Modeling Population Growth and Extinction
It sometimes seems that there are daily news reports about the threats of global
warming leading to the extinction of species.
However, most of the introductory
treatments of models for population dynamics in mathematics classes typically consider
only population growth models. In this article, we will construct a pair of more general
models that encompass the possibility of both growth and extinction.
Population Growth Models
We begin with a brief review of the two most common population growth models.
The simplest model that arises frequently across the mathematics curriculum at many
different levels is the exponential growth model for a population P. At the calculus and
higher levels, it is expressed as the differential equation
P’ = aP, a > 0.
At the precalculus or algebra level, it is expressed discretely using the difference equation
∆Pn = aPn , a > 1,
where
∆Pn = Pn+1 - Pn.
However, because no population can continue to grow in an exponential pattern
indefinitely, we typically consider the inhibited growth or logistic model, which is
expressed either as the differential equation
P’ = aP - bP2,
or discretely as the difference equation
∆Pn = aPn - bPn2.
In both cases, a > 1 and b is a positive constant that is considerably smaller than a. In
either form, the solution has the S-shaped logistic pattern shown in Figure 1. The
horizontal asymptote L is called the limit to growth, the maximum sustainable population,
or the carrying capacity of the environment. We solve for L algebraically from either the
differential equation or the difference equation to get L = a/b.
We can use this value for L to rewrite both models if we factor the parameter b
from the equation:
P’ = aP - bP2 = bP(L – P)
∆Pn = aPn - bPn2 = b Pn (L - Pn).
Both model highlight the fact that the change in the population depends on both the size
of the population P and the difference between the current population and the maximum
sustainable level, which we can think of as the space remaining for additional members
of the species.
Furthermore, the inflection point seen in Figure
1, where the population is growing most rapidly, turns
out to occur at a height of P = ½L = ½(a/b) because
both P’ and ∆P are quadratic functions of P with
negative leading coefficients.
Figure 1: Graph of a logistic function
Figure 1
Both models can be solved algebraically when either P’ = 0 or ∆Pn = 0 to give P
= 0 and P = L. These solutions divide the t-P plane or the n-Pn plane into three distinct
regions: P > L, 0 < P < L, and P < 0. Whenever P > L, both P’ < 0 and ∆Pn < 0, so that
any solution that starts above the maximum sustainable population must decay down
toward it. For any solution that starts between P = 0 and P = L, both P’ > 0 and ∆Pn > 0,
so that the solution must be increasing. Also, whenever P < 0, both P’ and ∆Pn are also
negative, and so any solution that starts below the
horizontal axis must decay toward -∞ as time passes.
We illustrate all of these possibilities in Figure 2.
Figure 2: Different solution patterns for the
logistic equation
The two horizontal asymptotes, including the
horizontal axis, in Figure 2 are the equilibrium levels
for the solutions. The upper one at the height of L = a/b is a stable equilibrium since any
solution that starts near that level converges to that
Figure 2
equilibrium; the lower equilibrium at P = 0 is an unstable equilibrium because any
solution that starts near it diverges from that level.
Modeling the Extinction of Species
We now turn to the question of how to model the extinction of species. We tend to think
that, whenever a species is present in an environment, it will grow. Even if there is a
catastrophic change in the local conditions – say a major volcanic eruption (such as
Mount Saint Helen’s), or a significant rise in temperature to make the habitat
uncomfortably warm, or even a meteor impact such as the one that apparently brought an
end to the age of the dinosaurs – our image of population dynamics would suggest that
the survivors will continue to breed and so expand the population. Obviously, since
species do become extinct, there must be an additional biological mechanism that comes
into play to change the dynamics of such a situation.
In addition to a maximum sustainable population, biologists also talk about a
minimum sustainable population – a level K below which there are not enough members
of the species to sustain it. Thus, if the population drops below this level for whatever
reason, we should expect the population to begin to decay toward zero. At a very
simplistic level, one could simply assume that the process follows an exponential decay
model. However, as we discuss below, this is not a reasonable pattern for realistic
populations.
We now attempt to extend the logistic growth model to take this decay toward
extinction into account as well. To do so, we will
take the above development of the logistic model
and essentially reverse the steps to create the model.
We now need three equilibrium levels, one for a
zero population, another corresponding to the
maximum sustainable population L, and a third
corresponding
to
the
minimum
sustainable
population K, as illustrated in Figure 3. These three
equilibria divide the plane into four distinct regions.
