Question

Write and describe the meaning of the equation for population growth using the variables \(G, r,\) and \(N\)

Short answer

The population growth equation is given by \(G = rN\), where \(G\) represents the growth rate of the population, \(r\) is the per capita growth rate, and \(N\) is the current population size. This equation helps us calculate the net increase in population, taking into account both births and deaths. A positive per capita growth rate (\(r\)) results in an increasing population, while a negative per capita growth rate leads to a declining population. By using this equation, we can predict population dynamics, which is essential for fields such as ecology, conservation, and public health.

Step-by-step solution

Understanding the Population Growth Equation

The equation for population growth is given by the formula \(G = rN\), where: - \(G\) represents the growth rate of the population - \(r\) represents the per capita growth rate - \(N\) represents the current population size The population growth equation is used to calculate the growth rate of a population over time. This growth rate depends on the current population size and the per capita growth rate.

Breaking Down the Variables

Let's further break down each variable: 1. \(G\) (Growth Rate): The growth rate shows the net increase in the population size, taking into account both births and deaths. A positive growth rate indicates an increasing population, while a negative growth rate indicates a declining population. 2. \(r\) (Per Capita Growth Rate): The per capita growth rate represents the average number of new individuals produced per individual during a specific time period. It's the difference between the birth rate and the death rate. The per capita growth rate can be positive, negative, or zero, depending on whether the population is growing, shrinking, or remaining stable. 3. \(N\) (Current Population Size): The current population size is the number of individuals in the population at a specific time. This number can change over time as births, deaths, and migration occur.

Interpreting the Equation

To make sense of the population growth equation, \(G = rN\), we can think of it as: - When the per capita growth rate, \(r\), is positive, the population will grow, leading to a positive value of \(G\). - When the per capita growth rate, \(r\), is negative, the population will shrink, leading to a negative value of \(G\). - When the per capita growth rate, \(r\), is zero, the population will remain stable, leading to a value of \(G\) equal to zero. By using this equation, we can predict the growth or decline of a population based on the current population size and the per capita growth rate. This information is essential for understanding the dynamics of populations and can be used in various settings including ecology, conservation, and public health.