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

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)

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

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)

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

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

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

● Graphically: dx /dt=sin(x )

● dx/dt=0 -> no flow -> fixed points (FP)● Two types: stable and unstable

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

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

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

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

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)

What if df/dx=0?

unstable FPstable FP

half-stable FP non-isolated FP

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

±∞

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?

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.

Snails (3)

● What can we say about

dp /dt∝ p(1−p)( p−1/2)=f ( p)

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

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

Stability -- Snails

● Plot dp/dt vs. p

Numerical Integration of Snails

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)

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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