# Volterra-Lotka system.

#### charlottewill

##### New member
've been given the Lotka-Volterra system of differential equations

dW/dt= αW−βV,
dV/dt= −γV+δW

and the question is as follows:

a) Suppose we have two species interacting on the same environment. After observations we find that the size of their population obey the above set of differential equations where α,β,γ,δ are positive constants.

Determine what variables represents the predators and what variables represent the prey and WHY

b) Solve the system by proposing
W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

Find the value of the constants A,B,C,D in terms of the parameters α,β,γ,δ ω

There obviously must be a relationship between the equations given in part b) with the main dW/dt dV/dt equations above but how would you answer these questions?

#### chisigma

##### Well-known member
Re: 3rd Year Tutorial Task which I'm having really trouble solving!

've been given the Lotka-Volterra system of differential equations

dW/dt= αW−βV,
dV/dt= −γV+δW

...

b) Solve the system by proposing

W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

Find the value of the constants A,B,C,D in terms of the parameters α,β,γ,δ ω...

It seems that there is something wrong: we have a system of linear ODE of order 1 and You can have periodic solutions only if the system of linear ODE is at least of order 2...

Kind regards

$\chi$ $\sigma$

#### awkward

##### Member
Re: 3rd Year Tutorial Task which I'm having really trouble solving!

For a), just look at the equations. I think you are supposed to assume alpha, beta, etc. are positive. Wolves eat the prey, so if there are more wolves you should expect to see what happen to the prey?

As to chisigma's remark, I think you can see periodic behavior in a system of first-degree equations.

#### CaptainBlack

##### Well-known member
Re: 3rd Year Tutorial Task which I'm having really trouble solving!

It seems that there is something wrong: we have a system of linear ODE of order 1 and You can have periodic solutions only if the system of linear ODE is at least of order 2...

Kind regards

$\chi$ $\sigma$
Any ODE can be reduced to a first order system.

Consider:

$y''+y=0$

which we know has oscillatory solutions.

Putting $$u=y$$, and $$v=y'$$ reduces the second order ODE to the system:

\begin{aligned} v' & =- u \\ u' & = \phantom{-} v \end{aligned}

In fact it is standard practice when numerically integrating ODEs to reduce a higher order ODE to a system of first order ODEs.

CB

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#### chisigma

##### Well-known member
Re: 3rd Year Tutorial Task which I'm having really trouble solving!

Any ODE can be reduced to a first order system.

Consider:

$y''+y=0$

which we know has oscillatory solutions.

Putting $$u=y$$, and $$v=y'$$ reduces the second order ODE to the system:

\begin{aligned} v' & =- u \\ u' & = \phantom{-} v \end{aligned}

In fact it is standard practice when numerically integrating ODEs to reduce a higher order ODE to a system of first order ODEs.

CB
Yesterday evening I was tired and it was better of me to do some else, for example to read a roman...

The system of linear ODE...

$\displaystyle \frac{d W}{d t}= \alpha\ W - \beta\ V$

$\displaystyle \frac{ d V}{d t}=\delta\ W - \gamma\ V$ (1)

... written in terms of Laplace Transform with 'initial conditions' $\displaystyle W(0)=V(0)=0$ is...

$\displaystyle w(s)\ (s-\alpha) - \beta\ v(s)=0$

$\displaystyle w(s) - (s+\gamma)\ v(s)=0$ (2)

The linear system (2) has solution different from the 'trivial' $\displaystyle w(s)=v(s)=0$ only if the determinant vanishes and that means that it must be...

$\displaystyle s^{2} + (\alpha-\gamma)\ s -\alpha\ \gamma - \beta=0$ (3)

Condition of periodic solution is that (3) has roots $\displaystyle s= \pm i\ \omega$ and that is true for...

