Gompertz Growth Model This model is sometimes used to study the growth of a population

Chapter 8, Problem 5

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Gompertz Growth Model This model is sometimes used to study the growth of a population for which the per capita growth rate is density dependent. Denote the size of a population at time t by N(t); then, for N 0, dN dt = kN(ln K ln N) with N(0) = N0 (8.90) (a) Show that N(t) = K exp _ _ ln K N0 _ ekt _ is a solution of (8.90). To do this, differentiate N(t) with respect to t and show that the derivaive can be written in the form (8.90). Dont forget to show that N(0) = N0. Use a graphing calculator to sketch the graph of N(t) for N0 = 100, k = 2, and K = 1000. The function N(t) is called the . (b) Use lHospitals rule to show that lim N0 N ln N = 0 and use this equation to show that limN0 dN/dt = 0. Are there any other values of N where dN/dt = 0? (c) Sketch the graph of dN/dt as a function of N for k = 2 and K = 1000. Find the equilibria, and use your graph to and discuss their stability. Explain the meaning of K.

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