Mathematics

Mathematics

Problem

A wildlife refuge models a deer population with dydt=k y (300−y)\dfrac{dy}{dt}=k\,y\,(300-y). If the current population is 150150 deer and the parameter kk is suddenly doubled while the population size remains unchanged, how does the instantaneous growth rate change? a) It quadruples. b) It remains the same. c) It doubles. d) It is halved.
Intuitive explanation

This is a logistic growth problem where the instantaneous rate of change depends on both the current population and the model parameter. The key insight is that the growth rate dydt\frac{dy}{dt} is determined entirely by plugging in the current values of yy and kk into the differential equation—you don't need to solve it. When kk doubles but yy stays at 150, you're simply evaluating the same function at a new parameter value, so the growth rate scales by the same factor.

Intuitive explanation

This is a logistic growth problem where the instantaneous rate of change depends on both the current population and the model parameter. The key insight is that the growth rate dydt\frac{dy}{dt} is determined entirely by plugging in the current values of yy and kk into the differential equation—you don't need to solve it. When kk doubles but yy stays at 150, you're simply evaluating the same function at a new parameter value, so the growth rate scales by the same factor.

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