$\displaystyle \alpha= \gamma$

$\displaystyle -\beta- \alpha^{2}= -\beta - \gamma^{2}= \omega^{2}$ (4)

Kind regards

$\chi$ $\sigma$

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#### HallsofIvy

##### Well-known member
MHB Math Helper
've been given the Lotka-Volterra system of differential equations

dW/dt= αW−βV,
dV/dt= −γV+δW

Just to be precise lets assume "W" is wolves and "V" is voles.

and the question is as follows:
$dW/dt= \alpha W- \beta V$
$dV/dt= -\gamma V+ \delta W$

a) Suppose we have two species interacting on the same environment. After observations we find that the size of their population obey the above set of differential equations where α,β,γ,δ are positive constants.

Determine what variables represents the predators and what variables represent the prey and WHY
Now suppose W increases. What effect will that have on dV/dt? Will V increase or decrease? Suppose V increases. What effect will that have on dW/dt? Will W increase or decrease? Wow, those voles are really eating up the wolves, aren't they!
b) Solve the system by proposing
W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

Find the value of the constants A,B,C,D in terms of the parameters α,β,γ,δ ω
There obviously must be a relationship between the equations given in part b) with the main dW/dt dV/dt equations above but how would you answer these questions?

#### chisigma

##### Well-known member
There is a minor not well clear detail in this discussion. As far as I remember the Volterra-Lotka equations were non linear equation and that is confirmed in...

http://mathworld.wolfram.com/Lotka-VolterraEquations.html

... where, setting x the prey population and y the predator population, it is written...

$\displaystyle \frac{dx}{dt}= A\ x - B\ x\ y$

$\displaystyle \frac{dy}{dt}= -C\ y + D\ x\ y$ (1)

Kind regards

$\chi$ $\sigma$

#### charlottewill

##### New member

We've been asked to solve this system of equations

dW/dt = = αW−βV
dV dt = −γV+δW

by proposing that

W (t) = A sin(ωt) + B cos(ωt)
V (t) = C sin(ωt) + D cos(ωt).

so I've differentiated both W(t) and V(t) to give W'(t) = ωA cos(ωt) - ωB sin(wt) and V'(t) = ωC cos(ωt) - ωD sin(wt) and replaced with the dW/dt and dV/dt in the system to give the expressions

ωA cos(ωt) - ωB sin(wt) = αW−βV

ωC cos(ωt) - ωD sin(wt) = −γV+δW

and now I've replaced the W and V's on the RHS with W (t) = A sin(ωt) + B cos(ωt)
V (t) = C sin(ωt) + D cos(ωt) (originally given in the question) and I've factored out the constant and the parameters of cos(ωt) and sin (ωt) to give two simultaneous equations equal to zero which are

(ωA-αB+βD) cos(ωt) + (βC-αA-ωB) sin(ωt) = 0

(ωC+γD-δB) cos(ωt) + (γC-ωD-δA) sin(ωt) = 0

So how do I find the value of the constants A,B,C and D in terms of the parameters α,β,γ,δ ω?

Thanks,

Charlotte

#### chisigma

##### Well-known member

We've been asked to solve this system of equations

dW/dt = = αW−βV
dV dt = −γV+δW

by proposing that

W (t) = A sin(ωt) + B cos(ωt)
V (t) = C sin(ωt) + D cos(ωt).

so I've differentiated both W(t) and V(t) to give W'(t) = ωA cos(ωt) - ωB sin(wt) and V'(t) = ωC cos(ωt) - ωD sin(wt) and replaced with the dW/dt and dV/dt in the system to give the expressions

ωA cos(ωt) - ωB sin(wt) = αW−βV

ωC cos(ωt) - ωD sin(wt) = −γV+δW

and now I've replaced the W and V's on the RHS with W (t) = A sin(ωt) + B cos(ωt)
V (t) = C sin(ωt) + D cos(ωt) (originally given in the question) and I've factored out the constant and the parameters of cos(ωt) and sin (ωt) to give two simultaneous equations equal to zero which are

(ωA-αB+βD) cos(ωt) + (βC-αA-ωB) sin(ωt) = 0

(ωC+γD-δB) cos(ωt) + (γC-ωD-δA) sin(ωt) = 0

So how do I find the value of the constants A,B,C and D in terms of the parameters α,β,γ,δ ω?

