5 Parameter, Bifucation, Phase Diagram

Question 1

What determines if the population will have an epidemic outbreak?

Answer 1

R0 if bigger than 1 = epidemic outbreak
So 1/R0 = less than 1

Question 2

How does changing the parameters change the fixed points?

Answer 2

Changing the parameters will change the stability of the fixed points

Question 3

What determines linear system stability?

Answer 3

The signs of the constants/parameters

Question 4

Name some qualitative changes to system stability

Answer 4

Number and stability of fixed points

Question 5

What does it mean to treat a parameter as invariable?

Answer 5

Rate of change = 0

x dot = 0
Because the parameter of x dot here never changes, we don’t need to write down an ODE for it

Question 6

Describe a positive auto-regulation with ultrasensitive and constant external stimulus in 2 equations

Answer 6

x dot = r + x^2
r dot = 0

Positive = sign
autoregulation x dot has x in it
Ultrasensitivity ^2
Constant external stimulus = the constant derivative = 0

Question 7

When can we draw a 2D phase portrait for a parameter?

Answer 7

When we treat it as a variable = can test a whole range of r

Question 8

What happens when r is larger than 0?
(x dot = r + x^2)

Answer 8

As r increases the U shape moves upwards

There are no fixed points because it does not intersect with x = 0

Question 9

What happens when r is EQUAL to 0?
(x dot = r + x^2)

Answer 9

The quadratic curve sits at x dot = 0 because
0 = x^2 and the square root of x = 0

We get a SEMI-STABLE fixed point

Question 10

Why does a semi-stable fixed point occur?

Answer 10

Because of the gradient change

Question 11

What happens when r is less than 0?
(x dot = r + x^2)

Answer 11

Semi-stable fixed point = splits in two

This is a saddle-node bifurcation = one status became 2

When x is less than 0 = stable
When x is more than = unstable

Question 12

Is saddle-node bifurcation symmetric?

Answer 12

NO, not in its dynamic or trajectory

Stable and unstable fixed points will always be the same distance away from centre line

Question 13

Describe a saddle-node bifurcation

Answer 13

Aka blue-sky bifurcation because the pair of fixed points appear out of the blue

Saddle-node bifurcation = creation or annihilation of fixed points

Question 14

What is this equation and what does it create? x dot = rx - x^2

Answer 14

It is a logistic model
The quadratic is negative
It creates a transcritial bifurcation

Question 15

What happens when r is larger than 0?
(x dot = rx - x^2)

Answer 15

In this case, changing r moves the NEGATIVE quadratic curve left and right

When x dot and x are 0 = stable fixed point
Where x dot = 0 there is an unstable fixed point

Question 16

What happens when r is EQUAL to 0?
(x dot = rx - x^2)

Question 17

What happens when r is less than 0?
(x dot = rx - x^2)

Answer 17

The quadratic curve shifts to the right but because the gradients remain the same = fixed point stability is the same

Question 18

Describe the bifurcation diagram of
(x dot = rx - x^2)

Answer 18

When r is less than 0 there is a stable point at x = 0 and an unstable line when x is less than 0

When r = 0 is a transition point

The opposite when r is larger than =

Question 19

Describe transcritial bifurcation

Answer 19

Exchange of stability

Question 20

Is transcritical bifurcation symmetric?

Answer 20

No because if you reflect it in the X axis the vectors and trajectories would not be the same

Question 21

What is this equation and what does it create? x dot = rx - x^3

Answer 21

Pitchfork bifurcation

Negative cubic curve = has negative gradient

Question 22

What happens when r is greater than 0?
(x dot = rx - x^3)

Answer 22

It oscilates more crossing through the x dot = 0 at 3 points

2 stable fixed points
1 unstable fixed point = origin
Look at gradients

Question 23

What happens when r is EQUAL to 0?
(x dot = rx - x^3)

Answer 23

Horizontal flat line around the origin

But still one stable fixed point

Question 24

What happens when r is less than 0?
(x dot = rx - x^3)

Answer 24

Compressed narrow curve

With one stable fixed point at origin

Question 25

Is pitchfork bifurcation symmetric?

Answer 25

YES, because reflecting it in the X axis will mean the arrows will still point the same way

Question 26

Describe the difference between a negative and positive cubic curve for bifurcation?

Answer 26

Supercritical pitchfork = negative cubic
1 fixed point becomes 3

Subcritical pitchfork = positive cubic
3 fixed points becomes 1 unstable fp

Question 27

How do we break the symmetry of pitchfork bifurcation?

Answer 27

Introduce imperfection by moving graph up/down

x dot = h + rx - x^3

Question 28

What feature does the imperfectio parameter (h) give?

Answer 28

Bistability

Question 29

What happens when h=0 and r is less than or equal to 0?
(x dot = h + rx - X^3)

Answer 29

No bifurcation

Question 30

What happens when h≠0 and r is less than or equal to 0?
(x dot = h + rx - X^3)

Answer 30

The graph will move up or down
No bifurcation
Stable fixed point will move

Question 31

What happens when h=0 and r is greater than or equal to 0?
(x dot = h + rx - X^3)

Answer 31

Symmetry breaking
Causes semi-stable fixed points

Saddle-node bifurcation occurs because when it moves up 2 fixed points become 1

Question 32

What does bistability cause?

Answer 32

Non-continuous change (jump)
Hysteresis