Dynamics at the Horsetooth Volume 2, 2010.
Heteroclinic Cycles in Competitive Populations
Benjamin Cooper
Department of Mathematics
Colorado State University
colostate.edu
cooper@math.
Report submitted to Prof. I. Oprea for Math 640, Fall 2013
Abstract. We investigate heteroclinic cycles of the famous LotkaVolterra system modeling a system of 3 competitive populations.
Keywords: keyword 1, keyword 2, keyword 3
1
Introduction
Robust heteroclinic cycles in competitive population models have been the
subject of many papers in fields ranging through chemistry, ecology, network
flows, just to name a few. We are mainly concerned with systems of ordinary
differential equations of the form:
x01 = f1 (x1 , x2 , . . . xn )
x02 = f2 (x1 , x2 , . . . xn )
..
.
x0n = fn (x1 , x2 , . . . xn ).
Let us begin the discussion by reviewing some terminology first.
2
Preliminaries
Consider the following system of ordinary differential equations f:
x01 = f1 (x1 , x2 , . . . xn )
Heteroclinic Cycles
Cooper
x02 = f2 (x1 , x2 , . . . xn )
..
.
x0n = fn (x1 , x2 , . . . xn )
The ith nullcline is the set of points (x1 , x2 , . . . xn ) for which
fi ((x1 , x2 , . . . xn ) = 0.
The intersection(s) of all the nullclines i.e the point(s) where x0i = 0 for i ∈
{1 . . . n} are called the fixed points of the system. We are concerned with
the behavior of the system at the fixed points. In particular, we desire to find
the stable, unstable, and center manifolds of the system at the fixed points.
It is the interplay between the stable and unstable manifolds of different fixed
points that suggest the existence of heteroclinic cycles.
Our main tools for analysis of these systems lie within the realm of stability
theory. Before we begin, we review these tools. These methods can be found
in []
2.1
The Flow
Consider the system
x0 = f(x). (∗)
x(0) = x0 .
The definition of the flow of the system (∗) is given below:
Let E be an open subset of Rn and let f ∈ C 1 (E). For x0 ∈ E, let φ(t, x0 )
be solution of the initial value problem (*) defined on its maximal interval of
existence I(x0 ). Then for t ∈ I(x0 ), the set of mappings φt defined by
φt (x0 ) = φ(t, x0 )
is called the flow of the differential equation (*).
The standard approach for analysis of the behavior of the system (∗) at a
fixed point x0 is to near the fixed points is to compute the stable, unstable,
and center manifolds tangent to x0 . For the sake of simplicity, we assume
that x0 = 0 for the remainder of the discussion. The open set E, of the
preceding definition decomposes as:
E = Es ⊕ Eu ⊕ Ec
Dynamics at the Horsetooth
2
Vol. 2, 2010
Heteroclinic Cycles
Cooper
where
E s = {x ∈ E : limt→∞ φt (x) = x0 },
E u = {x ∈ E : limt→−∞ φt (x) = x0 },
3
Stable and Unstable Manifolds
Theorem Let E be an open subset of Rn containing the origin, let f ∈ C 1 (E),
and let φt be the flow of the system (*). Suppose that f(0) = 0 and that
Df(0) has k eigenvalues with negative real part and n - k eigen values with
positive real part. Then there exists a k - dimensional differentiable manifold
S tangent to the stable subspace E s of the linear system (*) at 0 such that
for all t ≥ 0, φt (S) ⊂ S and for all x0 ∈ S,
limt→∞ φt (x0 ) = 0;
and there exists an n - k dimensional differentiable manifold U tangent to
the unstable subspace E u of (*) at 0 such that for all t ≥ 0, φt (U ) ⊂ U and
for all x0 ∈ U ,
limt→−∞ φt (x0 ) = 0.
4
Limit Sets
[1] gives the the definition of a limit set as: Let E be an open subset of Rn .
A point p ∈ E is an ω limit point of the trajectory φ(·, x) of the system (*)
if there is a sequence
limn → ∞ φ(tn , x) = p.
The set of all ω-limit points of a trajectory is called a ω limit set.
5
Heteroclinic Cycles
Let x1 , . . . xm represent relative equilibria of a vector field. If there exists
trajectories {y1 (t), . . . , ym (t)} such that yj (t) is forward asymptotic to xj+1
and backward asymptotic t xj then the collection of trajectories {xj , yj (t)}
is called a heteroclinic cycle.
Dynamics at the Horsetooth
3
Vol. 2, 2010
Heteroclinic Cycles
6
Cooper
Competitive Population Model ( 2 species model)
We consider competitive population models in the case of two populations.
We focus our attention to a special case called the Lotka-Volterra equations
given for two populations x and y by:
x0 = x(x0 + ax + by)
y 0 = y(y0 + cx + dy).
• x0 and y0 correspond to the initial populations of x and y, respectively.
• The parameter a corresponds to the effect of population x on itself.
