Question

If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (r) of the population?

Short answer

The intrinsic rate of increase (r) of the population is approximately 0.2310.

Step-by-step solution

Recognize the Exponential Growth Equation

The general formula for exponential growth of a population can be expressed as: \( P(t) = P_0e^{rt} \), where:\( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population size, \( r \) is the intrinsic rate of increase, and \( t \) is the time in years.

Set Up the Equation

Since the population doubles in 3 years, the final population \( P(t) \) is twice the initial population \( P_0 \), so we have \( 2P_0 = P_0e^{3r} \).

Simplify the Equation

Divide both sides of the equation by \( P_0 \) to get \( 2 = e^{3r} \).

Solve for the Intrinsic Rate of Increase (r)

Take the natural logarithm of both sides to obtain \( \ln(2) = 3r \). Then, divide both sides of the equation by 3 to solve for \( r \), yielding \( r = \frac{\ln(2)}{3} \).

Calculate the Intrinsic Rate of Increase

Use a calculator to compute the value of \( r \) by plugging in the natural logarithm of 2, which gives \( r = \frac{\ln(2)}{3} \approx 0.2310 \).

Key concepts

Exponential Population Growth

Exponential population growth is a pattern of increase in numbers based on constant growth rates over equal time intervals. Unlike linear growth, which increases by a fixed amount each period, exponential growth multiplies the population by a certain factor in each passing period. This type of growth is often represented mathematically by the equation \( P(t) = P_0e^{rt} \), where \( P(t) \) represents the population at time \( t \), \( P_0 \) is the initial population size, \( e \) is the base of the natural logarithm, \( r \) is the intrinsic rate of increase, and \( t \) is the time over which the population grows.

In the context of our exercise, if a population doubles (\( 2P_0 \) being the final population) in a set period (\( 3 \) years for instance), one can use the exponential growth formula to derive the intrinsic rate of increase (\( r \)). Through mathematical manipulations involving natural logarithms, we can isolate \( r \) to ascertain how quickly the population is growing exponentially.

Population Dynamics

Population dynamics refer to the patterns and processes through which populations change over time. These may include variations in population size, composition, distribution, age structure, and the birth and death rates driving these changes. Understanding the dynamics of a population is crucial for many fields, such as ecology, conservation biology, and even urban development.

One key aspect of population dynamics is understanding the conditions under which a population grows or shrinks. Exponential growth, for example, may occur when the resources are abundant, and there are few limiting factors like predators or disease. However, this type of growth cannot continue indefinitely, as resources may become scarce, and other ecological factors come into play, potentially leading to a leveling off (logistic growth) or even decline. Through mathematical concepts and formulas, scientists predict population sizes and growth rates, helping to inform decision-making in wildlife management, public policy, and environmental preservation.

Biological Mathematics

Biological mathematics, often known as mathematical biology or biomathematics, is the interdisciplinary field that applies mathematical techniques and principles to solve biological problems. Using equations, modeling, statistics, and computational methods, researchers in this field translate biological processes into mathematical terms.

An integral component of biological mathematics is population modeling, which can provide insights on how populations interact with each other and their environment. The equation from our exercise is a fundamental example of how mathematics assists in quantifying biological growth and predicting future population trends based on current conditions. The calculation of the intrinsic rate of increase \( r \) gives us a numerical value that quantifies the speed at which a population will grow in a given environment, under the assumption that the conditions conducive to their growth will remain constant. Predictive models like these allow biologists to understand complex ecological interactions and to forecast changes in population sizes, which is vital in making conservation decisions and sustaining biodiversity.