Equilibrium solutions in mathematical modelling

In summary, the conversation discusses a problem involving a function and the need to find all equilibrium points. The speaker is unsure of how to get started and is seeking guidance without revealing the exact problem. They mention the possibility of multiple solutions due to constants and brackets in the function.
  • #1
klinklindeman
2
0
Hi,
I don't want to list the exact problem I have to solve, as I would rather do it myself, all I would like is some guidance. The problem is a function like: Pk+1=a (Pk-1) with a few more constants and some brackets thrown about on the right hand side. I need to find all the equilibrium points. I am not exactly sure how to get started, and I have had a look in my textbook and couldn't find any examples similar to this. Do I just derive this function, and then let it equal zero to solve for the points?

Also, when I expand the function I will have a polynomial with Pk^3, so there will be multiple solutions I assume.

Thanks for the help!
 
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  • #2
welcome to pf!

hi klinklindeman! welcome to pf! :smile:
klinklindeman said:
I need to find all the equilibrium points.

it's a little difficult to tell without seeing the question, but equilibrium usually means that you can put Pk = Pk+1
Also, when I expand the function I will have a polynomial with Pk^3, so there will be multiple solutions I assume.

not so much multiple solutions, as a solution with multiple constants :wink:
 

FAQ: Equilibrium solutions in mathematical modelling

What is an equilibrium solution in mathematical modelling?

An equilibrium solution in mathematical modelling refers to a state where the system or model is in a balanced state with no net change over time. In other words, the system is at rest and all forces acting on it are balanced. This is often represented by a point on a graph where the slope is equal to zero.

How is an equilibrium solution determined in mathematical modelling?

An equilibrium solution is determined by finding the point at which the rate of change in the system is equal to zero. This can be done by setting the derivative of the system's equations equal to zero and solving for the variables. The resulting values represent the equilibrium solution.

What are the different types of equilibrium solutions in mathematical modelling?

There are three types of equilibrium solutions in mathematical modelling: stable, unstable, and semi-stable. A stable equilibrium is when the system returns to its equilibrium state after a small disturbance. An unstable equilibrium is when the system moves away from its equilibrium state after a small disturbance. A semi-stable equilibrium is when the system returns to its equilibrium state after a large disturbance, but not after a small one.

How do equilibrium solutions affect the behavior of a system in mathematical modelling?

Equilibrium solutions play a crucial role in determining the behavior of a system in mathematical modelling. They can help identify critical points, such as maximum and minimum values, and provide information about the long-term behavior of the system. The stability of the equilibrium solutions also affects the stability of the system itself.

What are some real-world applications of equilibrium solutions in mathematical modelling?

Equilibrium solutions have numerous applications in various fields, including economics, biology, chemistry, and physics. For example, they are used to model supply and demand in economics, predict population growth in biology, and study chemical reactions in chemistry. In physics, equilibrium solutions are used to understand the motion of objects and the stability of systems.

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