biosim4 - Nonlinear Discrete Equation Read: Chapter 2.1 ~...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
onlinear Discrete Equation Nonlinear Discrete Equation Read: hapter 2.1 ~ 2.5 Chapter 2.1 2.5 Mathematical Models in Biology 005) Leah Edelstein eshet (2005), Leah Edelstein-Keshet
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Definition • A nonlinear difference equation takes the form •O n l y now the recursion function x n+1 = f(x n , x n-1 , …) depends nonlinearly on its arguments. • For nonlinear discrete equations, analytical solutions can only be found in rare cases.
Background image of page 2
How Do Solutions Behave? quilibrium points (steady states) and • Equilibrium points (steady states) and their stability provide information about e behavior of the solutions the behavior of the solutions.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Equilibrium Points • Also called steady states or fixed points elate to the absence of change in a Relate to the absence of change in a system epresent constant solutions or static • Represent constant solutions or static situations • If the system starts at a steady state it will remain there forever.
Background image of page 4
Finding Equilibrium Points • Analytically – To find an equilibrium point of x n+1 =f(x n ) –Se t x n+1 =x n = x e – Then equilibrium points are the solution of: – Because the recursion function is nonlinear, this x e = f(x e ) equation may not be easy to solve!
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Finding Equilibrium Points • Graphically – An equilibrium point is the intersection of two graphs: and x n+1 =x n x n+1 =f(x n ) x n+1 x n+1 n In this case ere are 3 x n+1 n ) there are 3 fixed points. x n
Background image of page 6
Stability of Fixed Points • Questions: What happens if you slightly erturb the value of way from perturb the value of x n away from x e ? Will the orbits return to x e or move away from x ? e • Analogy 3 Which positions re teady 2 4 are steady states? 1 Which steady states are table? Positions 1, 3 and 4 are steady states. Positions 1 and 4 are stable. Position 3 is unstable. stable?
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Determining Stability • Graphically: Cobweb Diagrams – Check to see if successive iterates move towards or away from x e . x n+1 x 1 = f(x 0 ) x 2 = f(x 1 ) x n x 0 x 1 x 2
Background image of page 8
Determining Stability • Can we use information about f(x n ) to determine the stability of an equilibrium point without sorting to cobweb diagrams? f’(x 3 ) < 1 x n+1 resorting to cobweb diagrams? • A closer look reveals – At stable fixed points f’(x e ) < 1 ’(x >1 – At unstable fixed points f’(x e ) > 1 • The stability of fixed points ppears to be determined by f’(x 1 ) < 1 f(x 2 ) > 1 appears to be determined by the slope of the recursion function.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/08/2011 for the course BM 501 taught by Professor Kop during the Spring '11 term at Bloomsburg.

Page1 / 38

biosim4 - Nonlinear Discrete Equation Read: Chapter 2.1 ~...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online