Equilibrium Analysis in 1d
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Aims of the lecture:
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Understand ways how to “extract” the qualitative
behaviour out of a 1d system without solving it
Understand the idea of equilibrium analysis and
stability
Be able to apply methods to explore the stability of
fixed points (graphical/analytical)
A good reference book to follow up material in
this lecture and the next is
S. Strogatz, “Nonlinear Dynamics and Chaos”,
Westview Press
Equilibrium Analysis
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Differential equations (especially if they are
non-linear!) often difficult to solve analytically
Can we make statements about solutions
without solving the equations?
Say ... we are not interested in the initial
transient behaviour but only worry about what
happens in the long run?
dx / dt =f ( x , t )
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Look for stationary points at which the system
stat
does not change, i.e.: 0=f ( x , t )
Example: Bacterial Growth
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Staphylococcus aureus can cause food poisoning, it
is important to understand the growth of the
bacterium in an organism
Experiments have been carried out measuring the
concentration of the bacterium over time in cultures
(with optical density measurements), trajectories
look like the following
carrying “capacity” K
not exponential, but initial
phase fits exponential
growth
● then growth saturates
-> food constraint?
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Bacterial Growth (2)
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Equation for exponential growth was
dP / dt=rP
with a growth rate that is constant
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Observation: the system has a “capacity” K
A (food/crowding) constraint will reduce the
growth rate, especially if populations are large
compared to capacity.
First modelling attempt might be that growth
rate decreases linearly with P, i.e. r(P)=r(1-P/K)
Logistic equation
dP / dt=r ( P) P=rP (1−P / K)
Logistic Equation
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Well ... could solve this equation (how?) but
let's attempt something simpler
Equilibrium states:
dP / dt=0
rP
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(1−P
stat
/ K )=0
Two solutions:
P
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stat
stat
=0
or
P
stat
=K
So: for this system we can identify where the
system will end up in the long run. Just: in
which of these states?
Stability
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With different initial conditions the system might
end up in either of these states
To analyze in which state the system will end up it
is useful to analyse phase portraits and investigate
the stability of the stationary states
Loosely speaking: a state is stable if the system
relaxes back to the state after a perturbation
stable
unstable
Logistic Equation (2)
dP/dt
unstable equilibrium
stable equilibrium
Unless we start at exactly P=0 we always end up at Pstat=K!
Logistic Equation (3)
Numerical integration of some sample trajectories confirms this.
Another Example
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Graphically: dx / dt=sin( x )
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dx/dt=0 -> no flow -> fixed points (FP)
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Two types: stable and unstable
Fixed points and stability (1)
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General System
dx/dt=f(x)
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Imagine fluid flowing
along real line with
local velocity dx/dt
Fixed points are equilibrium solutions with
dx/dt=0=f(x*) such that if x0=x* -> x(t)=x* all t
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Stable: small perturbations damp out
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Unstable: small perturbations grow
Fixed points and stability (2)
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2
Consider dx /dt = x −1=f ( x)
Classify the dynamics of (1) by analyzing fixed
points and their local and global stability!
x 1/ 2=±1
Fixed points: f (x )=0
Stability: x1 unstable, x2 locally stable, but not
globally
What kind of perturbation could destabilize x 2?
Linear Stability Analysis (1)
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Consider a FP x* (i.e. f(x*)=0) and the fate of a
small perturbation ε=x(t)-x* from it:
d ϵ/ dt=dx / dt =f ( x +ϵ)
Expand f into a Taylor series around x*:
2
d ϵ/dt=f ( x )+ϵ df /dx +O( ϵ )
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Perturbation ε:
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Grows exponentially if df/dx>0
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Declines exponentially if df/dx<0
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If df/dx=0 more analysis is needed
charact. timescale
1/∣df / dx∣
Logistic Growth (4)
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Let's come back to dP / dt=rP (1−P / K )=f ( P)
with P stat =0 and P stat =K
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Linearize f(P) around both:
f (ϵ)≈r ϵ
r>0, i.e. P=0 is unstable
f ( K + ϵ)≈f ( K )+ϵ df /dP( K)
=−r ϵ
r>0, i.e. P=K is stable
Linear Stability Analysis (2)
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A simple example: dx /dt =sin( x )
That is f ( x )=sin ( x )
x=k π
f ( x )=0
FP:
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Stability? df /dx=cos ( x )=cos( k π)
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Stable for odd k and unstable for even k
What if df/dx=0?
stable FP
half-stable FP
unstable FP
non-isolated FP
Impossibility of Oscillations in 1d
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So far: all trajectories tend to ±∞ or are FP
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Why?
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These are the only possible dynamics for one
dimensional differential equation on the real line
Topological reason: 1d system corresponds to a
flow on the real line. If you flow monotonically on a
line you never come back to starting position
What other types of behaviour are possible in
higher dimensions? Roughly:
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Linear oscillations
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Limit cycles
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Chaos
Another Example: Sinistral and
Dextral Snails
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There are two types of snails, such with left and
others with right handed patterns
Can we understand the
relative prevalence of
right and left handed
snails?
Snails (2)
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Under some assumptions
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Likelihood of a sinistral snail breeding with a dextral
snail is proportional to the product of their numbers
Breeding between like snails produces their own
type
Breeding sinistral-dextral produces both types with
equal likelihood
Let's denote the likelihood that a randomly picked
snail is sinistral by p
one can derive (*): dp / dt ∝ p( 1− p)( p−1/ 2)
(*) see: C. H. Taubes, Modeling Differential Equations in Biology, Prentice Hall, 2001.
Snails (3)
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What can we say about
dp / dt ∝ p( 1− p)( p−1/ 2)=f ( p)
Snails (3)
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What can we say about
dp / dt ∝ p( 1− p)( p−1/ 2)=f ( p)
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Stationary points:
f (p
stat
1
stat
)=0
stat
2
p =0 p =1
stat
3
p =1/2
Snails (3)
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What can we say about
dp / dt ∝ p( 1− p)( p−1/ 2)=f ( p)
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Stationary points:
f (p
stat
1
stat
)=0
stat
2
p =0 p =1
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Stability?
stat
3
p =1/2
Stability -- Snails
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Plot dp/dt vs. p
Numerical Integration of Snails
Analytically
dp / dt ∝ p( 1− p)( p−1/ 2)=f ( p)
2
f ( p)=−1/2p +3/2p − p
3
stat
1
df / dp( 0)=−1/ 2
stable
stat
2
df /dp(1/ 2)=1/4
unstable
stat
3
df /dp(1)=−1 /2
stable
p =0
p =1/2
p =1
No coexistence between dextral and sinistral snails
In our model
This is in fact the case for most species of gastropods
(see http://en.wikipedia.org/wiki/Gastropod_shell)
Summary
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For 1d (autonomous) ODE's on the real line we
have the following types of asymptotic
behaviour
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Exponential divergence to +/- infinity
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Convergence to fixed points
Can analyse asymptotic behaviour with
equilibrium analysis
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Calculate equilibria by setting derivatives to zero
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Analyse their strability by:
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Graphical methods
Linearization
Higher dimensions? -> next lecture.