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 Scientific Computing I Part III: Analysis of ODE Models Module 3: Population Modelling – Continuous Models (Parts II and III) Michael Bader Lehrstuhl Informatik V Winter 2007/2008

Scientific Part II: More Than One Species – Computing I Michael Bader Systems of ODE Outlines Part II: More Than One Species – Systems of ODE Part III: Analysis of ODE Models A Linear Model 1 First Example: Arms Race Second Example: Competition A Non-Linear Model 2 The Non-Linear Competition Model Predator-Prey Open Questions 3

Scientific Part III: Discussion and Analysis of Computing I Michael Bader ODE Models Outlines Critical Points 4 Part II: More Than One Species – Systems of ODE Points of Equilibrium Part III: Analysis of ODE Models Critical Points Direction Fields 5 Critical Points in 1D Critical Points in 2D 2D Direction Fields Summary Analysis of Systems of ODE 6 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 Part II A Non-Linear Model The Non-Linear Competition Model More Than One Species – Systems Predator-Prey Open Questions of ODE

Scientific A Linear Model Computing I Michael Bader A Linear Model similar to Verhulst’s saturation model First Example: Arms Race Second Example: Competition additional growth term proportional to other A Non-Linear species Model The Non-Linear Competition Model leads to system of differential equations: Predator-Prey Open Questions ˙ p ( t ) = b 1 + a 11 p ( t )+ a 12 q ( t ) ˙ q ( t ) = b 2 + a 21 p ( t )+ a 22 q ( t ) typically: b 1 > 0 , b 2 > 0 (growth term) a 11 < 0 , a 22 < 0 (saturation) a 12 , a 21 ?

Scientific First Example: Arms Race Computing I Michael Bader A Linear Model First Example: Arms Race Second Example: Competition armament of two (hostile) countries A Non-Linear Model our suspicion: a 12 > 0, a 21 > 0 The Non-Linear Competition Model Predator-Prey Observation: Open Questions long-time behaviour depends on size of parameters steady-state solutions exist solutions exist that show unlimited growth

Scientific Second Example: Competition Computing I Michael Bader A Linear Model First Example: Arms Race Second Example: two species sharing a common natural habitat Competition A Non-Linear competition: a 12 < 0, a 21 < 0 Model The Non-Linear Competition Model Predator-Prey Observation: Open Questions long-time behaviour depends on size of parameters steady-state solutions exist some scenarios are physically incorrect! (negative population size)

Scientific A Non-Linear Model Computing I Michael Bader A Linear Model similar to Verhulst’s logistic growth model First Example: Arms Race Second Example: Competition additional growth term proportional to other A Non-Linear species Model The Non-Linear Competition Model leads to system of differential equations: Predator-Prey Open Questions ˙ p ( t ) = ( b 1 + a 11 p ( t )+ a 12 q ( t )) p ( t ) ˙ q ( t ) = ( b 2 + a 21 p ( t )+ a 22 q ( t )) q ( t ) typically: b 1 > 0 , b 2 > 0 (growth term) a 11 < 0 , a 22 < 0 (saturation) a 12 , a 21 ?

Scientific The Non-Linear Competition Model Computing I Michael Bader A Linear Model First Example: Arms Race Second Example: Competition two species sharing a common natural habitat A Non-Linear Model The Non-Linear competition: a 12 < 0, a 21 < 0 Competition Model Predator-Prey Open Questions Possible Scenarios: steady-state one species dies out (extinction) no obvious nonsense

Scientific Competition – Steady State Computing I Michael Bader system of differential equations: A Linear Model � √ √ � First Example: Arms Race 5 24 − 5 3 3 ˙ p ( t ) = 2 + 8 p ( t ) − 24 q ( t ) p ( t ) Second Example: Competition � √ √ � A Non-Linear 7 8 + 3 2 − 3 3 8 p ( t ) − 7 3 q ( t ) ˙ = 8 q ( t ) q ( t ) Model The Non-Linear Competition Model solution for p 0 = 1 Predator-Prey 4 , q 0 = 3: Open Questions 4 3 p 2 1 0 2 4 6 8 10 t

Scientific Competition – Extinction Computing I Michael Bader system of differential equations: A Linear Model � � First Example: Arms Race 71 8 − 23 12 p ( t ) − 25 ˙ p ( t ) = 12 q ( t ) p ( t ) Second Example: Competition � � A Non-Linear 73 8 − 25 12 p ( t ) − 23 q ( t ) ˙ = 12 q ( t ) q ( t ) Model The Non-Linear Competition Model solution for p 0 = 1 4 , q 0 = 1 Predator-Prey 4 : Open Questions 4 p 3 2 1 0 0 2 4 6 8 10 t

Scientific Predator-Prey Computing I Michael Bader A Linear Model First Example: Arms Race Second Example: Competition two species: predator p and prey q A Non-Linear Model predator eats prey: a 12 > 0 The Non-Linear Competition Model prey is eaten by predator: a 21 < 0 Predator-Prey Open Questions Possible Scenarios: stable oscillations one species dies out (what happens with the other, then?)

Scientific Predator-Prey by Lotka & Volterra Computing I Michael Bader system of differential equations: A Linear Model First Example: Arms Race � � − 1 1 p ( t ) ˙ = 2 + 200 q ( t ) p ( t ) Second Example: Competition � 1 � 5 − 1 A Non-Linear q ( t ) ˙ = 50 p ( t ) q ( t ) Model The Non-Linear Competition Model solution for p 0 = 6, q 0 = 50: Predator-Prey Open Questions 160 120 p 80 40 0 20 40 60 80 100 t

Scientific Open Questions Computing I Michael Bader A Linear Model First Example: Arms Race Methods to Analyse a Given Model? Second Example: Competition predict approximate solution or shape of A Non-Linear Model solution? The Non-Linear Competition Model Predator-Prey predict possible steady states? Open Questions 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 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 Malthus 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