Goals: introduce the idea of models of population growth, and common symbols used in population growth.  Introduce exponential growth, and both differentiated and integrated forms of the exponential growth equation.  Discuss methods of estimating r.

The Lecture:

Models of population growth are basically hypotheses of how a population grows.  By presenting them mathematically, we can work out predictions mathematically, then see how well they fit natural populations.

In making models, there is a tradeoff between:

• general models that have features that fit basic trends, but not details, of many species
• accurate models for one species that wonít fit others at all.

There are two basic kinds of population growth models:

•  finite models look at change in population over a discrete time interval.  These are useful for species where all reproduction occurs at once, and most mortality is seasonal, over short time.
•  instantaneous models look at population change at any instant in time.  These are useful for species that reproduce and die over longer time intervals, and to look at longer term population change where we want to know the overall pattern, not annual fluctuation.

Before developing our models, we need to define some standard symbols. These are:
    N=population size.
    t = time.
    N_t = N at time t.
    (N₂-N₁)/(t₂-t₁) = finite growth rate of the population
    R = N_t+1/N_t = the finite per capita (per individual) growth rate of a population
    r = the instantaneous per capita (per individual) growth rate of a population
    dN/dt=the instantaneous population growth rate, where if youíve had calculus, you should recognize this as derivative of N with respect to t and remember that it means change in N with respect to t; if you havenít, note that dN stands for "instantaneous change in N" and dt stands for "instantaneous change in t" so the d is a symbol meaning instantaneous change, itís not something multiplied by N.

The instantaneous per capita growth rate r = b-d where b and d are the per capita birthrate and per capita deathrate, respectively.

EXPONENTIAL GROWTH: a model of population growth that assumes birthrate and deathrate remain constant over time.

If b and d are constant over time, then dN/dt = Nb - Nd = (b-d)N = rN.  r is constant since b and d are constant; r = dN/Ndt

The result of exponential growth, as shown here, is that if b>d, r >0, population increases at faster and faster rate, as shown in the following graph of population size (N) versus time (t):

We can consider how the growth rate of the population, dN/dt, changes as N increases. Since dN/dt = rN; the relationship between dN/dt and N is linear (a straight line.)  Remembering the formula for a straight line, Y=mX+b, note that if we let our Y variable be dN/dt and our X variable be N, that a plot of dN/dt versus N will be a straight line that starts at 0 (b, the intercept, is 0) and increases with slope r (m, the slope, is r.)  This is shown here:

It will be interesting to compare the relationship of the per capita population growth rate, r, to population size, N, between the exponential growth models and other models of population growth.  Since r is constant, for exponential growth the plot just shows a horizontal line:

Typically, exponential growth does not occur in the long term (although for one species, humans, it does currently model population growth reasonably well.  This can not occur indefinitely.)  Exponential growth does occur in the short term, however.  When populations are first growing in areas, they will grow according to exponential growth.  We can use the model for short term change in populations.  To do this, we need to use it in a different form.  We can not use the equation in the form dN/dt to solve anything.  If we apply some calculus (if we remember any) to this equation and integrate it, we will get it into a form we can use.

NOTE: I will NOT expect you to do the following derivation -- after all, calculus is not a prerequisite for this course.  I am presenting it to show you that ecologists do need to be able to do some math!   You will need to use the integrated form of this equation, given at the end of the derivation, to solve problems about exponential growth.

Integration of dN/dt = rN:

Step 1: Collect terms with N in them on the same side of the equation:

            dN/N = rdt

Step 2: Take the integral of each side.  The integral of dN/N = ln(N) + c1 where c1 is a constant of integration.  The integral of rdt =rt + c2 where c2 is a constant of integration.  So:

            lnN + c₁ = rt + c₂

Step 3: Combine the constants of integration into a single constant c.  Let c=c₂-c₁:
            lnN = rt + (c₂ - c₁)
            lnN = rt+c

Step 4: Take each side and raise e to that power, to get rid of the logarithms:

            e^lnN = e^rt+c

Step 5: Simplify equation; apply rules of exponents

           N = e^ce^rt

Step 6: determine what e^c means.  Note that since e and c are both constants, it must be some constant -- its value will not change with changing time or population size.  It turns out we can figure out what it means if we solve for a specific time, the initial time when we observe our population.  No time has passed at that point so t=0.    Since N will change with time we will label it N₀ to show it is at time t=0.  Now we solve the equation for t=0 and get:

        N₀= e^ce^r(0)

        N₀= e^ce⁽⁰⁾  and anything raised to the 0 power is 1, so

        N₀= e^c
So e^c is the initial population size.

Step 7: Substitute N₀ for e^c inthe equation from step 5.  To clarify that there is an initial value of N,  N₀ , as well as another value of N that will depend on how much time has passed, we will call the value of N that depends on how much time has passed N_t to indicate that it is the population size at time t.

        N_t  = N₀ e^rt

This is the form of the exponential growth equation that is useful for solving equations!  Remember it!

Note that we could use the equation above to estimate the value of r, if we censused a population at two different times.  We could also estimate r another way based on a larger number of samples of a population, as follows:

Consider the formula N_t  = N₀ e^rt .  If we take the natural log of each side, we get:

             lnN_t = ln(N₀ e^rt)        Simplifying this, we get:
            lnN_t  = ln(N₀ ) +lne^rt
            lnN_t = ln(N₀ ) +rt

If we plot lnN_t versus time, note that the relationship between lnN_t and time is linear (a straight line); to see this, again remember the formula for a straight line, Y=mX+b.  In this case we're letting lnN_t be our Y variable and time be our X variable.  So the intercept on the Y axis is ln(N₀ ) and the slope is r.  So we get the following graph:
 

This is what we predict for exponential growth.  If we have taken population censuses at several different times and plot the natural log of population size versus time, we predict that if the population is growing exponentially the points will fall on a straight line.  If they do, we have support for a hypothesis that the population is growing exponentially.  Further, we can estimate the slope of the line.  We could potentially do this statistically; in class, you should just be able to do it by drawing the straight line that best fits the points you get by eye, and then estimating the slope of the line from the graph.
 

Now you have seen two ways of estimating r.  When we look at life tables in a couple of weeks, we'll look at a third estimate of r.