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By B. Wanamaker, K. Massey

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9. (Calculus) Redo the preceding problem, but use derivatives to determine the stability of the equilibria of Pt+1 = Pt + r Pt (1 − Pt ). You should, of course, get the same answers. 10. (Calculus) Here is a slightly different approach to the relationship between derivatives and stability: Find the tangent line approximations to f (P) = P + r P(1 − P) at the equilibria P ∗ = 0 and 1. Then 30 Dynamic Modeling with Difference Equations replace P + r P(1 − P) by these approximations in Pt+1 = Pt + r Pt (1 − Pt ).

At ﬁrst this increase seems slow, then it speeds up and then it slows again. 2 shows the sigmoid-shaped pattern that often appears in data from carefully controlled laboratory experiments in which populations increase in a limited environment. ) Biologically, then, we have made some progress; we have a more realistic model to describe population growth. Mathematically, things are not so nice, though. Unlike with the linear model, there is no obvious formula for Pt that emerges from our table. In fact, the only way to get the value of P100 seems to be to create a table with a hundred entries in it.

Note that, although a product like The fact that for matrices multiplication is not commutative – that order matters in a product – is a signiﬁcant difference from the algebra of ordinary numbers. It is very important to always be aware of this when using matrices.