Download Population Growth Model: Exponential Growth and Recursive Equations and more Study Guides, Projects, Research Introduction to Business Management in PDF only on Docsity!
Project Grant Team
John S. Pazdar Patricia L. Hirschy Project Director Principal Investigator Capital Community College Asnuntuck Community College Hartford, Connecticut Enfield, Connecticut
This project was supported, in part, by the Peter A. Wursthorn National Science Foundation Principal Investigator Opinions expressed are those of the authors Capital Community College and not necessarily those of the Foundation Hartford, Connecticut
SPINOFFS
Spinoffs are relatively short learning modules inspired by the LTAs. They can be easily implemented to support student learning in courses ranging from prealgebra through calculus. The Spinoffs typically give students an opportunity to use mathematics in a real world context.
LTA - SPINOFF 15A The Capture-Recapture Method
LTA - SPINOFF 15B Florida Scrub-Jay Populations and Habitat
LTA - SPINOFF 15C Population Models with Recursive Equations
Brian Smith - AMATYC Writing Team Member McGill University, Montreal, Quebec, Canada
Mario Triola - AMATYC Writing Team Member Dutchess Community College, Poughkeepsie, New York
Janet Rebmann - NASA Scientist/Engineer Kennedy Space Center, Florida
NASA - AMATYC - NSF
SPINOFF 15C
Population Models with Recursive Equations
Mathematical models are important tools for monitoring and forecasting future population sizes. Four key factors affecting population size are:
- Birth rate
- Death rate
- Immigration rate
- Emigration rate
These components are affected by habitat conditions such as human activity, fire, predators, drought, vegetation, etc. In the NASA Kennedy Space Center surroundings, long-term effects include the environmental impact of “multiple launches and continuing land use changes.” (http://atlas.ksc.nasa.gov/program.html).
Mathematical models incorporating all relevant components are extremely complex, but useful models may be constructed by including the four key factors listed above. Because births and immigration increase the population size, and deaths and emigration decrease it, a simple model can be described as follows:
New Population Size = (Previous Population Size) + (Births) − (Deaths) + (Immigration) − (Emigration).
Each term in this equation is based on a given time interval (a day, a year, etc.). We can represent the above terms with the following mathematical symbols:
N (^) t+ 1 = New population size (population size at time t + 1) N (^) t = Previous population size (population size at time t) B = Number of births in one time unit D = Number of deaths in one time unit I = Number of new immigrants entering the population in one time unit E = Number of emigrants leaving the population in one time unit
Using these symbols, we can rewrite the equation as follows: Nt+1 = N + Bt − D + I −E
In a closed system we assume that there is no migration into or out of the system so that the equation becomes: N (^) t+1 = N (^) t+ B −D
In this model we may define the rate of growth as r = (B − D)/N 0 , where N 0 is the population at time
t = 0. As a result, we assume a constant rate of growth during the time interval. If the number of births
NASA - AMATYC - NSF
N (^) t = 1000(1.06t^ )
This formula is an example of exponential growth. Using EXCEL we can easily create a table representing 20 years of population growth (see Table 2). The accompanying graph is typical of exponential growth.
Table 2 Year Population 0 1000 1 1060 2 1124 3 1191 4 1262 5 1338 6 1419 7 1504 8 1594 9 1689 10 1791 11 1898 12 2012 13 2133 14 2261 15 2397 16 2540 17 2693 18 2854 19 3026 20 3207
Mouse Population Growth Chart
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3500
0 2 4 6 8 10 12 14 16 18 20 Year
Exercises
Consider a population of birds which is threatened due to damage to its natural habitat. At the beginning of 1985 the population was estimated to be 5800 birds. The annual birth rate is estimated to be 3% of the population, and the death rate is estimated to be 8% of the population.
Determine an exponential growth formula that yields the population size at any time t.
Create a table of population sizes from the beginning of 1985 to the beginning of 2000.
Draw a graph representing the bird population from 1985 to 2000.
Estimate the population size at the beginning of 2010.
In what year will the population be one half of its size at the beginning of 1985?
NASA - AMATYC - NSF
- How long will it take for the population to become extinct?
