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 ( ) Chemist, ecologist, mathematician Ukrainian immigrant to the USA Vito Volterra ( ) Mathematical Physicist Italian, refugee of fascist Italy

LOTKA AND VOLTERRA (Pioneers of two-species models) Alfred J. Lotka ( ) Chemist, ecologist, mathematician Ukrainian immigrant to the USA Vito Volterra ( ) 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. In what ways can the species be different?

The two species might have different maximal rates of growth. time per year

When alone each species might follow the logistic growth model: For species 1: For species 2:

When alone each species might follow the logistic growth model: 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.

N 1 N2N2 The effect of the small purple pecies on the growth rate of the large green species: N2N2 N 1 The effect of the large orange species on the growth rate of the small blue species:

The Lotka-Volterra two-species competition model: Two state variables: N 1 and N 2, which change in response to one another. 6 parameters: r 1, K 1, ,r 2,K 2,  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: (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 Definition of the zero-growth isocline: The set of all {N 1,N 2 } pairs that make the rate of change for either N1 or N2 equal to zero. defines the N 1 isocline defines the N 2 isocline

GRAPHICAL ANALYSIS OF TWO-DIMENSIONAL SYSTEMS: State space graph: a graph with the two state variables on the axes: N1N1 N2N2 Use this graph to plot zero- growth isoclines, which satisfy: “N 1 isocline” “N 2 isocline”

N1N1 N2N2 This is called a state space graph. N 2 isocline N 1 isocline K2K2 K2K2 K1K1 K1K1 ISOCLINES:

N 2 isocline N 1 isocline The equilibrium! N1N1 N2N2 K2K2 K2K2 K1K1 K1K1

N1N1 N2N2 The N 1 isocline dN 1 /dt = 0 K1K1 K1K1 dN 1 /(N 1 dt) < 0 dN 1 /(N 1 dt) > 0

N1N1 N2N2 The N 2 isocline dN 2 /dt = 0 K2K2 K2K2 dN 2 /(N 2 dt) < 0 dN 2 /(N 2 dt) > 0

N2N2 N 2 isocline N 1 isocline This equilibrium is stable! N1N1 K2K2 K2K2 K1K1 K1K1

N1N1 N2N2 N 2 isocline N 1 isocline K2K2 K2K2 K1K1 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 N1N1 N2N2 K2K2 K2K2 K1K1 K1K1

Case 4: no two-species equilibrium species 2 always wins N1N1 N2N2 K2K2 K2K2 K1K1 K1K1

K2K2 K2K2 K1K1 K1K1 K2K2 K2K2 K1K1 K1K1 Case 3: K 2 K 2 /  Case 4: K 2 >K 1 /  and K 1 <K 2 /  N2N2 N1N1 K2K2 K2K2 K1K1 K1K1 Case1 : K 2 <K 1 /  and K 1 <K 2 /  N1N1 N2N2 K2K2 K2K2 K1K1 K1K1 Case 2: K 2 >K 1 /  and K 1 >K 2 / 

GENERALIZED STABILITY ANALYSIS 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: x 1 (t)= N 1 (t) – N 1 * x 2 (t)= N 2 (t) – N 2 *, Step 2: We “linearize” the rates of change at the equilibrium: Or, in matrix script: J is called the Jacobian matrix or community matrix in ecology.

Stability identified by determining all partial derivatives, evaluated at the equilibrium N 1 *, N 2 *: Step 3: We find the Jacobian Matrix by finding the partial derivatives of all differential equations with respect to all state variables:

We already know that the eigenvalues of such a matrix can be determined by solving: x 1 = a 11 x 1 +a 12 x 2 x 2 = a 21 x 1 +a 22 x 2 As in Leslie matrix analysis, the eigenvalues determine the stability of the equilibrium.

Recall that eigenvalues (roots of polynomials) have the form  = a + bi, where i = StabilityReal (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 and  2 Saddle point (unstable) 1 2 Stable focus 1 and  2 Unstable focus 1 and  2 Linear stability analysis insufficient 1 and  2

STABLE NODE: Equilibrium is attracting. The pathway of approach is monotonic (straight) N1N1 N2N2 N 1 isocline N 2 isocline 1 and  2 are both real and negative

N1N1 N2N2 N 1 isocline N 2 isocline SADDLE POINT: Equilibrium is unstable. The saddle point is attracting in one direction and repelling in another. 1 and  2 are both real and one is negative, the other is positive

N1N1 N2N2 N 1 isocline N 2 isocline STABLE FOCUS: Equilibrium is stable. The pathway of approach is oscillatory. 1 and  2 are complex and the real part is negative.

N1N1 N2N2 N 1 isocline N 2 isocline UNSTABLE FOCUS: Equilibrium is unstable. The pathway away from the equilibrium is oscillatory. 1 and  2 are complex and the real part is positive.

N1N1 N2N2 N 1 isocline N 2 isocline NEUTRAL STABILITY: Equilibrium is neither stable nor unstable. The pathway is oscillatory and unchanging. 1 and  2 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.