Figure 3
Figure 3: Different regions of the t-P plane for logistic-type growth and extinction
Let’s think about the behavior of the solutions in each of these regions. In Region
IV, where P < 0, we should expect that the solutions decay toward -∞. In Region I,
where P > L, we should expect that the solutions decay toward L, as we have with the
logistic model. Similarly, in Region II, where K < P < L, we should expect that the
solutions rise toward L, eventually in an asymptotic manner, much as they behave with
the logistic model. Finally, in Region III, where 0 < P < K, we should expect that the
solutions decay toward zero.
But, what should be the actual pattern for the decay in Region III? Should it be a
purely concave up pattern, as with exponential decay? Picture what happens in the
logistic model with solutions that start on either side of the equilibrium level L. If the
initial population is significantly above L¸ the solution will drop very rapidly at first and
eventually slow down as it approaches L asymptotically. If the initial population is very
close to, but above, L, it will decay very slowly toward L. If the initial population is
equal to L, it will remain at that level indefinitely. If the initial population is slightly
below L, and above the inflection point at ½L, it rises slowly in a concave down pattern
as it approaches L asymptotically. Finally, if the initial population is well below L, and
also below the inflection point, it rises ever more rapidly until it passes the inflection
point and then begins to slow. The change from one behavior pattern to the next happens
continuously.
Consequently, we should expect that, in the
extended logistic model we are trying to create, there
should also be a continuous change in the behavior of
solutions depending on where they start with regard to
the new equilibrium level at the minimum sustainable
population.
In particular, the assumption that the
behavior of the solutions in Region III automatically
follows
an
exponential
decay
pattern
seems
inappropriate since that presumes a sudden drop in the
population if the value is slightly below K. Rather, as
Figure 4
shown in Figure 4, it makes more sense to expect that the solutions will display a
vertically reversed logistic shape and that there will be an inflection point level in this
region as well.
Figure 4: Solution patterns for logistic-type growth and for extinction
We show the right-half of the t-P plane in Figure 5 along with the signs of P’ or
∆Pn needed to produce the patterns of behavior we would expect in each region. In
particular, we need to analyze the signs of the factors needed for either P’ or ∆Pn in each
of the four regions that will lead to these behavior patterns. Clearly, in Region I, we need
P’ < 0 or ∆Pn < 0. In Region II, we need P’ > 0 or ∆Pn > 0. In Region III, we need P’ <
0 or ∆Pn < 0. Finally, in Region IV, we need P’ < 0 or ∆Pn < 0.
Figure 5: Signs of P’ in each region of the t-P plane
We next see how we might get these sign
combinations based on our experience above with the
logistic model. The extra equilibrium level of P = K
suggests that we might include an additional linear
factor of the form K – P in either the differential
equation or the difference equation. If we do so, then
in Region I, we have P > 0, L – P < 0, and K – P < 0
and their product will be positive, which does not
Figure 5
give the correct behavior. We might “fix” this problem, however, by introducing a
negative leading coefficient, so that we can try a differential equation of the form
P’ = -αP(L – P)(K – P),
where α is a positive constant.
In Region I, this expression must be negative because P > 0, L – P < 0, and K – P
< 0. Next, in Region II, we have P > 0, L – P > 0, and K – P < 0 and their product times
–α will be positive, as desired. Similarly, in Region III, we have P > 0, L – P > 0, and K
– P > 0, so that -αP(L – P)(K – P) < 0, as desired. Finally, in Region IV, we have P < 0,
L – P > 0, and K – P > 0, so that -αP(L – P)(K – P) > 0, which does not give the behavior
pattern we expect. We note that this cubic model is known to ecologists as the logistic
model with Allee Effect [1, 2] after the biologist Warder Clyde Allee who developed this
cubic model in the 1930’s and 1940’s.
In order to get the signs to match up appropriately, we need to extend the Allee
model by using a quartic model instead. In particular, we use P2 instead of P as the first
factor in the differential equation. When we do so, we have the differential equation
P’ = -αP2(L – P)(K – P)
and it is easy to verify that the sign of P’ is the desired one in each of the four regions.
The same is true of the corresponding difference equation model,
∆Pn = -αPn2(L – Pn)(K – Pn).
Finding the Inflection Points
With the logistic model, we know that the inflection point occurs when the
population passes a height of ½L. We now consider the possible inflection points that
arise in our quartic growth and extinction model. We expand the expression for P’ as a
fourth degree polynomial in P to get
P’ = -α[P4 –(K + L)P3 + KLP2].
The inflection points correspond to the points where the derivative of this quartic
function of P achieves its maximum or minimum, so we need to solve where the
derivative is zero:
-α[4P3 - 3(K + L)P2 + 2KLP] = 0.
