Analyzing Swedish Wolf Population Data from 1980 to 2001

1. Stochastic Birth-Death Processes in Wolf
Species
Analyzing Swedish Wolf Population Data from 1980 to 2001
Calvin Korponai1 and Daniel Granados2
Final Project for MAP4484
Modeling in Mathematical Biology
Department of Mathematics
University of Florida
Gainesville, Florida
April 22nd, 2020
1ckorponai@ufl.edu
2danymillos@ufl.edu

2. Abstract
Few theoretical studies have examined modeling population growth and extinction alto-
gether. Carlos A. Braumann, from the Universidade de Évora in Portugal, and Sheldon P.
Gordon from Farmingdale State College have published successful journals within this field.
However, there is constant debate on the type of modeling techniques used for population
growth and extinction. One can argue that an environmental stochastic model of population
growth and spread is more accurate than a logistic model for growth, since it accounts for
multiple variables that affect the survival of wolf populations. We were motivated to use the
deterministic Logistic Model to gain insight on the nature of the parameters that control popu-
lation growth and population extinction. Then we applied stochastic simulations of the logistic
model to approximate demographic stochastic processes and determine the mean time to ex-
tinction if the Russian wolf population was never introduced, and applied the same model
considering different parameters to determine the mean and variance of the wolf population 36
years after the Russian wolf population was first introduced in 1991.
1 Introduction
One of the common challenges faced in mathematical ecology is understanding a species’ risk of
extinction when considering both environmental and demographic stochasticity. Stochastic pro-
cesses refer to the uncertainty of outcomes due to randomness. While environmental stochasticity
alludes to the variation in environmental conditions that a population finds itself in, demographic
stochasticity is the unpredictability of behavior of the individuals that make up a population.
By 1800, there was abundant documentation of a substantial population of native swedish
wolves (Canis Lupus) occurring all over the Scandinavian Peninsula. However, from 1920-1921, a
major plan to hunt the Swedish wolf population resulted in the cumulative eradication of over 6000
wolves that year alone. This decision came about after many human families lost much of their
livestock and even some family members, after an incident where a single wolf killed nine chil-
dren. Thus only a fraction of the previously abundant wolf population survived and was forced to
migrate to the north of Sweden. High rates of wolf hunting continued to occur until 1966 when the
wolves were official declared a protected species in Scandinavia as many believed they had gone
extinct. It was not until 1980 that Swedish locals reported a pack of at least two wolves in South-
ern Sweden that ecologists now believe they had migrated from the Finnish/Russian population.
Since their initial appearance, diligent reporting continued each winter until the winter starting in
the year 2000. Minimum, mean, and maximum estimations of the wolf population are plotted in
Figure 1. Three years after the first sighting, a breeding event was reported in 1983. For the follow-
ing years, Scandinavian wolves struggled to maintain their population size. Ecologists believe this
comes as a result of allee effects, where significantly low population size results in lowered growth
rates, independent of environmental constraints such as limited food supply, reduced territory, and
hunting. Allee effects include the inability to find a mate and inbreeding depression, where lack of
genetic diversity increases the risk of stillbirths and less fit offspring that don’t grow old enough to
reproduce themselves.
As shown in Figure 1, there was a sudden burst in population growth from 1991 to 2000. Ecol-
ogists believe that this came as a result of the arrival of a single Russian Wolf to the Scandinavian
peninsula around 1990. This is thought to have returned genetic diversity into the gene pool and
eradicating the inbreeding depression that the Swedish wolves were experiencing.
1

3. The allee effects that predetermined the behavior of the wolf population between 1980 and
1980 1985 1990 1995 2000
020406080100
Year
Population
Wolf Population Maximum
Wolf Population Mean
Wolf Population Minimum
Figure 1: Population size reported at the start of each winter from 1980 to 2000
1991 are archetypal examples of demographic stochasticity. There is no realistic way to predict
wolf population behavior using deterministic models, but stochastic models can provide power-
ful insight considering random events. Considering this history, and the essential function of the
Scandinavian wolf in its ecosystem, ecologists are motivated to understand how the population
will behave in the future so that they can make informed decisions regarding population control or
fostering population growth.
2