• The parameter b corresponds to the effect of population y on population
x.
• The parameter c corresponds to the effect of population x on population
y.
• The parameter d corresponds to the effect of population y on itself.
Since we are considering competitive populations, we assume that each of the
parameters a, b, c, d are all negative. We consider the special case when a
and d = -1. Equivalently, we may write the system as:
x0 = x(x0 − x − by)
y 0 = y(y0 − cx − y).
6.1
Find the Nullclines
We find the nullclines by finding the values of x and y such that x0 = 0 and
y 0 = 0. First, if x0 = 0, then
x0 = x(x0 − x − by) ⇒
•
x = 0, or
•
(x0 − x − by) = 0.
Dynamics at the Horsetooth
4
Vol. 2, 2010
Heteroclinic Cycles
Cooper
Then x0 = 0 on the line
y =
−1
x0
x +
.
b
b
Similarly, if y 0 = 0, then
y 0 = y(y0 − cx − y) ⇒
•
y = 0, or
•
(y0 − cx − y) = 0.
Then y 0 = 0 on the line
y = −cx + y0 .
The nullclines are the lines Ny (through the points (0, y0 ) and ( yc0 , 0), and Nx
(through the points (x0 , 0) and (0, yb0 ). To find the fixed point of the system,
we simply compute where the two nullclines intersect.
−cx + y0 =
then
x∗ =
−1
x0
x +
,
b
b
x 0 − y0 b
.
1 − cb
640 graphic 1.jpg
7
Competitive Population Model (3 species )
We now consider competitive population models in the case of three
populations. We focus our attention to a special case of the Lotka-Volterra
equations given for three populations x, y, and z by:
x0 = x(1 − x − ay − bz)
y 0 = y(1 − bx − y − az)
Dynamics at the Horsetooth
5
Vol. 2, 2010
Heteroclinic Cycles
Cooper
z 0 = z(1 − ax − by − z).
we also require that:
1.
0 < b < 1 < a,
2.
2 < a + b.
7.1
Compute the Fixed Points and Nullclines
We begin by finding the x of R3 such that
fi (x) = 0 f or i = 1, 2, 3.
The fixed points are given by:
x1 = (1, 0, 0),
x2 = (0, 1, 0),
x3 = (0, 0, 1),
1
1
1
x4 = (
,
,
),
1+a+b 1+a+b 1+a+b
. Here are some of the nullclines:
x = 0,
y = 0,
z = 0,
x + y + z = 1.
640 graphic 2.jpg
Dynamics at the Horsetooth
6
Vol. 2, 2010
Heteroclinic Cycles
7.2
Cooper
Analysis of f at x1
We compute
Df(x1 )
to arrive at the following eigenvalues and eigenvectors:
−1
v1 (x1 ) = (1, 0, 0)T ,
,
a
λ2 (x1 ) = 1 − b, v2 (x1 ) = (
, 1, 0)T ,
b−2
b
, 0, 1)T .
λ3 (x1 ) = 1 − a, v3 (x1 ) = (
a−2
By the restrictions given to a and b, we know that λ2 (x1 ) > 0 and λ3 (x1 ) <
0. This allows us to compute:
a
E s (x1 ) = {(x, 0, z) · (
, 1, 0)T : x, z ∈ R} ∪ {(x, 0, 0) · (1, 0, 0)T ,
b−2
a
E u (x1 ) = {(0, y, 0) · (
, 1, 0) : y ∈ R},
b−2
λ1 (x1 ) =
7.3
Analysis of f at x2
We compute
Df(x2 )
to arrive at the following eigenvalues and eigenvectors:
λ1 (x2 ) = −1, v1 (x2 ) = (0, 1, 0)T ,
a−2
, 1, 0)T ,
b
b
λ3 (x2 ) = 1 − b, v3 (x2 ) = (0,
, 1)T .
a−2
By the restrictions given to a and b, we know that λ3 (x2 ) > 0 and λ2 (x2 ) <
0. This allows us to compute:
a
E s (x2 ) = {(x, y, 0) · (
, 1, 0)T : x, y ∈ R} ∪ {(0, y, 0) · (0, 1, 0)T ,
b−2
a
E u (x2 ) = {(0, 0, z) · (0,
, 1)T : z ∈ R}.
b−2
λ2 (x2 ) = 1 − a, v2 (x2 ) = (
Dynamics at the Horsetooth
7
Vol. 2, 2010
Heteroclinic Cycles
7.4
Cooper
Analysis of f at x3
We compute
Df(x3 )
to arrive at the following eigenvalues and eigenvectors:
λ1 (x3 ) = −1, v1 (x3 ) = (0, 0, 1)T ,
a−2 T
, 1) ,
b
b
λ3 (x3 ) = 1 − b, v3 (x3 ) = (
, 0, 1)T .
