Scientific Computing I Part III: Analysis of ODE Models Module 3: - - PowerPoint PPT Presentation

Scientific Computing I Michael Bader Outlines

Part II: More Than One Species – Systems of ODE Part III: Analysis of ODE Models

Scientific Computing I

Module 3: Population Modelling – Continuous Models (Parts II and III) Michael Bader

Lehrstuhl Informatik V

Winter 2007/2008

Scientific Computing I Michael Bader Outlines

Part II: More Than One Species – Systems of ODE Part III: Analysis of ODE Models

Part II: More Than One Species – Systems of ODE

1

A Linear Model First Example: Arms Race Second Example: Competition

2

A Non-Linear Model The Non-Linear Competition Model Predator-Prey

3

Open Questions

Scientific Computing I Michael Bader Outlines

Part II: More Than One Species – Systems of ODE Part III: Analysis of ODE Models

Part III: Discussion and Analysis of ODE Models

4

Critical Points Points of Equilibrium Critical Points

5

Direction Fields Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

6

Analysis of Systems of ODE Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Part II More Than One Species – Systems

• f ODE

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

A Linear Model

similar to Verhulst’s saturation model additional growth term proportional to other species leads to system of differential equations: ˙ p(t) = b1 +a11p(t)+a12q(t) ˙ q(t) = b2 +a21p(t)+a22q(t) typically:

b1 > 0,b2 > 0 (growth term) a11 < 0,a22 < 0 (saturation) a12,a21?

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

First Example: Arms Race

armament of two (hostile) countries

• ur suspicion: a12 > 0, a21 > 0

Observation:

long-time behaviour depends on size of parameters steady-state solutions exist solutions exist that show unlimited growth

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Second Example: Competition

two species sharing a common natural habitat competition: a12 < 0, a21 < 0

Observation:

long-time behaviour depends on size of parameters steady-state solutions exist some scenarios are physically incorrect! (negative population size)

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

A Non-Linear Model

similar to Verhulst’s logistic growth model additional growth term proportional to other species leads to system of differential equations: ˙ p(t) = (b1 +a11p(t)+a12q(t))p(t) ˙ q(t) = (b2 +a21p(t)+a22q(t))q(t) typically:

b1 > 0,b2 > 0 (growth term) a11 < 0,a22 < 0 (saturation) a12,a21?

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

The Non-Linear Competition Model

two species sharing a common natural habitat competition: a12 < 0, a21 < 0

Possible Scenarios:

steady-state

• ne species dies out (extinction)

no obvious nonsense

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Competition – Steady State

system of differential equations: ˙ p(t) =

• 5

2 + √ 3 24 − 5 8p(t)− √ 3 24 q(t)

• p(t)

˙ q(t) =

• 7

8 + 3 √ 3 2 − 3 √ 3 8 p(t)− 7 8q(t)

• q(t)

solution for p0 = 1

4, q0 = 3:

2 1 t 10 8 6 4 p 2 4 3

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Competition – Extinction

system of differential equations: ˙ p(t) =

• 71

8 − 23 12p(t)− 25 12q(t)

• p(t)

˙ q(t) =

• 73

8 − 25 12p(t)− 23 12q(t)

• q(t)

solution for p0 = 1

4, q0 = 1 4:

t 10 p 8 4 6 3 2 4 1 2

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Predator-Prey

two species: predator p and prey q predator eats prey: a12 > 0 prey is eaten by predator: a21 < 0

Possible Scenarios:

stable oscillations

• ne species dies out (what happens with the
• ther, then?)

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Predator-Prey by Lotka & Volterra

system of differential equations: ˙ p(t) =

• −1

2 + 1 200q(t)

• p(t)

˙ q(t) = 1

5 − 1 50p(t)

• q(t)

solution for p0 = 6, q0 = 50:

t 100 80 p 60 160 40 120 80 20 40

Scientific Computing I Michael Bader A Linear Model

First Example: Arms Race Second Example: Competition

A Non-Linear Model

The Non-Linear Competition Model Predator-Prey

Open Questions

Open Questions

Methods to Analyse a Given Model?

predict approximate solution or shape of solution? predict possible steady states? predict critical points? (species on edge of extiction?)

