# Week 3: equilibrium points

1. Equilibrium points in 1D

1. Equilibrium points in 1D

logistic equation:

• In some species, a minimal number of animals is necessary to ensure the survival of the group.
• For example, African hunting dogs, require the help of others to bring up their young. As a result, their reproductive success declines at low population levels, and a population that’s too small may go extinct.
• This decline in per capita population growth rates at low population sizes is called the Allee effect.

1. Equilibrium points in 1D

logistic equation with allee effect

k = 100

Where is the equilibrium?

Is this the only equilibrium?

r= 0.1, k = 100, a = 5

1. Equilibrium points in 1D

logistic equation with allee effect

k = 100

r= 0.1, k = 100, a = 5

1. Equilibrium points in 1D

more simple: logistic equation

What are the equilibrium points?

X = 0 and X = k

What is the stability of the equilibria?

X = 0: unstable

X = k: stable

1. Equilibrium points in 1D

different methods to determine stability:

1. Test points or analyzing the equation

2. Linear stability analysis

compute derivatives in the points

>0 unstable

<0 stable

1. Equilibrium points in 1D

different methods to determine stability:

2. Linear stability analysis

compute derivatives in the points

>0 unstable

<0 stable

Deep truth: the stability of an equilibrium point of a vector field is determined by the linear approximation to the vector field at the equilibrium point.

This principle, called the Hartman–Grobman theorem, enables us to use linearization to determine the stability of equilibria.

1. Equilibrium points in 1D

We will generalize this for higher dimensions!

Some exercises:

1. Equilibrium points in 1D

1. What are the equilibria?

X = 0; X = k; X = a

2. Stability via derivatives? (a<k)

1. Equilibrium points in 1D

• Growth of tumors
• Some special cases

Some exercises (3.1):

2. Equilibrium points in 2D

2. Equilibrium points in 2D

X' = X and Y' = Y

X' = X and Y' = 2Y

Different types

2. Equilibrium points in 2D

X' = X and Y' = Y

X' = X and Y' = 2Y

unstable node

Different types

2. Equilibrium points in 2D

X' = -X and Y' = -Y

X' = -X and Y' = -2Y

stable node

Different types

2. Equilibrium points in 2D

stable or unstable?

The only way to approach the equilibrium point is to start exactly on the stable axis.

Since the typical trajectory does not lie exactly on the stable axis, a saddle point is considered unstable.

Different types

2. Equilibrium points in 2D

X' = V

V' = -X

spring

spring with friction

X' = V

V' = -X - V

spring with negative friction

X' = V

V' = -X + V

Different types

2. Equilibrium points in 2D

X' = V

V' = -X

spring

spring with friction

X' = V

V' = -X - V

spring with negative friction

X' = V

V' = -X + V

stable spiral

unstable spiral

center

Different types

2. Equilibrium points in 2D

Determine type linear system

simplest case:

matrix differentiation:

solution:

what are the eigenvalues and eigenvectors?

what can you conclude about the solutions?

2. Equilibrium points in 2D

Determine type linear system

2. Equilibrium points in 2D

Determine type linear system

Decompose system into its eigenvalues and eigenvectors.

How to solve this case?

General case:

If you start at a certain         (length of eigenvector can be chosen)

M

2. Equilibrium points in 2D

Determine type linear system

make exercise 3.2.1 first bullet point

2. Equilibrium points in 2D

What if eigenvalues are complex?

Determine type linear system

General case:

remark: still has (0,0) as equilibrium

Eigenvalues will be complex conjugates: example

Eigenvectors will also be complex: leading to these solutions

2. Equilibrium points in 2D

Determine type linear system

Eigenvectors will also be complex: leading to these solutions

We are only interested in the real solutions

2. Equilibrium points in 2D

Determine type linear system

Eigenvectors will also be complex: leading to these solutions

We are only interested in the real solutions

The general solution is therefore:

(Same if you start from other eigenvalue)

2. Equilibrium points in 2D

Determine type linear system

The general solution is therefore:

If a < 0:

stable spiral

unstable spiral

center

If a > 0:

If a = 0:

2. Equilibrium points in 2D

Non-linear system

X' = f(X,Y)

Y' = g(X,Y)

The Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation -a natural simplification of the system- is effective in predicting qualitative patterns of behaviour.

2. Equilibrium points in 2D

Non-linear system

Example: the Rayleigh Oscillator

Rayleigh modeled the reed of the clarinet as a thin, flexible wand attached to a solid object, with a mass on its end. The clarinetist supplies energy to the system by blowing along the long axis of the wand.

2. Equilibrium points in 2D

Non-linear system

Example: the Rayleigh Oscillator

If we bend the reed up or down, it will oscillate in a damped manner and eventually return to the equilibrium position.

X' = V

V' = -X - V

take m = 1

2. Equilibrium points in 2D

Non-linear system

Example: the Rayleigh Oscillator

When the clarinetist blows on the reed, the situation is changed!

Blowing supplies energy to the system and therefore acts like the opposite of friction

2. Equilibrium points in 2D

Non-linear system

Example: the Rayleigh Oscillator

Of course, a trajectory that spirals out forever isn’t realistic.

