Consider the system x? = x(1 ? ?x ? 0.5y), y? = y(?0.75
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Consider the system x? = x(1 ? ?x ? 0.5y), y? = y(?0.75 + 0.25x), where ? > 0. Observe that this system is a modificatio
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Consider the system x? = x(1 ? ?x ? 0.5y), y? = y(?0.75 + 0.25x), where ? > 0. Observe that this system is a modification of the system (2) in Example 1. (a) Find all of the critical points. How does their location change as ? increases from zero? Observe that there is a critical point in the interior of the first quadrant only if ? < 1 3 . (b) Determine the type and stability property of each critical point. Find the value ?1 <1 3 where the nature of the critical point in the interior of the first quadrant changes. Describe the change that takes place in this critical point as ? passes through ?1. (c) Draw a direction field and phase portrait for a value of ? between zero and ?1; for a value of ? between ?1 and 1 3 . (d) Describe the effect on the two populations as ? increases from zero to 1 3 .
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