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