# 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**.

2. The saddle node bifurcation

2. The saddle node bifurcation

The lac operon

a = 0.006

Exercise!

2. The saddle node bifurcation

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**

2. The 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?**

2. The saddle node bifurcation

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

2. The saddle node bifurcation

Equilibria? X' = 0

2. The saddle node bifurcation

Equilibria

low k

2. The saddle node bifurcation

larger k: we increase r

refuge

outbreak

2. The saddle node bifurcation

Exercise!

2. The saddle node bifurcation

- 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

2. The saddle node bifurcation

- 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