Scientific Computing I Module 3: Population Modelling Continuous - - PowerPoint PPT Presentation

Scientific Computing I Michael Bader

Scientific Computing I

Module 3: Population Modelling – Continuous Models (Part III) Michael Bader

Lehrstuhl Informatik V

Winter 2005/2006

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 Maltus

˙ 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

x→∞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(t)) = 0 for analysis of critical points: linearization ˙ x = f(x) ≈ f(t,xc)

=0

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