# The Harvest Effect on Dynamics of Northern Fur Seal Population: Mathematical Modeling and Data Analysis Results

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## Abstract

**:**

## 1. Introduction

- Address the question, how did harvest methods affect the number and age structure of harvested seals over time?
- Analyze the dynamics of visual sex ratio or observed average harem size;
- Then, using a matrix population model for the observed part (males), we estimate the average growth rate and stationary age composition of this population part on the base of average estimates of birth and survival rates for periods of different dynamics. In order to account for annual changes in the survival and birth rates, we supplement the deterministic estimates of the population growth rate with stochastic ones;
- Assess females in the population by comparing the observational results of their numbers at the rookery with the past period for which information on their physiological state from marine samples is available. Analyze the dynamics of the sex ratio of that period and estimate this parameter in the late period using the known number of bulls, pups, and the proportion of pregnant females available from marine samples.

## 2. Materials and Methods

#### 2.1. Dynamics of Numbers and Ages of the Males Harvested on Land for Various Periods of Time

#### 2.2. Dynamics of the Average Harem Size and Average Number of PUPS per Bull (or the Reproductive Efficiency of Bull)

#### 2.3. Matrix Model of the Male Part of the Population

_{M}(n + 1) = A⋅S

_{M}(n),

**(n) = (M**

_{M}_{0}(n), M

_{1}(n), M

_{2}(n), M

_{3}(n), M

_{4}(n), M

_{5}(n) M

_{6}(n), M(n))

^{T}describes the age structure of the considered population part, where M

_{i}(n) is the number of males of age i in the nth year, the reproductive group or the bulls (M) is represented by males of age 7 years and older; w

_{ii+}

_{1}(n) is the survival rate of males being i years old in the nth year between i and i + 1 year of age; w(n) is the survival rate of bulls between nth and (n + 1)th years. We suppose M

_{0}(n) = P(n)/2; the validity of this equation is confirmed by many studies: the sex ratio of fur seal embryos almost equals 1:1 [19]. Although by birth, there may be a deviation from the 1:1 ratio for both ways in different populations [12,27,28], these deviations are insignificant. During the first year of life, the fur seal pup is known from the biology of this species to have the lowest survival rate. Then up to seven years, when males start breeding, it does not practically change; therefore

**,**we suggest w

**= w**

_{12}**= … w**

_{23}**= w**

_{67}**. Thus, we determine the survival rate of males from birth till one-year-old (w**

_{27}_{01}) using previously calculated parameters (w

_{02}and w

_{27}) as w

_{01}= w

_{02/}w

_{27}. Fertility B is half the bull reproductive efficiency: B = b/2, given the sexual ratio. A projection matrix A takes the following form:

#### 2.4. Effect of Changing Environments (Introducing Stochasticity to the Model)

#### 2.5. Characteristics of the Data of Females from Marine Samples (Obtained during Observations in 1958–1988): Dynamics of Pregnant Females’ Proportion and Sex Ratio

#### 2.6. Rough Estimates of Mature Female Numbers in the Late Period

#### 2.7. Partitioning All Observation Period on Several Parts Taking into Account Changes in Bull’s Reproductive Efficiency and the INTENSITY of the harvest

## 3. Results

#### 3.1. Dynamics of Numbers and Ages of the Animals Harvested on Land in Different Years

#### 3.2. Dynamics of the Observed Average Harem Size and the Reproductive Efficiency of Bulls

^{2}= 87.5%. Chow test showed that there was no change in the trend, i.e., although the size of the harems clearly decreased after 1987, the production remained at the same level.

#### 3.3. Changes in Population Growth Rate and Stable Age Structure, Results of Deterministic and Stochastic Modeling

_{e}in the late periods are far from the early ones, i.e., the expected rate of population growth has declined significantly lately.

#### 3.4. Actual vs. Observed Sex Ratio (Including the Calculated One)

_{obs =}F

_{obs}(n)/M(n) for 1980–2013, supplemented by calculated ones tF

_{obs}(n)/M(n) for 1958–1979 based on Equation (5). One can see the number of mature females in the population is always greater than that observed on the rookery, and the difference varies from year to year. The minimum value of ρ/ρ

_{obs}= 1.28 corresponds to 1987, the maximum ρ/ρ

_{obs}= 2.67 to 1972, and the average ρ/ρ

_{obs}= 1.96.

