Final answer:
To find the time at which the initial population subject to logistic growth has doubled, one must integrate the logistic equation and solve for the doubling time, using properties of logarithms for approximation when the growth rate is small.
Step-by-step explanation:
The student is working with the logistic growth equation, which is a model used to describe populations as they approach a carrying capacity. The given differential equation dy/dt = ry(1-y/k) describes the change in a population over time, where r is the growth rate, y is the current population size, k is the carrying capacity of the environment, and y0 is the initial population size.
To find the time τ at which the initial population has doubled, we need to solve this differential equation for y = 2y0, where y0 = k/10. Substituting the given values and integrating, we can find τ as a function of r and the initial and final populations. For r=0.021 per year, we need to find the specific value of τ that corresponds to a doubling of the population from k/10 to 2(k/10).
Substituting the values and integrating, the function representing y involves a logarithmic component based on the initial and final populations and the growth rate. For small values of the growth rate (e.g. r ≈ 0.021), the time τ can be approximated using properties of the natural logarithm, giving us an expression for τ in terms of r. As the growth rate is within a range where it's approximately equal to the percentage growth (r ≈ p for small p), we can use the approximation ln(1 + p) ≈ p to simplify the calculation.