5 Parameter, Bifucation, Phase Diagram Flashcards
What determines if the population will have an epidemic outbreak?
R0 if bigger than 1 = epidemic outbreak
So 1/R0 = less than 1
How does changing the parameters change the fixed points?
Changing the parameters will change the stability of the fixed points
What determines linear system stability?
The signs of the constants/parameters
Name some qualitative changes to system stability
Number and stability of fixed points
What does it mean to treat a parameter as invariable?
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
Describe a positive auto-regulation with ultrasensitive and constant external stimulus in 2 equations
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
When can we draw a 2D phase portrait for a parameter?
When we treat it as a variable = can test a whole range of r
What happens when r is larger than 0?
(x dot = r + x^2)
As r increases the U shape moves upwards
There are no fixed points because it does not intersect with x = 0
What happens when r is EQUAL to 0?
(x dot = r + x^2)
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
Why does a semi-stable fixed point occur?
Because of the gradient change
What happens when r is less than 0?
(x dot = r + x^2)
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
Is saddle-node bifurcation symmetric?
NO, not in its dynamic or trajectory
Stable and unstable fixed points will always be the same distance away from centre line
Describe a saddle-node bifurcation
Aka blue-sky bifurcation because the pair of fixed points appear out of the blue
Saddle-node bifurcation = creation or annihilation of fixed points
What is this equation and what does it create? x dot = rx - x^2
It is a logistic model
The quadratic is negative
It creates a transcritial bifurcation
What happens when r is larger than 0?
(x dot = rx - x^2)
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
What happens when r is EQUAL to 0?
(x dot = rx - x^2)
What happens when r is less than 0?
(x dot = rx - x^2)
The quadratic curve shifts to the right but because the gradients remain the same = fixed point stability is the same
Describe the bifurcation diagram of
(x dot = rx - x^2)
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 =
Describe transcritial bifurcation
Exchange of stability
Is transcritical bifurcation symmetric?
No because if you reflect it in the X axis the vectors and trajectories would not be the same
What is this equation and what does it create? x dot = rx - x^3
Pitchfork bifurcation
Negative cubic curve = has negative gradient
What happens when r is greater than 0?
(x dot = rx - x^3)
It oscilates more crossing through the x dot = 0 at 3 points
2 stable fixed points
1 unstable fixed point = origin
Look at gradients
What happens when r is EQUAL to 0?
(x dot = rx - x^3)
Horizontal flat line around the origin
But still one stable fixed point
What happens when r is less than 0?
(x dot = rx - x^3)
Compressed narrow curve
With one stable fixed point at origin
Is pitchfork bifurcation symmetric?
YES, because reflecting it in the X axis will mean the arrows will still point the same way
Describe the difference between a negative and positive cubic curve for bifurcation?
Supercritical pitchfork = negative cubic
1 fixed point becomes 3
Subcritical pitchfork = positive cubic
3 fixed points becomes 1 unstable fp
How do we break the symmetry of pitchfork bifurcation?
Introduce imperfection by moving graph up/down
x dot = h + rx - x^3
What feature does the imperfectio parameter (h) give?
Bistability
What happens when h=0 and r is less than or equal to 0?
(x dot = h + rx - X^3)
No bifurcation
What happens when h≠0 and r is less than or equal to 0?
(x dot = h + rx - X^3)
The graph will move up or down
No bifurcation
Stable fixed point will move
What happens when h=0 and r is greater than or equal to 0?
(x dot = h + rx - X^3)
Symmetry breaking
Causes semi-stable fixed points
Saddle-node bifurcation occurs because when it moves up 2 fixed points become 1
What does bistability cause?
Non-continuous change (jump)
Hysteresis