^{2}= 0.0506). Thus, the sex ratio did not contribute significantly to the proportion of pregnant females during the entire observation period. An increase in the average for this index seems to be related to the aging of the female population, which was shown in our previous paper [24], given older females are more successful in reproduction (Figure 10).

#### 3.5. Results of Calculation: Rough Estimates of Mature Females’ Numbers in the Late Period

_{min}), median F(λ

_{me}) and lower F(λ

_{max}) estimates of the total number of mature females in the population for each observation year. The dynamics of females’ number estimates from marine samples F(n) for 1958–1988 is also given. It falls within the given limits and gravitates toward the median estimate exceeding it somewhat in the early period and dropping slightly lower in the late period. Such a deviation is quite consistent with the dynamics of the average proportion of pregnant females, which tended to increase. In addition, Figure 11a shows the dynamics of bull abundance, which shows a marked increase in the late period of observations (after 1988). At the same time, if we assume the average proportion of pregnant females does not change, being at the level of the median λ

_{me}, then the females’ number has not increased by now and lies even under its average value of the early period. A slight increase in the females’ number could only occur if the proportion of pregnant females decreased to the lowest of the observed λ

_{min}.

_{me}level, we see only about nine females per bull, which is more than three times lower than the ratio of the initial period and more than two times lower than that of the period of females’ number depression (1974–1988). Even with the proportion of pregnant females reduced to the lowest of the observed values λ

_{min}, the change in the sex ratio remains significant: about 15:1.

_{me}, n) and that observed at the rookery falls within the range of the period for which the observations are available (Figure 11c). Assuming lower pregnant females’ proportion λ

_{min}gives a higher difference between the observed onshore and real sex ratio with the values of this indicator beyond the range of the previous period, which could occur due to females beginning to spend more time in the sea searching for food.

## 4. Discussion and Conclusions

^{2}= 87.5%. Using this correlation, we were able to substitute the known number of newborn pups to obtain theoretical values of the observed number of females (tF

_{obs}) for the early period and estimate the average harem size observed on the rookery (Figure 6). Our results show that the observed sizes of the harems in the early period were significantly greater than those in the later period, with their maximum values reaching almost 50, but still lower than at the beginning of the last century (60–100, according to [14,34]. Note that the early period is also heterogeneous in the estimated observed harem size: in a number of years, values are not different from those observed in 1980–1987, while in other years, there are significantly higher numbers of females per male than could be found in the set of field data. The observed sex ratio (F

_{obs}/M) has declined after 1987 to the level that does not overlap with that of recovered values for the period prior to 1980.

_{min}, the change in the sex ratio remains significant: about 15:1, and thus the number of females only slightly exceeds the average values of the initial period. This begs the question of how natural selection is acting on population processes such that population growth become so different in the male and female parts of the population?

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Review of Methods Used to Estimate Male Survival Rates

#### Appendix A.1. Iteration Method

_{02L}and w

_{02U}(Equations (A1) and (A2)) were developed in [23], which was based on Trites’ approach [61] without any assumptions about harvest. At that, the larger the number of harvested animals the narrower estimated intervals (w

_{02L}, w

_{02U}) for the value of w

_{02}. Given that the harvest level decreased significantly in the late period, the intervals became wide. To narrow them down, the iteration method uses estimates of subadults’ survival as additional information.

_{0}= P/2, P is the number of newborn pups; R

_{i}denotes the number of harvested males of age i (i = 2,…,6) from the considered generation.

_{02}(1958)), we use M

_{0}(1958), R

_{2}(1960), R

_{3}(1961) etc., and M

_{7}(1965).

