using stella to explore dynamic single species models: the magic of making humpback whales thrive in...

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 4Figure 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). 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). 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). 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). 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). 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). 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). 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.



References

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Baker CS, Clapham PJ. 2004. Modelling the past and future of whales and whaling. Trends in Ecology & Evolution 19(7):365-371.

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Clapham PJ, Mayo CA. 1987. Reproduction and recruitment of individually identified humpback whales, Megaptera novaeangliae, observed in Massachusetts Bay, 1979–1985. Canadian Journal of Zoology 65(12):2853-2863.

Clapham PJ, Young SB, Brownell RL. 1999. Baleen whales: conservation issues and the status of the most endangered populations. Mammal Review 29(1):37-62.

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Gotelli NJ. 2008. A primer of ecology. Sunderland, MA: Sinauer Associates, Inc.isee. 2010. STELLA, systems thinking for education and research. 9.X ed.: isee.Jackson JA, Patenaude NJ, Carroll EL, Baker CS. 2008. How few whales were there after whaling?

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and Policy: CSUMB. p. 2.Resnick M. 2003. Thinking Like a Tree (and Other Forms of Ecological Thinking). International Journal of

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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 2000.25 202.09 202.09 202.09 202.090.5 204.18 204.18 204.18 204.180.75 206.29 206.29 206.29 206.29

1 208.4 208.4 208.4 208.41.25 210.52 210.52 210.52 210.521.5 212.65 212.65 212.65 212.651.75 214.78 214.78 214.78 214.78

2 216.93 216.93 216.93 216.932.25 219.08 219.08 219.08 219.082.5 221.23 221.23 221.23 221.232.75 223.4 223.4 223.4 223.4

3 225.56 225.56 225.56 225.563.25 227.74 227.74 227.74 227.743.5 229.92 229.92 229.92 229.923.75 232.1 232.1 232.1 232.1

4 234.29 234.29 234.29 234.294.25 236.49 236.49 236.49 236.494.5 238.69 238.69 238.69 238.694.75 240.89 240.89 240.89 240.89

5 243.1 243.1 243.08 243.085.25 245.32 245.32 245.24 245.245.5 247.54 247.54 247.36 247.365.75 249.77 249.77 249.44 249.44

6 252.01 252.01 251.49 251.496.25 254.25 254.25 253.5 253.56.5 256.5 256.5 255.47 255.476.75 258.76 258.76 257.41 257.41

7 261.02 261.02 259.31 259.317.25 263.29 263.29 261.17 261.177.5 265.56 265.56 262.99 262.997.75 267.84 267.84 264.78 264.78

8 270.13 270.13 266.52 266.528.25 272.42 272.42 268.24 268.24




Whales Model 2

Whales Model 3

Whales Model 4

8.5 274.71 274.71 269.91 269.918.75 277.01 277.01 271.55 271.55

9 279.31 279.31 273.15 273.159.25 281.62 281.62 274.71 274.719.5 283.93 283.93 276.24 276.249.75 286.25 286.25 277.74 277.7410 288.56 288.56 279.2 279.2

10.25 290.88 290.88 280.62 280.6210.5 293.21 293.21 282.01 282.0110.75 295.53 295.53 283.37 283.37

11 297.86 297.86 284.69 284.6911.25 300.19 300.19 285.98 285.9811.5 302.52 302.52 287.23 287.2311.75 304.85 304.87 288.46 288.46

12 307.17 307.22 289.65 289.6512.25 309.49 309.6 290.81 290.8112.5 311.81 311.98 291.95 291.9512.75 314.12 314.38 293.05 293.05

13 316.43 316.79 294.12 294.1213.25 318.73 319.21 295.16 295.1613.5 321.03 321.64 296.18 296.1813.75 323.32 324.09 297.17 297.17

14 325.61 326.55 298.13 298.1314.25 327.89 329.03 299.06 299.0614.5 330.17 331.51 299.97 299.9714.75 332.43 334.01 300.85 300.85