• If the wand is moving slowly (V is small), then blowing on it will actually accelerate it, so the force of the breath is in the same direction as the motion and adds energy to the system.
• But if the velocity of the wand is high, the blowing produces conventional friction (due to air resistance), which retards the motion.

2. Equilibrium points in 2D

Non-linear system

Example: the Rayleigh Oscillator

make exercise 3.2.1 second bullet point

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

1. None of the eigenvalues of the linearized system can be zero!

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

1. None of the eigenvalues of the linearized system can be zero!

eigenvector

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

1. None of the eigenvalues of the linearized system can be zero!

eigenvector (X, Y) = (1, 3)

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

1. None of the eigenvalues of the linearized system can be zero!

eigenvector (X, Y) = (1, 3) = V

eigenvector (X, Y) = (2, 1) = U

A system like this is not robust!

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

1. None of the eigenvalues of the linearized system can be zero!

2. Equilibrium points in 2D

Non-linear system

When Linearization Fails

2. Purely imaginary eigenvalues

center

Similar as before: a center will become stable spiral or unstable spiral

3. Equilibrium points in N dim

3. Equilibrium points in N dim

eigenvalues:

3. Equilibrium points in N dim

Example:

• unstable spiral in X and Y ,
• a stable node in Z
• => 3D unstable equilibrium point

4. The method with the nullclines

4. The method with the nullclines

Example: competition between moose and deer

• Deer = D, Moose = M
• No environmental limitations: deer population would grow at a per capita rate 3, moose at 2
• Each animal competes for resources within its own species: -D^2 for deer; -M^2 for moose
• Deer compete with moose: impact for deer is 1
• Moose compete with deer: impact for moose is 0.5

4. The method with the nullclines

Example: competition between moose and deer

What are the equilibrium points?

Set D' = 0; M' = 0:  (D*, M*) =  (0, 0); (0, 2); (3, 0); (2, 1)

Easy way to find them: nullclines

D' = 0:    D = 0  and M = 3 - D

M' = 0:   M = 0 and M = 2 - D/2

4. The method with the nullclines

Example: competition between moose and deer

4 sections

Equilibrium is stable!

4. The method with the nullclines

Example: competition between moose and deer

4 sections

Equilibrium is stable!

4. The method with the nullclines

4. The method with the nullclines

Why bother with nullclines?

In case we do not know exact parameters, we cannot simulate

This is a vertical line and a line from (0, rD/kD) to (rD/cD, 0)

In this way, plotting nullclines can allow us to sketch an approximate vector field and get a sense of the system’s dynamics without having numerical parameter values

4. The method with the nullclines

switch

E. Coli

• E. coli can use extracellular lactose for energy,
• In order to import extracellular lactose into the cell, cell needs a transport protein -> lactose permease, to transport the extracellular lactose across the cell boundary.
• Making lactose permease costs a lot of resources. Only advantageous for cell to make this protein in large amounts only when lactose concentrations are high.
• In that case, it wants to “switch on” lactose permease production.

4. The method with the nullclines

switch

X' = lactose import − lactose metabolism

X = concentration internal lactose

kX

4. The method with the nullclines

switch

X' = lactose import − lactose metabolism

X = concentration internal lactose

kX

sigmoid

4. The method with the nullclines

switch

X' = lactose import − lactose metabolism

X = concentration internal lactose

4. The method with the nullclines

switch

depending on startconcentrations: switch

4. The method with the nullclines

The Collins Genetic Toggle Switch

• Phage lambda = virus -> infects the bacterium E. coli
• It faces an uncertain environment:
1. In a healthy cell: virus -> into the genome of E. Coli -> passed the progeny (nakomelingen) = lysogenic growth = default mode.
2. Sick or damaged cell: virus -> lytic growth = virus uses host to produce hundreds of copies of the virus -> cell bursts
• How does the viral cell sense unhealthy host -> lytic growth?
• Two genes in virus:
repressor: produces protein R
control of repressor: produces protein C
• R and C form feedback loops that inhibit their own production as well as that of the other.

switch

4. The method with the nullclines

The Collins Genetic Toggle Switch

switch

4. The method with the nullclines

The Collins Genetic Toggle Switch

switch

lytic state

lysogenic state (default)

d_r = 0.02

4. The method with the nullclines

The Collins Genetic Toggle Switch

switch

lytic state

d_r = 0.2 when the cell is damaged

4. The method with the nullclines

The Collins Genetic Toggle Switch

• In 2000, a group at Boston University led by James Collins
using genetic engineering techniques in the bacterium E. coli to construct two genes that neatly repressed each other:

Exercise!  Investigate this model! Bistable behavior!

switch

4. The method with the nullclines

switch

The concept of a switch plays in important role in many biologically processes:

• Hormone or enzyme production is “switched on” by regulatory mechanisms when certain signals pass “threshold” values.
• Cells in development pass the switch point, after which they are irreversibly committed to developing into a particular type of cell (say, a neuron or a muscle cell). This is of critical importance in both embryonic development and in the day-to-day replacement of cells.
• In neurons and cardiac cells, the voltage V is stable unless a stimulus causes V to pass a “threshold,” which switches on the action potential.

#### Week 3: equilibria

By Nele Vandersickel

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