Thanks,

Charlotte
what You have done till now is probably correct. You have two equations and four unknown variables A,B,C,D so that you need two more equations... what are these 'missing' equations?...

Kind regards

$\chi$ $\sigma$

#### charlottewill

##### New member
I don't know about the two missing equations because that is all the information I have given in the question so unless I do
ωA-αB+βD = 0
βC-αA-ωB = 0
ωC+γD-δB = 0
γC-ωD-δA = 0

because sin AND cos can never = 0 so that makes sense but now how do I solve these so that I have A,B,C,D in terms of the 4 parameters?

Kind Regards,

Charlotte

##### New member
Hey

i am having the exactly the same problem with this question.
Did you ever discover how to like the parameters to A B C D?
If you did do you mind exampling it to me?

#### CaptainBlack

##### Well-known member
've been given the Lotka-Volterra system of differential equations

dW/dt= αW−βV,
dV/dt= −γV+δW

and the question is as follows:

a) Suppose we have two species interacting on the same environment. After observations we find that the size of their population obey the above set of differential equations where α,β,γ,δ are positive constants.

Determine what variables represents the predators and what variables represent the prey and WHY

b) Solve the system by proposing
W (t) = A sin(ωt) + B cos(ωt)

V (t) = C sin(ωt) + D cos(ωt).

Find the value of the constants A,B,C,D in terms of the parameters α,β,γ,δ ω

There obviously must be a relationship between the equations given in part b) with the main dW/dt dV/dt equations above but how would you answer these questions?
If these are predator prey equations and supposedly are realistic both B and D must be independent of the birth death coefficients since they are the initial populations.

Then trivially from the set of equations in charlotte's last post (assuming correct algebra):

$A=\frac{\alpha B-\beta D}{\omega}$
$C=\frac{\delta B-\omega D}{\omega}$

CB

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##### New member
ok that helps a little but my question is a bit different and it doesn't really help me solve it

i have a system of:
dW/dt= αW−βVW,
dV/dt= −γV+δWV

I then have
W (t) = A +B sin(ωt)

V (t) = C + D cos(ωt).

I know the values of ABCD and ω but i am trying to find the parameters α,β,γ,δ ω.

When i rearrange to let them equal zero i keep on getting them to cancel out and simplify. when i solve for lets say alpha, when i substitute back in i get them to cancel out.

ωB cos(ωt) - (α(A + B sin(ωt))+β[(A + B sin(ωt))( C + D cos(ωt))[ = 0
- ωD sin(wt) + γ( C + D cos(ωt))-δ[(A + B sin(ωt)) (C + D cos(ωt))] = 0

I just need to know how to find the values of α,β,γ,δ ω?

If you can help me establish the link i will be highly grateful !

#### CaptainBlack

##### Well-known member
ok that helps a little but my question is a bit different and it doesn't really help me solve it

i have a system of:
dW/dt= αW−βVW,
dV/dt= −γV+δWV

I then have
W (t) = A +B sin(ωt)

V (t) = C + D cos(ωt).

I know the values of ABCD and ω but i am trying to find the parameters α,β,γ,δ ω.

When i rearrange to let them equal zero i keep on getting them to cancel out and simplify. when i solve for lets say alpha, when i substitute back in i get them to cancel out.

ωB cos(ωt) - (α(A + B sin(ωt))+β[(A + B sin(ωt))( C + D cos(ωt))[ = 0
- ωD sin(wt) + γ( C + D cos(ωt))-δ[(A + B sin(ωt)) (C + D cos(ωt))] = 0

I just need to know how to find the values of α,β,γ,δ ω?