15 334.7 336.52 301.73 301.7315.25 336.95 339.05 302.62 302.6215.5 339.2 341.58 303.54 303.5415.75 341.44 344.13 304.48 304.48

16 343.67 346.69 305.45 305.4516.25 345.89 349.27 306.44 306.4416.5 348.11 351.85 307.46 307.4616.75 350.31 354.45 308.5 308.5

17 352.51 357.06 309.58 309.5817.25 354.7 359.68 310.68 310.6817.5 356.87 362.32 311.81 311.8117.75 359.04 364.97 312.97 312.97

18 361.2 367.63 314.16 314.1618.25 363.35 370.31 315.39 315.3918.5 365.5 372.99 316.65 316.65




Whales Model 2

Whales Model 3

Whales Model 4

18.75 367.64 375.7 317.95 317.9519 369.77 378.41 319.28 319.28

19.25 371.9 381.14 320.65 320.6519.5 374.02 383.88 322.05 322.0519.75 376.14 386.64 323.5 323.5

20 378.25 389.4 324.99 324.9920.25 380.35 392.19 326.53 326.5320.5 382.44 394.98 328.11 328.1120.75 384.53 397.79 329.73 329.73

21 386.61 400.61 331.41 331.4121.25 388.68 403.44 333.13 333.1321.5 390.74 406.29 334.91 334.9121.75 392.8 409.14 336.74 336.74

22 394.85 412.02 338.63 338.6322.25 396.89 414.9 340.57 340.5722.5 398.92 417.8 342.58 342.5822.75 400.94 420.71 344.65 344.65

23 402.95 423.63 346.79 346.7923.25 404.96 426.55 349 34923.5 406.95 429.48 351.28 351.2823.75 408.94 432.41 353.63 353.63

24 410.92 435.35 356.07 356.0724.25 412.89 438.29 358.58 358.5824.5 414.84 441.23 361.19 361.1924.75 416.79 444.18 363.83 363.83

25 418.73 447.13 366.48 366.4825.25 420.66 450.09 369.15 369.1525.5 422.58 453.05 371.84 371.8425.75 424.48 456.01 374.53 374.53

26 426.38 458.97 377.24 377.2426.25 428.27 461.94 379.97 379.9726.5 430.15 464.9 382.7 382.726.75 432.03 467.87 385.45 385.45

27 433.89 470.85 388.21 388.2127.25 435.74 473.82 390.99 390.9927.5 437.58 476.79 393.78 393.7827.75 439.41 479.77 396.58 396.58

28 441.23 482.74 399.39 399.3928.25 443.04 485.71 402.22 402.2228.5 444.84 488.67 405.06 405.0628.75 446.63 491.62 407.91 407.91




Whales Model 2

Whales Model 3

Whales Model 4

29 448.41 494.57 410.78 410.7829.25 450.18 497.51 413.66 413.6629.5 451.93 500.44 416.55 416.5529.75 453.68 503.36 419.46 419.46

30 455.42 506.28 422.37 422.3730.25 457.14 509.18 425.29 425.2930.5 458.86 512.08 428.22 428.2230.75 460.56 514.97 431.15 431.15

31 462.25 517.85 434.08 434.0831.25 463.93 520.72 437.02 437.0231.5 465.6 523.57 439.97 439.9731.75 467.26 526.42 442.91 442.91

32 468.91 529.26 445.86 445.8632.25 470.54 532.09 448.82 448.8232.5 472.16 534.9 451.77 451.7732.75 473.78 537.71 454.73 454.73

33 475.38 540.5 457.7 457.733.25 476.97 543.29 460.66 460.6633.5 478.55 546.09 463.63 463.6333.75 480.11 548.89 466.6 466.6

34 481.67 551.7 469.57 469.5734.25 483.21 554.51 472.54 472.5434.5 484.74 557.33 475.51 475.5134.75 486.26 560.15 478.49 478.49

