# IVP involving Eigenvalues and Population Dynamics

In summary: Since they are real and distinct, the general solution can be written as x(t)=c1e^(λ1t)v1+c2e^(λ2t)v2 and y(t)=d1e^(λ1t)v1+d2e^(λ2t)v2 where v1 and v2 are the corresponding eigenvectors. Then use the initial values to solve for the constants. In this case, x(0)=9 and y(0)=7. Substitute the values into the general solution and solve for c1, c2, d1, and d2. This will give you the unique solution for x(t) and y(t).

## Homework Statement

Consider the interaction of two species of animals in a habitat. We are told that the change of the populations and can be modeled by the equations

$\frac{dx}{dt}$= 0.1x-0.8y
$\frac{dy}{dt}$=-0.2x+0.7y

Find the solution to the above equations with initial values x(0)=9 and y(0)=7
x(t)=
y(t)=

## The Attempt at a Solution

We are asked to compute the eigenvalues and I know then you have to find the corresponding eigenvectors. I did this but i am not sure I'm doing it correctly.
This was my process: p.s. not sure how to use a template for a matrix so I'm just going to wing it...

(A-λ)V = (R1) 0.1-λ -0.8
(R2) -0.2 0.7-λ

So I compute the characteristic polynomial and get λ2-0.8λ-0.9 which gives some pretty funky values for λ...

I actually found the solutions to this exact question, except with different coefficients and it seems like there is a formula I am missing in order to get the particular solution for y(t).
The general solution is supposed to look like (I'll do this using the vector Y=(x,y) for simplicity):
Y=k1eλ1t*V1+k2eλ2t*V2
where V is the eigenvector.

The fishy part comes when I go to plug in the IV's and solve for the constants k1 and k2. It seems like for y(t) they are using an equation that looks like
y(t)= k11-a)eλ1t+k22-a)eλ2t
where a is the first coefficient in the dx/dt equation...or something along those lines...
If anyone knows what I'm missing here, please just tell me straight up what's going on. I've been investigating this for two days and its due tomorrow at 2pm. I have a feeling it has to do with the fact that my coefficients are very small. Also, it's a population model so maybe it has something to do with the interactions of the two species..?
Let me know if you want more information. any help is really appreciated.
Thanks****Click on the link to see the same question (different coefficients) including the answers. Scroll down to page 7, question#2 and you'll see the complete question. http://math.la.asu.edu/~cheng/MAT274F2010/T3P2.pdf

Last edited:

## Homework Statement

Consider the interaction of two species of animals in a habitat. We are told that the change of the populations and can be modeled by the equations

$\frac{dx}{dt}$= 0.1x-0.8y
$\frac{dy}{dt}$=-0.2x+0.7y

Find the solution to the above equations with initial values x(0)=9 and y(0)=7
x(t)=
y(t)=

## The Attempt at a Solution

We are asked to compute the eigenvalues and I know then you have to find the corresponding eigenvectors. I did this but i am not sure I'm doing it correctly.
This was my process: p.s. not sure how to use a template for a matrix so I'm just going to wing it...

(A-λ)V = (R1) 0.1-λ -0.8
(R2) -0.2 0.7-λ

So I compute the characteristic polynomial and get λ2-0.8λ-0.9 which gives some pretty funky values for λ...

I actually found the solutions to this exact question, except with different coefficients and it seems like there is a formula I am missing in order to get the particular solution for y(t).
The general solution is supposed to look like (I'll do this using the vector Y=(x,y) for simplicity):
Y=k1eλ1t*V1+k2eλ2t*V2
where V is the eigenvector.

The fishy part comes when I go to plug in the IV's and solve for the constants k1 and k2. It seems like for y(t) they are using an equation that looks like
y(t)= k11-a)eλ1t+k22-a)eλ2t
where a is the first coefficient in the dx/dt equation...or something along those lines...
If anyone knows what I'm missing here, please just tell me straight up what's going on. I've been investigating this for two days and its due tomorrow at 2pm. I have a feeling it has to do with the fact that my coefficients are very small. Also, it's a population model so maybe it has something to do with the interactions of the two species..?
Let me know if you want more information. any help is really appreciated.
Thanks

****Click on the link to see the same question (different coefficients) including the answers. Scroll down to page 7, question#2 and you'll see the complete question. http://math.la.asu.edu/~cheng/MAT274F2010/T3P2.pdf

For one thing your characteristic polynomial λ2-0.8λ-0.9 is just a little wrong. There is a decimal point in the wrong place. Fix it and you'll get nice eigenvalues.

Dick said:
For one thing your characteristic polynomial λ2-0.8λ-0.9 is just a little wrong. There is a decimal point in the wrong place. Fix it and you'll get nice eigenvalues.

Thank you. Anything about the unique solution part? Thats where I really need help
Thanks.

Thank you. Anything about the unique solution part? Thats where I really need help
Thanks.

First find the eigenvalues and eigenvectors.

## What is an IVP involving Eigenvalues and Population Dynamics?

An IVP (initial value problem) involving eigenvalues and population dynamics is a mathematical model used to study the growth or decline of a population over time, taking into account factors such as birth rates, death rates, immigration, and emigration. The eigenvalues in this context represent the rates of change for each factor.

## What is the significance of eigenvalues in population dynamics?

Eigenvalues play a crucial role in understanding the behavior of a population over time. They determine the stability of the population, with positive eigenvalues indicating growth and negative eigenvalues indicating decline. The size of the eigenvalues also reflects the magnitude of the population change.

## How is the IVP solved for population dynamics?

The IVP can be solved using techniques from linear algebra and differential equations. First, the population dynamics model is represented as a system of linear differential equations. Then, the characteristic equation is formed by setting the determinant of the coefficient matrix to zero. The eigenvalues are then found by solving the characteristic equation, and the corresponding eigenvectors are used to construct the general solution to the system of equations.

## What are the limitations of using IVP involving Eigenvalues and Population Dynamics?

While IVP involving eigenvalues and population dynamics can provide valuable insights into population behavior, there are some limitations to this approach. It assumes a constant growth rate and does not account for external factors that may affect the population. Additionally, it may not accurately model populations with complex dynamics.

## What are some real-world applications of IVP involving Eigenvalues and Population Dynamics?

IVP involving eigenvalues and population dynamics has various applications in fields such as biology, ecology, economics, and epidemiology. It can be used to study the growth of animal populations, economic trends, and the spread of diseases. It can also help predict and manage population changes, making it a valuable tool for decision-making in various industries.

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