(Note that this is not P”, which is the second derivative with respect to t.) One immediate
solution is P = 0 and we note that the concavity of P’ as a function of P changes on either
size of the horizontal axis, as seen in Figure 4. The remaining two inflection points
correspond to the roots of the quadratic equation
4P2 - 3(K + L)P + 2KL = 0.
Using the quadratic formula, we find that
P
3( K  L)  9 K 2  14 KL  9 L2
.
8
Unfortunately, this expression provides little insight into the locations of the inflection
points. To gain a better feel, let’s consider an example with specific values for the
parameters. Suppose that the population in question has a maximum sustainable level of
L = 2000 and a minimum sustainable level of K = 200. With these values, the above
quadratic formula gives P = 131.73 and P =
1518.27 as the two roots, correct to two decimal
P'
P
1.00E+08
places.
We see that the first inflection
point corresponds to a height of roughly ⅔
-200
-4.00E+08
Figure
6 800
300
1300
1800
-9.00E+08
of the minimum sustainable population and the
-1.40E+09
second inflection point corresponds to a height
-1.90E+09
of roughly 3/4 of the maximum sustainable
Figure 6
population. We show the graph of the
corresponding cubic function of P’ versus P in Figure 6 without the –α coefficient.
Figure 6: Graph of P´ versus P with L = 2000 and K = 200
Some Solution Curves
We next consider some specific solution curves based on different initial values
for the population. Doing this is considerably simpler using the discrete formulation,
because we can rewrite
∆Pn = Pn+1 - Pn = -αPn2(L – Pn)(K – Pn)
as
Pn+1 = Pn - αPn2(L – Pn)(K – Pn)
and calculate all successive values given any desired starting value for P0. Alternatively,
if we attempted to use the continuous formulation, we could integrate the differential
equation in closed form using partial fractions, but that would produce a complicated
combination of transcendental expressions and a rational expression in P that is equal to a
multiple of t. Since it is not possible to solve the resulting expression for P as a function
of t, we would face a complicated implicit function and the best that could be done with it
is to approximate the coordinates of a large number of points along each solution curve.
Consequently, in what follows, we work with the discrete models.
A little exploration shows that, in order to produce reasonable values for Pn based
on the difference equation model, it is necessary for the parameter α to take on extremely
small values, roughly on the order of 10-10. If we use values for α that are larger than
this, the successive values generated by the
difference equation for Pn+1 become
extremely large very quickly; rather than falling into any reasonably smooth pattern, the
successive values typically jump from one of the four regions far into a different region
and may bounce around dramatically.
Suppose we then select α = 10-10 and
2000
again use K = 200 and L = 2000, as above. We
1600
show the results of starting with initial values of
1200
P0 = 500 and P0 = 1200 in Figure 7. Notice that
800
both curves approach the horizontal asymptote
400
of 2000, which is the maximum sustainable
0
0
population level, very quickly – the upper curve
5
10
15
20
25
30
Figure 7
starting at P0 = 1200 essentially reaches it in about
n = 8 time periods and the lower curve starting at P0 = 500 gets there in about n = 28 time
periods.
Figure 7: Solutions starting from P0 = 500 and P0 = 1200
Moreover, as we indicated above,
2000
the inflection point for the increasing part
of the solution curve occurs at a height of
about P = 1518. If we consider the the
1600
initial condition P0 = 1600, the solution
starts at above this level and its shape is
1200
0
5
10
exclusively concave down, as shown in
Figure 8. On the other hand, the solution
Figure 8
starting at P0 = 500 begins below this level
and its shape, as seen in Figure 7 above, follows a roughly logistic pattern.
15
Figure 8: Solution starting from P0 = 1600
In comparison, in Figure 9, we show
200
the results of using initial values of P0 =
180, P0 = 75, and P0 = -50. The inflection
point in Region III occurs at a height of
0
50
100
150
200
250
300
350
roughly P = 132. The solution starting at P0
= 75 begins below this level and its shape
-200
follows a decreasing, concave up pattern. In
contract, the solution starting at P0 = 180
Figure 9
begins above this level and its shape is obviously decreasing and concave down at first,
but clearly changes to decreasing and concave up at around n = 250, where P is roughly
130, as predicted. Finally, the solution starting at P0 = -50 is obviously decreasing and
concave down in the view shown; beyond about n = 250 it drops extremely rapidly as it
decreases toward -∞.
