Suppose that the discrete logistic model for population growth is models as follows: $P _ { n + 1 } = P _ { n } + 0.351 P _ { n } \left( 1 - \frac { P _ { n } } { 2.849 \times 10 ^ { 8 } } \right)$ What is the natural rate of increase when the population is small?

Question

Quizplus · Accepted Answer

When the population is small, we can use the linear approximation of the discrete logistic model, which is:

$$P_{n+1}\approx P_n+rP_n$$

where $r=0.351-0.351\frac{P_n}{2.849	imes 10^8}$. When the population is small, we can assume that $P_n\ll 2.849	imes 10^8$, so $r\approx 0.351$. Therefore, the natural rate of increase when the population is small is approximately 0.351. Choice A is the answer.

Suppose That the Discrete Logistic Model for Population Growth Is Pn+1=Pn+0.351Pn(1−Pn2.849×108)P _ { n + 1 } = P _ { n } + 0.351 P _ { n } \left( 1 - \frac { P _ { n } } { 2.849 \times 10 ^ { 8 } } \right)Pn+1​=Pn​+0.351Pn​(1−2.849×108Pn​​)

Suppose That the Discrete Logistic Model for Population Growth Is $P _ { n + 1 } = P _ { n } + 0.351 P _ { n } \left( 1 - \frac { P _ { n } } { 2.849 \times 10 ^ { 8 } } \right)$