Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

On the basis of data collected during an experiment, a biologist found that the growth of a fruit fly (Drosophila) with a limited food supply could be approximated by the exponential model $$ N(t)=\frac{400}{1+39 e^{-0.16 t}} $$ where \(t\) denotes the number of days since the beginning of the experiment. a. What was the initial fruit fly population in the experiment? b. What was the population of the fruit fly colony on the 20th day?

Step-by-Step Solution

Answer

The initial fruit fly population in the experiment was 10. On the 20th day, the population of the fruit fly colony was approximately 155.

Step 1: Understand the Variables

In this exercise, the equation provided to represent the growth of the fruit fly population is: $$ N(t)=\frac{400}{1+39 e^{-0.16 t}} $$ Here, \(N(t)\) is the population at time \(t\), expressed in days.

Step 2: (a) Evaluate at t=0 for Initial Population

We are asked to find the initial population of the fruit flies when the experiment starts. To do this, we substitute \(t=0\) into the given equation. $$ N(0)=\frac{400}{1+39 e^{-0.16 (0)}} $$

Step 3: (a) Simplify the Equation

Now, we simplify the equation for \(N(0)\) by performing the operations step by step. $$ N(0)=\frac{400}{1+39 e^{0}} $$ Since, \(e^0 = 1\): $$ N(0)=\frac{400}{1+39(1)} $$ $$ N(0)=\frac{400}{1+39} $$ $$ N(0)=\frac{400}{40} $$

Step 4: (a) Calculate Initial Population

Finally, we can find the value of \(N(0)\). $$ N(0)=10 $$ Therefore, the initial population of the fruit fly was 10.

Step 5: (b) Evaluate at t=20 for Population on the 20th Day

Now, we want to find the population of the fruit fly colony on the 20th day of the experiment. To do this, we substitute \(t=20\) into the given equation. $$ N(20)=\frac{400}{1+39 e^{-0.16 (20)}} $$

Step 6: (b) Simplify the Equation

We simplify the equation for \(N(20)\) by performing the operations step by step. $$ N(20)=\frac{400}{1+39 e^{-3.2}} $$

Step 7: (b) Calculate Population on the 20th Day

Now, we can use a calculator or software to find out the precise value of \(N(20)\). $$ N(20) \approx \frac{400}{1+39 (0.040762203})\approx\frac{400}{1+1.587528157}\approx\frac{400}{2.587528157} $$ $$ N(20)\approx 154.62 $$ Consequently, the population of the fruit fly colony on the 20th day is approximately 155, considering we cannot have a fraction of a fruit fly.

Key Concepts

Biology

Biology is the study of living organisms and their vital processes, and it encompasses a broad range of complex topics. One essential aspect of biology is understanding how populations of organisms change over time. In this context, a biologist is investigating the population dynamics of fruit flies (Drosophila) when food is limited, which can provide valuable insights into ecosystem management and species survival.
This biological experiment explores how a population grows over time, even when resources are constrained. Understanding how organisms adapt to their environment is crucial for fields such as conservation biology and agriculture, where optimal resource use is necessary.
Given a mathematical model that predicts how populations change, researchers can better manage natural resources and predict changes in biodiversity.

Population Dynamics

Population dynamics refers to the changes in the size and composition of populations over time. It is a vital area of study within ecology and biology that examines the factors influencing population growth and decline.

Understanding population dynamics helps biologists:

• Predict how populations grow or shrink, which can be critical for conservation efforts.
• Identify trends and patterns in reproductive rates, mortality, and migration.
• Develop strategies for wildlife management and the sustainable use of natural resources.

In our exercise, the fruit fly population is modeled using an exponential growth model. This model helps simulate real-world scenarios where populations grow rapidly when conditions are favorable but slow when resources become limited. By recognizing these patterns, scientists and policymakers can better manage ecosystems and improve conservation efforts.

Mathematical Modeling

Mathematical modeling is a powerful tool that enables scientists to describe and analyze complex systems. It allows them to make predictions about how these systems behave under different conditions. In the context of population dynamics, mathematical models can predict how a population will grow or decline over time.

The model provided for the fruit fly population is an example of an exponential growth model, which is a type of mathematical model used to describe populations that grow rapidly. The equation given is:

• \(N(t)=\frac{400}{1+39 e^{-0.16 t}}\)

The equation allows us to calculate the population size \(N(t)\) at any given time \(t\). The initial population and how it evolves can be calculated by substituting different values of \(t\) into the model.

Models like this are fundamental in biology because they help in understanding how populations are likely to change, thereby guiding decision-making in research and conservation strategies. By using mathematics, biologists can qualify the growth trends and make informed predictions, which are essential for strategic planning in fields such as ecology and environmental science.