_{02L}) the survival rates of older ages are assumed to be 1: w

_{U}= w

_{27U}= 1; then w

_{27}= 1 is substituted in Equation (A1), and the number of new bulls is calculated as M

_{7L}= M(n) − w

_{U}(M(n − 1) − R(n − 1)). Here n is the year when the considered generation became seven-year-olds; M(n) is the total number of bulls in the n-th year, and R(n − 1) denotes the number of harvested bulls in the previous year. Then, the estimated value of M

_{7L}was compared with the total harvest from the generation in subsequent years R

_{7+}(i.e., R

_{7+}= R

_{7}(n) + R

_{8}(n + 1) + …). If M

_{7L}< R

_{7+}, then we increase M

_{7L}to R

_{7+}, thus, avoiding obtaining fewer seven-year-old animals than were subsequently killed). Similar verification applies to the M

_{7U}.

_{27}) is replaced by their minimum possible values (w

_{27L}= (w

_{02U})

^{0.5}, w

_{L}= (w

_{02U})

^{0.5}), and the number of new bulls is calculated as M

_{7U}(n) = M(n) − (w

_{02U})

^{0.5}(M(n − 1) − R(n − 1)).

_{02}= w

_{01}·w

_{12}, and survival rate from birth to one year is a minimal one, we assume w

_{12}= w

_{27}. Then, w

_{02U}= w

_{01U}·w

_{27L}= (w

_{27L})

^{2}and therefore w

_{27L}= (w

_{02U})

^{0.5}. Similarly, we get the lower bound of the bulls’ survival: w

_{L}≥ w

_{01}⇒ w

_{L}= w

_{27L}= (w

_{02U})

^{0.5}.

_{27U}= 1, w

_{27L}= (w

_{02})

^{1/2}) in Equation (A1) to recalculate the bounds w

_{02L}and w

_{02U}. The maximum possible survival of bulls is considered the lower bound of the subadults’ survival: w

_{27L}= w

_{U}, and this restriction is substituted in Equation (A1) to refine the lower bound of the juvenile survival rate:

_{L}= w

_{01U}= w

_{02U}/w

_{27L}. Then, Equation (A2) takes the following form:

_{02U}:

_{02L(1)}and w

_{02U(1)}. Note that even the first step of such a procedure narrows the bounds (w

_{02L}, w

_{02U}) considerably. The next step uses the updated values of the juvenile survival rates w

_{02(1)}to recalculate the bounds of the subadults’ survival rates (w

_{27L}, w

_{27U}) and then using Formulas (A3) and (A5) obtain the next approximation of the values (w

_{02L(2)}, w

_{02U(2)}), and so on. After a few such iterations, the estimates (w

_{02L(i)}, w

_{02U(i)}) stabilize, with narrow intervals being obtained only for a few generations, as shown in Figure A1. The initial bounds of the juvenile survival rate (w

_{02L}, w

_{02U}) obtained by Formulas (A1) and (A2) stabilized bounds of both juvenile and the subadults’ survival rates and the number of iterations (i) required for their stabilization (with an accuracy of 0.0001).

**Figure A1.**Dynamics of the estimates of juvenile survival rate: initial (w

_{02L}, w

_{02U}) and refined (w

_{02L(i)}, w

_{02U(i)}) after i iterations (number of iterations on the right axis), as well as bounds of the subadults’ survival rate (w

_{27L(i)}, w

_{27U(i)}).

#### Appendix A.2. Fitting Method

_{7}) one can express in terms of survived two-year-old males (M

_{2}), the yearly survival rate w

_{27}and the number of harvested animals:

_{7}= (((((M

_{2}– R

_{2})w

_{27}– R

_{3})w

_{27}– R

_{4})w

_{27}– R

_{5})w

_{27}– R

_{6})w

_{27}.

_{7}), i.e., in the n-th year (the year of birth of the considering generation is n – 7), their number can be also expressed in terms of the number of bulls (M(n)), and their yearly survival rate w and the number of bulls in the previous year escaped harvest (M(n–1) – R(n–1)) is:

_{max}= min{(M(n) − R

_{7+})/(M(n − 1) − R(n − 1)), 1}, only correct values of M

_{7}can be obtained.

_{2}= M

_{0}·w

_{02}:

_{01}= w

_{02}/w

_{27}:

_{01}depends on w

_{27}and M

_{7}, and it is a function of two variables. When calculating, one has to hold the condition 0 < w

_{01}< w < w

_{27}< 1.

