By S. Lipschutz, et al.

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2jnπ k + dj sin 2jnπ k , 42 1. 24 [23]. 5) which describes a population with a propensity to simple exponential growth at low densities and a tendency to decrease at high densities. The quantity λ = exp(r(1 − x(n))) could be considered the densitydependent reproductive rate of the population. This model is plausible for a single-species population that is regulated by an epidemic disease at high density. The nontrivial fixed point of this equation is given by x∗ = 1. Now, ′ f (1) = 1 − r. Hence x∗ = 1 is asymptotically stable if 0 < r ≤ 2 (check r = 2).

For most biological species, however, none of the above cases is valid as the population increases until it reaches a certain upper limit. Then, due to the limitations of available resources, the creatures will become testy and engage in competition for those limited resources. This competition is proportional to the number of squabbles among them, given by y 2 (n). A more reasonable model would allow b, the proportionality constant, to be greater than 0, y(n + 1) = µy(n) − by 2 (n). 3) x(n + 1) = µx(n)(1 − x(n)) = f (x(n)).

Consider Baker’s map defined by ⎧ 1 ⎪ ⎨2x for 0 ≤ x ≤ , 2 B(x) = 1 ⎪ ⎩2x − 1 for < x ≤ 1. 2 (i) Draw the function B(x) on [0,1]. (ii) Show that x ∈ [0, 1] is an eventually fixed point if and only if it is of the form x = k/2n , where k and n are positive integers,2 with 0 ≤ k ≤ 2n − 1. 13. Find the fixed points and the eventually fixed points of x(n + 1) = f (x(n)), where f (x) = x2 . 14. 7 that is not in the form k/2n . 15. 7. Show that if x = k/2n , where k and n are positive integers with 0 < k/2n ≤ 1, then x is an eventually fixed point.

### Beginning Finite Mathematics [Theory and Problems of] by S. Lipschutz, et al.

by Donald

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