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There's been a little bit of confusion about the different types of graphs
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that arise when solving these differential equations, so I thought it might
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be good to go over the solution to quiz 1 in a bit more detail to highlight this.
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So, we're solving differential equations of this sort: dX/dt = f(X)
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and I've given you the graph of f(X), and we can come up with the phase line for this,
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we can also sketch solutions of X as a function of t,
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and to make this discussion a little bit more concrete
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I think it will help to reinterpret this as a population growth problem, so let me do that...
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So, all I'll do is replace the variable X by the variable P,
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and we'll think of P as the population size,
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and since we're measuring population in these funny units,
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maybe this is population in metric tons of biomass, or something,
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so it's not like there's 1 (we're talking about rabbits), I don't think there's 1 rabbit, 2 rabbits, 3 rabbits,
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so we won't worry too much about the units here,
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but the idea is that we have a population that's changing in time
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and this says the growth rate - if it's growing or shrinking, how fast,
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is a function of the current population.
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so then this graph, the axis here, is population P
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and then this is the growth rate - how the population is changing.
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So we can see if the population is between 1 and 9, then the population will increase
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because the growth rate is positive - the population is growing.
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If the population is greater than 9, then the population will decrease
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because the growth rate is negative,
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and if the population is less than 1, the growth rate will decrease,
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again, because the population is negative.
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So from this information, I can sketch possible solution curves
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for this differential equation, so let me do that...
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OK, so here are my axes - this is now time on the horizontal axis,
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and population on the vertical axis.
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I've sketched the fixed points, the two stable points, 1 and 9, just as dotted lines,
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- these are fixed points because the growth rate is 0 - when the growth rate is 0,
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the population P doesn't change.
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If I start, say, at 2, I'll grow, I'll increase, until I get to 9, so I could sketch that here...
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- I'm increasing, and I'm approaching 9.
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If I started at 5, I would also increase, and approach 9 - it's going to look something like this.
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If I start at 12, I'll decrease, and hit 9 - approach 9
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and if I started a little less than 1, I would decrease and go 0.
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OK, so these purple curves, I would say these are solutions to this differential equation,
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for four different initial conditions, so these purple curves, are P as a function t.
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So let me just a little bit again about these two types of graphs,
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because it's really easy to get them confused.
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So this is dP/dT, as a function of P - Is the population growing or shrinking for a given population?,
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so there's not really, there's no time, on this axis.
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This plot is P, as a function of time - how does the population vary as time goes forward.
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Right so, we can see that solutions approach this stable fixed point at 9,
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and are pushed away from an unstable fixed point at 1.
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So lastly, another type of graph that we can draw is the phase line for this, and we do that...
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Again we can see that there's a stable fixed point at 9, and an unstable fixed point at 1,
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and in fact, you could draw the phase line right here on this axis, if you wanted,
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- I'll do that in a different colour, so it stands out - grab a bright red pen,
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so in this region, between 1 and 9, the population is growing, its moving to the right,
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Why? - because this graph is positive, and when this graph is positive,
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that means the growth rate is positive.
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So the arrows here, have to go like this...
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If I'm over here, the population will get smaller, because the growth rate is negative,
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- the value of this function, dP/dT, is negative, and same story over here,
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and we have fixed points at the intersection, where the growth rate is 0, the population is fixed.
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So it's very easy to get these two types of graph mixed up - I think a few people were
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asking really questions about this in the forum,
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so I thought it would be good to underscore this just one more time.