Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of

# Linear applications of systems of 1 st order des

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Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Two Species Competition Model 1 Two Species Competition Model : Let X ( t ) be the density of one species of yeast and Y ( t ) be the density of another species of yeast. Assume each species follows the logistic growth model for interactions within the species. Model has a Malthusian growth term . Model has a term for intraspecies competition . The differential equation for each species has a loss term for interspecies competition . Assume interspecies competition is represented by the product of the two species. If two species compete for a single resource, then 1. Competitive Exclusion - one species out competes the other and becomes the only survivor 2. Coexistence - both species coexist around a stable equilibrium Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (37/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Two Species Competition Model 2 Two Species Competition Model : The system of ordinary differential equations (ODEs) for X ( t ) and Y ( t ) : dX dt = a 1 X - a 2 X 2 - a 3 X Y = f 1 ( X, Y ) dY dt = b 1 Y - b 2 Y 2 - b 3 Y X = f 2 ( X , Y ) First terms with a 1 and b 1 represent the exponential or Malthusian growth at low densities The terms a 2 and b 2 represent intraspecies competition from crowding by the same species The terms a 3 and b 3 represent interspecies competition from the second species Unlike the logistic growth model , this system of ODEs does not have an analytic solution, so we must turn to other analyses. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (38/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Model – Analysis 1 Competition Model : Analysis always begins finding equilibria , which requires: dX dt = 0 and dY dt = 0 , in the model system of ODEs. Thus, a 1 X e - a 2 X 2 e - a 3 X e Y e = 0 , b 1 Y e - b 2 Y 2 e - b 3 X e Y e = 0 . Factoring gives: X e ( a 1 - a 2 X e - a 3 Y e ) = 0 , Y e ( b 1 - b 2 Y e - b 3 X e ) = 0 . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (39/68) Introduction Linear Applications of Systems of 1 st Order DEs Nonlinear Applications of Systems of DEs Model of Glucose and Insulin Control Glucose Tolerance Test Competition Model Competition Model – Analysis 2 The equilibria of the competition model satisfy: X e ( a 1 - a 2 X e - a 3 Y e ) = 0 , Y e ( b 1 - b 2 Y e - b 3 X e ) = 0 . This system of equations must be solved simultaneously. The first equation gives: X e = 0 or a 1 - a 2 X e - a 3 Y e = 0.

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