Methods to Improve Modeling?

predict failure of the model? tune parameters to model a specific situation?

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Part III Discussion and Analysis of ODE Models

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Analysing the Slope of a Solution

Example: Model of Malthus

˙ p(t) = αp(t) for a sensible solution: p(t) > 0 α decides slope of solution:

α > 0: growing population (accelerated growth) α < 0: receding population (decelerated reduction)

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Points of Equilibrium

Example: Model of Verhulst (saturation)

˙ p(t) = α −βp(t) equilibrium: ˙ p(t) = 0

• nly, if p(t) = α

β

Example: Logistic Growth

˙ p(t) = α

• 1− p(t)

β

• p(t)

constant solution, if p(t) = β or p(t) = 0

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Critical Points

Observation on Logistic Growth:

constant solution p(t) = β, if p(0) = β constant solution p(t) = 0, if p(0) = 0 equilibrium at p = β is reached for nearly all initial conditions ⇒ attractive (stable) equilibrium equilibrium at p = 0 is not reached for any

• ther initial conditions (“repulsive”)

⇒ unstable equilibrium

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Critical Points – Derivatives

Examine derivatives:

critical point p = ¯ p attractive equilibrium (asymptotically stable): ˙ p < 0 for p = ¯ p+ε ˙ p > 0 for p = ¯ p−ε unstable equilibrium: ˙ p > 0 for p = ¯ p+ε ˙ p < 0 for p = ¯ p−ε

• therwise: saddle point

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Direction Field

plot derivatives vs. time and size of population:

Example: Logistic Growth

˙ p(t) = α

• 1− p(t)

β

• p(t)

p(t) 3 2,5 2 1,5 1 t 0,5 10 8 6 4 2

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Direction Field (2)

Example: Logistic Growth with Threshold

˙ p(t) = α

• 1− p(t)

β

• 1− p(t)

δ

• p(t)

p(t) t 5 10 4 3 8 2 1 6 4 2

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Identifying Critical Points

attractive equilibrium: unstable equilibrium saddle point

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Critical Points in 2D

Example: Arms Race

system of differential equations equilibrium: ˙ p = 0, ˙ q = 0 ˙ p(t) = b1 +a11p(t)+a12q(t) = 0 ˙ q(t) = b2 +a21p(t)+a22q(t) = 0 solution of a linear system of equations: a11p(t)+a12q(t) = −b1 a21p(t)+a22q(t) = −b2 in most cases one critical point critical line, if system matrix is singular

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Direction Field for a System of ODE

example: 2D system of differential equations: ˙ p(t) = b1 +a11p(t)+a12q(t) ˙ q(t) = b2 +a21p(t)+a22q(t) natural exension: 3D plot: t vs. p vs. q 1D direction field for p vs. t or q vs. t not sufficient: what values to chose for q (or p resp.)? but: stationary problem ⇒ independent of t thus: plot directions depending on p and q

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

2D Direction Field – Arms Race

system of differential equations: ˙ p(t) =

3 2 −p(t)+ 1 2q(t)

˙ q(t) = 0+ 1

2p(t)−q(t)

direction field – with critical point at (2,1):

1 4 q 3 p 0,5 2 1,5 2 1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Arms Race – unlimited growth

system of differential equations: ˙ p(t) =

1 2 − 3 4p(t)+q(t)

˙ q(t) = −5

4 +p(t)− 3 4q(t)

direction field – with critical point at (2,1):

q 4 p 2 3 5 5 2 1 3 1 4

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Arms race – the peaceful neighbour

system of differential equations: ˙ p(t) = 0− 3

4p(t)+q(t)

˙ q(t) =

5 2 −p(t)− 3 4q(t)

direction field – with critical point at

• 8

5, 6 5

• :

1 0,5 2,5 p 0,5 q 3 1 1,5 3 2,5 2 1,5 2

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Nonlinear System – Competition

system of differential equations: ˙ p(t) =

• 5

2 + √ 3 24 − 5 8p(t)− √ 3 24 q(t)

