Using STELLA to Explore Dynamic Single Species Models: The Magic of Making Humpback Whales Thrive in a Lab

Using STELLA to Explore Dynamic Single Species Models:
The Magic of Making Humpback Whales Thrive in a Lab
Lisa A. Jensen
Division of Science & Environmental Policy, California State University Monterey Bay, Seaside, CA, USA.
Abstract
The use of formal, mathematical models allows stakeholders, decision makers and scientists to
better visualize interactions and relationships within ecological systems. This study uses
STELLA, a modeling tool, to simulate simple population dynamics for the humpback whale
(Megaptera novaengliae) to better understand the impacts of reproductive and mortality rates as
well as alternative solution algorithms used to drive the model. A wide range of population
dynamics occurred as a result of varying time increments for calculating populations and use of
available solution algorithms. Populations are most likely to achieve equilibrium when
reproduction and mortality result in approximately the same number of individuals.
Introduction
Scientiﬁc models provide a mechanism to explore and examine relationships between
organisms and their environment. This process often leads to more questions along with an
improved understanding of the complex nature of the relationships we study. The use of
computers and software enables us to model and test our understanding of the relationships
between and within different communities (Doerr 1996, Lindholm 2008). STELLA, Structural
Thinking Experimental Learning Laboratory with Animation (Doerr 1996, isee 2010), is a
visually oriented model development tool which allows the user to readily build and modify
models (Lindholm 2008). The ease of rapidly changing relationships, inputs and interactions
enables the scientist to explore complex systems and identify gaps in understanding more
readily (Doerr 1996, Resnick 2003, Lindholm 2008).
While computer models are less complex than the systems they represent, they offer the
opportunity to test theories regarding relationships, introduce new information and grow the
investigator’s conceptual understanding of the system under study (Doerr 1996). It is this ability
to shift viewpoints and rapidly test ideas where software modeling is a powerful tool available
to science (Resnick 2003). At the same time the investigator needs to remain clear that modeling
tools do not fully describe the systems being reviewed, models frequently hold constant some
number of inﬂuencing factors to examine the systemic response to other factors (Lindholm
2008).
When examining at-risk populations, the use of computer modeling is an easy mechanism to
explore questions of exploitation, recovery, opportunities available for sustainability and
ENVS545, 2012 Jensen
Using STELLA to Explore Dynamic Single Species Models:! 1

improved management practices (Baker et al. 1987). Modeling systems allows decision makers
and stakeholders to deepen their understanding of the system and the variables which provide
impacts (Costanza and Ruth 1998). This exercise focuses on the use of STELLA (isee 2010) to
explore population dynamics of the humpback whale (Megaptera novaeangliae) with a simple
model encompassing reproduction and mortality. Humpback whales are a commercially
valuable resource and have been hunted nearly to the point of extinction (Clapham and Mayo
1987). Utilizing simple models, as are created within this exercise, will allow the investigator the
opportunity to explore the relationships between reproductive and mortality rates.
Methods
The exploratory models used for this study were informed by published information on
reproductive and mortality rates for the humpback whale (Baker et al. 1987, Clapham and Mayo
1987, Straley et al. 1994, Barlow and Clapham 1997, Steiger and Calambokidis 2000, Gabriele et
al. 2001) as well as modeling and the use of STELLA (Doerr 1996, Ruth and Lindholm 2002,
Scott Baker and Clapham 2004, isee 2010).
Data Collection
Data for this study was generated within the STELLA models with an initial population size of
200 humpback whales being studied over a period of forty years . This study examined
population dynamics looking initially at a closed system (no immigration, emigration, or
mortality) and exploring the changes in population size when density dependence was
considered, was not considered and recovery following sudden decreases in reproduction rates
(Table 1, Table 2, Appendix A). The model was modiﬁed to incorporate a mortality rate for the
population as a whole (Ruth and Lindholm 2002) (Table 3, Appendix A).
Research Questions
This study asked several questions prior to development and implementation of the models.
These included:
• How does altering the graphical relationship between population size and reproductive rate
impact the population over time?
H0: N(R0a) = N(R0b) = N(R0c) … = N(R0n)
H1: N(R0a) ≠ N(R0b) = N(R0c) … = N(R0n)
Hn+1: N(R0a) = N(R0b) ≠ N(R0c) … = N(R0n)
...
H2: N(R0a) ≠ N(R0b) ≠ N(R0c) … ≠ N(R0n)
where population size (N) is a function of the reproductive rate (R0) for the species. The null
hypothesis states the modifying the graphical relationship between reproductive rate and

population has no effect on population size, the alternative hypotheses state population size
between models is affected by reproductive rate.
• What impact does reproductive rate have on population size over time?
H0: N(R0a) = N(R0b) = N(R0c) … = N(R0n)
H1: N(R0a) ≠ N(R0b) = N(R0c) … = N(R0n)
Hn+1: N(R0a) = N(R0b) ≠ N(R0c) … = N(R0n)
...
H2: N(R0a) ≠ N(R0b) ≠ N(R0c) … ≠ N(R0n)
where population size (N) is a function of the reproductive rate (R0) for the species. The null
hypothesis states reproductive rate has no effect on population size, the alternative
hypotheses state population size between models is affected by reproductive rate. As the
reproductive rate increases the population size increases more quickly and as the rate
decreases, population size increases more slowly.
• What are the interactions between reproduction and mortality rates on population size over
time?
H0: N(D0a, R0a) = N(D0b, R0b) = N(D0c, R0c) … = N(Dn, R0n)
H1: N(D0a, R0a) ≠ N(D0b, R0b) = N(D0c, R0c) … = N(Dn, R0n)
H1: N(D0a, R0a) = N(D0b, R0b) ≠ N(D0c, R0c) … = N(Dn, R0n)
...
H2: N(D0a, R0a) ≠ N(D0b, R0b) ≠ N(D0c, R0c) … ≠ N(Dn, R0n)
where population size (N) is a function of both mortality (D0) and reproductive (R0) rates for
the species. The null hypothesis states there is no effect on population size, the alternative
hypotheses state population size between models is affected by mortality and reproductive
rates. When the mortality and reproductive rates are approximately the same the population
maintains a steady state condition, if the mortality rate is greater than the reproductive rate
the population will decline.
• How do altering the time step (DT) and solution algorithm effect appropriate model selection?
Granularity of the time step (DT) will have the effect of driving down the difference
between solution algorithms.
Assumptions
Models are by their nature a simpliﬁcation of real world systems (Lindholm 2008). The use of a
simple, closed loop model violates several assumptions found within an ecosystem. These
include:
• No immigration or migration.
• All members of the population give birth.
• No age-structure dependence for either reproductive or mortality rates (Gotelli 2008).
• No genetic structure (Gotelli 2008).
• No time lags (Gotelli 2008).
• No Allee effect for small populations (Jackson et al. 2008).

