Mathematics

Problem

A wildlife refuge monitors the biomass

W(t)

, measured in kilograms, of a newly introduced plant species. The rate of change of

W

with respect to time

t

(in months) is given by

\frac{dW}{dt} = r\,W\,(C - W),

where

r

and

C

are positive constants. Initially,

W(0) = 40\,\text{kg}

, and the refuge records that after one month the biomass has grown to

55\,\text{kg}

. (a) Show that the recorded information implies the equation

r(55)(C - 55) = 15

. (b) The refuge also observes that when

W = 120\,\text{kg}

, the biomass is increasing at a rate of

96\,\text{kg}

per month. Use this additional observation together with your result from part (a) to determine the numerical values of

r

and

C

. (c) Verify that the biomass increases most rapidly at

W = \frac{C}{2}

, and calculate this maximal growth rate using the constants you found. (d) Discuss, in one or two sentences, the ecological significance of the parameter

C

for long-term management of the plant species.

Intuitive explanation

This is a logistic growth model, where a population grows exponentially at first but slows as it approaches a carrying capacity $C$ . The key insight is to recognize this as a separable differential equation: rearrange it so all $W$ terms are on one side and all $t$ terms on the other, then integrate using partial fractions. Once you solve for $W(t)$ , you'll have a formula with unknown constants $r$ and $C$ that you can determine using your two initial conditions: $W(0) = 40$ and $W(1) = 55$ .

Intuitive explanation

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