# The Population Dynamics of Extinction by alicejenny

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Dr. J.M.Halley, Extinction Models; Part 3.

3. The Population Dynamics of Extinction.
This section is all about the fact that in order to conserve a species it may not be enough just
to leave it alone and stop hunting it or destroying/polluting its habitat. There are various things
that happen once the population’s size has been reduced seriously. The figure below shows the
various mechanisms that are important at various sizes.

Healthy
populati on

*Hunting
*Habitat Fragmentation
*Global catas trophes

Vulnerable
e.g. Heath Hen;
Si ngle location or
Blue Whale;
wide spread low densi ty                N. Elephant s eal.

The above plus …
*Environmental stochas ticity
*Local catastrophes

Critical
Very small numbe rs
or very low de nsi ty

All the above plus …
*Allee effects
*Demographic stochas ticity

Extinction

Figure 3-1. Chart showing the major possible factors involved in the road to extinction, including those causing
increased vulnerability to extinction in small populations.

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Dr. J.M.Halley, Extinction Models; Part 3.

3.1. Extinction in an Impoverished Environment.
The most obvious way in which we can model the extinction of a species is when an organism
is placed in a poor environment. This is one in which the number of deaths per unit time is
greater than the number of births. Let N be the number of organisms in the population, let N0
be the initial number of organisms, t be time, and let B and D be births and deaths respectively
per adult in the population. Then the population will develop according to the equation:

N t +1 = (1 + B − D) N t ,   t=0,1,2,3….                     …(3-1a)

So the replacement coefficient is

λ = 1+ B − D = 1+ r                                    …(3-1b)

The r constant is the growth rate. Clearly if D>B, then the replacement rate is less than unity
(and r<1), and so the population is exponentially declining, as you saw in Fig. 2-2a. Unless
action is taken, extinction will follow.
This system is relatively elementary but it may arise in various natural situations. For
example, we might introduce a colony, of size N0, into a patch of poor habitat. Alternatively, a
population of size N0, at a given time may experience a change in the conditions prevailing in
its habitat, going from that of Figure 1b from that of 1a.

3.2. Extinction through Allée Effects
Sometimes populations are subject to additional strains simply from being very small through the
Allée effect that leads to a reduced λ at low numbers. If such an effect is operating in a
population, reducing the numbers of an organism may lead to a chain reaction resulting in
extinction. An Allée effect can arise in a number of ways.
• The community structure may collapse. The passenger pigeon is believed to have suffered
extinction for this reason.
• The population density (number per area) may be very low so animals may have difficulty
locating each other to achieve a mating. This has been suggested (though unproven) in the
case of the blue whale (Balaenoptera musculus).
• Inbreeding depression may induce extra mortality. When a large population is suddenly
reduced to a small size, many deleterious recessive genes are "exposed" by the increased
inbreeding, leading to reduced fitness. However, not all inbred populations suffer from
inbreeding depression.
Models nicely describe the influence of the Allée effect. We can set up the model according to
the Eq. (3-1) but with an extra mortality term that is small but gets large when the number of
animals is small:
δ
Λ( N ) = 1 + r −                                    …(3-2)
N

This means that exponential growth is modified by an extra factor:
N t +1 = (1 + r − δ / N t ) ⋅ N t

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Dr. J.M.Halley, Extinction Models; Part 3.

In the sheet I have given you, for the graphical method, I have already shown the replacement
curve, with this effect. Use your graphical skills to see what happens for different starting
populations, using the figure provided Fig 2.2(c).

What happens?
• Is there an equilibrium?

•    If the equilibrium exists what is its value?

We can also find the equilibrium value from the equation above. It is:

When there is an Allée effect operating the important consideration is that there is a minimum
viable population, below which the population cannot survive and above which it can survive.

3.3. Environmental Variability and Catastrophes.
These two mechanisms fall into the category of things that affect medium size populations.
The effect can be sweeping as the case of the heath hen showed.

Introducing environmental variability into a model requires that things like birthrate
must vary with time also, not just due to the randomness inherent in births and deaths, but in a
systematic way. Instead of staying at a value of K, the maximum value of the population will
vary from generation to generation.
Population

Time

Figure 3-2. Fluctuations about a carrying capacity due to environmental stochasticity in the replacement rate, and
(lower curve) due to occasional epidemics of varying sizes. Notice the slow recovery in case of the lower curve.