35 487.76 562.97 481.46 481.4635.25 489.26 565.8 484.44 484.4435.5 490.74 568.63 487.4 487.435.75 492.2 571.47 490.36 490.36

36 493.66 574.31 493.31 493.3136.25 495.1 577.16 496.25 496.2536.5 496.53 580.01 499.18 499.1836.75 497.95 582.86 502.11 502.11

37 499.35 585.72 505.03 505.0337.25 500.74 588.58 507.94 507.9437.5 502.12 591.45 510.84 510.8437.75 503.49 594.32 513.73 513.73

38 504.85 597.19 516.61 516.6138.25 506.19 600.06 519.48 519.4838.5 507.52 602.95 522.35 522.3538.75 508.84 605.84 525.2 525.2

39 510.14 608.75 528.04 528.04




Whales Model 2

Whales Model 3

Whales Model 4

39.25 511.44 611.68 530.87 530.8739.5 512.72 614.62 533.69 533.6939.75 513.99 617.57 536.5 536.5

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 2020.5 213.22 226.24 226.24 200.42 2040.75 219.75 240.28 240.28 200.63 206.01

1 226.2 254.72 254.72 200.84 208.021.25 232.55 269.26 269.26 201.04 210.031.5 238.81 283.79 283.79 201.25 212.051.75 244.98 298.23 298.23 201.46 214.07

2 251.25 312.54 312.54 201.67 216.12.25 257.61 326.7 326.7 201.88 218.122.5 264.07 340.66 340.66 202.09 220.152.75 270.62 354.33 354.33 202.3 222.18

3 277.27 367.56 367.56 202.51 224.213.25 284.01 379.42 379.42 202.72 226.243.5 290.84 389.88 389.88 202.93 228.273.75 297.75 399.02 399.02 203.14 230.3

4 304.74 406.95 406.95 203.35 232.334.25 311.69 413.78 413.78 203.56 234.374.5 318.59 419.64 419.64 203.77 236.44.75 325.43 424.89 424.89 203.99 238.43

5 332.19 430.01 430.01 204.2 240.465.25 338.88 435.03 435.03 204.41 242.495.5 345.48 439.94 439.94 204.62 244.525.75 351.99 444.75 444.75 204.83 246.56

6 358.39 449.44 449.44 205.04 248.616.25 364.7 454.03 454.03 205.25 250.676.5 370.92 458.51 458.51 205.46 252.73




(Ruth and Lindholm

2002)

Model 2 (R0 = 0.37)

(Baker et al. 1987)


1987)


Calambokidis 2000)

Whales Model 5


Calambokidis 2000)6.75 377.05 462.88 462.88 205.68 254.79

7 383.09 467.14 467.14 205.89 256.877.25 389.03 471.28 471.28 206.1 258.957.5 394.87 475.32 475.32 206.31 261.037.75 400.6 479.25 479.25 206.52 263.12

8 406.22 483.08 483.08 206.74 265.218.25 411.73 486.84 486.84 206.95 267.318.5 417.12 490.53 490.53 207.16 269.418.75 422.4 494.15 494.15 207.37 271.52

9 427.57 497.7 497.7 207.59 273.639.25 432.64 501.18 501.18 207.8 275.759.5 437.6 504.59 504.59 208.01 277.879.75 442.46 507.93 507.93 208.23 28010 447.21 511.19 511.19 208.44 282.12

10.25 451.85 514.39 514.39 208.65 284.2510.5 456.38 517.52 517.52 208.87 286.3910.75 460.8 520.59 520.59 209.08 288.53

11 465.11 523.58 523.58 209.29 290.6711.25 469.31 526.51 526.51 209.51 292.8111.5 473.41 529.37 529.37 209.72 294.9611.75 477.39 532.16 532.16 209.93 297.11

12 481.26 534.89 534.89 210.15 299.2612.25 485.05 537.55 537.55 210.36 301.4112.5 488.78 540.15 540.15 210.58 303.5612.75 492.43 542.68 542.67 210.79 305.71