• No difference between adult and juvenile mortality rates (Gabriele et al. 2001).
• Constant calving intervals (Baker et al. 1987).
• Fractional increases in population allowed by specific solution algorithms (Lindholm 2008).
• Population growth over a period of forty or sixty-five years was representative of growth over
multiple generations.
Results
A simple model built utilizing reproduction as a function of population size at a given point in
time and reproductive rate (Fig. 1). In utilizing this model I explored modification of the
graphical relationship and alteration of reproduction rate to examine the effect on population
size.
Within the original model I modified the graphical relationship to reflect strict density
dependence (reproductivity goes to zero as the population reaches maximum size, 600 whales),
or not (reproductivity does not go to zero), and examined the role of sudden decreases in the
rate of reproduction (Fig.2, Table 2). Retaining a similar curve and turning on or off density
dependence indicated without density dependence, the population will continue growing in
spite of dramatic drops in the reproductive rate. The models with density dependence (1 and 4)
become asymptotic to population sizes near the maximum defined population. Models without
strict density dependence demonstrated a continued growth in the population. I explored
variations in density dependence due to discussion in the literature stating an insufficient
Figure 1. Simple closed loop model examining the relationship of reproductive rate
on population size.

knowledge base exists to fully understand whether grey whales experience density dependence
(Baker et al. 1987).
In the next iteration of model design the rate of reproduction was altered (Fig. 3, Table 3) based
on reproductive rates for humpback whales in other studies (Baker et al. 1987, Clapham and
! Model 1! Model 2
! Model 3! Model 4
Figure 2. Effect of modiﬁed graphical relationship between reproductive rate and population.

Mayo 1987, Steiger and Calambokidis 2000), maintaining the graphical relationship. These
models used strict density dependence looking only at varied reproductivity rates (0.20, 0.37,
0.43, 0.006, 0.059 respectively). Models 1, 2 and 3 each trend towards a steady state between
population sizes of 600 and 625. Models 4 and 5 do not exhibit a clear steady state condition
within the time frame of forty years.
Figure 3. Effect of altering reproductive rates while the graphical relationship, time step and
years remain constant.

The base graphical model was enhanced to incorporate mortality (Fig. 4, Table 4). This created a
slightly more complex, closed loop model and the opportunity to look at the relationship
between population size impacted by both mortality and reproduction. The initial model
Figure 4. Model for humpback whale population dynamics reﬂecting both reproductive and mortality
rates.
Figure 5. Chart for humpback whale population dynamics reﬂecting both
reproductive and mortality rates for two models.

created (R0 = 0.20, D = 0.03) shows a steady decrease in population size over time while the
second model (R0 = 0.20, D = 0.43) achieves a population steady state with the curve becoming
asymptotic to a population size of 600 whales within 25 years.
The final model iteration examined altering the time step (DT) and changing the solution
algorithm for each model (Fig. 6, Table 5). As the granularity of time step decreases the line of
population growth becomes less smooth. This is most apparent with a time step of 20 where two
straight lines and an angle are evident. Although all models become asymptotic, reaching a
steady state, the final values range between 600 for the most granular time steps to 870 for the
least granular.
Figure 6. Graphs reflecting altered time steps, holding reproductive and mortality rates
constant. All models were run using the Runge-Kutta 4 solution algorithm.

Table 1. Simulations for years 0 - 40 generated using Runge-Kutta 4, Runge-Kutta 2, and Euler
solution algorithm over a period of 40 years with varying time steps (DT).
Incremental Time Step (DT)Incremental Time Step (DT)Incremental Time Step (DT)Incremental Time Step (DT)Incremental Time Step (DT)Incremental Time Step (DT)Incremental Time Step (DT)
Study Year 0.125 (year) 0.25 (year) 0.5 (year) 1 (year) 5 (years) 10 (years) 20 (years)
Final RK4 599.94 599.94 599.94 599.94 599.93 625.1 681.09
Final RK2 599.94 599.93 599.93 599.93 599.59 627.71 681.44
Final Euler 599.94 599.95 599.96 599.98 602.88 681.44 870.58
Discussion
The first set of models (Fig. 2) offer an interesting perspective, in order to reach equilibrium in
this simple model which accounts only for reproduction, density dependence appears to be a
requirement. This is logical as density dependence has an implied assumption of limited
resources for a given population. An interesting point is the relative lack of impact shown by
drastic decreases in the reproductive rate (models 3 and 4). Regardless of whether or not density
dependence was considered, the population recovered and continued the growth trajectory.
Model 4, during which the whale population experienced severe decreases in reproductive rate
and included density dependence, recovered more quickly that the simpler model 1. This
appears to be due to a reproductive rate which is greater in model 4 than model 1 following the
decreased reproduction rates.
In the next set of whale population dynamic models, I examined 5 different reproductive rates
for humpback whales based on existing literature (Baker et al. 1987, Clapham and Mayo 1987,
Steiger and Calambokidis 2000, Ruth and Lindholm 2002). For each study, with the exception of
Ruth and Lindholm (2002), the authors indicated uncertainty in obtaining accurate reproductive
rates due in part to the challenges with sighting a given female following a birth and following
migration with calf. Each study utilized photo identification of flukes for individual animals.
Models 4 and 5 have the lowest suggested reproductive rate (0.006 - 0.059) may be the result of
early weanings or a sampling technique which precluded good sight lines and ready visibility
(Steiger and Calambokidis 2000). Models 1 and 2 (Baker et al. 1987, Ruth and Lindholm 2002)
appear to be steadily increasing, model 3 (Clapham and Mayo 1987) reaches equilibrium and
remains constant with a population size of approximately 600 whales. The relative agreement
between models 1 through 3 would suggest a higher degree of accuracy.
When mortality rates were added to the model it increased the level of complexity but
incorporated a real world approach. Model 1 (R0 = 0.2, D = 0.03) (Ruth and Lindholm 2002)
drives to extinction relatively rapidly which is not an intuitive conclusion when compared with
model 2 (R0 = 0.2, D = 0.43) which achieves equilibrium approximately at year 25. This begs the