_{max}= min{(M(n) − R

_{7+})/(M(n − 1) − R(n − 1)), 1}, w

_{27}from w to 1 and comparing the obtained w

_{01}(Equation A9) with w (0 <w

_{01}≤ w). As w and w

_{27}increase, w

_{01}decreases, i.e., the maximum w

_{27}gives the minimum w

_{01}, and the minimum w

_{27}provides the maximum w

_{01}.

**Figure A2.**(

**a**)—Dynamics of the estimates of upper and lower bounds of the survival rate of males from birth to one year (w

_{01L}, w

_{01U}), subadults (w

_{27L}, w

_{27U}), and bulls (w

_{L}, w

_{U}). (

**b**)—Average values of the survival rates.

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**Figure 1.**Field count of bulls before the harvest and number of males (including bulls and subadults from 2 to 6 years old) taken in the commercial harvest (on the left axis), and pups born (on the right axis) on Tyuleniy Island.

**Figure 3.**Structure of harvest. (

**a**) The numbers of males harvested on land accordingly to their ages (R2–R6), bulls are included in the group of seven-year-olds and older (R7+). (

**b**) The share of bulls (by their ages) in the total number of killed bulls, as well as that of killed bulls from their total number (the line with markers, R7 + (n)/M(n)). (

**c**) The share of subadults killed of each age in the total number of subadults killed.

**Figure 4.**(

**a**) Average observed number of females per male (or average observed harem size, on the left axis) and observed number of females (on the right axis) at the rookery. (

**b**) Box plots of average observed harem size corresponding with two periods: 1980–1987 and 1988–2013.

**Figure 5.**(

**a**) Average number of pups per bull (P(n)/M(n − 1), or average reproductive efficiency, on the left axis, and the number of bulls at the rookery (M(n − 1)), on the right axis. (

**b**) Box plots of average reproductive efficiency corresponding to the periods: 1959–1973, 1974–1978, 1979–1988, and 1989–2013.

**Figure 6.**Average number of pups per bull (P(n + 1)/M(n) dependence on average observed harem size (F

_{obs}(n)/M(n)) (theoretical values of tF

_{obs}(n) are used for the period 1958–1979).

**Figure 7.**Stable age structure of the male part of the northern fur seal population (left axis) and annual growth rate of the population (on the right axis), corresponding to the periods: 1958–1971, 1972–1977, 1978–1987, and after 1987; here, (

**a**) corresponds to the parameter estimates obtained by the iteration and (

**b**) refers to the fitting method.

**Figure 8.**Expected rate of population growth λ

_{e}distributions in random samples (1000 iterations) for each period; here, (

**a**) corresponds to parameter estimates obtained by the iteration and (

**b**) refers to the fitting method.

**Figure 9.**(

**bottom**) Dynamics of actual sex ratio ρ = F(n)/M(n) for the period 1958–1988, visual sex ratio ρ

_{obs =}F

_{obs}(n)/M(n) for 1980–2013, supplemented by calculated ones tF

_{obs}(n)/M(n) for 1958–1979 based on Equation (5) (

**on the right axis**); and ρ/ρ

_{obs}(

**on the left axis**). (

**top**) Dynamics of pregnant females’ proportion in the next year λ(n + 1) (

**on the left**), and a box plot of λ(n + 1) for the periods corresponding to different reproductive efficiency of bulls (

**on the right**).

**Figure 11.**(

**a**) Dynamics of the upper F(λ

_{min}), median F(λ

_{me}) and lower F(λ

_{max}) estimates of the total number of mature females in the population for each observation year, Equation (3); dynamics of females’ number estimates from marine samples F(n) for 1958–1988 (on the left axis), and the number of bulls at the rookery M(n) (on the right axis). (

**b**) Dynamics of sex ratio F(n)/M(n) according to observations for the period 1958–1988 and sex ratios corresponding to the upper, median, and lower estimates of the females’ number for the entire period. (

**c**) The difference dynamics between calculated sex ratios and that observed at the rookery.

**Table 1.**Model coefficient values (adapted from [25]), annual growth rate of the population (λ) and its stable age structure (x*).