13 496.01 545.13 545.09 211.01 307.8613.25 499.52 547.5 547.42 211.22 310.0113.5 502.97 549.8 549.66 211.43 312.1513.75 506.34 552.03 551.82 211.65 314.29

14 509.64 554.19 553.89 211.86 316.4314.25 512.87 556.27 555.88 212.08 318.5714.5 516.04 558.29 557.79 212.29 320.714.75 519.13 560.25 559.62 212.51 322.83

15 522.16 562.14 561.38 212.73 324.9515.25 525.11 563.96 563.06 212.94 327.0815.5 528.01 565.73 564.68 213.16 329.1915.75 530.83 567.44 566.23 213.37 331.3

16 533.59 569.09 567.72 213.59 333.41




(Ruth and Lindholm

2002)

Model 2 (R0 = 0.37)

(Baker et al. 1987)


1987)


Calambokidis 2000)

Whales Model 5


Calambokidis 2000)16.25 536.28 570.68 569.15 213.8 335.5216.5 538.91 572.22 570.51 214.02 337.6116.75 541.48 573.7 571.82 214.24 339.71

17 543.97 575.14 573.07 214.45 341.7917.25 546.38 576.52 574.27 214.67 343.8717.5 548.71 577.85 575.42 214.89 345.9517.75 550.97 579.14 576.52 215.1 348.02

18 553.16 580.39 577.57 215.32 350.0818.25 555.29 581.58 578.58 215.54 352.1418.5 557.34 582.74 579.54 215.75 354.1918.75 559.32 583.86 580.46 215.97 356.23

19 561.24 584.93 581.34 216.19 358.2619.25 563.1 585.97 582.19 216.4 360.2919.5 564.89 586.96 582.99 216.62 362.3119.75 566.63 587.93 583.76 216.84 364.33

20 568.3 588.85 584.5 217.06 366.3420.25 569.92 589.75 585.2 217.27 368.3420.5 571.49 590.61 585.87 217.49 370.3420.75 573 591.44 586.52 217.71 372.34

21 574.46 592.23 587.13 217.93 374.3321.25 575.86 593 587.72 218.15 376.3121.5 577.22 593.74 588.28 218.36 378.2921.75 578.53 594.46 588.81 218.58 380.26

22 579.8 595.14 589.32 218.8 382.2222.25 581.02 595.8 589.81 219.02 384.1822.5 582.19 596.44 590.28 219.24 386.1322.75 583.33 597.05 590.72 219.46 388.07

23 584.42 597.64 591.15 219.68 390.0123.25 585.48 598.21 591.56 219.89 391.9423.5 586.49 598.75 591.94 220.11 393.8623.75 587.47 599.28 592.31 220.33 395.78

24 588.41 599.78 592.67 220.55 397.6824.25 589.32 600.27 593 220.77 399.5824.5 590.2 600.76 593.33 220.99 401.4724.75 591.04 601.24 593.63 221.21 403.36

25 591.86 601.73 593.93 221.43 405.2325.25 592.64 602.22 594.21 221.65 407.125.5 593.39 602.71 594.47 221.87 408.96




(Ruth and Lindholm

2002)

Model 2 (R0 = 0.37)

(Baker et al. 1987)


1987)


Calambokidis 2000)

Whales Model 5


Calambokidis 2000)25.75 594.12 603.2 594.73 222.09 410.81

26 594.82 603.69 594.97 222.31 412.6526.25 595.49 604.18 595.2 222.53 414.4926.5 596.14 604.67 595.43 222.75 416.3126.75 596.76 605.16 595.64 222.97 418.13

27 597.36 605.65 595.84 223.19 419.9427.25 597.94 606.14 596.03 223.41 421.7327.5 598.49 606.63 596.21 223.63 423.5127.75 599.03 607.13 596.39 223.85 425.28