In such cases it is common to build a Logistic or Beverton-Holt equation, for example, with
the parameters fluctuating. How can we do this? One possibility is actually to go out and
measure both r and K for several years to see how much they vary. We then include this in the
model and try to see how often the population goes extinct.

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Dr. J.M.Halley, Extinction Models; Part 3.
I have included a file on the computer in EXCEL with which you can analyse the
behaviour of such a model.

Figure 3-3. Copy of the excel spreadsheet when you open it. You can use this to investigate the effects of
environmental stochasticity on the growth rate and the carrying capacity, and of changing other parameters. You
should try σr=1%, 5%, 10%, 30%, 98%and 60% with σK=0, and vice versa, for each of r=0.3 and r=0.03.

Although this method helps us to understand the mechanism of extinction, it has
limitations. The main problem is that the environmental variability is assumed to be constant in
variance. However, we know that variance increases with time: more timeàmore variation.
Because, as we take in more time, all sorts of things which seemed “stationary” on short
timescales start to “kick in”. For example ice-age cycles are not noticeable on short timescales,
but they dominate fluctuations on timescales of 10-thousand years to 100 thousand years. If
we want to predict extinction on longer timescales we must include such effects in our models.

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Dr. J.M.Halley, Extinction Models; Part 3.

0.30
0.4
T      T       T
0.2                                                       0.25
Rel. Temp. (Degrees C)

0
0.20
-0.2

StDev
-0.4                                                      0.15
-0.6
0.10
-0.8
-1                                                       0.05
-1.2
1200   1400        1600     1800      2000             0.00
1   10       100   1000
Date
T

Figure 3-4. More time more variation for environmental stochasticity! On the left is a reconstruction of
termperature in the Northern hemisphere over the last 750 years (heavy black line is the smoothed version). We find
that the amount of variability (variance) over an interval of 100 years tends to be greater than that for 10 years.

Changing weather patterns could be described as “environmental stochasticity” and epidemics
might be described as “catastrophes”. However, both are essentially part of the same complex
process that we might call “the environment”, since they are extrinsic to the population rather
than intrinsic. The complexity of this is enormous, but we hope that some patterns may help to
put some order in the modelling of the environment.

3.4. Fragmentation of Populations and Immigration.
One of the characteristics of Man's current effect on the environment is an increasing
level of habitat fragmentation. As cities, motorways and agricultural areas expand, those areas
left to "Nature" contract and become fragmented. This has harmful effects:

•                 Smaller population are more likely to be exposed to the harmful effects of demographic
stochasticity (next section). (“Together we stand, divided we fall”).
•                 Inbreeding is more serious problem in smaller populations.
•                 Usually the borders of habitats suffer from various edge effects that effectively reduce the
effective population in each patch.

On the other hand, fragmentation may be a good thing. If there is plenty of distance between
the patches but still reasonable migration, there will be a
• built-in protection against epidemics
• Better protection against geographically localised natural disasters.

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Dr. J.M.Halley, Extinction Models; Part 3.

Figure 3-4. Two types of “metapopulation” geometry. Island archipelago (left) and mainland-island (right).

Trying to quantify the relative merits of these options, in conservation biology, is
called the SLOSS (single large or several small) problem. This problem continues to be the
subject of research and it has not been solved entirely. In cases where there is reasonable
immigration into the outlying patches from other populations is significant there can be
restoration of populations. The outlying islands become part of a metapopulation, a
fragmented population, linked by dispersal and recolonisation. If the island cannot survive by
itself, it may still appear to do so, having a small population. Carry out the graphical procedure
for Fig 2-2(d). Such a population is referred to as a sink population, and the population that
supports it as a source population. In such studies, local extinction refers to the loss of a
population in a single island or patch. Global extinction means that the population is lost from
all islands or patches, so that there is no chance of recolonisation.