• p(t)

˙ q(t) =

• 7

8 + 3 √ 3 2 − 3 √ 3 8 p(t)− 7 8q(t)

• q(t)

direction field – critical points at (4,1),...:

4 4 5 2 3 1 2 1 3 p q

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Nonlinear System – Extinction

system of differential equations: ˙ p(t) =

• 71

8 − 23 12p(t)− 25 12q(t)

• p(t)

˙ q(t) =

• 73

8 − 25 12p(t)− 23 12q(t)

• q(t)

critical points at (0,4.76...),(4.63...,0),...:

0,5 3 q 1,5 p 3 4 1 2 2 1 5 2,5

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Lotka & Volterra

system of differential equations: ˙ p(t) =

• −1

2 + 1 200q(t)

• p(t)

˙ q(t) = 1

5 − 1 50p(t)

• q(t)

direction field – with critical point at (10,100):

q 40 30 150 200 100 50 20 10 p

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

2D Critical Points – Summary

Different types of critical points in 2D: attractive/stable equilibrium (arms race – steady state) unstable equilibrium saddle point (arms race – unlimited growth) attractive “spiral point” (“peaceful neighbour”) unstable “spiral point” centre of “rotation” (Lotka-Volterra) ⇒ How to discriminate between these types?

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Homogeneous Systems of ODE

Homogeneous System in matrix-vector-notation: ˙ x = Ax x : R → Rn, A ∈ Rn×n example: x(t) = (p(t),q(t)) Solutions: let xλ be an eigenvector: Axλ = λxλ then xλeλt is a solution: Axλeλt = λxλeλt = d dt

• xλeλt

q.e.d.

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Eigenvectors and Eigenvalues

Corollaries: the solutions of the homogeneous system ˙ x = Ax are linear combinations of the respective eigen-solutions: xhom(t) = ∑

λ

aλxλeλt, aλ ∈ R the solutions of the inhomogeneous system ˙ x = Ax+b are x(t) = −A−1b+xhom(t)

• bservation: xc = −A−1b is a critical point!

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Eigenvalues and Critical Points

the ODE system ˙ x = Ax+b is solved by x(t) = xc +∑

λ

aλxλeλt xc attractive equilibrium, lim

t→∞x(t) = xc,

• nly if eλt → 0 for all eigenvalues λ

λ ∈ R ⇒ λ < 0 λ = µ +iν ⇒ µ < 0 (eiνt = cosνt+isinνt)

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of Linear Systems

Overview:

• eigenval. (λj = µj +iνj)

critical point stability real, all λ < 0 node stable, attr. real, all λ > 0 node unstable real, λk > 0,λl < 0 saddle point unstable complex, all µ < 0 spiral point stable, attr. complex, all µ > 0 spiral point unstable complex, all µ = 0 centre stable

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Real Eigenvalues:

λ1 < 0, λ2 < 0, attractive equilibrium

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Real Eigenvalues:

λ1 > 0, λ2 > 0, unstable equilibrium

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Real Eigenvalues:

λ1 > 0, λ2 < 0, saddle point

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Complex Eigenvalues:

µ1 < 0, µ2 < 0, spiral point (asympt. stable)

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Complex Eigenvalues:

µ1 > 0, µ2 > 0, spiral point (unstable)

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of 2D Systems

Complex Eigenvalues:

µ1 = µ2 = 0, centre of oscillation

x2 x1 eig2 eig1

Scientific Computing I Michael Bader Critical Points

Points of Equilibrium Critical Points

Direction Fields

Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary

Analysis of Systems of ODE

Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Stability of Non-Linear Systems

2D system of ODE: ˙ x(t) = f(x(t)), f : Rn → Rn nonlinear critical point at xc: f(xc) = 0 for analysis of critical points: linearization ˙ x(t) = f(x(t)) ≈ f(xc)

=0

+Jf(xc)(x(t)−xc) examine eigenvalues of Jf(xc)