A bacterial culture growing in a flask with limited nutrient follows the logistic equation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Ordinary differential equations · Ordinary differential equations

Mathematical Methods for Physics, Ch First-order differential equations and physical modelling

Problem

A bacterial culture growing in a flask with limited nutrient follows the logistic equation y˙=ry(1−y/K)\dot{y} = ry\left(1-y/K\right), where y(t)y(t) is the population (in units of 10610^6 cells), r=0.20 hour−1r=0.20\,\mathrm{hour^{-1}} and the carrying capacity is K=500K=500 (in the same units). At t=0t=0 the population is y0=50y_0 = 50. ((a)) Find the equilibria of the equation and use the sign of ∂y˙/∂y\partial\dot{y}/\partial y at each to state whether it is stable or unstable. ((b)) Near y=0y=0 the equation reduces to y˙≈ry\dot{y}\approx ry: state the effective time constant of this early growth. ((c)) Sketch the direction field for 0≤y≤6000 \le y \le 600 with three solution curves, starting at y=50y=50, 250250 and 600600, and say from it whether the culture that starts at y0=50y_0=50 grows towards KK or shrinks away from it.

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