biosim lec7

biosim lec7 - Phase Plane Analysis PhasePlane Analysis...

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hase lane Analysis Phase Plane Analysis Read: hapter .1 ~ .10 Chapter 5.1 5.10 Mathematical Models in Biology 005) Leah E elstein eshet (2005), Leah E delstein Keshet
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Phase Portraits • A graphical picture that summarizes the behavior of a system of two ODEs. • Chemostat Model dn dt = a 1 c 1 + c n n dc dt = c 1 + c n c + a 2
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Phase Plane Analysis nc-plane is called phase plane We can graph (n, c) as a single curve in the nc-plane. Such a curve is called trajectory (orbit or path ). Fill the phase plane with trajectories is called phase portrait The tangent vector (n’, c’) is specified by vector field . Plot many vectors in the (n, c) plane, sometimes just consider the direction of vector field, not the magnitude, which yields a direction field. Note: do not actually solve ODE -1.5 -1 -0.5 0 0.5 1 1.5 -2 -1.5 -0.5 0 0.5 1 1.5 2 x y Direction field of an OED 1 12 2 '5 2 '7 4 x xx x = =
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Nullclines • Nullclines are curves of zero slope – That is curves for which – Therefore steady states are located at the intersection of nullclines dn dt = 0 and dc dt = 0
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n-Nullclines • On these lines, the slope (velocity) in the n- direction (vertical) is zero • The only movement can be in the c-direction (ie right or left) n a 1 c 1 + c 1 = 0 n = 0 c = 1 a 1 1 dn dt = a 1 c 1 + c n n = 0
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c-Nullclines • On this curve, the slope (velocity) in the c- direction (horizontal) is zero • The only movement can be in the n-direction (ie up or down) n = a 2 c ( ) 1 + c ( ) c dc dt = c 1 + c n c + a 2 = 0
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biosim lec7 - Phase Plane Analysis PhasePlane Analysis...

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