28 599.54 607.62 596.56 224.07 427.0328.25 600.04 608.11 596.72 224.29 428.7728.5 600.52 608.61 596.87 224.51 430.528.75 601.01 609.1 597.01 224.74 432.2

29 601.5 609.59 597.15 224.96 433.929.25 601.99 610.09 597.28 225.18 435.5729.5 602.48 610.58 597.41 225.4 437.2429.75 602.97 611.08 597.53 225.62 438.88

30 603.46 611.58 597.64 225.84 440.5230.25 603.94 612.07 597.75 226.06 442.1330.5 604.43 612.57 597.86 226.29 443.7330.75 604.93 613.07 597.96 226.51 445.32

31 605.42 613.56 598.05 226.73 446.8931.25 605.91 614.06 598.14 226.95 448.4531.5 606.4 614.56 598.23 227.18 449.9931.75 606.89 615.06 598.31 227.4 451.52

32 607.38 615.56 598.39 227.62 453.0332.25 607.88 616.06 598.46 227.84 454.5232.5 608.37 616.56 598.54 228.06 45632.75 608.86 617.06 598.6 228.29 457.47

33 609.36 617.56 598.67 228.51 458.9233.25 609.85 618.06 598.73 228.73 460.3533.5 610.35 618.56 598.79 228.96 461.7733.75 610.84 619.06 598.85 229.18 463.18

34 611.34 619.57 598.9 229.4 464.5734.25 611.84 620.07 598.95 229.63 465.9434.5 612.33 620.57 599 229.85 467.334.75 612.83 621.08 599.05 230.07 468.65

35 613.33 621.58 599.09 230.3 469.98




(Ruth and Lindholm

2002)

Model 2 (R0 = 0.37)

(Baker et al. 1987)


1987)


Calambokidis 2000)

Whales Model 5


Calambokidis 2000)35.25 613.82 622.08 599.13 230.52 471.335.5 614.32 622.59 599.17 230.74 472.635.75 614.82 623.09 599.21 230.97 473.88

36 615.32 623.6 599.25 231.19 475.1636.25 615.82 624.11 599.28 231.42 476.4136.5 616.32 624.61 599.32 231.64 477.6636.75 616.82 625.12 599.35 231.86 478.89

37 617.32 625.63 599.38 232.09 480.137.25 617.82 626.13 599.41 232.31 481.337.5 618.32 626.64 599.43 232.54 482.537.75 618.82 627.15 599.46 232.76 483.68

38 619.33 627.66 599.49 232.99 484.8538.25 619.83 628.17 599.51 233.21 486.0138.5 620.33 628.68 599.53 233.44 487.1638.75 620.84 629.19 599.55 233.66 488.3

39 621.34 629.7 599.57 233.89 489.4239.25 621.84 630.21 599.59 234.11 490.5439.5 622.35 630.72 599.61 234.34 491.6539.75 622.85 631.23 599.63 234.56 492.75

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.


(R0 = 0.2, D = 0.03)

Whales Model 2

(R0 = 0.2, D = 0.43)0 200 200

0.25 199.54 208.50.5 199.07 217.220.75 198.62 226.17

1 198.16 235.341.25 197.71 244.721.5 197.26 254.331.75 196.82 264.17




(R0 = 0.2, D = 0.03)

Whales Model 2

(R0 = 0.2, D = 0.43)2 196.38 274.24

2.25 195.94 284.542.5 195.5 295.042.75 195.07 305.74

3 194.64 316.43.25 194.21 326.953.5 193.79 337.373.75 193.37 347.63

4 192.95 357.74.25 192.54 367.64.5 192.13 377.444.75 191.72 387.22

5 191.32 396.95.25 190.91 406.495.5 190.51 415.965.75 190.12 425.32

6 189.72 434.686.25 189.33 444.036.5 188.94 453.376.75 188.55 462.68

7 188.17 471.957.25 187.79 481.177.5 187.41 489.97.75 187.04 497.86

8 186.66 505.088.25 186.29 511.628.5 185.92 517.528.75 185.56 522.83

9 185.19 527.69.25 184.83 531.879.5 184.47 535.699.75 184.12 539.110 183.76 542.18

