Module 19: Growth Models

Module 19: growth modelsImagine you are a happy bacteria, or rabbit, with lots of food, land and no predators. Your population growth curve might look like this:

This is called exponential growth, or "J shaped growth"Note that it has no end, or limiting factor.
Small r is the growth rate. If you have had physics (yay!) this is usually "k" in some examples, or related to RC decay/growth.Learning this equation is VERY useful.Note that it depends on two things: the starting amount in the population (No) and the growth rate (r)Here is an example:
You might also find this link useful:https://www.khanacademy.org/science/ap-biology/ecology-ap/population-ecology-ap/a/exponential-logistic-growth

Many systems follow J shaped exponential growth until they run out of food or space, then there is overshoot and die-off:
A more ideal version of this is the S shaped curve, called logistic growth:

Here is a formula for logistic growth that we'll discuss:
Don't be intimidated by this formula...dN/dt is just delta N over delta t, or the ∆ in number over the ∆ in time, (∆ means change)or the rate of population growth (some of you may see this as the slope of the S curve)

Note that when the ratio N/K is very small or close to zero, the stuff in the parentheses becomes 1, so the formula is rate = rN, or J curved exponential growth.

Recall: J curved exponential growth only depends on starting population and reproductive rate.

As N/K nears one (number of critters equals carrying capacity) the term in the parentheses becomes zero, so no growth.

Note also that if N/K is GREATER than one, the growth rate (slope of the curve) become negative. This is overshoot and die off.

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Logistic growth worksheet

First: exponential growth:

Imagine 10 imaginary rabbits (No=10)

Assume r = 0.5 (50 percent growth rate, or each rabbit makes 0.5 rabbits per year)

Find the population 2 years later:

Nt = Noe^rt

Nt = 10e^0.5*2

Nt = 27 rabbits

After 4 years:

After 10 years:

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Next, use the logistic growth formula, same data, with a carrying capacity (K) of 100:

Small population: 10 rabbits

∆N/∆t = rN(1-N/K)

= 0.5*10(1-10/100)

= 5(1-0.1)

= 5(0.9)

= 4.5 rabbits per year

Find the rabbits per year for these populations:

Medium population: 27 rabbits

Near K: 74 rabbits

Above K: 1489 rabbits

quiz (you may use your worksheet only)

1. 30 rabbits live on an island with carrying capacity 200. They reproduce at a rate of 0.5 per year. How many rabbits will be on the island after 3 years? (hint: you would do the calculation three times-this is only an estimate though, the true formula would be a bit more complex)
2. What will be the slope of the growth curve at this point?

Next: predator/prey phase diagramsCheck this out:
Note the phase (timing) relationship between the abundance of the food and the population of the prey, then the predator.
Predator Prey Lab:

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