question, did this investigator use the right numbers for the first model? Intuitively, model 1
should maintain an increase in population over time. The population should reach an
equilibrium state over a period of sixty years when reproduction and mortality are relatively
similar (Alava and Felix 2006).
Determining the frequency of sampling the population of humpback whales under study has
financial as well as accuracy implications. This is not an easy population to reach given the large
migration range (Baker et al. 1987) and multiple challenges with data collection and verification.
The associated costs of launching a research effort which may span thousands of miles further
inhibit extensive efforts which may drive towards accuracy. The final exercise for this study was
experimenting with different time steps (DT) and solution algorithms to identify an appropriate
combination which would give the investigator a degree of confidence in the model. Based on
provided information (Ruth and Lindholm 2002) the solution algorithm selected for the
previous exercises was Runge-Kutta 4 (RK4), it offers the highest degree of accuracy due to the
use of 4 intermediate steps to calculate F(t , X(t), . ) where X(t) is the population at a given point
in time t, F(t , X(t), . ) are the net flows depending on time.
Decreasing the granularity on DT results in decreased fidelity within the resulting simulated
data and on the graph (Fig. 6). This makes intuitive sense as well, when you increase the time
between data generation some loss is to be expected. At the highest level of granularity (DT =
0.125) the three solution algorithms are in agreement. As the granularity decreases to generation
of data once every 20 years the three algorithms begin to diverge with Euler diverging the
fastest and leading to significantly more whales in the simulated population than seen in RK4.
Runge-Kutta 2 diverges more slowly and remains within a couple of whales of RK4. The trade-
off for the degree of accuracy between the algorithms is computational time (Ruth and
Lindholm 2002), more accuracy demands increased time. As processing speed and RAM
increase this may not be as much of a consideration as it was previously but it should be
considered during selection of the algorithm. For this small data set there were no obvious
performance issues.
Conclusion
Although the models created for these simulations were very simple they offered the
investigator the opportunity to explore use of modeling and the implications for use within real-
world situations such as the development of policy. The International Whaling Commission
(IWC) indicates a strong recovery and a lower historic population than existing research would
indicate (Clapham et al. 1999, Steiger and Calambokidis 2000, Baker and Clapham 2004, Alava and Felix
2006, Jackson et al. 2008) as a result the use of models, especially when different studies drive
towards the same conclusion, may prove beneficial to policy development leading to population
recovery for these magnificent animals.

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Appendix A – Data
Table 2. Data generated modifying the graphical relationship between whale population size and
reproduction rate. These models look at alterations in density dependence and sudden decreases in
reproduction rates. Study year is point at which each whale population is counted, each of these
models used 0.25 years as time step (DT). Whales is the population size in number of whales at each
time step.
Study Year Whales
Model 1
Whales
Model 2
Whales
Model 3
Whales
Model 4
0 200 200 200 200
0.25 202.09 202.09 202.09 202.09
0.5 204.18 204.18 204.18 204.18
0.75 206.29 206.29 206.29 206.29
1 208.4 208.4 208.4 208.4
1.25 210.52 210.52 210.52 210.52
1.5 212.65 212.65 212.65 212.65
1.75 214.78 214.78 214.78 214.78
2 216.93 216.93 216.93 216.93
2.25 219.08 219.08 219.08 219.08
2.5 221.23 221.23 221.23 221.23
2.75 223.4 223.4 223.4 223.4
3 225.56 225.56 225.56 225.56
3.25 227.74 227.74 227.74 227.74
3.5 229.92 229.92 229.92 229.92
3.75 232.1 232.1 232.1 232.1
4 234.29 234.29 234.29 234.29
4.25 236.49 236.49 236.49 236.49
4.5 238.69 238.69 238.69 238.69
4.75 240.89 240.89 240.89 240.89
5 243.1 243.1 243.08 243.08
5.25 245.32 245.32 245.24 245.24
5.5 247.54 247.54 247.36 247.36
5.75 249.77 249.77 249.44 249.44
6 252.01 252.01 251.49 251.49
6.25 254.25 254.25 253.5 253.5
6.5 256.5 256.5 255.47 255.47
6.75 258.76 258.76 257.41 257.41
7 261.02 261.02 259.31 259.31
7.25 263.29 263.29 261.17 261.17
7.5 265.56 265.56 262.99 262.99
7.75 267.84 267.84 264.78 264.78
8 270.13 270.13 266.52 266.52
8.25 272.42 272.42 268.24 268.24

Study Year Whales
Model 1
Whales
Model 2
Whales
Model 3
Whales
Model 4
8.5 274.71 274.71 269.91 269.91
8.75 277.01 277.01 271.55 271.55
9 279.31 279.31 273.15 273.15
9.25 281.62 281.62 274.71 274.71
9.5 283.93 283.93 276.24 276.24
9.75 286.25 286.25 277.74 277.74
10 288.56 288.56 279.2 279.2
10.25 290.88 290.88 280.62 280.62
10.5 293.21 293.21 282.01 282.01
10.75 295.53 295.53 283.37 283.37
11 297.86 297.86 284.69 284.69
11.25 300.19 300.19 285.98 285.98
11.5 302.52 302.52 287.23 287.23
11.75 304.85 304.87 288.46 288.46
12 307.17 307.22 289.65 289.65
12.25 309.49 309.6 290.81 290.81
12.5 311.81 311.98 291.95 291.95
12.75 314.12 314.38 293.05 293.05
13 316.43 316.79 294.12 294.12
13.25 318.73 319.21 295.16 295.16
13.5 321.03 321.64 296.18 296.18
13.75 323.32 324.09 297.17 297.17
14 325.61 326.55 298.13 298.13
14.25 327.89 329.03 299.06 299.06
14.5 330.17 331.51 299.97 299.97
14.75 332.43 334.01 300.85 300.85
15 334.7 336.52 301.73 301.73
15.25 336.95 339.05 302.62 302.62
15.5 339.2 341.58 303.54 303.54
15.75 341.44 344.13 304.48 304.48
16 343.67 346.69 305.45 305.45
16.25 345.89 349.27 306.44 306.44
16.5 348.11 351.85 307.46 307.46
16.75 350.31 354.45 308.5 308.5
17 352.51 357.06 309.58 309.58
17.25 354.7 359.68 310.68 310.68
17.5 356.87 362.32 311.81 311.81
17.75 359.04 364.97 312.97 312.97
18 361.2 367.63 314.16 314.16
18.25 363.35 370.31 315.39 315.39
18.5 365.5 372.99 316.65 316.65

