**OBJECTIVES**

- Model population growth exponentially and logistically
- Graph exponential and logistic curves
- Describe factors affecting different types of growth

**INTRODUCTION**

Populations that breed continuously exhibit two types of growth: exponential and logistic. Most populations will experience exponential growth at some point, especially when food supplies are abundant. However, exponential growth is typically kept in check with **density-dependent factors** which affect populations as their population increases such as predation, lack of food sources, lack of space, and disease. These density-dependent factors push populations into logistic growth in which populations reach a carrying capacity. When mathematically modeling logistic growth, the population appears to reach a finite value, but in reality, the population will fluctuate around the carrying capacity as environmental conditions change. **Density-independent factors** affect both exponential and logistic populations no matter the size of the population. Some examples include natural disasters such as hurricanes, tornadoes, and wildfires. Today, we will explore the mathematical models of exponential growth and logistic using Microsoft Excel. With these models, we can make predictions about population sizes of harmful bacteria, recovering populations of endangered species, and the impact of a growing human population.

**Submission Note: **You will be turning in a Microsoft Excel file; you do not need to copy and paste data tables or graphs into this worksheet.

**CASE STUDY**

This lab will explore logistic and exponential growth in a black rat population, which has a severe impact on farming in Southeast Asia. We will be watching various parts of the PBS Nova Documentary ‘Rat Attack’ on YouTube which focuses on Mizoram, a state in eastern India. To begin, navigate to the video: https://www.youtube.com/watch?v=tSGxVx0uT8Y and watch from the beginning to 2:29. Keep this video open as we will refer to different parts of the video during the lab.

These ‘rat attacks’ do not happen every year. Watch the video from 11:02 – 13:46 and answer the following questions.

- How often do the rat plagues occur?

The rat plague have been observed in India at a predictable timeframe of after 48 years. They first appeared in 1863, then 1911, and 1959

- The rat plagues occur during mautam. What is mautam?

In order to calculate black rat growth over many generations, we need to figure out the per-capita growth rate (r) during both a regular season and during mautam. We can figure out *r *by understanding the life history of black rats in these two periods, as seen in the table below.

**Table 1: **Black rat life history table during a regular season and during mautam

Regular Season | Mautam | |

Lifespan | 1 year | |

Number of litters during lifetime | ||

Length of one generation | ||

Number of offspring per litter | ||

Number of surviving offspring per litter | (assume <50%) |

- Use the following clips of the film to learn a little bit more about the life history of black rats during a regular season and during mautam. Then fill in the table with their traits and answer the follow-up questions.
- 49:15 – 50:36
- 31:49 – 33:15
- 16:41 – 17:37

In another part of the video, the scientist estimates that there are about 100 rats outside of a local farm. Assuming there are 50 males and 50 females that can all reproduce and are evenly distributed in ages, answer the following questions. Be sure to show all work.

- During a
**regular**year, how many offspring will be created in one generation?

- During a
**regular**year, how many rats will have died in one generation (assume 50% of the adults die)?

- Using this information, calculate the per-capita growth rate (r) for rats during the regular season.

- During a
**mautam**year, how many offspring will be created in one generation?

- During a
**mautam**year, how many rats will have died in one generation (assume 25% of the adults die)? - Using this information, calculate the per-capita growth rate (r) for rats during a mautam season.

- Why were 50% and 25% of the adults assumed to have died in questions 5 & 8? (*hint, review the info in the paragraph preceding question 4 and in the life history table).

As you may have noticed, the per-capita growth rate (r) is different for these two time periods. R can fluctuate based on environmental conditions, though in exponential and logistic growth modeling, r remains constant for the entirety of the model. One way we can make a distinction between different Rs in a single population is denoting r as r_{max}. This metric, r_{max}, is the maximum per-capita growth rate that a population can undergo. This assumes the maximum number of offspring are being created, maximum offspring survival, and maximum adult survival the population can sustain given unlimited resources.

**LOGISTIC GROWTH DURING THE REGULAR SEASON **

Most of the time in southeast Asia is *not *in mautam. Numerous species of rats live in the area but farmers and locals do not need to worry about their population growth wreaking havoc on their livelihoods. The black rats and other species live in an ecosystem balance. Let’s imagine that near a rice farm there are 100 black rats and see how their population changes over 10 generations if there is a carrying capacity of 1,000 on the population.

- Using Microsoft Excel, create a new workbook and save it as “LastName_PopulationGrowthLab”. Within this workbook, create a new worksheet called “Logistic”. Create a data table and use formulas to determine the black rat population size outside this farm over 10 generations using the r that you calculated earlier.
- All you need are simple equations to make this table. Refer back to your notes on what you need to calculate and use formulas to do this. DO NOT CALCULATE ROW BY ROW.

- There are many skills from Excel Tutorial 1 that you will find useful, particularly the worksheets: Filling Cells, Absolute Reference, and Copying Formulae & Numbers

- What is the population size at the beginning of the 10
^{th}generation?

- Although you plotted 10 generations, how much time does this cover?

- In this population, describe two ways that this carrying capacity could change. Describe one way how it could decrease and another way it could increase.

**EXPONENTIAL GROWTH DURING MAUTAM**

Mautam can be a catastrophic time for communities in southeast Asia. The devastation the black rat population can do can cause widespread crop failure, leading to less food and the need for humanitarian intervention. Understanding how these black rat populations grow can help farmers plan the timing of their crops to be harvested just before the rat population explodes to prevent losing their crops.

Similar to logistic growth, we will calculate exponential growth starting with 100 rats and the r_{max} calculated earlier in this activity.

- In a Microsoft Excel workbook, create a new worksheet called “Exponential”. Create a data table and use formulas to determine the black rat population size outside this farm over 10 generations using the r that you calculated earlier.

- In this worksheet, create a scientific graph of the population growth over time.

- What is the population size at the beginning of the 10
^{th}generation?

- Although you plotted 10 generations, how much time does this cover?

- Specifically, how could farmers use the information from this growth model to make decisions related to their livelihood?

- This black rat population likely wouldn’t keep growing at this rate for 10+ generations. Describe why not and what would eventually happen.

- There are two reasons why we cannot graph both of these growth curves on the same graph. Describe these reasons.
**Hint: try to plot them in Excel on the same graph to see what happens. Also refer to questions 14 & 19.**

- In reviewing the growth curves, data, and graphs, what do you notice? If you were a farmer in an area plagued by black rats, what would your thoughts and feelings be from seeing this data?