e^{kt}$$

Then rearrange:

$$P(t)P_0\cdot \left(e^{kt}-1\right)=N (P_0 \cdot e^{kt} - P(t))$$

$$P_0$$ is the initial production level and use least squares to solve for N. I'm still not convinced oil production should be logistic. The only reason it should be logistic is because the production is driven by the demand which is driven by the population.

Last edited:
You need some code to do this least squares fitting in case one wants to calculate N, P0 and k at the same time. When this is done you will end up with N=14.9395; P0=0.23349 and k=0.102758 for the US and N=116.435; P0=0.04219 and k=0.12401 for the world production. This in case the time is considered the center of the intervals as is usually done in statistics. I used my own code for this, however I assume that this must be possible in some other mathematical package available. I can't help you in this though. One final remark is the following, I'm not giving any advice on whether this is a good curve for describing the data, this is something you need to do yourself. Other curves or equations might be more suitable. If you look at the one for the US, it is strange data, it's going down. Interpret this very carefully.

1. What is a logistic differential equation?

A logistic differential equation is a mathematical model that describes the growth or decay of a population over time. It takes into account the limiting factors that affect the population, such as resource availability and carrying capacity.

2. How is a logistic differential equation different from a regular differential equation?

A logistic differential equation includes a parameter called the carrying capacity, which represents the maximum population size that can be sustained in the given environment. This makes it more realistic for modeling population growth in real-life scenarios.

3. What data is needed to build a logistic differential equation?

To build a logistic differential equation, we need data on the initial population size, the growth rate, and the carrying capacity. These values can be obtained through observations, experiments, or previous studies.

4. How do you solve a logistic differential equation?

There are several methods for solving a logistic differential equation, such as separation of variables, Euler's method, and numerical integration. The specific method used will depend on the complexity of the equation and the desired level of accuracy.

5. What are some real-life applications of logistic differential equations?

Logistic differential equations have many applications in fields such as biology, ecology, economics, and epidemiology. They can be used to model population growth of species, the spread of diseases, and the dynamics of market demand. They are also useful for predicting and managing resource depletion and environmental sustainability.

• Differential Equations
Replies
25
Views
2K
• Differential Equations
Replies
65
Views
3K
• Calculus and Beyond Homework Help
Replies
7
Views
622
• Differential Equations
Replies
2
Views
1K
• Differential Equations
Replies
3
Views
2K
• Differential Equations
Replies
6
Views
1K
• Differential Equations
Replies
4
Views
2K
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
11
Views
2K
• Differential Equations
Replies
1
Views
2K