Study Year Whales
Model 1
Whales
Model 2
Whales
Model 3
Whales
Model 4
18.75 367.64 375.7 317.95 317.95
19 369.77 378.41 319.28 319.28
19.25 371.9 381.14 320.65 320.65
19.5 374.02 383.88 322.05 322.05
19.75 376.14 386.64 323.5 323.5
20 378.25 389.4 324.99 324.99
20.25 380.35 392.19 326.53 326.53
20.5 382.44 394.98 328.11 328.11
20.75 384.53 397.79 329.73 329.73
21 386.61 400.61 331.41 331.41
21.25 388.68 403.44 333.13 333.13
21.5 390.74 406.29 334.91 334.91
21.75 392.8 409.14 336.74 336.74
22 394.85 412.02 338.63 338.63
22.25 396.89 414.9 340.57 340.57
22.5 398.92 417.8 342.58 342.58
22.75 400.94 420.71 344.65 344.65
23 402.95 423.63 346.79 346.79
23.25 404.96 426.55 349 349
23.5 406.95 429.48 351.28 351.28
23.75 408.94 432.41 353.63 353.63
24 410.92 435.35 356.07 356.07
24.25 412.89 438.29 358.58 358.58
24.5 414.84 441.23 361.19 361.19
24.75 416.79 444.18 363.83 363.83
25 418.73 447.13 366.48 366.48
25.25 420.66 450.09 369.15 369.15
25.5 422.58 453.05 371.84 371.84
25.75 424.48 456.01 374.53 374.53
26 426.38 458.97 377.24 377.24
26.25 428.27 461.94 379.97 379.97
26.5 430.15 464.9 382.7 382.7
26.75 432.03 467.87 385.45 385.45
27 433.89 470.85 388.21 388.21
27.25 435.74 473.82 390.99 390.99
27.5 437.58 476.79 393.78 393.78
27.75 439.41 479.77 396.58 396.58
28 441.23 482.74 399.39 399.39
28.25 443.04 485.71 402.22 402.22
28.5 444.84 488.67 405.06 405.06
28.75 446.63 491.62 407.91 407.91

Study Year Whales
Model 1
Whales
Model 2
Whales
Model 3
Whales
Model 4
29 448.41 494.57 410.78 410.78
29.25 450.18 497.51 413.66 413.66
29.5 451.93 500.44 416.55 416.55
29.75 453.68 503.36 419.46 419.46
30 455.42 506.28 422.37 422.37
30.25 457.14 509.18 425.29 425.29
30.5 458.86 512.08 428.22 428.22
30.75 460.56 514.97 431.15 431.15
31 462.25 517.85 434.08 434.08
31.25 463.93 520.72 437.02 437.02
31.5 465.6 523.57 439.97 439.97
31.75 467.26 526.42 442.91 442.91
32 468.91 529.26 445.86 445.86
32.25 470.54 532.09 448.82 448.82
32.5 472.16 534.9 451.77 451.77
32.75 473.78 537.71 454.73 454.73
33 475.38 540.5 457.7 457.7
33.25 476.97 543.29 460.66 460.66
33.5 478.55 546.09 463.63 463.63
33.75 480.11 548.89 466.6 466.6
34 481.67 551.7 469.57 469.57
34.25 483.21 554.51 472.54 472.54
34.5 484.74 557.33 475.51 475.51
34.75 486.26 560.15 478.49 478.49
35 487.76 562.97 481.46 481.46
35.25 489.26 565.8 484.44 484.44
35.5 490.74 568.63 487.4 487.4
35.75 492.2 571.47 490.36 490.36
36 493.66 574.31 493.31 493.31
36.25 495.1 577.16 496.25 496.25
36.5 496.53 580.01 499.18 499.18
36.75 497.95 582.86 502.11 502.11
37 499.35 585.72 505.03 505.03
37.25 500.74 588.58 507.94 507.94
37.5 502.12 591.45 510.84 510.84
37.75 503.49 594.32 513.73 513.73
38 504.85 597.19 516.61 516.61
38.25 506.19 600.06 519.48 519.48
38.5 507.52 602.95 522.35 522.35
38.75 508.84 605.84 525.2 525.2
39 510.14 608.75 528.04 528.04

Study Year Whales
Model 1
Whales
Model 2
Whales
Model 3
Whales
Model 4
39.25 511.44 611.68 530.87 530.87
39.5 512.72 614.62 533.69 533.69
39.75 513.99 617.57 536.5 536.5
40 515.25 620.54 539.3 539.3
Table 3. Models created looking at varying reproduction rates over a period of 40 years with constant
DT of 0.25 (3 months). Study year is point at which each whale population is counted. Whales is the
population size in number of whales at each time step. R0 is the reproductive rate used for the model.
Study Year Model 1
(R0 = 0.2)
(Ruth and
Lindholm
2002)
Model 2
(R0 = 0.37)
(Baker et al.
1987)
Model 3
(R0 = 0.43)
(Clapham
and Mayo
1987)
Model 4
(R0 = 0.006)
(Steiger and
Calambokidis 2000)
Whales
Model 5
(R0 = 0.059)
(Steiger and
Calambokidis 2000)
0 200 200 200 200 200
0.25 206.63 212.81 212.81 200.21 202
0.5 213.22 226.24 226.24 200.42 204
0.75 219.75 240.28 240.28 200.63 206.01
1 226.2 254.72 254.72 200.84 208.02
1.25 232.55 269.26 269.26 201.04 210.03
1.5 238.81 283.79 283.79 201.25 212.05
1.75 244.98 298.23 298.23 201.46 214.07
2 251.25 312.54 312.54 201.67 216.1
2.25 257.61 326.7 326.7 201.88 218.12
2.5 264.07 340.66 340.66 202.09 220.15
2.75 270.62 354.33 354.33 202.3 222.18
3 277.27 367.56 367.56 202.51 224.21
3.25 284.01 379.42 379.42 202.72 226.24
3.5 290.84 389.88 389.88 202.93 228.27
3.75 297.75 399.02 399.02 203.14 230.3
4 304.74 406.95 406.95 203.35 232.33
4.25 311.69 413.78 413.78 203.56 234.37
4.5 318.59 419.64 419.64 203.77 236.4
4.75 325.43 424.89 424.89 203.99 238.43
5 332.19 430.01 430.01 204.2 240.46
5.25 338.88 435.03 435.03 204.41 242.49
5.5 345.48 439.94 439.94 204.62 244.52
5.75 351.99 444.75 444.75 204.83 246.56
6 358.39 449.44 449.44 205.04 248.61
6.25 364.7 454.03 454.03 205.25 250.67
6.5 370.92 458.51 458.51 205.46 252.73

