equilibrium analysis in 1d - university of southamptonmb1a10/intro_diff4.pdfdifferential equations...
TRANSCRIPT
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Equilibrium Analysis in 1d
● Aims of the lecture:● 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
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Equilibrium Analysis
● 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?
● Look for stationary points at which the system does not change, i.e.:
dx /dt=f ( x , t)
0=f (x stat , t)
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Example: Bacterial Growth● 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
● not exponential, but initial phase fits exponential growth
● then growth saturates -> food constraint?
carrying “capacity” K
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Bacterial Growth (2)● Equation for exponential growth was
with a growth rate that is constant● 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)
dP /dt=rP
Logistic equation dP /dt=r (P)P=rP (1−P /K)
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Logistic Equation
● Well ... could solve this equation (how?) but let's attempt something simpler
● Equilibrium states:
● Two solutions:
● So: for this system we can identify where the system will end up in the long run. Just: in which of these states?
dP /dt=0rPstat (1−P stat /K )=0
P stat=0 P stat=Kor
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Stability● 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
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Logistic Equation (2)
dP/dt
unstable equilibrium stable equilibrium
Unless we start at exactly P=0 we always end up at Pstat=K!
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Logistic Equation (3)
Numerical integration of some sample trajectories confirms this.
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Another Example
● Graphically: dx /dt=sin(x )
● dx/dt=0 -> no flow -> fixed points (FP)● Two types: stable and unstable
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Fixed points and stability (1)
● General System
dx/dt=f(x)● 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
● Stable: small perturbations damp out● Unstable: small perturbations grow
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Fixed points and stability (2)
● Consider● Classify the dynamics of (1) by analyzing fixed
points and their local and global stability!● Fixed points:● Stability: x
1 unstable, x
2 locally stable, but not
globally● What kind of perturbation could destabilize x
2?
dx /dt= x2−1=f (x)
f (x )=0 x1/2=±1
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Linear Stability Analysis (1)
● Consider a FP x* (i.e. f(x*)=0) and the fate of a small perturbation ε=x(t)-x* from it:
● Expand f into a Taylor series around x*:
● Perturbation ε:● Grows exponentially if df/dx>0● Declines exponentially if df/dx
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Logistic Growth (4)
● Let's come back to
with and ● Linearize f(P) around both:
dP /dt=rP (1−P /K )= f (P)
P stat=0 P stat=K
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
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Linear Stability Analysis (2)
● A simple example:● That is● FP:● Stability?● Stable for odd k and unstable for even k
dx /dt=sin(x )
f (x )=sin (x )f (x )=0 x=k π
df /dx=cos (x )=cos(k π)
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What if df/dx=0?
unstable FPstable FP
half-stable FP non-isolated FP
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Impossibility of Oscillations in 1d● So far: all trajectories tend to or are FP
● These are the only possible dynamics for one dimensional differential equation on the real line
● Why?● 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:● Linear oscillations● Limit cycles● Chaos
±∞
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Another Example: Sinistral and Dextral Snails
● There are two types of snails, such with left and others with right handed patterns
Can we understand therelative prevalence ofright and left handedsnails?
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Snails (2)
● Under some assumptions● 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.
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Snails (3)
● What can we say about
dp /dt∝ p(1−p)( p−1/2)=f ( p)
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Snails (3)
● What can we say about
● Stationary points:
dp /dt∝ p(1−p)( p−1/2)=f ( p)
f ( pstat)=0
p1stat=0 p2
stat=1 p3stat=1/2
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Snails (3)
● What can we say about
● Stationary points:
● Stability?
dp /dt∝ p(1−p)( p−1/2)=f ( p)
f ( pstat)=0
p1stat=0 p2
stat=1 p3stat=1/2
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Stability -- Snails
● Plot dp/dt vs. p
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Numerical Integration of Snails
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Analytically
p1stat=0
dp /dt∝ p(1−p)( p−1/2)=f ( p)
df /dp(0)=−1/2
f ( p)=−1/2p+3/2p2−p3
stable
p2stat=1/2 df /dp(1/2)=1/4 unstable
p3stat=1 df /dp(1)=−1 /2 stable
No coexistence between dextral and sinistral snailsIn our model
This is in fact the case for most species of gastropods(see http://en.wikipedia.org/wiki/Gastropod_shell)
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Summary
● For 1d (autonomous) ODE's on the real line we have the following types of asymptotic behaviour● Exponential divergence to +/- infinity● Convergence to fixed points
● Can analyse asymptotic behaviour with equilibrium analysis● Calculate equilibria by setting derivatives to zero● Analyse their strability by:
– Graphical methods– Linearization
● Higher dimensions? -> next lecture.
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