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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.