Mathematical Modeling Introduction to MatLab Qualitative Behavior of

Mathematical modeling introduction to matlab

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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Taylor’s Theorem Let y e be an equilibrium solution of the DE dy dt = f ( y ) , so f ( y e ) = 0. Theorem (Taylor Series) If for a range about y e , the function, f , has infinitely many derivatives at y e , then f ( y ) satisfies the Taylor Series f ( y ) = f ( y e ) + f 0 ( y e )( y - y e ) + f 00 ( y e ) 2! ( y - y e ) 2 + ... Since f ( y e ) = 0, then the dominate term near y e is the linear term f 0 ( y e )( y - y e ). Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (23/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Linearization The next step is finding the local behavior near each of the equilibrium solutions of the DE dy dt = f ( y ) . Theorem (Linearization about an Equilibrium Point) Let y e be an equilibrium point of the DE above and assume that f has a continuous derivative near y e . If f 0 ( y e ) < 0 , then y e is an asymptotically stable equilibrium. If f 0 ( y e ) > 0 , then y e is an unstable equilibrium. If f 0 ( y e ) = 0 , then more information is needed to classify y e . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (24/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Example: Logistic Growth Model 1 Example: Logistic Growth Model Consider the logistic growth equation: dP dt = f ( P ) = 0 . 05 P 1 - P 2000 Equilibria satisfy f ( P e ) = 0, so P e = 0, the extinction equilibrium P e = 2000, the carrying capacity It is easy to compute f 0 ( P ) = 0 . 05 - 0 . 1 P 2000 Since f 0 (0) = 0 . 05 > 0, P e = 0 is an unstable equilibrium or repeller Since f 0 (2000) = - 0 . 05 < 0, P e = 2000 is a stable equilibrium or attractor Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (25/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Example: Logistic Growth Model 2 Geometric Local Analysis : Equilibria are P e = 0 and P e = 2000 The graph of f ( P ) gives more information To the left of P e = 0, f ( P ) < 0 Since dP dt = f ( P ) < 0, P ( t ) is decreasing Note that this region is outside the region of biological significance For 0 < P < 2000, f ( P ) > 0 Since dP dt = f ( P ) > 0, P ( t ) is increasing Population monotonically growing in this area For P > 2000, f ( P ) < 0 Since dP dt = f ( P ) < 0, P ( t ) is decreasing Population monotonically decreasing in this region Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (26/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Example: Logistic Growth Model 3 Phase Portrait Use the above information to draw a Phase Portrait of the
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