Suppose that a certain pair of competing species are des
Question and Solution
Suppose that a certain pair of competing species are described by the system dx dt = x(4 ? x ? y), dy dt = y(2 + 2? ? y
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Suppose that a certain pair of competing species are described by the system dx dt = x(4 ? x ? y), dy dt = y(2 + 2? ? y ? ?x), where ? > 0 is a parameter. (a) Find the critical points. Note that (2, 2) is a critical point for all values of ?. (b) Determine the nature of the critical point (2, 2) for ? = 0.75 and for ? = 1.25. There is a value of ? between 0.75 and 1.25 where the nature of the critical point changes abruptly. Denote this value by ?0; it is also called a bifurcation point. (c) Find the approximate linear system near the point (2, 2) in terms of ?. (d) Find the eigenvalues of the linear system in part (c) as functions of ?. Then determine ?0. (e) Draw phase portraits near (2, 2) for ? = ?0 and for values of ? slightly less than, and slightly greater than, ?0. Explain how the transition in the phase portrait takes place as ? passes through ?0.
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