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 Part III Direction Fields Critical Points in 1D Critical Points in 2D 2D Direction Fields Discussion and Analysis of ODE Summary Analysis of Systems of ODE Models Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems

Scientific Analysing the Slope of a Solution Computing I Michael Bader Critical Points Points of Equilibrium Example: Model of Maltus Critical Points Direction Fields Critical Points in 1D p ( t ) = α p ( t ) ˙ Critical Points in 2D 2D Direction Fields Summary Analysis of Systems of ODE for a sensible solution: p ( t ) > 0 Homogeneous Systems Eigenvalues and Critical α decides slope of solution: Points Stability of Linear Systems α > 0: growing population (accelerated Stability of Non-Linear Systems growth) α < 0: receding population (decelerated reduction)

Scientific Points of Equilibrium Computing I Michael Bader Example: Model of Verhulst (saturation) Critical Points Points of Equilibrium Critical Points ˙ p ( t ) = α − β p ( t ) Direction Fields Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary equilibrium: ˙ p ( t ) = 0 Analysis of Systems of ODE only, if p ( t ) = α β Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Example: Logistic Growth Stability of Non-Linear Systems � � 1 − p ( t ) p ( t ) = α ˙ p ( t ) β constant solution, if p ( t ) = β or p ( t ) = 0

Scientific Critical Points Computing I Michael Bader Critical Points Points of Equilibrium Observation on Logistic Growth: Critical Points Direction Fields constant solution p ( t ) = β , if p ( 0 ) = β Critical Points in 1D Critical Points in 2D 2D Direction Fields constant solution p ( t ) = 0, if p ( 0 ) = 0 Summary Analysis of equilibrium at p = β is reached for nearly all Systems of ODE initial conditions Homogeneous Systems Eigenvalues and Critical Points ⇒ attractive (stable) equilibrium Stability of Linear Systems Stability of Non-Linear equilibrium at p = 0 is not reached for any Systems other initial conditions (“repulsive”) ⇒ unstable equilibrium

Scientific Critical Points – Derivatives Computing I Michael Bader Critical Points Examine derivatives: Points of Equilibrium Critical Points critical point p = ¯ p Direction Fields Critical Points in 1D attractive equilibrium (asymptotically stable): Critical Points in 2D 2D Direction Fields Summary p < 0 ˙ for p = ¯ p + ε Analysis of Systems of ODE ˙ p = ¯ p > 0 for p − ε Homogeneous Systems Eigenvalues and Critical Points unstable equilibrium: Stability of Linear Systems Stability of Non-Linear Systems ˙ p = ¯ p > 0 for p + ε ˙ p = ¯ p < 0 for p − ε otherwise: saddle point

Scientific Direction Field Computing I Michael Bader plot derivatives vs. time and size of population: Critical Points Example: Logistic Growth Points of Equilibrium Critical Points � � 1 − p ( t ) Direction Fields ˙ p ( t ) = α p ( t ) Critical Points in 1D β Critical Points in 2D 2D Direction Fields Summary 3 Analysis of Systems of ODE Homogeneous Systems 2,5 p(t) Eigenvalues and Critical Points 2 Stability of Linear Systems Stability of Non-Linear Systems 1,5 1 0,5 0 0 2 4 6 8 10 t

Scientific Direction Field (2) Computing I Michael Bader Example: Logistic Growth with Threshold Critical Points Points of Equilibrium � �� � 1 − p ( t ) 1 − p ( t ) Critical Points ˙ p ( t ) = α p ( t ) Direction Fields β δ Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary 5 Analysis of Systems of ODE 4 p(t) Homogeneous Systems Eigenvalues and Critical Points 3 Stability of Linear Systems Stability of Non-Linear Systems 2 1 0 0 2 4 6 8 10 t

Scientific Identifying Critical Points Computing I Michael Bader attractive equilibrium: Critical Points Points of Equilibrium Critical Points Direction Fields Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary unstable equilibrium Analysis of Systems of ODE Homogeneous Systems Eigenvalues and Critical Points Stability of Linear Systems Stability of Non-Linear Systems saddle point

Scientific Critical Points in 2D Computing I Michael Bader Example: Arms Race Critical Points Points of Equilibrium system of differential equations Critical Points Direction Fields equilibrium: ˙ p = 0, ˙ q = 0 Critical Points in 1D Critical Points in 2D 2D Direction Fields ˙ p ( t ) = b 1 + a 11 p ( t )+ a 12 q ( t ) = 0 Summary Analysis of ˙ q ( t ) = b 2 + a 21 p ( t )+ a 22 q ( t ) = 0 Systems of ODE Homogeneous Systems Eigenvalues and Critical solution of a linear system of equations: Points Stability of Linear Systems Stability of Non-Linear Systems a 11 p ( t )+ a 12 q ( t ) − b 1 = a 21 p ( t )+ a 22 q ( t ) = − b 2 in most cases one critical point critical line, if system matrix is singular