Study Year Model 1
(R0 = 0.2)
(Ruth and
Lindholm
2002)
Model 2
(R0 = 0.37)
(Baker et al.
1987)
Model 3
(R0 = 0.43)
(Clapham
and Mayo
1987)
Model 4
(R0 = 0.006)
(Steiger and
Calambokidis 2000)
Whales
Model 5
(R0 = 0.059)
(Steiger and
Calambokidis 2000)
6.75 377.05 462.88 462.88 205.68 254.79
7 383.09 467.14 467.14 205.89 256.87
7.25 389.03 471.28 471.28 206.1 258.95
7.5 394.87 475.32 475.32 206.31 261.03
7.75 400.6 479.25 479.25 206.52 263.12
8 406.22 483.08 483.08 206.74 265.21
8.25 411.73 486.84 486.84 206.95 267.31
8.5 417.12 490.53 490.53 207.16 269.41
8.75 422.4 494.15 494.15 207.37 271.52
9 427.57 497.7 497.7 207.59 273.63
9.25 432.64 501.18 501.18 207.8 275.75
9.5 437.6 504.59 504.59 208.01 277.87
9.75 442.46 507.93 507.93 208.23 280
10 447.21 511.19 511.19 208.44 282.12
10.25 451.85 514.39 514.39 208.65 284.25
10.5 456.38 517.52 517.52 208.87 286.39
10.75 460.8 520.59 520.59 209.08 288.53
11 465.11 523.58 523.58 209.29 290.67
11.25 469.31 526.51 526.51 209.51 292.81
11.5 473.41 529.37 529.37 209.72 294.96
11.75 477.39 532.16 532.16 209.93 297.11
12 481.26 534.89 534.89 210.15 299.26
12.25 485.05 537.55 537.55 210.36 301.41
12.5 488.78 540.15 540.15 210.58 303.56
12.75 492.43 542.68 542.67 210.79 305.71
13 496.01 545.13 545.09 211.01 307.86
13.25 499.52 547.5 547.42 211.22 310.01
13.5 502.97 549.8 549.66 211.43 312.15
13.75 506.34 552.03 551.82 211.65 314.29
14 509.64 554.19 553.89 211.86 316.43
14.25 512.87 556.27 555.88 212.08 318.57
14.5 516.04 558.29 557.79 212.29 320.7
14.75 519.13 560.25 559.62 212.51 322.83
15 522.16 562.14 561.38 212.73 324.95
15.25 525.11 563.96 563.06 212.94 327.08
15.5 528.01 565.73 564.68 213.16 329.19
15.75 530.83 567.44 566.23 213.37 331.3
16 533.59 569.09 567.72 213.59 333.41

Study Year Model 1
(R0 = 0.2)
(Ruth and
Lindholm
2002)
Model 2
(R0 = 0.37)
(Baker et al.
1987)
Model 3
(R0 = 0.43)
(Clapham
and Mayo
1987)
Model 4
(R0 = 0.006)
(Steiger and
Calambokidis 2000)
Whales
Model 5
(R0 = 0.059)
(Steiger and
Calambokidis 2000)
16.25 536.28 570.68 569.15 213.8 335.52
16.5 538.91 572.22 570.51 214.02 337.61
16.75 541.48 573.7 571.82 214.24 339.71
17 543.97 575.14 573.07 214.45 341.79
17.25 546.38 576.52 574.27 214.67 343.87
17.5 548.71 577.85 575.42 214.89 345.95
17.75 550.97 579.14 576.52 215.1 348.02
18 553.16 580.39 577.57 215.32 350.08
18.25 555.29 581.58 578.58 215.54 352.14
18.5 557.34 582.74 579.54 215.75 354.19
18.75 559.32 583.86 580.46 215.97 356.23
19 561.24 584.93 581.34 216.19 358.26
19.25 563.1 585.97 582.19 216.4 360.29
19.5 564.89 586.96 582.99 216.62 362.31
19.75 566.63 587.93 583.76 216.84 364.33
20 568.3 588.85 584.5 217.06 366.34
20.25 569.92 589.75 585.2 217.27 368.34
20.5 571.49 590.61 585.87 217.49 370.34
20.75 573 591.44 586.52 217.71 372.34
21 574.46 592.23 587.13 217.93 374.33
21.25 575.86 593 587.72 218.15 376.31
21.5 577.22 593.74 588.28 218.36 378.29
21.75 578.53 594.46 588.81 218.58 380.26
22 579.8 595.14 589.32 218.8 382.22
22.25 581.02 595.8 589.81 219.02 384.18
22.5 582.19 596.44 590.28 219.24 386.13
22.75 583.33 597.05 590.72 219.46 388.07
23 584.42 597.64 591.15 219.68 390.01
23.25 585.48 598.21 591.56 219.89 391.94
23.5 586.49 598.75 591.94 220.11 393.86
23.75 587.47 599.28 592.31 220.33 395.78
24 588.41 599.78 592.67 220.55 397.68
24.25 589.32 600.27 593 220.77 399.58
24.5 590.2 600.76 593.33 220.99 401.47
24.75 591.04 601.24 593.63 221.21 403.36
25 591.86 601.73 593.93 221.43 405.23
25.25 592.64 602.22 594.21 221.65 407.1
25.5 593.39 602.71 594.47 221.87 408.96

