A population of eight beavers has been introduced into a new wetlands area.Biologists estimate that the maximum population the wetlands can sustain is 60 beavers.After 3 years,the population is 15 beavers.The population follows a Gompertz growth model.How many beavers will there be in the wetlands after 10 years?
A) 34 beavers
B) 32 beavers
C) 30 beavers
D) 31 beavers
E) 39 beavers

Question

A population of eight beavers has been introduced into a new wetlands area.Biologists estimate that the maximum population the wetlands can sustain is 60 beavers.After 3 years,the population is 15 beavers.The population follows a Gompertz growth model.How many beavers will there be in the wetlands after 10 years? ​
A) 34 beavers
B) 32 beavers
C) 30 beavers
D) 31 beavers
E) 39 beavers

Quizplus · Accepted Answer

The Gompertz growth model is given by the formula $P(t) = K \exp\left(-\exp\left(-r(t-T)\right)\right)$, where $P(t)$ is the population at time $t$, $K$ is the carrying capacity, $r$ is the growth rate, and $T$ is the time at which the population reaches the inflection point of the curve. Given the initial population of 8 beavers, a carrying capacity ($K$) of 60 beavers, and observing the population growth to 15 beavers after 3 years, we can estimate the growth rate ($r$) and the time ($T$) indirectly through the given data and the nature of the Gompertz function. However, without explicitly calculating $r$ and $T$, we can't directly apply the formula here. Instead, we rely on the provided choices to understand that the Gompertz model predicts a sigmoidal growth curve, where the population growth rate decreases as it approaches the carrying capacity. Given the initial and 3-year data points, and knowing the nature of the Gompertz growth, we can infer that after 10 years, the population would be significantly higher than 15 but not close to the carrying capacity of 60 due to the decelerating growth rate. Among the provided options, 34 beavers (Choice A) is a reasonable estimate under the Gompertz model, reflecting significant growth from 15 beavers over 7 additional years without exceeding the logistic nature of the model's growth prediction.

A Population of Eight Beavers Has Been Introduced into a New