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Let’s write down our simple model for population growth one more time.
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And let’s suppose that the birth rate is equal to two.
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In the previous subunit I said that this simple model is linear
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when you plot the population at time t plus 1
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versus the population at time t.
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This is because individual bunnies didn’t interact with one another
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and there was no limit to growth.
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Another way to describe linearity is to say that
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the whole is the sum of the parts.
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In the simple population growth model we can see this.
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For example, suppose we start off our initial population with 1 bunny
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and a birth rate of 2
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and we reproduce for 3 years.
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We end up with 8 bunnies.
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Now, let’s set our initial population to 10.
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That is 10 times the original.
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Setup
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Reproduce 1, 2, 3
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We end up with 80 bunnies.
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10 times the original.
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So if we had taken 10 different runs of initial population at 1
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and run it for 3 years and summed up the results
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We get the same thing as taking an initial population of 10 for 3 years.
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That is 80 bunnies in either case.
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So here the whole is equal to the sum of the parts.
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But this is not the case if the system is nonlinear.
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We’ll make our system nonlinear by putting in
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nonlinear interactions between the individuals.
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Here’s our model of nonlinear interaction.
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Here we have n sub t plus 1 is equal to the birth rate times n sub t
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But now we’re gonna subtract the number of offspring who die due to overcrowding.
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We’re gonna assume that overcrowding occurs due to limits on food or space
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and that there is some maximum population that can live in this particular habitat.
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So we’re gonna define the number who die due to overcrowding is equal to n sub t squared.
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The square of the number in the current population divided by this maximum possible population.
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This model is a nonlinear interaction.
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The n sub t squared term comes from the fact that there are on the order of n squared
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possible pairwise interactions between the individuals.
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This is a simplification of course from reality.
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But this model is highly simplified model of real world population growth.
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It’s meant to capture the effects of nonlinear interactions
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and it’s only very approximate to what happens in reality.
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Let’s extend the model and make even more general by allowing
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some of offspring bunnies to die of causes not due to overcrowding.
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We’ll call this the death rate.
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Our extended model is the same except
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now we’re gonna have birth rate minus death rate.
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This is called the logistic model
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and was developed by the mathematician Pierre Verhulst
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in the early eighteen hundreds.
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Now let’s see this run in Netlogo
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The model was called the Logistic model dot nlogo.
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And as usual you can download it from the link below
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or from the course material page.
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Here’s what the model looks like.
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We have an initial population slider,
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and sliders for here birth rate and death rate.
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Notice that these are now real numbers not integers
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and if the model comes up with a real numbered value
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for the population size at the next generation,
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it just rounds off that real number to produce an integer.
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There is also a slider for carrying capacity
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which is the scientific name for the maximum population
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that the habitat can support.
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Ok so let’s set the initial population here to 1
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The birth rate to 2 point 0
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Let’s leave the death rate at 0
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And let’s set the carrying capacity to 50
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Do set up and now let’s just reproduce several times in a row
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So the population is growing, growing, growing
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and notice here that it has fixed at 25
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that is if I do another run of reproduce
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it just stays at 25
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now note that this population versus time plot
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is no longer exponential in its shape
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it doesn’t go up really fast
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instead, starts out by going up quickly but then
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starts slowing down in its growth until it flattens out.
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This is called the logistic function instead of an exponential function
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and the plot of this year’s population versus last year’s population
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is no longer a straight line.
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Because we no longer have a linear system.
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Now let’s go back to the netlogo model.
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An obvious question is why we didn’t the number of bunnies go to 50
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which is the carrying capacity.
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Well, this is just the way the model works
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if there are 25 individuals then 25 squared divided by 50
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equals 12 point 5 of those individuals will die off
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and the remainder have a birth rate of 2 to create offspring
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for the next generation so we get 25.
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It’s a mathematical model not necessarily a realistic model of population growth
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especially with these small numbers.
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It’s meant to capture the nonlinear nature of how population
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actually grows in the real world but not necessarily to be precise.
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So what we’re interested in here is not how realistic the model is
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but what effect the nonlinear aspect of it has on its behavior.
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Let’s go back to the model.
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So our question is
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Is the whole the sum of the parts?
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as we saw it was in the linear model.
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Let’s do what we did before.
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Run, reproduce for 3 time steps
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Here we get 7 bunnies, because of rounding off.
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But now let’s set the initial population to 5
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That is 5 times the original.
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Do set up and again run for 3 time steps.
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And you see here that the number of bunnies is not 5 times 7.
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Which would have been sum of the parts of the first version of this.
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That is one individual produced 7 individuals after 3 time steps.
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So you think that 5 individuals would reproduce 5 times that
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or 35 individuals after 3 times steps.
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But instead that, we get 21
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So in this nonlinear case
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the whole is different from the sum of the parts.
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So now you’ve seen the difference between linear and nonlinear systems.
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That is what happens when the parts interact in a nonlinear way.
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This is a key concept in complex systems that will come back to again and again
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Now it’s time for you to do a few short exercises.