Study Year Model 1
(R0 = 0.2)
(Ruth and
Lindholm
2002)
Model 2
(R0 = 0.37)
(Baker et al.
1987)
Model 3
(R0 = 0.43)
(Clapham
and Mayo
1987)
Model 4
(R0 = 0.006)
(Steiger and
Calambokidis 2000)
Whales
Model 5
(R0 = 0.059)
(Steiger and
Calambokidis 2000)
25.75 594.12 603.2 594.73 222.09 410.81
26 594.82 603.69 594.97 222.31 412.65
26.25 595.49 604.18 595.2 222.53 414.49
26.5 596.14 604.67 595.43 222.75 416.31
26.75 596.76 605.16 595.64 222.97 418.13
27 597.36 605.65 595.84 223.19 419.94
27.25 597.94 606.14 596.03 223.41 421.73
27.5 598.49 606.63 596.21 223.63 423.51
27.75 599.03 607.13 596.39 223.85 425.28
28 599.54 607.62 596.56 224.07 427.03
28.25 600.04 608.11 596.72 224.29 428.77
28.5 600.52 608.61 596.87 224.51 430.5
28.75 601.01 609.1 597.01 224.74 432.2
29 601.5 609.59 597.15 224.96 433.9
29.25 601.99 610.09 597.28 225.18 435.57
29.5 602.48 610.58 597.41 225.4 437.24
29.75 602.97 611.08 597.53 225.62 438.88
30 603.46 611.58 597.64 225.84 440.52
30.25 603.94 612.07 597.75 226.06 442.13
30.5 604.43 612.57 597.86 226.29 443.73
30.75 604.93 613.07 597.96 226.51 445.32
31 605.42 613.56 598.05 226.73 446.89
31.25 605.91 614.06 598.14 226.95 448.45
31.5 606.4 614.56 598.23 227.18 449.99
31.75 606.89 615.06 598.31 227.4 451.52
32 607.38 615.56 598.39 227.62 453.03
32.25 607.88 616.06 598.46 227.84 454.52
32.5 608.37 616.56 598.54 228.06 456
32.75 608.86 617.06 598.6 228.29 457.47
33 609.36 617.56 598.67 228.51 458.92
33.25 609.85 618.06 598.73 228.73 460.35
33.5 610.35 618.56 598.79 228.96 461.77
33.75 610.84 619.06 598.85 229.18 463.18
34 611.34 619.57 598.9 229.4 464.57
34.25 611.84 620.07 598.95 229.63 465.94
34.5 612.33 620.57 599 229.85 467.3
34.75 612.83 621.08 599.05 230.07 468.65
35 613.33 621.58 599.09 230.3 469.98

Study Year Model 1
(R0 = 0.2)
(Ruth and
Lindholm
2002)
Model 2
(R0 = 0.37)
(Baker et al.
1987)
Model 3
(R0 = 0.43)
(Clapham
and Mayo
1987)
Model 4
(R0 = 0.006)
(Steiger and
Calambokidis 2000)
Whales
Model 5
(R0 = 0.059)
(Steiger and
Calambokidis 2000)
35.25 613.82 622.08 599.13 230.52 471.3
35.5 614.32 622.59 599.17 230.74 472.6
35.75 614.82 623.09 599.21 230.97 473.88
36 615.32 623.6 599.25 231.19 475.16
36.25 615.82 624.11 599.28 231.42 476.41
36.5 616.32 624.61 599.32 231.64 477.66
36.75 616.82 625.12 599.35 231.86 478.89
37 617.32 625.63 599.38 232.09 480.1
37.25 617.82 626.13 599.41 232.31 481.3
37.5 618.32 626.64 599.43 232.54 482.5
37.75 618.82 627.15 599.46 232.76 483.68
38 619.33 627.66 599.49 232.99 484.85
38.25 619.83 628.17 599.51 233.21 486.01
38.5 620.33 628.68 599.53 233.44 487.16
38.75 620.84 629.19 599.55 233.66 488.3
39 621.34 629.7 599.57 233.89 489.42
39.25 621.84 630.21 599.59 234.11 490.54
39.5 622.35 630.72 599.61 234.34 491.65
39.75 622.85 631.23 599.63 234.56 492.75
40 623.36 631.75 599.65 234.79 493.84
Table 4. Model data incorporating both reproductive and mortality rates, the second model reﬂects a
steady state equilibrium. Study year is point at which each whale population is counted, these studies
used a constant DT of 0.25 year. Whales is the population size in number of whales at each time step.
R0 is the reproductive rate in births of whales/year for the model and D is the mortality rate in deaths/
year.
Study Year Whales
Model 1
(R0 = 0.2, D = 0.03)
Whales
Model 2
(R0 = 0.2, D = 0.43)
0 200 200
0.25 199.54 208.5
0.5 199.07 217.22
0.75 198.62 226.17
1 198.16 235.34
1.25 197.71 244.72
1.5 197.26 254.33
1.75 196.82 264.17

Study Year Whales
Model 1
(R0 = 0.2, D = 0.03)
Whales
Model 2
(R0 = 0.2, D = 0.43)
2 196.38 274.24
2.25 195.94 284.54
2.5 195.5 295.04
2.75 195.07 305.74
3 194.64 316.4
3.25 194.21 326.95
3.5 193.79 337.37
3.75 193.37 347.63
4 192.95 357.7
4.25 192.54 367.6
4.5 192.13 377.44
4.75 191.72 387.22
5 191.32 396.9
5.25 190.91 406.49
5.5 190.51 415.96
5.75 190.12 425.32
6 189.72 434.68
6.25 189.33 444.03
6.5 188.94 453.37
6.75 188.55 462.68
7 188.17 471.95
7.25 187.79 481.17
7.5 187.41 489.9
7.75 187.04 497.86
8 186.66 505.08
8.25 186.29 511.62
8.5 185.92 517.52
8.75 185.56 522.83
9 185.19 527.6
9.25 184.83 531.87
9.5 184.47 535.69
9.75 184.12 539.1
10 183.76 542.18
10.25 183.41 545.12
10.5 183.06 547.92
10.75 182.71 550.58
11 182.37 553.13
11.25 182.03 555.55
11.5 181.69 557.86
11.75 181.35 560.06

