DB 5: Briefly describe the situation in which population growth/decline is being considered and modeled

Modeling Population Growth

You can generally model population growth as either linear by an arithmetic sequence, or exponential by a geometric sequence. These can be defined by one of the general forms:

Arithmetic Sequence

an = a0 + dn where,

• d is the common difference (change per year for population increase or decrease)
• n is a natural number indicating the years since the initial period
• a0 is the population in the initial period

Geometric Sequence

an = a0rn where,

• r is the common ratio (multiple for population increase or decrease per year)
• n is a natural number indicating the years since the initial period
• a0 is the population in the initial period

Post 1: Initial Response

Suppose you are working as an IT consultant and are known to have keen mathematical skills in addressing real-world situations. You have recently been contacted by a city council to present a plan for a population growth/decline initiative. The council has in recent years experienced considerably unexpected changes in population due to obsolete models and no use of technology. This has left them scrambling to find remedies for the city (e.g., new roads if the population is growing, repurpose vacant lots if the population is declining).

You will spearhead this initiative, and you are charged with planning a mathematical model for future use and prioritizing the city council’s use of technology. Your goal is to apply your knowledge of sequences and create an appropriate mathematical model for the city’s population growth/decline. You will also share the technological tools that can be used to employ this model well into the future in the form of a computer algorithm.

In planning for your goal, you will want to briefly describe the city population growth/decline scenario and identify what factors might contribute to the growth/decline. This initial stage of planning is critical in creating an accurate mathematical model. You also want to present how your model can be codified as a computer algorithm so that the council’s staff can use it in the future.

Draft a detailed response to the council to demonstrate what you propose for more effectively modeling and preparing for population changes in the future.

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Include all of the following in your response.

1. Provide a narrative introduction to the planning initiative, where you:
   • Briefly describe the situation in which population growth/decline is being considered and modeled
   • State whether you project the population growth/decline to be linear or geometric in nature
   • Identify at least three factors you believe might be contributing to the unexpected population growth/decline (i.e., birth, disease, innovations, trends, migration)
   • Discuss and prioritize which of these factors you believe has possibly had the greatest impact on the population growth/decline
   • Reflect on how that factor may relate to one of the inputs in the model
2. Present your proposed mathematical model for how the council estimates the future population each year, using either an arithmetic or a geometric sequence. Clearly state the following:
   • Initial population size, a0
   • Your change variable, d or r
   • The general equation for your model with these two inputs substituted into the formula
3. Write an algorithm that will take as input the number of years (n) since the initial period, and that will output the population size for each year from now until the nth year. Your algorithm must use an iterative operation (such as a while or for-next loop).

Post 2: Reply to a Classmate

Review a classmate’s population model scenario, equation, and algorithm. If no other classmate has yet responded, review their model for accuracy by providing an input value for n 5 and perform a trace of the algorithm.

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Address the following:

• Write out the outputs in a trace table.
• Select two future population estimates (two output values of an) and discuss whether the output produced is appropriate for the scenario. What limitations, or problems, may there be in using this model to estimate the population indefinitely?
• Based on your assessment of the output and potential limitations of the model, suggest a change to the general formula or algorithm for their sequence to represent a potentially better long-term model for the scenario. (You may consider the factors that were described in the initial planning phase or other biological limitations on growth/decline of populations.)

Post 3: Reply to Another Classmate
Review a different classmate’s population model scenario, equation, and algorithm. You will improve their model to account for a potential event that could affect the population growth/decline.

Modify your classmate’s population model by adjusting their algorithm with a conditional statement (if-then or if-then-else) for some event that leads the population to slow its growth or decline (or reverse between growth versus decline) at a specific population size (i.e., you choose a specific value of an).

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Address the following:

• Copy their original algorithm into your post and insert a conditional operation into an appropriate place to account for this change in the model. (You will need to incorporate a second formula with an updated value for d or r, for use when your conditional operation begins taking effect.)
• Select an input value for n, large enough so that it will trigger your conditional statement to calculate some population estimates using your revised model, or second formula.
• Perform a trace of the algorithm using this value of n and write out the outputs in a trace table.
• Describe and explain the results, commenting specifically in your role as a consultant on how you believe this is now a good long-term modeling algorithm for the scenario.