Use differential equations to estimate the population of bacteria in a toxic environment

by RoRi

January 31, 2016

A strain of bacteria in a medium is destroyed by a toxin at a rate jointly proportional to the population of the bacteria and to the amount of the toxin. If there were no toxins, the bacteria would have a growth rate proportional to the amount of bacteria present. Let $x$ be the population of bacteria at time $t$ , and assume the amount of the toxin is increasing at a constant rate starting with 0 toxins at $t = 0$ . Find a differential equation for $x$ and solve this equation.

From the problem statement we have that $x$ satisfies the differential equation

$x' - k(1-at)x = 0$

with $x(0) = x_0$ . This is a linear first-order differential equation so we may use Theorem 8.3 (page 310 of Apostol) with

$P(t) = -k(1-at), \quad Q(t) = 0 \quad \implies \quad A(t) = - \int_0^t (1-au) \, du = -kt + \frac{kat^2}{2}.$

Therefore,

$x = x_0e^{kt-\frac{kat^2}{2}} = x_0 e^{k\left( t - \frac{at^2}{2} \right)}.$

The correct curve (see the image on page 322 of Apostol for the curves) is (d) since

$e^{k \left( t - \frac{at^2}{2} \right)} \to 0 \quad \text{as} \quad t \to +\infty$

implies

$x \to 0 \quad \text{as} \quad t \to +\infty.$

Furthermore, $x$ is never 0. Finally, we know there is initial growth in $x$ at time $t = 0$ since $x' = kx$ is positive.

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