Study Year Whales
Model 1
(R0 = 0.2, D = 0.03)
Whales
Model 2
(R0 = 0.2, D = 0.43)
12 181.01 562.15
12.25 180.67 564.13
12.5 180.34 566.02
12.75 180 567.82
13 179.66 569.53
13.25 179.33 571.15
13.5 178.99 572.68
13.75 178.66 574.14
14 178.34 575.53
14.25 178.01 576.84
14.5 177.68 578.09
14.75 177.36 579.27
15 177.04 580.39
15.25 176.72 581.45
15.5 176.41 582.46
15.75 176.09 583.41
16 175.78 584.31
16.25 175.47 585.17
16.5 175.16 585.98
16.75 174.86 586.74
17 174.55 587.47
17.25 174.25 588.15
17.5 173.95 588.8
17.75 173.65 589.42
18 173.35 590
18.25 173.06 590.55
18.5 172.76 591.07
18.75 172.47 591.56
19 172.18 592.02
19.25 171.89 592.46
19.5 171.61 592.88
19.75 171.32 593.27
20 171.04 593.64
20.25 170.76 594
20.5 170.48 594.33
20.75 170.2 594.64
21 169.92 594.94
21.25 169.65 595.22
21.5 169.38 595.48
21.75 169.1 595.73

Study Year Whales
Model 1
(R0 = 0.2, D = 0.03)
Whales
Model 2
(R0 = 0.2, D = 0.43)
22 168.83 595.97
22.25 168.57 596.19
22.5 168.3 596.41
22.75 168.03 596.61
23 167.77 596.79
23.25 167.51 596.97
23.5 167.25 597.14
23.75 166.99 597.3
24 166.73 597.45
24.25 166.47 597.59
24.5 166.22 597.73
24.75 165.97 597.85
25 165.71 597.97
25.25 165.46 598.08
25.5 165.21 598.19
25.75 164.97 598.29
26 164.72 598.39
26.25 164.48 598.48
26.5 164.23 598.56
26.75 163.99 598.64
27 163.75 598.72
27.25 163.51 598.79
27.5 163.27 598.86
27.75 163.03 598.92
28 162.8 598.98
28.25 162.57 599.04
28.5 162.33 599.09
28.75 162.1 599.14
29 161.87 599.19
29.25 161.64 599.23
29.5 161.41 599.28
29.75 161.19 599.32
30 160.96 599.36
30.25 160.74 599.39
30.5 160.52 599.43
30.75 160.29 599.46
31 160.07 599.49
31.25 159.85 599.52
31.5 159.64 599.54
31.75 159.42 599.57

Study Year Whales
Model 1
(R0 = 0.2, D = 0.03)
Whales
Model 2
(R0 = 0.2, D = 0.43)
32 159.2 599.59
32.25 158.99 599.62
32.5 158.78 599.64
32.75 158.56 599.66
33 158.35 599.68
33.25 158.14 599.69
33.5 157.93 599.71
33.75 157.73 599.73
34 157.52 599.74
34.25 157.31 599.76
34.5 157.11 599.77
34.75 156.91 599.78
35 156.7 599.8
35.25 156.5 599.81
35.5 156.3 599.82
35.75 156.1 599.83
36 155.9 599.84
36.25 155.71 599.85
36.5 155.51 599.86
36.75 155.32 599.86
37 155.12 599.87
37.25 154.93 599.88
37.5 154.74 599.89
37.75 154.55 599.89
38 154.36 599.9
38.25 154.17 599.9
38.5 153.98 599.91
38.75 153.79 599.91
39 153.6 599.92
39.25 153.42 599.92
39.5 153.24 599.93
39.75 153.05 599.93
40 152.87 599.94

Table 5. Data for years 0 - 39.75 generated using Runge-Kutta 4 solution algorithm over a period of 40
years with varying time steps (DT). Final data for Runge-Kutta 4, Runge-Kutta 2, and Euler provided.
0 200 200 200 200 200 200 200
0.125 204.22
0.25 208.5 208.5
0.375 212.83
0.5 217.22 217.22 217.22
0.625 221.67
0.75 226.17 226.17
0.875 230.73
1 235.34 235.34 235.34 235.34
1.125 240.01
1.25 244.72 244.72
1.375 249.5
1.5 254.33 254.33 254.33
1.625 259.22
1.75 264.17 264.17
1.875 269.18
2 274.24 274.24 274.24 274.25
2.125 279.36
2.25 284.53 284.54
2.375 289.76
2.5 295.04 295.04 295.04
2.625 300.38
2.75 305.73 305.74
2.875 311.07
3 316.39 316.4 316.38 316.41
3.125 321.68
3.25 326.95 326.95
3.375 332.17
3.5 337.37 337.37 337.36
3.625 342.52
3.75 347.62 347.63
3.875 352.68
4 357.7 357.7 357.69 357.71
4.125 362.66
4.25 367.6 367.6
4.375 372.52
4.5 377.44 377.44 377.44
4.625 382.33
4.75 387.21 387.22

4.875 392.06
5 396.89 396.9 396.9 396.94 400.05
5.125 401.7
5.25 406.48 406.49
5.375 411.23
5.5 415.95 415.96 415.95
5.625 420.64
5.75 425.32 425.32
5.875 430
6 434.68 434.68 434.69 434.69
6.125 439.35
6.25 444.03 444.03
6.375 448.7
6.5 453.36 453.37 453.37
6.625 458.02
6.75 462.67 462.68
6.875 467.31
7 471.95 471.95 471.96 471.96
7.125 476.57
7.25 481.17 481.17
7.375 485.64
7.5 489.9 489.9 489.99
7.625 493.98
7.75 497.86 497.86
7.875 501.56
8 505.08 505.08 505.15 504.99
8.125 508.43
8.25 511.62 511.62
8.375 514.65
8.5 517.52 517.52 517.57
8.625 520.24
8.75 522.83 522.83
8.875 525.28
9 527.6 527.6 527.64 527.53
9.125 529.79
9.25 531.87 531.87
9.375 533.83
9.5 535.69 535.69 535.72
9.625 537.44
9.75 539.1 539.1
9.875 540.67

10 542.18 542.18 542.21 542.2 548.13 521.39
10.125 543.67
10.25 545.12 545.12
10.375 546.53
10.5 547.92 547.92 547.94
10.625 549.27
10.75 550.59 550.58
10.875 551.87
11 553.13 553.13 553.15 553.14
11.125 554.36
11.25 555.55 555.55
11.375 556.72
11.5 557.86 557.86 557.88
11.625 558.97
11.75 560.06 560.06
11.875 561.12
12 562.15 562.15 562.17 562.16
12.125 563.15
12.25 564.14 564.13
12.375 565.09
12.5 566.02 566.02 566.04
12.625 566.93
12.75 567.82 567.82
12.875 568.68
13 569.53 569.53 569.54 569.53
13.125 570.35
13.25 571.15 571.15
13.375 571.93
13.5 572.68 572.68 572.7
13.625 573.42
13.75 574.14 574.14
13.875 574.85
14 575.53 575.53 575.54 575.53
14.125 576.19
14.25 576.84 576.84
14.375 577.47
14.5 578.09 578.09 578.1
14.625 578.69
14.75 579.27 579.27
14.875 579.84
15 580.39 580.39 580.4 580.39 582