Fitting Method | ||||||||

Years | w_{27} | w_{01} | w | b | λ | |||

1958–1971 | 0.87 | 0.45 | 0.72 | 10.43 | 1.22 | |||

1972–1977 | 0.84 | 0.36 | 0.59 | 22.40 | 1.24 | |||

1978–1987 | 0.85 | 0.40 | 0.61 | 8.12 | 1.13 | |||

after 1987 | 0.88 | 0.42 | 0.69 | 3.85 | 1.09 | |||

Perron Vector (x*) | ||||||||

1958–1971 | 0.45 | 0.17 | 0.12 | 0.08 | 0.06 | 0.04 | 0.03 | 0.05 |

1972–1977 | 0.53 | 0.16 | 0.11 | 0.07 | 0.05 | 0.03 | 0.02 | 0.03 |

1978–1987 | 0.43 | 0.15 | 0.12 | 0.09 | 0.07 | 0.05 | 0.04 | 0.06 |

after 1987 | 0.37 | 0.14 | 0.11 | 0.09 | 0.07 | 0.06 | 0.05 | 0.10 |

Iteration Method | ||||||||

Years | w_{27} | w_{01} | w | b | λ | |||

1958–1971 | 0.83 | 0.51 | 0.72 | 10.43 | 1.20 | |||

1972–1977 | 0.79 | 0.45 | 0.59 | 22.40 | 1.22 | |||

1978–1987 | 0.81 | 0.47 | 0.62 | 8.12 | 1.12 | |||

after 1987 | 0.84 | 0.49 | 0.70 | 3.85 | 1.08 | |||

Perron Vector (x*) | ||||||||

1958–1971 | 0.43 | 0.18 | 0.12 | 0.09 | 0.06 | 0.04 | 0.03 | 0.05 |

1972–1977 | 0.49 | 0.18 | 0.12 | 0.08 | 0.05 | 0.03 | 0.02 | 0.03 |

1978–1987 | 0.41 | 0.17 | 0.12 | 0.09 | 0.07 | 0.05 | 0.03 | 0.06 |

after 1987 | 0.35 | 0.16 | 0.12 | 0.10 | 0.07 | 0.06 | 0.04 | 0.10 |

Years | Min(F(n)/M(n)) | Me(F(n)/M(n)) | Max(F(n)/M(n)) |
---|---|---|---|

1958–1972 | 18.14580538 | 31.45767579 | 71.41407582 |

1973–1977 | 45.20987382 | 70.09244868 | 122.1069182 |

1978–1988 | 14.18993578 | 20.01973463 | 31.46966894 |

Min(F(n)/M(n)) | Me(F(n)/M(n)) | Max(F(n)/M(n)) | |

1989–2013 | 11.78511348 | 14.95958214 | 24.06312045 |

Min(F(λ_{me})/M) | Me(F(λ_{me})/M) | Max(F(λ_{me})/M) | |

1989–2013 | 7.31092815 | 9.280218674 | 14.92762416 |

Min(F(λ_{max})/M) | Me(F(λ_{max})/M) | Max(F(λ_{max})/M) | |

1989–2013 | 6.243618746 | 7.925416047 | 12.74836685 |

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## Share and Cite

**MDPI and ACS Style**

Zhdanova, O.; Kuzin, A.; Frisman, E.
The Harvest Effect on Dynamics of Northern Fur Seal Population: Mathematical Modeling and Data Analysis Results. *Mathematics* **2022**, *10*, 3067.
https://doi.org/10.3390/math10173067

**AMA Style**

Zhdanova O, Kuzin A, Frisman E.
The Harvest Effect on Dynamics of Northern Fur Seal Population: Mathematical Modeling and Data Analysis Results. *Mathematics*. 2022; 10(17):3067.
https://doi.org/10.3390/math10173067

**Chicago/Turabian Style**

Zhdanova, Oksana, Alexey Kuzin, and Efim Frisman.
2022. "The Harvest Effect on Dynamics of Northern Fur Seal Population: Mathematical Modeling and Data Analysis Results" *Mathematics* 10, no. 17: 3067.
https://doi.org/10.3390/math10173067