4. 2 Wolf Population Model
2.1 Exponential Growth Model
To gain a better understanding of Swedish wolf population dynamics, and the stochastic nature of
birth and death events, we first consider the most basic birth/death model and add restrictions to
define the most accurate model that fits the data we have for Swedish wolves plotted in Figure 1.
As seen in the data, the change in population size experiences multiple oscillations from 1980 to
1990 due to the allee effects of small population sizes. However, after the introduction of Russian
wolves to the population in 1991, the population growth rate resembles that of exponential growth
until 2000. An exponential growth model inherits the following differential equation,
dW
dt
= rW (1)
where r is the rate of change r = b−d and t is measured in years. Here, b represents the intrinsic
birth rate and d represents the intrinsic death rate. Considering this model, we attempted to find
the growth rate from 1991 to 2000 by first plotting our data with the logarithm of each population
data point (Figure 2).
1992 1994 1996 1998 2000
2.53.03.54.04.5
Year
Log(Population)
Population Maximum
Population Mean
Population Minimum
Fitted Population Mean
R2
= 0.9845
1992 1994 1996 1998 2000
020406080100
Year
Population
Data
Exponential Growth (r=0.2)
Figure 2: (Left)The logarithm of each population datapoint is plotted on a logarithmic
scale in order to use linear regression. Notice that R2 = 0.9845 meaning the ordinary
least squares regression accurately fits the data.
Figure 3: (Right)Using the slope of the linear regression used in Figure 2, we find that
setting r = 0.2 best fits our data.
In such a simple model, if r is positive the population is increasing (Figure 3), Conversely, if r
is negative then the population is decreasing.
When solving this differential equation, we get the equation W(t) = W0ert, where W0 is our
initial population size W(0) = W0. The change in W over time is directly proportional to the
size of the population in the present. This model represents unbounded exponential growth over
3

5. time which is unrealistic in the real world. There are various environmental limitations on any
population that makes the exponential growth model unfitting of our situation. The next section
explains the need to implement a carry capacity (K) and a minimum viable population (m) using a
modification to the Verhulst model, a model derived by Pierre François Verhulst (more commonly
known as the Logistic Growth Model), that includes a bottom bound for an initial population size
to approaches its carrying capacity over time.
2.2 Logistic Growth Model
Consider the following Logistic Growth Model
dW
dt
= rW(1−
W
K
) (2)
where W is the wolf population size, r is the intrinsic growth/decay rate of the population, and K is
the carrying capacity of the population. Clearly, if we start with a population of zero (W0 = 0), there
is no growth and the population does not change over time. Similarly, an initial population size
of 1 does not allow for population growth since the singular wolf has no opportunity to reproduce
with another wolf. Thus, it is necessary to establish a minimum viable population of 2. The data
in Figure 1 solidifies this assumption since the first wolf sighting in 1980 reported a mean of four
wolves, then a mean of three wolves in the following two years until the first recorded birth event
in the winter of 1983. Thus we add a new component to the Verhulst model that accounts for a
minimum viable population m,
dW
dt
= rW(1−
W
K
)(1−
m
W
). (3)
We think of m+1 as the minimum initial wolf population such that the model can show logistic
growth considering a positive growth rate (We later show the case that W0 = m). In other words,
if the population of wolves is less than the minimum number of Wolves (W0 < m), the population
will go extinct over a finite time t < ∞. We now analyze Equation 3 considering the following
cases:
If m < W < K,
dW
dt
= rW (1−
W
K
)(1−
m
W
)
dW
dt
= ⊕ ⊕ ⊕
dW
dt
= ⊕
we see that the growth rate is positive. Thus, the population increases toward the carrying capacity
over time.
If 0 < W < m
dW
dt
= rW (1−
W
K
)(1−
m
W
)
dW
dt
= ⊕ ⊕
dW
dt
=
4

