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2010-02-22 - MAT 1332 CALCULUS FOR LIFE SCIENCES JING LI...

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MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review : Equilibria & Graphical Display of Autonomous DEs 1 2. Stability Analysis of Equilibria 1 3. Example: A Model of a Disease 2 4. 5.5: Two-Dimensional Differential Equations 3 1. Review : Equilibria & Graphical Display of Autonomous DEs Equilibrium (a fixed point, steady state): A value m * of the state variable is called an equilibrium of the autonomous DE dm dt = f ( m ) if f ( m * ) = 0 . Algorithm for finding equilibria of an autonomous DE: (1) Make sure that the differential equation is autonomous; (2) Write the equation for the equilibria; (3) Factor; (4) Set each factor equal to 0 and solve for the equilibria; (5) Meditate upon the results. Graphical Display of Autonomous DE: the phase line diagram Example 1. Draw the phase-line diagram for the selection model dp dt = ( r a - r b ) p (1 - p ) with r a = 1 . 5 , and r b = 2 . 0 . Solution: Exercise 1. What about the case with r a = 2 . 0 , and r b = 1 . 5 . 2. Stability Analysis of Equilibria While finding equilibria for autonomous DEs is simply a matter of solving equations, determine the stability of each equilibrium can be more complex. In discrete-time dynamic system, we graphed the updating function and find patterns that predicted stability or instability. Here we can go through a
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