If you can help me establish the link i will be highly grateful !

CB

##### New member
I dont have a specific question. i am making a mathematic model on turtles and algae in a dam. i am given that there is 200 tutrles that fluctuate by 10% and that there is 1-2 micrograms/litre of algae in the dam for an oscillation of a 6 month period.

using W=1.5+0.5sin(pi/3*t) and V=200*10 cos(pi/3t) and substituting into the volterra lotka system as stated above i ahould be able to solve for the unknown parameters the problem is that they keep on cancelling out

so i basically need help with finding by two main predator-prey equations with the unknown parameters known so i can actually get my assignment going as this is like a "pre-assignment" thing. I have decided to model it this way but i still need to code it up and actually answer the assignment but i cant do that until i have these equations

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#### CaptainBlack

##### Well-known member
I dont have a specific question. i am making a mathematic model on turtles and algae in a dam. i am given that there is 200 tutrles that fluctuate by 10% and that there is 1-2 micrograms/litre of algae in the dam for an oscillation of a 6 month period.

using W=1.5+0.5sin(pi/3*t) and V=200*10 cos(pi/3t) and substituting into the volterra lotka system as stated above i ahould be able to solve for the unknown parameters the problem is that they keep on cancelling out

so i basically need help with finding by two main predator-prey equations with the unknown parameters known so i can actually get my assignment going as this is like a "pre-assignment" thing. I have decided to model it this way but i still need to code it up and actually answer the assignment but i cant do that until i have these equations
Then I can tell you that your proposed solution is not a solution except for degenerated situations that are unlikely to be of interest.

Plug the suggested solution into the DE and you get an expressions for the derivatives with 3 Fourier components corresponding to angular frequencies $$0$$, $$\omega$$ and $$2 \omega$$ while the suggested solution when differentiated give expressions with only the first two Fourier components.

(I am pretty sure from other sources that those coupled non-linear ODEs do not have sinusoidal solutions)

CB

#### chisigma

##### Well-known member
I apologize for having for long time not written posts in this [old] thread and I will try to repair now...

Let's write again the pair of linear differential equations...

$\displaystyle \frac{d W}{dt}= \alpha\ W - \beta\ V$

$\displaystyle \frac{d V}{dt}= \delta\ W - \gamma\ V$ (1)

... and search if there is a periodic solution of (1) like that...

$\displaystyle W= A\ \sin \omega t + B\ \cos \omega\ t$

$\displaystyle V = C\ \sin \omega t + D\ \cos \omega\ t$ (2)

Writing (1) in terms of Laplace Transform we have...

$\displaystyle (s-\alpha)\ w(s)+\beta\ v(s)= W(0)$

$\displaystyle (s+\gamma)\ v(s)- \delta\ w(s)= V(0)$ (3)

... where W(0) and V(0) are the so called 'initial values'. The (3) is a pair on linear algebraic equation the solution of which is...

$\displaystyle w(s)= \frac{(s+\gamma)\ W(0) -\beta\ V(0)}{s^{2}+ (\gamma - \alpha)\ s - \alpha\ \gamma + \beta\ \delta}$

$\displaystyle v(s)= \frac{(s-\alpha)\ V(0) + \delta\ W(0)}{s^{2}+ (\gamma - \alpha)\ s - \alpha\ \gamma + \beta\ \delta}$ (4)

Observing (4) we note that we have periodic solutions of (1) only if is $\gamma=\alpha$ and in this case is...

$\displaystyle \omega= \frac{2\ \pi}{T}= \sqrt{\beta\ \delta - \alpha^{2}}$ (5)

... and the constants A,B,C and D are...

$\displaystyle A= \frac{\alpha\ W(0) -\beta\ V(0)}{\omega}$

$B=W(0)$

$\displaystyle C= \frac{\delta\ W(0) -\alpha\ V(0)}{\omega}$

$D=V(0)$ (6)

Kind regards

$\chi$ $\sigma$