Could use real experimental verification of the assumptions 2 C H Taubes

Could use real experimental verification of the

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Could use real experimental verification of the assumptions [2] C. H. Taubes, Modeling Differential Equations in Biology , Prentice Hall, 2001. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (36/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Left Snail Model 2 Taubes Snail Model Let p ( t ) be the probability that a snail is dextral A model that qualitatively exhibits the behavior described on previous slide: dp dt = αp (1 - p ) p - 1 2 , 0 p 1 , where α is some positive constant What is the behavior of this differential equation? What does its solutions predict about the chirality of populations of snails? Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (37/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Left Snail Model 3 Taubes Snail Model This differential equation is not easy to solve exactly Qualitative analysis techniques for this differential equation are relatively easily to show why snails are likely to be in either the dextral or sinistral forms The snail model: dp dt = f ( p ) = αp (1 - p ) p - 1 2 , 0 p 1 , Equilibria are p e = 0 , 1 2 , 1 f ( p ) < 0 for 0 < p < 1 2 , so solutions decrease f ( p ) > 0 for 1 2 < p < 1, so solutions increase The equilibrium at p e = 1 2 is unstable The equilibria at p e = 0 and 1 are stable Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (38/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Left Snail Model 4 Phase Portrait : dp dt = αp (1 - p ) p - 1 2 0 0.2 0.4 0.6 0.8 1 -0.1 -0.05 0 0.05 0.1 > > > < < < p α p (1 - p )( p - 1 / 2) Phase Portrait for Snail Model ( α =1 . 5) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (39/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Left Snail Model 4 Diagram of Solutions for Snail Model Snail Model 0 0.2 0.4 0.6 0.8 1 p(t) 0 2 4 6 8 10 t Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (40/50)
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Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Left Snail Model Allee Effect Left Snail Model 5 Snail Model - Summary Figures show the solutions tend toward one of the stable equilibria , p e = 0 or 1 When the solution tends toward p e = 0, then the probability of a dextral snail being found drops to zero, so the population of snails all have the sinistral form When the solution tends toward p e = 1, then the population of snails virtually all have the dextral form This is what is observed in nature suggesting that this model exhibits the behavior of the evolution of snails This does not mean that the model is a good model!
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