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Dr. J.M.Halley, Extinction Models; Part 3.
3. 5. Extinction by Demographic Stochasticity.
When it happens, extinction usually involves “demographic stochasticity”. This means
that although the external conditions causing extinction may be well-defined, the exact time is
uncertain because the internal dynamics of individuals are chancy, since birth-rate, death-rate
and sex ratio only exist as averages. For example, a population is truly doomed, though it may
be “large” when all individuals are of the same sex, since there are no further offspring. There
are three different cases to consider the effects of demographic stochasticity.
1. Declining Population. Fig 3-4a, b shows the results of a Monte-Carlo simulation model
used to simulate the fate of 15 identical island populations that have been simultaneously
subjected to a degradation of their environment. Shown in Fig 3-1 are average population
(a) and number of surviving populations (b). Each population starts at 100 individuals.
Although the exact time of extinction is not certain, there is clearly a characteristic time
when the status of a population becomes critical, in this example it is when the population
falls to a value of about 10.
2. A Colony That Fails to get Established. The other two panels in Fig 3-4 show the reverse
process; colonisation by an organism of a favourable environment, subject to demographic
stochasticity. Although the average population (c) increases exponentially, the fate of
individual islands is chancy. Any colony may dribble along at very low numbers, before it
“takes off” into exponential growth. This transition is the establishment of a colony. Over
half the initial colonies (of 10 animals) in this model went extinct before they have a
chance to get established. One of them petered out after 50 years. For this reason, remote
islands, subject to such colonisation processes, often contain very different faunas and
floras although conditions might be very similar. A species may be fortunate to establish
itself on one island, yet fail to get established on another. A similar question, with the same
mathematical model, is whether a mutant gene can invade a population of otherwise
normal genes.
In the early stages of colonisation, a population is very vulnerable to demographic
extinction. For populations starting from a small number of individuals, there is a high
probability that we might have all males or all females in the population, or that the few
individuals in the population might suffer reproductive failure.

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Dr. J.M.Halley, Extinction Models; Part 3.

(a)                                   Deathrate>Birthrate (Exponential decline)               (c)                                 Deathrate>Birthrate (Exponential increase)
-0.1129x
y = 118.11e
50                                                                                          50

40                                                                                          40
Exponential
y = 4.8313e 0.0643x
Population size

model

Population size
30                                                                                          30

20                                                                                          20

10                                                                                                                           Exponential
10
model
0
0
0           20              40          60                                                   0      10           20          30        40
Time (years)                                                                             Time (years)

Extinctions begin when
average populations falls
below N~10
(b) 120                                                                                       (d)                                 120
Number of Surviving Populations

100                                                                                           100
Number of Surviving Populations

80                                                                                           80

60                                                                                           60

40                                                                                           40

20                                                                                           20

0                                                                                            0
1   11     21      31       41     51   61    71                                             1     11       21          31        41
Time (years)                                                                             Time (years)

Figure 3-5. Extinctions due to demographic stochasticity in an ensemble of 20 identical populations with (left)
negative growth-rate and (right) positive growth-rate.

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Dr. J.M.Halley, Extinction Models; Part 3.
What is the probability that a colony, beginning with N0 individuals in a favourable
environment, gets established? This problem has been solved mathematically for simple
population dynamics. Suppose that organisms breed (one offspring at a time) at any time with
probability B per individual per unit time and die (one at a time) at any time with probability D
per individual per unit time. Notice that the birth and death probabilities are independent of the
number of individuals present on the island and that we don’t distinguish between males and
females. Obviously, this is a fiction; however, the error is very small.
The answer is given by the following formula:

D
N0

PE =       , B > D
B            

 Initial colony size is N 0       …(3-3)
=1 ,    B≤D     




This formula may also hold true from a population recovering from a serious population loss.
So if we include demographic stochasticity, the population only survives with a probability, its
survival is not certain. Nevertheless, even if the rate of births is much greater than the number
of deaths, the chance of extinction is small.

3. Extinction of Established but Small Populations. Again because of demographic
stochasticity, there is a constant fluctuation about the ceiling or carrying capacity. Most of
these fluctuations are small, but occasionally there are large ones, big enough to cause
extinction. Even for an established population, if the “ceiling” or carrying capacity is low,
eventual extinction is possible by this mechanism. It is just a matter of waiting! The typical
probability of extinction, per unit time (provided rK>>1, remember r=B-D) is given by:

PE ≅ 2r 2 K ⋅ Exp[−rK ] ,    rK>>1                     …(3-4)

It says that there is no totally safe size of population, eventual extinction is certain for every
size of population, since every value of K and r leads to a finite PE. As Keynes once said “We
are all dead in the end”. Of course, the time may be very long, so it is a bit philosophical and
this demographic mechanism alone seems insufficient to explain why even large populations
suffer eventual extinction over periods of millions of years. The reason our calculations yield
unreasonable results is that we have missed out the effects of environmental stochasticity.
Thus extinction is a complex event involving the interaction of deterministic and
random forces.
• Even though the conditions imposed may be well-defined, the actual time of
extinction is uncertain.
• An organism may become extinct even in a “friendly” environment due to
demographic stochasticity if the population is very low.

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