Population growth is represented by the given logistic equation (see image), where t is measured in weeks. The population is 1/3 the carrying capacity initially. How many weeks does it take for the population to reach 2/3 of the carrying capacity? Your answer should be an integer multiple of ln2. The equation displayed is a differential equation, which describes the rate of change of a population $ P $ with respect to time $ t $. The equation is given by:\[\frac{dP}{dt} = \frac{P}{10} - \frac{P^2}{3000}\]### Explanation:- The term $\frac{P}{10}$ represents a linear growth rate of the population, suggesting that the population grows proportionally to its current size. - The term $-\frac{P^2}{3000}$ represents a quadratic decay rate, indicating that as the population increases, it faces a limiting factor that slows its growth, possibly due to limited resources or increased competition. This term prevents the population from growing indefinitely.This type of model is often used in biology to describe population dynamics, especially in scenarios where resources are limited and populations cannot grow indefinitely.

According to the given information, Consider the given differential equation:

Answered: Population growth is represented by the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Population growth is represented by the given logistic equation (see image), where t is measured in weeks. The population is 1/3 the carrying capacity initially. How many weeks does it take for the population to reach 2/3 of the carrying capacity? Your answer should be an integer multiple of ln2.

The equation displayed is a differential equation, which describes the rate of change of a population $ P $ with respect to time $ t $. The equation is given by:

\[
\frac{dP}{dt} = \frac{P}{10} - \frac{P^2}{3000}
\]

### Explanation:

- The term $\frac{P}{10}$ represents a linear growth rate of the population, suggesting that the population grows proportionally to its current size.

- The term $-\frac{P^2}{3000}$ represents a quadratic decay rate, indicating that as the population increases, it faces a limiting factor that slows its growth, possibly due to limited resources or increased competition. This term prevents the population from growing indefinitely.

This type of model is often used in biology to describe population dynamics, especially in scenarios where resources are limited and populations cannot grow indefinitely.

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