Two-species competition

The Lotka-Volterra Model

Working with differential equations to predict population dynamics

Testing the consequences of

species interactions: Georgii Frantsevich

Gause (b. 1910)

Paramecium caudatum

Paramecium aurelia

Gause’s competitive exclusion principle:

Two species competing for the exactly same resources cannot stably coexist if other ecological factors relevant to the organism remain

constant. One of the two competitors will always outgrow the other, leading to the extinction of one of the competitors: Complete

competitors cannot coexist.

If two species utilize sufficiently separate niches, the competitive effects of one species on another decline enough to allow stable

coexistence.

Overcoming Gause’s exclusion principle:

LOTKA AND VOLTERRA(Pioneers of two-species models)

Alfred J. Lotka (1880-1949)Chemist, ecologist, mathematicianUkrainian immigrant to the USA

Vito Volterra (1860-1940)Mathematical Physicist

Italian, refugee of fascist Italy

LOTKA AND VOLTERRA(Pioneers of two-species models)

Alfred J. Lotka (1880-1949)Chemist, ecologist, mathematicianUkrainian immigrant to the USA

Vito Volterra (1860-1940)Mathematical Physicist

Italian, refugee of fascist Italy

Let’s say, two species are competing for the same limited space:

The two species might have a different carrying capacities.

251 K 1002 K

In what ways can the species be different?

The two species might have different maximal rates of growth.

time

time

21 r

42 r

per year

per year

When alone each species might follow the logistic growth model:

1

1111

1

K

NKNr

dt

dN

2

2222

2

K

NKNr

dt

dN

For species 1:

For species 2:

When alone each species might follow the logistic growth model:

1

1111

1

K

NKNr

dt

dN

2

2222

2

K

NKNr

dt

dN

For species 1:

For species 2:

How do we express the effect one has on the other?

1 light blue square has the same effect as four dark blue squares.

1 dark blue squares has the same effect as 1/4 light blue square.

1

211

111 4

1

K

NNKNr

dt

dN N1N2

The effect of the small purple pecies on the growth rate of the large green species:

2

12222

2 4

K

NNKNr

dt

dNN2 N1

The effect of the large orange species on the growth rate of the small blue species:

The Lotka-Volterra two-species competition model:

1

21111

1

K

NNKNr

dt

dN

2

12222

2

K

NNKNr

dt

dN

Two state variables: N1 and N2, which change in response to one another.

6 parameters: r1, K1, ,r2 ,K2 ,which stay constant.

and are new to us: they are called interspecific competition coefficients.

The Lotka-Volterra Model is an example of a system of differential equations:

),...,(

.

.

),...,(

),...,(

21

212

211

mm

m

m

NNNqdt

dN

NNNgdt

dN

NNNfdt

dN

(differential equations)

What are the equilibria?What stability properties do the equilibria have? Are there complex dynamics and strange attractors for some parameter values?

Analysis tools for systems of two equations:

Isoclines

0),(

0),(

212

211

NNgdt

dN

NNfdt

dN

Definition of the zero-growth isocline:

The set of all {N1,N2} pairs that make the rate of change for either N1 or N2 equal to zero.

defines the N1 isocline

defines the N2 isocline

GRAPHICAL ANALYSIS OF TWO-DIMENSIONAL SYSTEMS:

State space graph: a graph with the two state variables on the axes:

N1

N2Use this graph to plot zero-growth isoclines, which satisfy:

0

0

2

1

dt

dNdt

dN“N1 isocline”

“N2 isocline”

N1

N2

This is called a state space graph.

N2 isocline

N1 isocline

K2

K2

K1

K1

112

NKN

122 NKN

ISOCLINES:

N2 isocline

N1 isocline

The equilibrium!

N1

N2

K2

K2

K1

K1

N1

N2

The N1 isocline

dN1 /dt = 0

K1

K1

dN1/(N1dt) < 0

dN1/(N1dt) > 0

N1

N2

The N2 isocline

dN2 /dt = 0

K2

K2

dN2/(N2dt) < 0

dN2/(N2dt) > 0

N2

N2 isoclineN1 isocline

This equilibrium is stable!

