Problem 21 - Calculus for Biology and Medicine

Question

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that quadruples in size every unit of time and that has five individuals at time 0 .

Step-by-Step Solution

Answer

The population size at time \( t \) is given by \( N_{t} = 5 \times 4^{t} \).

Step 1: Understand Exponential Growth

In exponential growth, a population grows by a consistent factor over regular intervals of time. The general formula for exponential growth is \( N_{t} = N_{0} imes r^{t} \), where \( N_{t} \) is the population at time \( t \), \( N_{0} \) is the initial population, \( r \) is the growth rate (factor), and \( t \) is the time elapsed.

Step 2: Identify Given Values

From the problem statement, we know the population quadruples (multiplies by 4) every unit of time, meaning \( r = 4 \). Additionally, the initial population at time \( t = 0 \) is given as 5 individuals, so \( N_{0} = 5 \).

Step 3: Substitute Values into the Exponential Growth Formula

Plug the known values into the general formula: \( N_{t} = N_{0} imes r^{t} \). Substitute \( N_{0} = 5 \) and \( r = 4 \) to get \( N_{t} = 5 imes 4^{t} \).

Step 4: Formulate Final Equation

Combine all substituted components to write the final equation for population as a function of time: \( N_{t} = 5 imes 4^{t} \). This is the exponential growth equation for the described scenario.

Key Concepts

Population Modeling

Population modeling is a useful technique in understanding how populations change over time. It helps predict future population sizes based on current data and certain assumptions. In mathematical terms, population modeling involves creating formulas to represent population dynamics. These models can be linear or non-linear, and they help us to project future scenarios based on different growth patterns.

When modeling population growth, we often encounter models that describe how rapidly numbers can increase over time. Exponential growth is one such model, where the population size increases by a certain multiplicative factor over equal time intervals. This is common in environments with abundant resources, allowing for rapid and sustained population growth.

Population modeling provides insight into scenarios involving resources, environmental impacts, and planning. By adjusting initial conditions and growth factors in the mathematical representations, we can simulate various growth scenarios to study their implications.

Growth Rate

The growth rate in population modeling represents the factor by which a population multiplies over time. To understand the concept of growth rate, think of it as a consistent multiplier that affects the population with each time unit. It's a crucial parameter in the exponential growth formula.

For instance, in the given exercise, the growth rate is 4, meaning the population quadruples every unit of time. Mathematically, this is represented in the formula as \( r = 4 \). The growth rate indicates how aggressive or rapid the population increase is. A higher rate means a population will rise dramatically within short periods.

Understanding the growth rate is essential for estimating future population sizes and planning resources accordingly. It enables us to predict how quickly a population might reach certain thresholds, potentially triggering environmental or economic impacts.

Initial Population

The initial population, often denoted as \( N_{0} \), is a critical starting point in any population model. It represents the number of individuals in a population at time \( t = 0 \). In our exercise, the initial population is 5 individuals, meaning our model starts its calculations from this baseline.

The initial population provides the base from which growth calculations are performed. In exponential growth models, the effect of the growth rate multiplies the initial population over time. This base number is affected by the growth rate, leading to exponential increases in the total population.

Initial populations are crucial for predicting future sizes accurately. A small change in \( N_{0} \) can lead to significant differences in predicted outcomes over time, especially in long-term planning scenarios. Therefore, careful estimation of the starting population ensures realistic and useful predictions from models.