# Week 4-5: Bifurcations

1. Transcritical bifurcation

1. Transcritical bifurcation

• A change in the parameters -> can result in a qualitative change in the equilibrium points of a system

Exercise!

1. Transcritical bifurcation

A pair of equilibrium points approach each other, collide, and exchange stability as a parameter smoothly varies

= transcritical bifurcation.

The lac operon

a = 0.006

Exercise!

The lac operon

One equilibrium. When r reaches a critical value, however, a new equilibrium point is born and immediately splits into two, one stable and one unstable = saddle node bifurcation

The spruce budworm: is a caterpillar that inhabits the forests of the northeastern United States

Typically: low numbers,

Sometimes: outbreak: dramatic increase population, to the point of defoliating large tracts of forest.

Why do these outbreaks happen?

X = budworm population

X is eaten by birds.

->  X small: little hunting

-> As X rise:  more hunting,

-> certain large X:  birds have eaten their

fill, hunting saturates

-> larger X: no more extra hunting anymore

Equilibria? X' = 0

Equilibria

low k

larger k: we increase r

refuge

outbreak

Exercise!

• lower k: defoliants
• lower r: the reproductive rate  (as they try with mosquitos, by breeding them with infertile females)
• lower X: insecticide
• option 1: lowest cost: follow red line
• option 2: follow red line partially and then lower X to stable equilibrium

• option 1: lowest cost: follow red line
• option 2: follow red line partially and then lower X to stable equilibrium

Exercise!

3. Pitchfork bifurcation

less common in biology

3. Pitchfork bifurcation

Opinion model

N = people with negative opinion

P = people with positive opinion

We set N + P = 2m (total population)

the imbalance toward positive:

• X = 0: P = N
• X = 1: P = 2m
• X −1: N = 2m

3. Pitchfork bifurcation

Opinion model

N + P = 2m

X  =  (2m - N) - N

2m

3. Pitchfork bifurcation

Opinion model

bandwagon effect: how sensitive is KNP to the degree of positive tilt

This says that the larger the per capita conversion rate, the more sensitive it is to the degree of positive tilt.

3. Pitchfork bifurcation

Opinion model

3. Pitchfork bifurcation

Opinion model

analyze this model!

3. Pitchfork bifurcation

Opinion model

analyze this model!

Bifurcations

Bifurcations: explaining system in a qualitative way

<-> quantitative description of many other subjects in physics

Very often in science: look for the qualitative behavior

•  Can deer–moose coexist? Do they sometimes not coexist?
•  Why does the lac operon have a bistable switch? What causes it to flip from mode A to mode B?
•  In the model of public opinion, why did the middle “balanced opinions” equilibrium become unstable and the two extremes become stable?
•  Why does the spruce budworm have outbreaks?

If you have time, you can make the extra exercises!

#### Week 4- 5: bifurcations

By Nele Vandersickel

• 33