N1

K2

K2

K1

K1

N1

N2

N2 isocline

N1 isocline

K2

K2

K1

K1

Case 2: • an unstable equilibrium• only one of the two species survives• which one survives depends on initial population densities.

Case 3: • no two-species equilibrium• species 1 always wins

N1

N2

K2

K2

K1

K1

Case 4: • no two-species equilibrium• species 2 always wins

N1

N2

K2

K2

K1

K1

K2

K2

K1

K1

K2

K2

K1

K1

Case 3:K2<K1/ and K1>K2/

Case 4:K2>K1/ and K1<K2/

N2

N1

K2

K2

K1

K1

Case1 :K2<K1/ and K1<K2/

N1

N2

K2

K2

K1

K1

Case 2:K2>K1/ and K1>K2/

GENERALIZED STABILITY ANALYSIS

),...,(

.

.

),...,(

),...,(

21

212

211

mm

m

m

NNNqdt

dN

NNNgdt

dN

NNNfdt

dN

Step 1: determine all equilibrium points by setting all rates of change to zero and solve for N.

Step2: Determine rates of change for each variable at the equilibrium.

Step3: Determine for every state variable, when in a position just off the equilibrium, if the are attracted to or repelled by the equilibrium.

Step 1: We rescale equations with respect to the equilibrium of interest:

Define: x1(t)= N1(t) – N1* x2(t)= N2(t) – N2* ,

Step 2: We “linearize” the rates of change at the equilibrium:

2221212

2121111

xaxadt

dx

xaxadt

dx

Or, in matrix script:

xJx

J is called the Jacobian matrix or community matrix in ecology.

*2

*1 ,1

11

NNN

fa

*2

*1 ,2

12

NNN

fa

*2

*1 ,1

21

NNN

ga

*2

*1 ,2

22

NNN

ga

Stability identified by determining all partial derivatives, evaluated at the equilibrium N1*, N2*:

Step 3: We find the Jacobian Matrix by finding the partial derivatives of all differential equations with respect to all state variables:

),(

),(

212

211

NNgdt

dN

NNfdt

dN

We already know that the eigenvalues of such a matrix can be determined by solving:

x1 = a11x1+a12x2

x2 = a21x1+a22x2

As in Leslie matrix analysis, the eigenvalues determine the stability of the equilibrium.

2221

1211

aa

aaJ

Recall that eigenvalues (roots of polynomials) have the form= a + bi, where i = 1

Stability Real (b=0) and a<0

Real (b=0) and a>0

Complex (b≠0) and a<0

Complex (b≠0) and a>0

Purely imaginary (a=0)

Stable node 1 and2

Saddle point (unstable)

1 2

Stable focus 1 and2

Unstable focus 1 and2

Linear stability analysis insufficient

1

and2

STABLE NODE:Equilibrium is attracting.

The pathway of approach is monotonic (straight)

N1

N2 N1 isocline

N2 isocline

1 and2

are both real and

negative

N1

N2

N1 isocline

N2 isocline

SADDLE POINT:Equilibrium is unstable.

The saddle point is attracting in one direction and repelling in another.

1 and2

are both real and one is

negative, the other is

positive

N1

N2

N1 isocline

N2 isocline

STABLE FOCUS:Equilibrium is stable.

The pathway of approach is oscillatory.

1 and2

are complex and the real

part is negative.

N1

N2

N1 isocline

N2 isocline

UNSTABLE FOCUS:Equilibrium is unstable.

The pathway away from the equilibrium is oscillatory.

1 and2

are complex and the real

part is positive.

N1

N2

N1 isocline

N2 isocline

NEUTRAL STABILITY:Equilibrium is neither stable nor unstable.

The pathway is oscillatory and unchanging.

1 and2

are purely imaginary.

Summary:

1. We search for equilibria to determine the long-term asymptotic behavior of dynamical systems. This is not limited to population models. We can ask this about all dynamic models.

2. We use local stability analysis to determine the stability of equilibrium points. This is done by linearizing the dynamical system near the equilibrium (or near each equilibrium).

3. The matrix of partial differentials that represent the linearized version of the dynamical system around a given equilibrium point is called the Jacobian, an n x n matrix for n differential equations.

4. The eigenvalues of this matrix determine the stability of the equilibrium.