5. Below we further analyze the nutrient-depletion model given in Section 4.1. (

Question

5. Below we further analyze the nutrient-depletion model given in Section 4.1. (a) Show that the equation a = K(C. - QN)N can be written in the form W - (1 - 4)w, (b) where r = Cox and B = Co/a. This is called a logistic equation. Inter- pret r and B. Show that the equation can be written dN (1 - N/B)N r dt, (c) and integrate both sides. Rearrange the equation in (b) to show that the solution thereby ob- tained is NOB N(e) = NOB N(C) = N. + (B - No)e" (d) Show that for t o the population approaches the density B. Also show that if N, is very small, the population initially appears to grow exponen- tially at the rate r. *Problems preceded by an asterisk are especially challenging. An Introduction to Continuous Models 153 (e) (f) Interpret the results in terms of the original parameters of the bacterial model. Find the values of B, No, and r in the curve that Gause (1934) fit to the growth of the yeast Schizosaccharomyces kephir (see caption of Figure 4.1c).

Explanation / Answer

b)

c) Rewrite

d) When t tends to infinity, the denominator becomes N0+0 hence N(t) = B as t tends to infinity.

N(t) = kert for some constant k.

Hence increases exponentially for small N0.

e)

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