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Chapter 9: Problem 7

How can the finite growth multiplier be used to predict future values of population density, \(N(t) ?\)

Short Answer

Expert verified
Answer: The finite growth multiplier (位) is used to predict future values of population density (N(t)) by working with the discrete-time population growth equation: N(t+1) = 位 脳 N(t). We calculate the population density after a certain number of time intervals n as N(t+n) = 位^n 脳 N(t). By applying this relationship and using the appropriate multiplier, we can estimate how a population will change over time for planning and decision-making purposes.

Step by step solution

01

Define the finite growth multiplier

The finite growth multiplier is a constant, denoted by lambda (位), that represents the factor by which a population will increase, or decrease, over a certain time interval 螖t. If 位 > 1, the population will grow; if 位 < 1, the population will decrease, and if 位 = 1, the population will remain constant.
02

Formulate the population growth equation

To predict the future values of population density, we will use the discrete-time population growth equation given by: \[N(t+1) = \lambda \cdot N(t)\] where: - \(N(t)\) is the population density at time t - \(N(t+1)\) is the population density at time (t + 1) - \(\lambda\) is the finite growth multiplier
03

Calculate the population density for a specific time

To predict the population density in the future, we will use the population growth equation and the finite growth multiplier. For example, if we want to find the population density after a certain number of time intervals 'n', we calculate \[N(t+n) = \lambda^n \cdot N(t)\] where: - \(N(t+n)\) is the population density at time (t + n) - \(n\) is the number of time intervals in the future - \(\lambda^n\) is the finite growth multiplier raised to the power of n.
04

Interpret the results

By using the finite growth multiplier to predict future values of population density, we can estimate how a population will change over time. This information can be valuable for making informed decisions about resource management, conservation, and other areas of planning and decision-making related to population growth and decline.

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