10.25 183.41 545.1210.5 183.06 547.9210.75 182.71 550.58

11 182.37 553.1311.25 182.03 555.5511.5 181.69 557.8611.75 181.35 560.06




(R0 = 0.2, D = 0.03)

Whales Model 2

(R0 = 0.2, D = 0.43)12 181.01 562.15

12.25 180.67 564.1312.5 180.34 566.0212.75 180 567.82

13 179.66 569.5313.25 179.33 571.1513.5 178.99 572.6813.75 178.66 574.14

14 178.34 575.5314.25 178.01 576.8414.5 177.68 578.0914.75 177.36 579.27

15 177.04 580.3915.25 176.72 581.4515.5 176.41 582.4615.75 176.09 583.41

16 175.78 584.3116.25 175.47 585.1716.5 175.16 585.9816.75 174.86 586.74

17 174.55 587.4717.25 174.25 588.1517.5 173.95 588.817.75 173.65 589.42

18 173.35 59018.25 173.06 590.5518.5 172.76 591.0718.75 172.47 591.56

19 172.18 592.0219.25 171.89 592.4619.5 171.61 592.8819.75 171.32 593.27

20 171.04 593.6420.25 170.76 59420.5 170.48 594.3320.75 170.2 594.64

21 169.92 594.9421.25 169.65 595.2221.5 169.38 595.4821.75 169.1 595.73




(R0 = 0.2, D = 0.03)

Whales Model 2

(R0 = 0.2, D = 0.43)22 168.83 595.97

22.25 168.57 596.1922.5 168.3 596.4122.75 168.03 596.61

23 167.77 596.7923.25 167.51 596.9723.5 167.25 597.1423.75 166.99 597.3

24 166.73 597.4524.25 166.47 597.5924.5 166.22 597.7324.75 165.97 597.85

25 165.71 597.9725.25 165.46 598.0825.5 165.21 598.1925.75 164.97 598.29

26 164.72 598.3926.25 164.48 598.4826.5 164.23 598.5626.75 163.99 598.64

27 163.75 598.7227.25 163.51 598.7927.5 163.27 598.8627.75 163.03 598.92

28 162.8 598.9828.25 162.57 599.0428.5 162.33 599.0928.75 162.1 599.14

29 161.87 599.1929.25 161.64 599.2329.5 161.41 599.2829.75 161.19 599.32

30 160.96 599.3630.25 160.74 599.3930.5 160.52 599.4330.75 160.29 599.46

31 160.07 599.4931.25 159.85 599.5231.5 159.64 599.5431.75 159.42 599.57




(R0 = 0.2, D = 0.03)

Whales Model 2

(R0 = 0.2, D = 0.43)32 159.2 599.59

32.25 158.99 599.6232.5 158.78 599.6432.75 158.56 599.66

33 158.35 599.6833.25 158.14 599.6933.5 157.93 599.7133.75 157.73 599.73

34 157.52 599.7434.25 157.31 599.7634.5 157.11 599.7734.75 156.91 599.78

35 156.7 599.835.25 156.5 599.8135.5 156.3 599.8235.75 156.1 599.83

36 155.9 599.8436.25 155.71 599.8536.5 155.51 599.8636.75 155.32 599.86

37 155.12 599.8737.25 154.93 599.8837.5 154.74 599.8937.75 154.55 599.89

38 154.36 599.938.25 154.17 599.938.5 153.98 599.9138.75 153.79 599.91

39 153.6 599.9239.25 153.42 599.9239.5 153.24 599.9339.75 153.05 599.93

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.

