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Understanding Exponential and Logarithmic Models in Population Dynamics

This section explores key exponential and logarithmic models used in mathematics and biology. It covers the bell-shaped Gaussian curve, logistic growth models for populations, and the principles of exponential growth and decay. Examples include modeling population growth with the formula P = 800e^(0.05t) and estimating the age of fossils using carbon dating. It also introduces how the Gaussian model pertains to normally distributed populations, illustrated with test scores. Discover practical applications and analyze scenarios through graphs and equations.

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Understanding Exponential and Logarithmic Models in Population Dynamics

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  1. Section 3.5 Exponential and Logarithmic Models

  2. Important Vocabulary • Bell-shaped curve The graph of a Gaussian model. • Logistic curve A model for describing populations initially having rapid growth followed by a declining rate of growth. • Sigmoidal curve Another name for a logistic growth curve.

  3. The exponential growth model The exponential decay model The Gaussian model The logistic growth model Logarithmic models Y = aebx Y = ae-bx y = ae-(x-b)^2/c Y = a/(1 + be-rx) Y = a + b ln x and y = a + b log10 x 5 Most Common Types of Models

  4. Section 3.5 Mathematical Models

  5. Exponential Growth • Suppose a population is growing according to the • model P = 800e0.05t, where t is given in years. • (a) What is the initial size of the population? • (b) How long will it take this population to double?

  6. Carbon Dating Model • To estimate the age of dead organic matter, scientists use the carbon dating model R = 1/1012e−t/8245 , which denotes the ratio R of carbon 14 to carbon 12 present at any time t (in years).

  7. Exponential Decay • The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil.

  8. Gaussian Models • The Gaussian model is commonly used in probability and statistics to represent populations that are normally distributed . • On a bell-shaped curve, the average value for a population is where the maximum y-value of the function occurs.

  9. Gaussian Models • The test scores at Nick and Tony’s Institute of Technology can be modeled by the following normal distribution y = 0.0798e-(x-82)^2/50 where x represents the test scores. • Sketch the graph and estimate the average test score.

  10. Logistic Growth Models • Give an example of a real-life situation that is modeled by a logistic growth model. • Use the graph to help with the explanation.

  11. Logistic Growth Examples • The state game commission released 100 pheasant into a game preserve. The agency believes that the carrying capacity of the preserve is 1200 pheasants and that the growth of the flock can be modeled by p(t) = 1200/(1 + 8e-0.1588t) where t is measured in months. How long will it take the flock to reach one half of the preserve’s carrying capacity?

  12. Logarithmic Models • The number of kitchen widgets y (in millions) demanded each year is given by the model y = 2 + 3 ln(x + 1) , where x = 0 represents the year 2000 and x ≥ 0. Find the year in which the number of kitchen widgets demanded will be 8.6 million.

  13. Homework • P. 232 – 236 28, 29, 32, 37, 39, 41

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