3 Exponential Growth We begin with the simple exponential...

3 Exponential Growth We begin with the simple exponential growth model. Assume all changes in the population result from either births or deaths. Furthermore, assume that the per-capita birth and death rates are constant. Then letting $r = b - d$, where, $r$ is known as the intrinsic growth rate, the simple exponential growth model is given by $\frac{dy}{dt} = ry = f(y)$. (3) Thus, we can see that the rate of growth for the population is directly proportional to the size of the population. As the population's size grows, the rate of change increases for $r > 0$. Problem: 1. Find all equilibrium solutions $y_e$, sketch the graph of $y$ vs. $f(y)$, and sketch the phase- line and phase portrait. Use Theorem 1 and phase-portrait to determine whether each equilibrium solution is stable or unstable. 2. Interpret from the phase-portrait whether the population is increasing or decreasing over time.

Transcript

00:01 Here i'll review two rate equations that determine models of population growth.

00:07 We'll look at the equations and then solve them and see what these mean for population growth.

00:14 So the first model, we'll call it number one, is exponential growth.

00:24 This basically states that the rate of population change.

00:28 So p is number of organisms.

00:34 This model states that.

00:36 That population rate of change is proportional to the number that are already there.

00:43 So the ideas they are reproducing and producing more, which reproduce more, etc.

00:51 So let's see how you would solve that.

00:53 But basically you would use separate and integrate.

01:01 That's usually the tool you use for a first order differential equation.

01:06 Separate means put all the p variable on one side.

01:11 And get the t variable on the other, so it's a cross -multplication.

01:17 And then you just do the integral on each side.

01:21 I usually like to add the arbitrary constant of integration to the right -hand side.

01:28 It will be determined by initial condition.

01:36 On the left -hand side, it's the natural logarithm, is the integral.

01:40 On the right, it is simply k -times t plus the constants.

01:46 And if we then invert this, we can take both sides and use them to raise the base e2, and the logarithm will cancel.

01:59 And give us p equals some other constant.

02:04 It's not the same c, but it's another constant.

02:08 Maybe we'll just call it c1, c prime, some other constant, c1.

02:16 And then we can put in for the initial value, p0 is c1, e to the 0, e to the 0 is just 1.

02:29 So c1 is p0.

02:32 And our final expression is p of t is p0, e to the kt.

02:41 What does that look like? it basically looks like a curve that swoops upward, x, exponentially, it starts at p0 and just goes up and up and up and up and up and up and up and up and up and up.

02:58 And at some point it has an infinite slope.

03:03 So this seems a little bit unrealistic, but this would be a population that has open borders, say, with no limits.

03:18 So just as the organisms keep reproducing, those organisms have someplace to go and enough food, enough space, et cetera.

03:32 And typically, if, say, we're looking at human population, if humans settle in an area, and they're the first settlers there, there are plenty of resources, and they will just keep creating families, larger and larger families, and spreading into the area until some limit gets reached.

03:55 Which brings us to the second, model, which is called logistic growth.

04:11 Okay.

04:12 And this rate is a rate equation, dp by d t.

04:21 Again, there is a rate constant.

04:25 Okay.

04:28 But then there is a product of two terms.

04:31 On the other side, there is a p, and then a 1 minus p over a second parameter.

04:46 That we call m, and m is known as the carrying capacity.

04:52 It's the maximum, which is why it's good to use m.

04:58 It's the maximum that the area will support for that population.

05:05 So let's see what happens.

05:07 Are there equilibrium solutions? what we mean by that is that the population stops changing.

05:15 The dp by dt is equal to zero.

05:22 The first model, there is no equilibrium.

05:26 But do we have dp by dt equal to zero? well, there is an equilibrium in the first situation.

05:35 It's the trivial result that p0 is zero.

05:40 So there are two solutions here, either population is zero, and it'll stay at zero, or one minus p over m equals zero.

05:56 Those are the two factors that either one can be zero.

06:05 And the second condition is an equilibrium where the population has reached its maximum.

06:16 So the logistic solution typically looks like this.

06:21 You start off at some p0 and you grow rapidly at first, exponentially at first.

06:29 And at some point, about halfway through, you have.

06:32 The maximum rate of change when you've reached half your maximum.

06:40 And then the curve starts to slow down.

06:46 So when you've reached half the limit, things will slow down after that.

06:53 And then you'll reach kind of a saturation point at the maximum...

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