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)0 200 200 200 200 200 200 2000.125 204.220.25 208.5 208.50.375 212.830.5 217.22 217.22 217.220.625 221.670.75 226.17 226.170.875 230.731 235.34 235.34 235.34 235.341.125 240.011.25 244.72 244.721.375 249.51.5 254.33 254.33 254.331.625 259.221.75 264.17 264.171.875 269.182 274.24 274.24 274.24 274.252.125 279.362.25 284.53 284.542.375 289.762.5 295.04 295.04 295.042.625 300.382.75 305.73 305.742.875 311.073 316.39 316.4 316.38 316.413.125 321.683.25 326.95 326.953.375 332.173.5 337.37 337.37 337.363.625 342.523.75 347.62 347.633.875 352.684 357.7 357.7 357.69 357.714.125 362.664.25 367.6 367.64.375 372.524.5 377.44 377.44 377.444.625 382.334.75 387.21 387.22



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)4.875 392.065 396.89 396.9 396.9 396.94 400.055.125 401.75.25 406.48 406.495.375 411.235.5 415.95 415.96 415.955.625 420.645.75 425.32 425.325.875 4306 434.68 434.68 434.69 434.696.125 439.356.25 444.03 444.036.375 448.76.5 453.36 453.37 453.376.625 458.026.75 462.67 462.686.875 467.317 471.95 471.95 471.96 471.967.125 476.577.25 481.17 481.177.375 485.647.5 489.9 489.9 489.997.625 493.987.75 497.86 497.867.875 501.568 505.08 505.08 505.15 504.998.125 508.438.25 511.62 511.628.375 514.658.5 517.52 517.52 517.578.625 520.248.75 522.83 522.838.875 525.289 527.6 527.6 527.64 527.539.125 529.799.25 531.87 531.879.375 533.839.5 535.69 535.69 535.729.625 537.449.75 539.1 539.19.875 540.67



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)10 542.18 542.18 542.21 542.2 548.13 521.3910.125 543.6710.25 545.12 545.1210.375 546.5310.5 547.92 547.92 547.9410.625 549.2710.75 550.59 550.5810.875 551.8711 553.13 553.13 553.15 553.1411.125 554.3611.25 555.55 555.5511.375 556.7211.5 557.86 557.86 557.8811.625 558.9711.75 560.06 560.0611.875 561.1212 562.15 562.15 562.17 562.1612.125 563.1512.25 564.14 564.1312.375 565.0912.5 566.02 566.02 566.0412.625 566.9312.75 567.82 567.8212.875 568.6813 569.53 569.53 569.54 569.5313.125 570.3513.25 571.15 571.1513.375 571.9313.5 572.68 572.68 572.713.625 573.4213.75 574.14 574.1413.875 574.8514 575.53 575.53 575.54 575.5314.125 576.1914.25 576.84 576.8414.375 577.4714.5 578.09 578.09 578.114.625 578.6914.75 579.27 579.2714.875 579.8415 580.39 580.39 580.4 580.39 582



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)15.125 580.9315.25 581.45 581.4515.375 581.9615.5 582.46 582.46 582.4715.625 582.9415.75 583.41 583.4115.875 583.8716 584.31 584.31 584.32 584.3116.125 584.7416.25 585.17 585.1716.375 585.5816.5 585.98 585.98 585.9816.625 586.3616.75 586.74 586.7416.875 587.1117 587.47 587.47 587.47 587.4717.125 587.8117.25 588.15 588.1517.375 588.4817.5 588.8 588.8 588.8117.625 589.1117.75 589.42 589.4217.875 589.7118 590 590 590 59018.125 590.2818.25 590.55 590.5518.375 590.8118.5 591.07 591.07 591.0718.625 591.3218.75 591.56 591.5618.875 591.7919 592.02 592.02 592.03 592.0319.125 592.2519.25 592.46 592.4619.375 592.6719.5 592.88 592.88 592.8819.625 593.0819.75 593.27 593.2719.875 593.4620 593.64 593.64 593.65 593.65 593.95 625.1 681.0920.125 593.82



