hw1(2) - constructing some cobwebs for non-decreasing f s)....

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Homework assignment 1 * January 6, 2006 1. Give 4 examples illustrating the 4 types of linearized stability that may occur as we discussed in class. 2. In class we showed the outcome of a population under contest competition and under scramble competition by constructing the cobweb in case the basic reproductive ratio R 0 > 1. Discuss in both cases what happens if R 0 (0 , 1]. Do you notice any diFerences in the outcomes? Also discuss linearized stability for all ±xed points. 3. Suppose a population is governed by N t +1 = f ( N t ) , where f (0) = 0. Assume that f : R + R + is non-decreasing (that is: x y implies that f ( x ) f ( y )), and that there is some M > 0 such that f ( N t ) M for all N t . Prove that every solution sequence N t converges as t → ∞ . 1 (Convince yourself ±rst by
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Unformatted text preview: constructing some cobwebs for non-decreasing f s). Notice that contest competition is a special case. Suppose that f is continuously diFerentiable and that the xed points are isolated and that f is never equal to 1 at each xed point. Show (geometrically) that the xed points alternate bewteen monotonically stable and monotonically unstable. 4. Verify the occurrence of a transcritical bifurcation for Hassels equation at the bifurcation point ( R ,x ) = (1 , 0). * MAP 4484 / 5489; Instructor: Patrick De Leenheer. 1 It may be helpful to review the chapter on Sequences as it is taught in your Calc course. 1...
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