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Chapter 9: Problem 1

What is the difference between a discrete \((\Delta N / \Delta t)\) and continuous \((\mathrm{d} N / \mathrm{d} t)\) model of population growth? What is the difference between geometric and exponential growth?

Short Answer

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Question: Explain the differences between a discrete and continuous model of population growth and the differences between geometric and exponential growth. Answer: A discrete model of population growth involves changes in population at fixed time intervals, with the population increasing by a constant proportion or factor at each interval, known as geometric growth. On the other hand, a continuous model of population growth involves continuous changes in population over time, where the rate of population growth is directly proportional to the population size at any given time, known as exponential growth.

Step by step solution

01

Discrete Model of Population Growth

In a discrete model of population growth, changes in population occur at specific time intervals, for example, per day, per month, or per year. The change in population during each time interval is denoted by \((\Delta N / \Delta t)\), where \(\Delta N\) represents the change in population and \(\Delta t\) represents the time interval.
02

Continuous Model of Population Growth

In a continuous model of population growth, the population changes continuously over time. The rate of change of population with respect to time is given by \(\frac{\mathrm{d} N}{\mathrm{d} t}\), where \(\mathrm{d} N\) represents the infinitesimal change in population and \(\mathrm{d} t\) represents the infinitesimal change in time.
03

Geometric Growth

Geometric growth occurs in a discrete model of population growth. In geometric growth, the population increases by a constant proportion or factor at each time interval. This can be represented by the equation \(N_t = N_0(1 + r)^t\), where \(N_t\) is the population at time \(t\), \(N_0\) is the initial population, and \(r\) is the per capita growth rate.
04

Exponential Growth

Exponential growth occurs in a continuous model of population growth. In exponential growth, the rate at which the population grows is directly proportional to the population size at any given time. This can be represented by the equation \(N_t = N_0e^{rt}\), where \(N_t\) is the population at time \(t\), \(N_0\) is the initial population, \(r\) is the per capita growth rate, and \(e\) is the base of the natural logarithm. In summary, a discrete model of population growth consists of changes in population at fixed time intervals and can be described by geometric growth. In contrast, a continuous model of population growth involves continuous changes in population over time and follows exponential growth.

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