a−2
By the restrictions given to a and b, we know that λ3 (x3 ) > 0 and λ2 (x3 ) <
0. This allows us to compute:
λ2 (x3 ) = 1 − a, v2 (x3 ) = (0,
E s (x3 ) = {(0, y, z) · (0,
a−2 T
, 1) : y, z ∈ R} ∪ {(0, 0, z) · (0, 0, 1)T ,
b
E u (x2 ) = {(0, 0, z) · (
7.5
b−2
, 0, 1)T : z ∈ R}.
a
Analysis of f at x4
We compute
Df(x4 )
to arrive at the following eigenvalues and eigenvectors:
λ1 (x4 ) =
−1
, v1 (x3 ) = (1, 1, 1)T ,
1+a+b
(λ2 (x4 )) = ω1 , v2 (x4 ),
λ3 (x4 ) = ω2 , v3 (x4 ).
By the restrictions given to a and b, we know that the real part of λ2 (x4 ) >
0 and λ3 (x4 ) > 0. This allows us to compute:
E s (x3 ) = {(x, y, z) · (1, 1, 1)T : x, y, z ∈ R}
E u (x2 ) = {(x, y, z) ∈
/ E s (x4 ).
Now observe that:
E u (x1 ) ⊂ E s (x2 ) ⊂ {(x, y, 0) : x, y ∈ R
Dynamics at the Horsetooth
8
Vol. 2, 2010
Heteroclinic Cycles
Cooper
E u (x2 ) ⊂ E s (x3 ) ⊂ {(0, y, z) : y, z ∈ R
E u (x3 ) ⊂ E s (x1 ) ⊂ {(x, 0, z) : x, z ∈ R.
Thus, all three fixed points are connected by stable and unstable manifolds.
The dynamics of the system can be described as follows: The orbit cycles
around the fixed points x1 , x2 , x3 in succession. It follows that the LotkaVolterra system with three competitive populations contains heteroclinic
cycles for the parameters given above. We now check the stability of this
cycle.
7.6
Stability of Heteroclinic Cycles
Let cj , ej , tj denote the real parts of the, strongest contracting eigenvalue,
the strongest expanding eigenvalue, the weakest contracting eigenvalue,
respectively. Then [2] has stated the following result:
Theorem Let cj , ej , tj denote the real parts of the, strongest contracting
eigenvalue, the strongest expanding eigenvalue, the weakest contracting
eigenvalue, respectively of a system having a heteroclinic cycle. Then the
heteroclinic cycle is stable provided that:
m
Y
min(cj , ej − tj ) >
j=1
7.7
m
Y
ej .
j=1
Finding Heteroclinic Cycles
Finding heteroclinic cycles in general is a very difficult problem. We
investigate the behavior of the Lotka-Volterra system with the parameters:
1
1
a = 1 + 100000
b = 1 - 100000
, with the initial condition
1 1 1
x0 = ( , , ).
3 6 6
The solution is given by:
√
1
3t
3t
x(t) = 1/9e
+ + e Cos(
),
3
100000
√
√
√
1
3t
3t
−3t
3t
y(t) = 1/9e
− + e Cos(
) − 3Sin(
,
3
100000
100000
√
√
√
1
3t
3t
−3t
3t
z(t) = 1/9e
− + e Cos(
) + 3Sin(
.
3
100000
100000
−3t
The trajectories for the solution are given by:
Dynamics at the Horsetooth
9
Vol. 2, 2010
Heteroclinic Cycles
Cooper
Figure 1: (t, x(t))
640 prj xt.jpg
Figure 2: (t , y(t))
640 project yt.jpg
Figure 3: (t, z(t))
640 project zt.jpg
Dynamics at the Horsetooth
10
Vol. 2, 2010
Heteroclinic Cycles
Cooper
Figure 4: 1,000,000 iterations
640 project 106 itrs.jpg
Figure 5: 108 iterations
640 project 108 itrs.jpg
Dynamics at the Horsetooth
11
Vol. 2, 2010
Heteroclinic Cycles
Cooper
Figure 6: 109 iterations
640 project 109.jpg
Figure 7: 5 × 1010 iterations
640 project 51010.jpg
Dynamics at the Horsetooth
12
Vol. 2, 2010
Heteroclinic Cycles
Cooper
Figure 8: 2.5 × 1011 iterations
640 project (2)1011 itrs.jpg
Of the system above, we see that the strongest contracting eigenvalue
1
1
is -1, and the strongest expanding eigenvalue is 1-(1- 100000
) = 100000
, and
1
1
the weakest contracting eigenvalue 1-(1+ 100000 ) = - 100000 for each of the
equilibria. Using the above theorem we see that this cycle is stable.
References
[1] Perko, L. Differential Equations and Dynamical Systems. Springer,
Berlin (2001). 107-108.
[2] Ladd. J. Heteroclinic Cycles in an Eight Dimensional Vector Field Derived
from Nematic Electroconvection. Thesis, Department of Mathematics,
Colorado State University, 2004.
Dynamics at the Horsetooth
13
Vol. 2, 2010