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## The Mathematics of Population Growth Pages 474-491

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**Mini-Excursion 3**The Mathematics of Population Growth Pages 474-491**Continuous vs. Discrete Growth**• Population growth is a time-dependent process. • Discrete growth occurs when there are gaps in the growth times. • Continuous growth occurs without gaps between growth times.**Discrete Growth**• During discrete population growth, there will be sudden changes in the population called a transition. • A mathematical model of population growth defines the transition rules in the form of an equation.**Discrete Growth**• A population sequence is a sequence of numbers that represent the population levels at each transition time. The notation for a population sequence is: • is called the initial population. • The transition rule determines the population sequence.**Discrete Growth**• A time-series graph shows the time of each transition on the horizontal axis and the size of the population on the vertical axis.**Linear Growth**• For a lineargrowthmodel, each generation of the population increases (or decreases) by adding (or subtracting) fixed amount called the commondifference. For example: Initial population = 1, common difference = 4**Linear Growth**• The time-series graph of a linear growth model makes a straight line.**Linear Growth**• If the common difference is denoted d, the linear growth model is • This is called an arithmetic sequence.**Linear Growth**• We can summarize the linear growth model with the formula:**Example**• Problem 4 on page 488**Exponential Growth**• For an exponential growthmodel, each generation of the population changes by multiplying a fixed amount called the commonratio. This is a geometric sequence. For example: Initial population = 1, common ratio = 5**Exponential Growth**• The time-series graph of an exponential growth model makes an exponential curve.**Exponential Growth**• If the common ratio is denoted r, the exponential growth model is • Or:**Example**• A population grows according to an exponential growth model with: • Find the common ratio r • Find • Give an explicit formula for (this example is similar to assigned problems 11 and 12)**Logistic Growth**• The linear growth model predicts unlimited growth as N gets larger if d is positive. • The exponential growth model predicts unlimited growth as N gets larger if r is larger than 1.**Logistic Growth**• In reality, resources are limited and populations do not grow without limit. • The logistic growth model predicts resource limited growth as N gets larger.**Logistic Growth**• In the logistic growth model represents the fraction of the population out of the total population allowed by the habitat called the carrying capacity of the habitat. • The value of will always be a number between 0 and 1.**Logistic Growth**• page 483**Logistic Growth**• Solution:**Logistic Growth**• Solution:**Logistic Growth**• Solution:**Logistic Growth**• page 484**Logistic Growth**• Solution**Logistic Growth**• Solution**Logistic Growth**• Solution**Logistic Growth**• page 484-485**Logistic Growth**• page 484-485**Logistic Growth**• page 485**Logistic Growth**• page 486**Logistic Growth**• No pattern (chaos)