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In this optional video, I'll give a derivation or motivation of the Lotka-Volterra equations.
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So the Lotka-Volterra Equations describe two populations
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and I'll discuss them in terms of rabbits and foxes
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but the key thing is that the rabbits are the prey--things that are eaten
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and the foxes are the predators.
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These are often called predator-prey dynamics.
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This is a particular example of a predator-prey dynamic and it applies to lots of things beyond rabbits and foxes.
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In fact, I don't know if it even applies to rabbits and foxes,
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who knows -- people apply these models to all sorts of things.
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So let me give a motivation for the model, and then say just a little bit more about it.
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So let's start by thinking about the fate of the rabbits...
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So our starting point........
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think about the growth rate of the rabbits.
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So we could start with a population model like this: this says that the dR/dt
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(that's the growth rate of the rabbits) is a function of just the number of rabbits there are.
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It's 'a,' some constant, times the number of rabbits.
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So for 'a' POSITIVE, the growth rate is always positive. The more rabbits there are the faster the growth rate is
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the faster the growth rate is, the more rabbits there are, and so on.
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And this leads to exponential growth.
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This solution to this differential equation is an exponential function.
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However, that's not the end of the story--there are also the presence of foxes
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and that could enter in like this.
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So 'a' is a model parameter, it's the rabbit growth rate if there are no foxes.
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And 'b' is another model parameter, we'll take 'b' to be positive
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and so this term is negative, and 'b' is some measure of how deadly the foxes are
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to the rabbits.
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Let's look at this term by term.
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The minus sign-that means that this term tends to decrease the growth rate.
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The presence of foxes makes the growth rate smaller, perhaps even negative
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if this term is bigger than that.
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This term on the right, R x F is just the simplest sort of interaction term that has the following properties:
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If we were to double the number of foxes, looking at just this term here, if we double the number of foxes,
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this term gets twice as large.
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So that makes sense--twice as many foxes should have twice the impact on the growth rate of the rabbits.
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But additionally, the number of rabbits also has an impact on the growth rate of the rabbits.
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If there are a lot of rabbits, it is easy for the foxes to find them
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and so there will be more rabbits eaten, leading to a greater impact HERE.
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So, separately, this term is linear in both R and F...if you double the number of rabbits, you double the number of rabbits that get eaten.
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That's what this says-if you double the number of foxes, you double the number of rabbits that get eaten.
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And then 'b' is some parameter that can be some measure of the deadliness of foxes.
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OK, so that's this term for the rabbits...let's look at the foxes.
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So dF/dt--that's the fox growth rate--and I'll start with this term HERE
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This says that the growth rate of foxes depends on cRF.
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'c' again is model parameter, some constant that we could change,
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In this case it represents the nutritional value of rabbits (from the point of view of a fox) or how easy it is for foxes to catch rabbits.
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Some sort of interaction strength term.
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So let's look at this--as similar properties to this term, this says if I doubled the number of rabbits,
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I've doubled the growth rate of the foxes --that sort of makes sense.
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The more rabbits there are, the more the foxes can eat.
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We double the number of rabbits, double the fox nutrition. That's going to double this growth rate.
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Also the more foxes there are, the more foxes there are to catch rabbits
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So that will also increase this growth rate.
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So RF is an interaction term. It's large when both R and F are large.
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So if this was the end of this equation, the foxes would take over the universe.
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The would keep growing and growing and growing.
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We know that foxes don't take over the universe, so we add one more term.
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dF. 'd' here is a parameter--this isn't a differential--'d" is another constant that represents the death rate of the foxes.
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So if there were no rabbits - --if we got rid of the cRF term,
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then we get rid of the interaction term between foxes and rabbits.
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This then says that foxes die at rate 'd.' The fox population sadly would exponentially decay down to zero because they have nothing to eat.
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So we have these two terms, rabbit growth moderated by being eaten by foxes
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and fox growth because they're eating the rabbits moderated by some natural death rate.
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So these two terms taken together have the form of a Lotka-Volterra model put forth by Lotka and Volterra about a century ago
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and again a, b, c, and d are parameters that can be adjusted.
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I'm not sure anybody takes this model tremendously seriously
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as a model of rabbit and fox dynamics, or whatever
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but like the Logistic Equation, either the Logistic map or the differential equation,
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this is more of a cartoon-style or a sketch designed to give a general feel
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for what sort of dynamics might happen if we have two interacting populations this way.
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In the field of mathematical ecology this is one of the very basic models
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and one uses this a building block for more complex and more realistic models.
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So again, these are Lotka-Volterra equations, and they're an example of predator-prey dynamics.
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And lastly I should mention these can also be used to model certain situations
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in economics, where you have two industries or sets of firms. One that's going to tend to grow
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but maybe there is something else that is hostile to it,
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and then this would be some firm or industry that depends on 'rabbits'
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and if there aren't any rabbits, whatever that firm would be, they would disappear, they would die.
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So one could see the use in economics also to model all sorts of cyclical behavior.
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I know less about economics than I do about biology, so I won't even try to make up a story for that
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but the bottom line is: these are very simple equations that model some sort of predator-prey interaction.
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