6. we see that the growth rate is negative. Thus, the population decreases overtime.
Considering W0 = m, we see in Equation 3 that W0 = m is a fixed point, meaning that the
population neither increases or decreases for that population size. However, this does not consider
the wolves’ finite lifespan, so in reality, the population also goes extinct since it is not reproducing.
Finally, we solve dW
dt in Equation 3 to find W(t):
dW
dt
= rW(1−
W
K
)(1−
m
W
)
dW
dt
= rW(1−
W
K
)(1−
m
W
)·
W
W
·
K
K
dW
dt
=
r
K
·(K −W)·(W −m)
Using partial fraction decomposition to solve dW
dt ,
m·|W −m|−m·|K −W|+C1 = (W −m)
r
K
·t +C2 for constants C1 and C2
Solving of W,
W −m
K −W
= De(K−m) r
K t
where D = ±eC
is a constant
For t = 0, and W(0) = W0, we finally get W as a fuction of time:
W(t) =
m(K −W0)+K(W0 −m)e(K−m) r
K t
K −W0 +(W0 −m)e(K−m) r
K t
(4)
Using Equations 3 and 4, we can use some outside research along with the data we were given
to solve for W and find realistic values for our coefficients K, m, and r.
2.3 Determining Coefficients
Now that we have our complete modified Logistic Growth Model, we can estimate coefficients to
fit our data and predict future wolf population sizes. The most difficult coefficient to estimate is
the carrying capacity K.
According to research done on Scandinavian wolf monitoring , wolf population has maintained
a steady state from 2007 to 2018, varing in a range between 230 and 340 individuals. Although this
does not necessarily imply that Scandinavian wolves have reached their carrying capacity, recent
the Swedish Government is working with the EU to set such that the wolf population maintains it-
self at about 250. Thus, we believe that 300 is a good estimate of the carrying capacity considering
this legislation and the previous population data.
5

7. To estimate our value of m, we previously mentioned that Swedish wolves were able to repro-
duce with a population of 3 wolves in 1983. Thus we will set m = 2 using the argument that we
used when we first introduced the minimum viable population (Section 2.2).
Lastly, When attempting to estimate r, we must differentiate a different rate of change for the
following two cases:
• r1: rate of change of population if the Russian wolves were never introduced in 1991 (decay
rate)
• r2: rate of change in population after the introduction of Russian wolves in 1991 (growth
rate)
Now that the carrying capacity and minimum population number have been established, we
can use the data from Figure 1 to get values for W0 and t. We used MATLAB to estimate r1 in
Equation 4 using the following values:
• t = 2, W0 = 10, W(2) = 8, m = 2, K = 300
Our estimated coefficient is r1 = −0.1483. By plotting the slope fields of this equation it is clear
that the population goes extinct as t → ∞. This will be apparent when we begin talking about our
stochastic model (See Figure 6).
Figure 4: Using GeoGebra and r2, it is clear that W → 0 if W < m and W → K if
W > m.
Repeating the same process with new values to estimate r2 following values:
• t = 10, W0 = 9, W(2) = 92, m = 2, K = 300
Our estimated coefficient is r2 = 0.2909. To see how our model behaves deterministically, see
Figure 4.
6