Scientific Direction Field for a System of ODE Computing I Michael Bader Critical Points example: 2D system of differential equations: Points of Equilibrium Critical Points Direction Fields p ( t ) ˙ = b 1 + a 11 p ( t )+ a 12 q ( t ) Critical Points in 1D Critical Points in 2D 2D Direction Fields q ( t ) ˙ = b 2 + a 21 p ( t )+ a 22 q ( t ) Summary Analysis of Systems of ODE natural exension: 3D plot: t vs. p vs. q Homogeneous Systems Eigenvalues and Critical 1D direction field for p vs. t or q vs. t not Points Stability of Linear Systems sufficient: Stability of Non-Linear Systems what values to chose for q (or p resp.)? but: stationary problem ⇒ independent of t thus: plot directions depending on p and q

Scientific 2D Direction Field – Arms Race Computing I Michael Bader system of differential equations: Critical Points 2 − p ( t )+ 1 3 ˙ Points of Equilibrium p ( t ) = 2 q ( t ) Critical Points 0 + 1 ˙ Direction Fields q ( t ) = 2 p ( t ) − q ( t ) Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary direction field – with critical point at ( 2 , 1 ) : Analysis of Systems of ODE Homogeneous Systems 2 Eigenvalues and Critical Points Stability of Linear Systems q 1,5 Stability of Non-Linear Systems 1 0,5 0 0 1 2 3 4 p

Scientific Arms Race – unlimited growth Computing I Michael Bader system of differential equations: Critical Points 2 − 3 1 ˙ Points of Equilibrium p ( t ) = 4 p ( t )+ q ( t ) Critical Points − 5 4 + p ( t ) − 3 ˙ Direction Fields q ( t ) = 4 q ( t ) Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary direction field – with critical point at ( 2 , 1 ) : Analysis of Systems of ODE Homogeneous Systems 5 Eigenvalues and Critical Points Stability of Linear Systems 4 q Stability of Non-Linear Systems 3 2 1 0 0 1 2 3 4 5 p

Scientific Arms race – the peaceful neighbour Computing I Michael Bader system of differential equations: Critical Points 0 − 3 ˙ p ( t ) = 4 p ( t )+ q ( t ) Points of Equilibrium Critical Points 2 − p ( t ) − 3 5 ˙ q ( t ) = 4 q ( t ) Direction Fields Critical Points in 1D Critical Points in 2D 2D Direction Fields � � 5 , 6 8 Summary direction field – with critical point at : 5 Analysis of Systems of ODE 3 Homogeneous Systems Eigenvalues and Critical Points 2,5 Stability of Linear Systems q Stability of Non-Linear Systems 2 1,5 1 0,5 0 0 0,5 1 1,5 2 2,5 3 p

Scientific Nonlinear System – Competition Computing I Michael Bader system of differential equations: Critical Points � √ √ � 5 24 − 5 3 3 p ( t ) ˙ = 2 + 8 p ( t ) − 24 q ( t ) p ( t ) Points of Equilibrium Critical Points � � √ √ Direction Fields 7 8 + 3 2 − 3 3 8 p ( t ) − 7 3 ˙ q ( t ) = 8 q ( t ) q ( t ) Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary direction field – critical points at ( 4 , 1 ) ,... : Analysis of Systems of ODE Homogeneous Systems 4 Eigenvalues and Critical Points Stability of Linear Systems q Stability of Non-Linear 3 Systems 2 1 0 0 1 2 3 4 5 p

Scientific Nonlinear System – Extinction Computing I Michael Bader system of differential equations: Critical Points � � 71 8 − 23 12 p ( t ) − 25 p ( t ) ˙ = 12 q ( t ) p ( t ) Points of Equilibrium Critical Points � � Direction Fields 73 8 − 25 12 p ( t ) − 23 ˙ q ( t ) = 12 q ( t ) q ( t ) Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary critical points at ( 0 , 4 . 76 ... ) , ( 4 . 63 ..., 0 ) ,... : Analysis of Systems of ODE Homogeneous Systems 5 Eigenvalues and Critical Points Stability of Linear Systems 4 q Stability of Non-Linear Systems 3 2 1 0 0 0,5 1 1,5 2 2,5 3 p