A bacterial population obeys the logistic equation d N d t = rN (1- N K )

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Ordinary differential equations

Computational Physics, Ch Letting the computer do the algebra

Problem

A bacterial population N(t)N(t) obeys the logistic equation dNdt=rN(1−NK),\frac{\mathrm{d}N}{\mathrm{d}t} = rN\left(1-\frac{N}{K}\right), with growth rate r=0.35 h−1r = 0.35\,\mathrm{h^{-1}}, carrying capacity K=5.0×105K = 5.0\times10^{5} cells, and initial population N(0)=N0=1.0×104N(0)=N_0=1.0\times10^{4} cells. (a) Separate the variables, using the partial-fraction decomposition 1N(1−N/K)=1N+1K−N\dfrac{1}{N(1-N/K)} = \dfrac{1}{N} + \dfrac{1}{K-N}, and integrate symbolically to obtain N(t)N(t) in closed form. (b) Verify by substitution that your N(t)N(t) satisfies the logistic equation. (c) Evaluate NN at t=10 ht=10\,\mathrm{h}, to three significant figures.

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