If X e 0 then from the second equation we have either the extinction

If x e 0 then from the second equation we have either

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If X e = 0, then from the second equation we have either the extinction equilibrium , ( X e , Y e ) = (0 , 0) or the competitive exclusion equilibrium (with Y winning): ( X e , Y e ) = 0 , b 1 b 2 , where Y e is at carrying capacity . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (40/68)
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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 3 Continuing the equilibria of the competition model : If a 1 - a 2 X e - a 3 Y e = 0 from the first equation, then from the second equation we have either the competitive exclusion equilibrium (with X winning): ( X e , Y e ) = a 1 a 2 , 0 , where X e is at carrying capacity or the nonzero equilibrium : ( X e , Y e ) = a 1 b 2 - a 3 b 1 a 2 b 2 - a 3 b 3 , a 2 b 1 - a 1 b 3 a 2 b 2 - a 3 b 3 . If X e > 0 and Y e > 0, then we obtain the cooperative equilibrium with neither species going extinct. Note: This last equilibrium could have a negative X e or Y e , depending on the values of the parameters. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (41/68) 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 Maple Equilibrium Maple can readily be used to find equilibria : Later we find the numerical values of the parameters, so Maple easily finds all equilibria: Note : The positive equilibrium is close to the late data points. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equ — (42/68) 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 Nullclines 1 Equilibrium analysis shows there are always the extinction and two competitive exclusion equilibria with the latter going to carrying capacity for one of the species. Provided a 2 b 2 - a 3 b 3 6 = 0, there is another equilibrium, and it satisfies: 1. X e 0 and Y e > 0 or 2. X e > 0 and Y e 0 or 3. X e > 0 and Y e > 0. We concentrate our studies on Case 3, where there exists a positive cooperative equilibrium . Finding equilibia can be done geometrically using nullclines . Nullclines are simply curves where dX dt = 0 and dY dt = 0 . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Equations: — (43/68) 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 Nullclines 2 For the competition model , the nullclines satisfy: dX dt = X ( a 1 - a 2 X - a 3 Y ) = 0 and dY dt = Y ( b 1 - b 2 Y - b 3 X ) = 0 , where the first equation has solutions only flowing in the Y -direction and the second equation has solutions only flowing in the X -direction .
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