Figure 9: Solutions starting from P0 = 180, P0 = 75 and P0 = -50
Estimating the Parameter Values
In any practical setting, one would normally start with a set of observed or
experimental data and seek to fit a function of the desired form to that data. When
working with data that falls into a logistic pattern, obtaining a logistic function that fits
the data extremely well is simple using the built-in logistic routine in most graphing
calculators. Failing in that, one could apply either of the following two approaches using
Excel, say, which does not have a logistic function routine. We begin with the logistic
difference equation
∆Pn = aPn - bPn2,
where a and b are two parameters whose values we need to estimate. If we have a set of
data consisting of n and Pn values, we can form a table in column form, say, and extend it
with an additional column containing the differences ∆Pn of successive values of Pn.
Since the difference equation basically says that ∆Pn is a quadratic function of Pn, we can
“hit” these two columns with the quadratic polynomial fit and find the best quadratic
function in the regression sense to match the data. This is one approach to estimating the
values for a and b. Alternatively, we could divide both sides of the difference equation
by Pn to get
∆Pn/Pn = a - bPn,
which is a linear function of Pn. Consequently, we could create still another column in
the table with values for the expression ∆Pn/Pn and “hit” that column and the column for
Pn with the linear regression routine to produce a different pair of estimates for a and b.
We note that both approaches typically lead to functions that are not as good as the one
obtained by the calculator’s logistic regression routine.
Both of the above approaches easily generalize to methods for estimating the
parameters in either the logistic model with the Allee effect or the difference equation
model we developed here. First, the model with the Allee effect is the differential
equation
P’ = -αP(L – P)(K – P)
and we can create the comparable difference equation model
∆Pn = -αPn (L – Pn)(K – Pn).
If we multiply this out, we have the equivalent discrete model
∆Pn = -α [KLPn - (K – L) Pn 2 + Pn3],
which clearly shows that ∆Pn is a cubic function of Pn. Therefore, we could simply use
the cubic polynomial fit routine of Excel or of any calculator applied to appropriate
columns of data values to estimate the values of the associated parameters. In turn, this
would allow us to estimate the values for both the maximum sustainable population L and
the minimum sustainable population K, assuming that these values are not evident from
the original data. Alternatively, we could divide both sides of the difference equation by
Pn to get
∆Pn/Pn = -α [KL - (K – L) Pn + Pn2],
and apply the quadratic regression routine to estimate the parameters. This is probably a
preferable approach because we have three parameters to estimate and the quadratic
function has three coefficients, while the cubic function has four.
Next, we consider the difference equation we have developed here,
∆Pn = -αPn2(L – Pn)(K – Pn).
If we multiply this out, we get the equivalent difference equation
∆Pn = -α [KLPn2 - (K – L) Pn 3 + Pn4],
so that ∆Pn is now a quartic function of Pn . We can therefore simply apply the quartic
polynomial routine of a calculator or Excel to estimate the values for the parameters and,
again, to estimate the minimum and maximum sustainable population levels.
Alternatively, we could divide both sides of the above difference equation by Pn to get
∆Pn/ Pn = -α [KLPn - (K – L) Pn 2 + Pn3],
which says that ∆Pn/ Pn is a cubic function of Pn . We would then apply cubic regression
to estimate the parameters. Alternatively, we could divide the difference equation by Pn 2
to get
∆Pn/ Pn 2 = -α [KL - (K – L) Pn + Pn2],
so that ∆Pn/ Pn 2 is a quadratic function of Pn . We could then apply quadratic regression
to estimate the parameters. Again, since there are three parameters to determine, it
probably makes the most sense to go with the quadratic fit to ∆Pn/ Pn 2 rather than the
quartic fit to ∆Pn/ Pn or the cubic fit to ∆Pn.
References
1. Allee, W. C. (1931). Animal Aggregations. A Study in General Sociology. University
of Chicago Press, Chicago. ISBN 0-404-14501-9.
2. Allee, W. C. (1949). Principles of Animal Ecology. W.B. Saunders Co., Philadelphia.
ISBN 0-7216-1120-6.
3. Clark, Colin W. (2005). Mathematical Bioeconomics: The Optimal Management of
Renewable Resources, 2nd Ed, Wiley-Interscience, New York, ISBN 0-471-75152-9
Acknowledgment The work described in this article was supported by the Division of
Undergraduate Education of the National Science Foundation under grants DUE0310123 and DUE-0442160. However, the views expressed are those of the author and
do not necessarily reflect those of the Foundation.
Keywords
Population growth, logistic model, extinction, Allee factor, differential
equation, difference equation
Abstract The exponential growth model and the logistic model typically introduced in
the mathematics curriculum presume that a population grows exclusively. In reality,
species can also die out and more sophisticated models that take the possibility of
extinction into account are needed. In this article, two extensions of the logistic model
are considered, one known as the logistic model with Allee effect, which is a cubic model,
and the other a further generalization, which is a quartic model. The approach used is
designed to illustrate some techniques of model-building.