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)20.25 594 59420.375 594.1620.5 594.33 594.33 594.3320.625 594.4920.75 594.64 594.6420.875 594.7921 594.94 594.94 594.94 594.9421.125 595.0821.25 595.22 595.2221.375 595.3521.5 595.48 595.48 595.4921.625 595.6121.75 595.73 595.7321.875 595.8522 595.97 595.97 595.97 595.9722.125 596.0822.25 596.19 596.1922.375 596.322.5 596.41 596.41 596.4122.625 596.5122.75 596.61 596.6122.875 596.723 596.79 596.79 596.8 596.7923.125 596.8823.25 596.97 596.9723.375 597.0623.5 597.14 597.14 597.1423.625 597.2223.75 597.3 597.323.875 597.3824 597.45 597.45 597.45 597.4524.125 597.5224.25 597.59 597.5924.375 597.6624.5 597.73 597.73 597.7324.625 597.7924.75 597.85 597.8524.875 597.9125 597.97 597.97 597.97 597.97 597.9925.125 598.0325.25 598.08 598.08



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)25.375 598.1425.5 598.19 598.19 598.1925.625 598.2425.75 598.29 598.2925.875 598.3426 598.39 598.39 598.39 598.3926.125 598.4326.25 598.48 598.4826.375 598.5226.5 598.56 598.56 598.5626.625 598.626.75 598.64 598.6426.875 598.6827 598.72 598.72 598.72 598.7227.125 598.7527.25 598.79 598.7927.375 598.8227.5 598.86 598.86 598.8627.625 598.8927.75 598.92 598.9227.875 598.9528 598.98 598.98 598.98 598.9828.125 599.0128.25 599.04 599.0428.375 599.0628.5 599.09 599.09 599.0928.625 599.1228.75 599.14 599.1428.875 599.1729 599.19 599.19 599.19 599.1929.125 599.2129.25 599.23 599.2329.375 599.2629.5 599.28 599.28 599.2829.625 599.329.75 599.32 599.3229.875 599.3430 599.36 599.36 599.36 599.36 599.33 625.130.125 599.3730.25 599.39 599.3930.375 599.41



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)30.5 599.43 599.43 599.4330.625 599.4430.75 599.46 599.4630.875 599.4731 599.49 599.49 599.49 599.4931.125 599.531.25 599.52 599.5231.375 599.5331.5 599.54 599.54 599.5431.625 599.5631.75 599.57 599.5731.875 599.5832 599.59 599.59 599.59 599.5932.125 599.632.25 599.62 599.6232.375 599.6332.5 599.64 599.64 599.6432.625 599.6532.75 599.66 599.6632.875 599.6733 599.68 599.68 599.68 599.6833.125 599.6933.25 599.69 599.6933.375 599.733.5 599.71 599.71 599.7133.625 599.7233.75 599.73 599.7333.875 599.7434 599.74 599.74 599.74 599.7434.125 599.7534.25 599.76 599.7634.375 599.7634.5 599.77 599.77 599.7734.625 599.7834.75 599.78 599.7834.875 599.7935 599.8 599.8 599.8 599.8 599.7835.125 599.835.25 599.81 599.8135.375 599.8135.5 599.82 599.82 599.82



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)35.625 599.8235.75 599.83 599.8335.875 599.8336 599.84 599.84 599.84 599.8436.125 599.8436.25 599.85 599.8536.375 599.8536.5 599.86 599.86 599.8636.625 599.8636.75 599.86 599.8636.875 599.8737 599.87 599.87 599.87 599.8737.125 599.8737.25 599.88 599.8837.375 599.8837.5 599.89 599.89 599.8937.625 599.8937.75 599.89 599.8937.875 599.8938 599.9 599.9 599.9 599.938.125 599.938.25 599.9 599.938.375 599.9138.5 599.91 599.91 599.9138.625 599.9138.75 599.91 599.9138.875 599.9239 599.92 599.92 599.92 599.9239.125 599.9239.25 599.92 599.9239.375 599.9339.5 599.93 599.93 599.9339.625 599.9339.75 599.93 599.9339.875 599.93Final 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