15.125 580.93
15.25 581.45 581.45
15.375 581.96
15.5 582.46 582.46 582.47
15.625 582.94
15.75 583.41 583.41
15.875 583.87
16 584.31 584.31 584.32 584.31
16.125 584.74
16.25 585.17 585.17
16.375 585.58
16.5 585.98 585.98 585.98
16.625 586.36
16.75 586.74 586.74
16.875 587.11
17 587.47 587.47 587.47 587.47
17.125 587.81
17.25 588.15 588.15
17.375 588.48
17.5 588.8 588.8 588.81
17.625 589.11
17.75 589.42 589.42
17.875 589.71
18 590 590 590 590
18.125 590.28
18.25 590.55 590.55
18.375 590.81
18.5 591.07 591.07 591.07
18.625 591.32
18.75 591.56 591.56
18.875 591.79
19 592.02 592.02 592.03 592.03
19.125 592.25
19.25 592.46 592.46
19.375 592.67
19.5 592.88 592.88 592.88
19.625 593.08
19.75 593.27 593.27
19.875 593.46
20 593.64 593.64 593.65 593.65 593.95 625.1 681.09
20.125 593.82

20.25 594 594
20.375 594.16
20.5 594.33 594.33 594.33
20.625 594.49
20.75 594.64 594.64
20.875 594.79
21 594.94 594.94 594.94 594.94
21.125 595.08
21.25 595.22 595.22
21.375 595.35
21.5 595.48 595.48 595.49
21.625 595.61
21.75 595.73 595.73
21.875 595.85
22 595.97 595.97 595.97 595.97
22.125 596.08
22.25 596.19 596.19
22.375 596.3
22.5 596.41 596.41 596.41
22.625 596.51
22.75 596.61 596.61
22.875 596.7
23 596.79 596.79 596.8 596.79
23.125 596.88
23.25 596.97 596.97
23.375 597.06
23.5 597.14 597.14 597.14
23.625 597.22
23.75 597.3 597.3
23.875 597.38
24 597.45 597.45 597.45 597.45
24.125 597.52
24.25 597.59 597.59
24.375 597.66
24.5 597.73 597.73 597.73
24.625 597.79
24.75 597.85 597.85
24.875 597.91
25 597.97 597.97 597.97 597.97 597.99
25.125 598.03
25.25 598.08 598.08

25.375 598.14
25.5 598.19 598.19 598.19
25.625 598.24
25.75 598.29 598.29
25.875 598.34
26 598.39 598.39 598.39 598.39
26.125 598.43
26.25 598.48 598.48
26.375 598.52
26.5 598.56 598.56 598.56
26.625 598.6
26.75 598.64 598.64
26.875 598.68
27 598.72 598.72 598.72 598.72
27.125 598.75
27.25 598.79 598.79
27.375 598.82
27.5 598.86 598.86 598.86
27.625 598.89
27.75 598.92 598.92
27.875 598.95
28 598.98 598.98 598.98 598.98
28.125 599.01
28.25 599.04 599.04
28.375 599.06
28.5 599.09 599.09 599.09
28.625 599.12
28.75 599.14 599.14
28.875 599.17
29 599.19 599.19 599.19 599.19
29.125 599.21
29.25 599.23 599.23
29.375 599.26
29.5 599.28 599.28 599.28
29.625 599.3
29.75 599.32 599.32
29.875 599.34
30 599.36 599.36 599.36 599.36 599.33 625.1
30.125 599.37
30.25 599.39 599.39
30.375 599.41

30.5 599.43 599.43 599.43
30.625 599.44
30.75 599.46 599.46
30.875 599.47
31 599.49 599.49 599.49 599.49
31.125 599.5
31.25 599.52 599.52
31.375 599.53
31.5 599.54 599.54 599.54
31.625 599.56
31.75 599.57 599.57
31.875 599.58
32 599.59 599.59 599.59 599.59
32.125 599.6
32.25 599.62 599.62
32.375 599.63
32.5 599.64 599.64 599.64
32.625 599.65
32.75 599.66 599.66
32.875 599.67
33 599.68 599.68 599.68 599.68
33.125 599.69
33.25 599.69 599.69
33.375 599.7
33.5 599.71 599.71 599.71
33.625 599.72
33.75 599.73 599.73
33.875 599.74
34 599.74 599.74 599.74 599.74
34.125 599.75
34.25 599.76 599.76
34.375 599.76
34.5 599.77 599.77 599.77
34.625 599.78
34.75 599.78 599.78
34.875 599.79
35 599.8 599.8 599.8 599.8 599.78
35.125 599.8
35.25 599.81 599.81
35.375 599.81
35.5 599.82 599.82 599.82

35.625 599.82
35.75 599.83 599.83
35.875 599.83
36 599.84 599.84 599.84 599.84
36.125 599.84
36.25 599.85 599.85
36.375 599.85
36.5 599.86 599.86 599.86
36.625 599.86
36.75 599.86 599.86
36.875 599.87
37 599.87 599.87 599.87 599.87
37.125 599.87
37.25 599.88 599.88
37.375 599.88
37.5 599.89 599.89 599.89
37.625 599.89
37.75 599.89 599.89
37.875 599.89
38 599.9 599.9 599.9 599.9
38.125 599.9
38.25 599.9 599.9
38.375 599.91
38.5 599.91 599.91 599.91
38.625 599.91
38.75 599.91 599.91
38.875 599.92
39 599.92 599.92 599.92 599.92
39.125 599.92
39.25 599.92 599.92
39.375 599.93
39.5 599.93 599.93 599.93
39.625 599.93
39.75 599.93 599.93
39.875 599.93
Final RK4 599.94 599.94 599.94 599.94 599.93 625.1 681.09
Final RK2 599.94 599.93 599.93 599.93 599.59 627.71 681.44
Final Euler 599.94 599.95 599.96 599.98 602.88 681.44 870.58

Using STELLA to Explore Dynamic Single Species Models: The Magic of Making Humpback Whales Thrive in a Lab

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