8. 2.4 Stochastic Simulation of the Logistic Model
Our modified Logistic Growth Model with estimated coefficients can tell us powerful information
about the behavior of the population depending on our choice of r. Using r1 our deterministic
model tells us that the population will go extinct after a finite time period. Meanwhile r2 predicts
that the population will increase to it’s carrying capacity over time. However, these models don’t
account for demographic and environmental stochasticity. Environmental stochasticity refers to the
variation and uncertainty in the environmental conditions that the wolf population finds itself in
such as temperature change, rainfall, living conditions, habitat loss, competition, and what seems
like an endless amount of variables. For the remainder of this paper, we will implement stochastic
simulations on our modified Logistic Model to simulate the role of demographic stochasticity in
the birth and death behaviors of our individual wolves.
Figure 5: A Markov Chain simulating births and deaths. Population size, denoted by
the circles, is considered the current state and a birth occurs with probability λi or a
death occurs with probability µj.
We used two stochastic processes to attempt to simulate the uncertainty of events, which we
define as a moment of birth or death. The first stochastic process we use is the Markov Chain.
Markov Chains attempt to simulate a movement from one state to another dependent on the prob-
ability of the specific movement to a new state as shown in Figure 5. Applying the markov chain
to birth and death processes, each state is a population size and The probability of a birth event
happening, taking the population from one size to a larger size, is determined by the intrinsic birth
rate b and the population size at the initial state. Alternatively, the probability of a death event,
taking the population from one size to a smaller size, is determined by the intrinsic death rate d
and the population size at the initial state.
We also used the Poisson process to determine the stochasticity of the time length between
each event. We used R Statistical Software to randomly select the time length between events from
the poisson distribution, where the poisson rate λ is determined by the birth rate, death rate, and
population size at the initial state. This process is used to show that the time length between events
decreases as the population gets larger.
3 Results
We can finally combine our modified Logistic Model with our simulation of demographic stochas-
ticity to show what happens to wolf populations in the two cases:
7

9. • No introduction of the Russian wolf r1
• Introduction of the Russian wolf r2
0
5
10
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Population
Simulation of Wolf Population prior to Russian Wolf Introduction
Figure 6: Using r1 and starting population of 10 in 1988, we determined that the
population goes to extinction by 2003 on average, with a mean time of extinction of
15.2 years and variance of 76.69.
100
200
1991
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Population
Simulation of Wolf Population after Russian Wolf Introduction
Figure 7: Using r2 and a starting population of 171 in 1991, we determined that the
average population size is 239.28 by 2027, with a variance of 66.0642.
We used R Statistical software to implement 50 different simulations of our Stochastic Logistic
Model and worked to find the mean time and variance of extinction considering the situation where
no Russian wolves were introduced (Figure 6).
We again applied another 50 simulations of our Stochastic Logistic Model considering the
situation where the Russian wolf was introduced and calculated the mean population and variance
36 years after the Russian wolf was initially introduced in 1991(Figure 7).
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10. 4 Conclusions
We are pleased to see that our results for our stochastic simulations confirmed the parametric
nature of the deterministic version of our modified Logistic Model. However, it may not be as
accurate as we hope. It is difficult to get accurate coefficients such as r1 and r2 because the data we
were provided is a small sample size of only 21 years of recorded population sizes. Moreover, it
would not be accurate to conclude any statistical significance of our results knowing that we only
have data from one trial. This is often a barrier when doing case studies, Moreover, many of our
coefficients can change over time. Some resources explained that death rates were decreasing from
1991 to 2001 and birth rates changed since the wolf population became less at risk of inbreeding
depression after the Russian Wolf was introduced. One last thing to consider is the element of
uncertainty of environmental stochasticity. Considering evironmental stochastic factors such as
food supply and habitat loss may make more accurate results, however, models that considers too
many variables aren’t that best at showing the underlying birth and death processes of a Logistic
Model which was our goal to begin with.
5 References
Facts about Wolves - WildSweden - wildlife adventures in Sweden. (n.d.). Retrieved from
https://www.wildsweden.com/about/facts-about-wolves
Ledin, A., Arnemo, J. M., Liberg, O., and Hellman, L. (2008). High plasma IgE levels within
the Scandinavian wolf population, and its implications for mammalian IgE homeostasis. Molecu-
lar Immunology, 45(7), 1976–1980. doi: 10.1016/j.molimm.2007.10.029
Räikkönen, J., Vucetich, J. A., Vucetich, L. M., Peterson, R. O., and Nelson, M. P. (n.d.). What the
Inbred Scandinavian Wolf Population Tells Us about the Nature of Conservation. Retrieved from
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0067218
Wabakken, P., Sand, H., Liberg, O., and Bjärvall, A. (2012). The recovery, distribution, and popu-
lation dynamics of wolves on the Scandinavian peninsula, 1978–1998. SSRN Electronic Journal.
doi: 10.2139/ssrn.1988201
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