A Three Dimensional Lotka-Volterra

SystemKliah Soto

Jorge MunozFrancisco Hernandez

Two-Dimensional Case

Extreme Cases

x 0 dx

dt0

x 0 dy

dt cy

and

y 0 dx

dtax

y 0 dy

dt0

Equilibria

dx

dtax bxy 0

dy

dt cy dxy 0

results in

(0,0)

and

(c

d,a

b)

Solution Curves Solve the system of equations:

dy

dxdy

dt/dx

dt

cy dxyax bxy

y( c dx)x(a by)

a byy

dy c dxx

dx

a byy

dy c dxx

dx

aln y by c ln x dx k

Solution Curve Solution curve with all parameters = 1

Pink: prey x

Blue: predator y

Three Dimensional Case

dx

dtax bxy

dy

dt cy dxy eyz

dz

dt fz gyz

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

dimensional case Case 2: y=0

dx

dtax

dy

dt0

dz

dt fz

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

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

a

f

Kxz

dxax

dzfz

ax

fz

dt

dx

dt

dz

dx

dz

11

/

fzdt

dz

axdt

dx

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 10

1

2

3

4

Case 3: x=0

dx

dt0

dy

dt cy eyz

dz

dt fz gyz

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

dy

dt cy eyz

dz

dt fz gyz

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

Kgyyfezzc

dyy

gyfzd

z

ezc

dyy

gyfdz

z

ezc

ezcy

gyfz

dt

dy

dt

dz

dy

dz

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:

dx

dtax bxy 0

dy

dt cy dxy eyz 0

dz

dt 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:

fzdt

dz

axdt

dx

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

dx

dtax bxy

dy

dt cy dxy eyz

dz

dt fz gyz

When we consider:

)( gyfzgyzfzdt

dz

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:

dx

dtas bsy s(a by) y

a

bdy

dt cy dsy eyz y( c ds ez ) z

ds ce

dz

dt fz gyz z( f gy) y

f

g

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

Linearize the System by finding the Jacobian

gyfzg

yeezdxcyd

xbbya

zyxJ

0

0

),,(

),,(

),,(

),,(

zyxhgyzfzdt

dz

zyxgeyzdxycydt

dy

zyxfbxyaxdt

dx

zz

hy

y

hxx

h

dt

dy

zz

gy

y

gx

x

g

dt

dy

zz

fy

y

fxx

f

dt

dx

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

f

c

a

J

00

00

00

)0,0,0( Eigenvalues:

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

Solution:

5 1 0 1 5 2 0

1 0 0 00

2 0 0 00

3 0 0 00

4 0 0 00

5 0 0 00

1 2 3 4 5

10

5

5

10

Unstable x-axis Stable yz-Plane

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

Eigenvectors:

bgaf

baebad

dbc

badcJ

/00

/0/

0/0

)0,/,/(

ac

aci

bfbga

/)(

}0,,1{

}1

)2(,)(,1{2

222222

bc

acid

ceabdgaabdfgdfbab

cb

dagfb 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 predator

Yellow 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.

dx

dtax bxy

dy

dt cy dxy eyz

dz

dt fz gyz

aci

bfbga

/)(

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