A Three Dimensional Lotka-Volterra

SystemKliah Soto

Jorge MunozFrancisco Hernandez

Two-Dimensional Case

Extreme Cases

x 0 dxdt

0

x 0 dydt

cy

and

y 0 dxdt

ax

y 0 dydt

0

Equilibria

dxdt

ax bxy 0

dydt

cy dxy 0

results in(0,0)and

(cd

,ab

)

Solution Curves Solve the system of equations:

dydx

dydt

/dxdt

cy dxyax bxy

y( c dx)x(a by)

a byy

dy c dxx

dx

a byy

dy c dxx

dx

a ln y by c ln x dx k

Solution Curve Solution curve with all parameters = 1

Pink: prey xBlue: predator y

Three Dimensional Case

dxdt

ax bxy

dydt

cy dxy eyz

dzdt

fz gyz

Extremities Case 1: if z=0 then we have the 2

dimensional case Case 2: y=0

dxdt

ax

dydt

0

dzdt

fz

In the absence of the middle predator y, we are left with:

We combine it to one fraction and use separation of variables:

af

Kxz

dxax

dzfz

axfz

dtdx

dtdz

dxdz

11

/

fzdtdz

axdtdx

species z approaches zero as t goes to infinity, and species x exponentially grows as t approaches infinity.

Phase Portrait and Solution Curve when y=0

The blue curve represents the prey, while the red curve represents the predator.

0 2 4 6 8 1 0

1

2

3

4

Case 3: x=0

dxdt

0

dydt

cy eyz

dzdt

fz gyz

In the absence of the prey x, we are left with:

dydt

cy eyz

dzdt

fz gyz

We combine it to one fraction and use separation of variables:

Kgyyfezzc

dyygyfzd

zezc

dyygyfdz

zezc

ezcygyfz

dtdy

dtdz

dydz

lnln

)()(/ species y and z will approach zero

as t approaches infinity.

Phase Portrait and Solution Curve when x=0

The blue curve represents the top predator, while the red curve represents the middle predator.

1 2 3 4 5t

0 .2

0 .4

0 .6

0 .8

1 .0

yz

Equilibria Set all three equations equal to zero to

determine the equilibria of the system:

dxdt

ax bxy 0

dydt

cy dxy eyz 0

dzdt

fz gyz 0

Cases of Equilibria When x=0: Either y=0 or z=-c/e z has to be positive so we

conclude that y=0 making the last equation z=0.

Equilibrium at (0,0,0) When y=0 System reduces to:

fzdtdz

axdtdx

x=0 and y=0 since a and f are positive. Again equilibrium (0,0,0).

dxdt

ax bxy

dydt

cy dxy eyz

dzdt

fz gyz

When we consider:

)( gyfzgyzfzdtdz

Either z= 0 or –f+gy =0. Taking the first case will result in the trivial solution again as well as the equilibrium from the two dimensional case.(c/d,a/b,0)

Using parameterization we set x=s and the last equilibrium is:

dxdt

as bsy s(a by) y ab

dydt

cy dsy eyz y( c ds ez ) z ds ce

dzdt

fz gyz z( f gy) y fg

Equilibrium point at (s,a/b=f/g,(ds-c)/e)

Linearize the System by finding the Jacobian

gyfzgyeezdxcyd

xbbyazyxJ

0

0),,(

),,(

),,(

),,(

zyxhgyzfzdtdz

zyxgeyzdxycydtdy

zyxfbxyaxdtdx

zzhy

yhx

xh

dtdy

zzgy

ygx

xg

dtdy

zzfy

yfx

xf

dtdx

Where the partial derivatives are evaluated at the equilibrium point

Center Manifold Theorem

Real part of the eigenvalues ◦ Positive: Unstable◦ Negative: Stable◦ Zero: Center

Number of eigenvalues:◦ Dimension of the

manifold Manifold is tangent to

the eigenspace spanned by the eigenvectors of their corresponding eigenvalues

Equilibrium at (0,0,0)

One-dimensional unstable manifold: curve x-axis

Two-dimensional stable manifold: surface yz- Plane

fc

aJ

000000

)0,0,0( Eigenvalues:

◦ a, -c, -f Eigenvectors: {1,0,0}, {0,1,0}, {0,0,1}

Solution:

5 10 15 20

10 00 0

20 00 0

30 00 0

40 00 0

50 00 0

1 2 3 4 5

10

5

5

10

Unstable x-axis Stable yz-Plane

Equilibrium at (c/d, a/b, 0) Eigenvalues

Eigenvectors:

bgafbaebad

dbcbadcJ

/00/0/

0/0)0,/,/(

ac

aci

bfbga

/)(

}0,,1{

}1)2(,)(,1{ 222222

2

bcacid

ceabdgaabdfgdfbab

cbdagfb cd

One-Dimensional invariant curve:◦ Stable if ga<fb◦ Unstable ga>fb

Two-Dimensional center manifold Three-dimensional center

manifold◦ If ga=fb

aci

bfbga

/)(

Stable Equilibrium ga<fb

All parameters equal 1 a = 0.8

Blue represents the prey.Pink is the middle predatorYellow is the top predator

(2,2,2)

a=1.2 , b=c=d=e=f=g=1

Unstable Equilibrium ga>fb

Blue represents the prey.Yellow is the middle predatorPink is the top predator

(2,2,2)

Blue represents the prey.Pink is the middle predatorYellow is the top predator

Three Dimensional Manifold ga=fb

All parameters 1 initial condition (1,2,4)

Conclusion The only parameters that have an

effect on the top predator are a, g, f and b. ◦ Large values of a and g are

beneficial while large values of f and b represent extinction.

The parameters that affect the middle predator are c, d and e. They do not affect the survival of z.

The survival of the middle predator is guaranteed as long as the prey is present.

The top predator is the only one tha faces extinction when all species are present.

dxdt

ax bxy

dydt

cy dxy eyz

dzdt

fz gyz

aci

bfbga

/)(

Eigenvalues for (c/d, a/b,0)