carrying capacity equilibrium X e Y e 0 5 the Jacobian satisfies J 0 5 05 15 2

Carrying capacity equilibrium x e y e 0 5 the

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carrying capacity equilibrium , ( X e , Y e ) = (0 , 5), the Jacobian satisfies: J (0 , 5) = 0 . 05 0 - 0 . 15 - 0 . 2 ! . This has eigenvalues λ 1 = 0 . 05 ( ξ 1 = [5 , - 3] T ) and λ 2 = - 0 . 2 ( ξ 1 = [0 , 1] T ). This is a saddle node . At the cooperative equilibrium , ( X e , Y e ) = (4 , 2), the Jacobian satisfies: J (2 , 4) = - 0 . 08 - 0 . 04 - 0 . 06 - 0 . 08 ! . This has eigenvalues λ 1 = - 0 . 129 ( ξ 1 = [1 , 1 . 2247] T ) and λ 2 = - 0 . 031 ( ξ 1 = [1 , - 1 . 2247] T ). This is a stable node . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (53/68)
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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 Phase Portrait The figure below was generated with pplane8 and shows that Example 2 exhibits cooperation with all solutions going toward the nonzero equilibrium , ( X e , Y e ) = (4 , 2). Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (54/68)
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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 Yeast Competition Model 1 Competition Model: Competition is ubiquitous in ecological studies and many other fields Craft beer is a very important part of the San Diego economy Researchers at UCSD created a company that provides brewers with one of the best selections of diverse cultures of different strains of the yeast, Saccharomyces cerevisiae Different strains are cultivated for particular flavors Often S. cerevisiae is maintained in a continuous chemostat for constant quality - large beer manufacturers Large cultures can become contaminated with other species of yeast It can be very expensive to start a new pure culture We examine a competition model for different species of yeast Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (55/68)
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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 Yeast Competition Model 2 Yeast Experiment: G. F. Gause 23 studied competing species of yeast, Saccharomyces cerevisiae and a common contaminant species Schizosaccharomyces kephir The experiments examined growth in monocultures for individual growth laws and in mixed cultures to observe competition Below is a table combining two experimental studies of S. cerevisiae Time (hr) 0 1.5 9 10 18 18 23 Volume 0.37 1.63 6.2 8.87 10.66 10.97 12.5 Time (hr) 25.5 27 34 38 42 45.5 47 Volume 12.6 12.9 13.27 12.77 12.87 12.9 12.7 Below is a table combining two experimental studies of S. kephir Time (hr) 9 10 23 25.5 42 45.5 66 87 111 135 Volume 1.27 1 1.7 2.33 2.73 4.56 4.87 5.67 5.8 5.83 2 G. F. Gause, Struggle for Existence , Hafner, New York, 1934. 3 G. F. Gause (1932), Experimental studies on the struggle for existence.
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