SIMULATION MODELING OF BLACK BEAR HARVEST STRATEGIES
GARY C. WHITE, Department of Fishery and Wildlife Biology, Colorado State University, Fort Collins, CO 80523
R. BRUCE GILL, Research Section, Colorado Division of Wildlife, 317 West Prospect, Fort Collins, CO 80526
THOMAS D. I. BECK, Research Section, Colorado Division of Wildlife, 317 West Prospect, Fort Collins, CO 80526
Abstract: Harvest strategies for a south-central Colorado black bear (Ursus americanus) population (Beck 1991) were evaluated with a Monte Carlo simulation model. The model included binomial variation for birth and death rates, plus two types of annual environmental variability: 1 year in 10 heavy harvest because of optimal hunting conditions, and 1 year in 10 failure of the berry crop, causing increased natural mortality and decreased reproduction. No density-dependent responses were included. Six harvest scenarios were simulated: no harvest, 2 spring seasons with 30 and 35% of the harvest consisting of females, and 3 fall seasons with 35, 40, and 45% of the harvest consisting of females. Harvest rates were 5, 10, 15, and 20% of bears >2 year old. Because spring harvest is disproportionately males, our simulations suggest more bears can be harvested during spring seasons. Without density-dependent compensation in the population, ~15% of the population >2 years old can be removed annually based on observed reproductive and survival rates, and the population would not decline. Increased environmental variation caused by berry crop failures and occasional heavy harvest can significantly reduce population growth rate, and thus affect long-term management.
Key Words: Binomial distribution, environmental variation, Monte Carlo simulations, population growth rate, spring harvest, temporal variation, Ursus americanus.
No where in their range can black bears be considered numerous. The species evolved as long-lived with exceptionally low reproductive rates and low natural mortality rates. Unlike many other species hunted for sport, rate of population increase for black bear populations is low. In addition, wildlife managers have had difficulty in developing harvest strategies for black bear populations because populations are difficult to monitor. Population estimation methods are expensive, and often ineffective, due to the secretive nature of the animal. Individuals are difficult to capture, and thus to obtain an adequate sample of marked animals for mark-recapture methods. Further, revenue from the sale of licenses is not adequate to justify extensive management. As of a result of these limitations, little research has been conducted on harvest strategies. Public controversies have revolved around when to hunt, how to hunt, how many bears should be harvested, how many hunters should be allowed, how nuisance bears should be controlled, and how illegal hunting should be eliminated.
We developed a Monte Carlo computer simulation model of a Colorado black bear population to evaluate possible harvest strategies. The stochastic model is patterned after the grizzly bear (U. arctos horribilis) model of Knight and Eberhardt (1985). Structure of their model was modified to be appropriate for Colorado black bears, with parameter estimates taken from research of Beck (1991) in south-central Colorado.
Six harvest regimes were simulated: no harvest, 2 spring seasons with 30 and 35% of the harvest consisting of females, and 3 fall seasons with 35, 40, and 45% of the harvest consisting of females. In addition, two types of additional environmental variability were considered: occasional heavy harvest because of optimal hunting conditions, and occasional failure of the berry crop, causing increased natural mortality and decreased reproduction.
Acknowledgments. Funding was provided by Colorado Division of Wildlife, Federal Aid in Wildlife Restoration Project W-153-R.
METHODS
The black bear population model (available from the senior author) was developed using the Statistical Analysis System (SAS 1985). Structure of the model is based on 3 arrays. The subadult array holds numbers of cubs and yearlings of both sexes whose mothers have died. The male array holds numbers of males of ages 2 to 20 (the maximum age attainable). The 2-dimensional female array holds numbers of females of ages 2 to 20, and also their reproductive status (current number of offspring). Reproductive status 0 means no cubs or yearlings, 1 means 1 cub, 2 means 2 cubs, 3 means 3 cubs, 4 means 1 yearling, 5 means 2 yearlings, and 6 means 3 yearlings. Although the yearlings are not actually with the mother, the mother is not allowed to breed in 2 consecutive years (unless the cubs are lost). Thus categories 4-6 provide a mechanism to store females for one year until they are ready to breed again. Parameter values for survival and reproduction rates (from Beck 1990) are presented in Tables 1 and 2. These rates represent a non-hunted population, but with some illegal harvest plus some emigration to areas where harvest is allowed.
Table 1. Survival rates for the various age and sex classes in the black bear simulation model, derived from Beck (1991). Only cub survival was affected by annual variation in berry crops. All other age and sex classes were modeled as shown, regardless of variation in berry crops.
| Age Class | Males | Females |
| All years, regardless of berry crops | ||
| Adults, > 4 years old | 0.70 | 0.96 |
| 4-year olds | 0.70 | 0.96 |
| 3-year olds | 0.76 | 0.94 |
| 2-year olds | 0.76 | 0.94 |
| Yearlings (with mother) | 0.94 | 0.94 |
| Yearlings (without mother) | 0.94 | 0.94 |
| Constant environment — no variation in berry crops | ||
| Cubs (with mother) | 0.56 | 0.56 |
| Cubs (without mother) | 0.56 | 0.56 |
| Variable environment — good berry crop years | ||
| Cubs (with mother) | 0.56 | 0.56 |
| Cubs (without mother) | 0.56 | 0.56 |
| Variable environment — bad berry crop years | ||
| Cubs (with mother) | 0.33 | 0.33 |
| Cubs (without mother) | 0.33 | 0.33 |
Table 2. Reproductive rates for females in the black bear simulation model, derived from Beck (1991). For simulations with no variation in years, the parameters listed for good berry years were used.
| Parameter Definition | Good Berry Years | Bad Berry Years |
| Prop. of 2 year olds producing first cubs | 0.001 | 0.001 |
| Prop. of virgin 3 year olds producing first cubs | 0.22 | 0.001 |
| Prop. of virgin 4 year olds producing first cubs | 0.29 | 0.001 |
| Prop. of virgin 5 year olds producing first cubs | 0.60 | 0.001 |
| Prop. of virgin 6+ year olds producing first cubs | 0.50 | 0.20 |
| Prop. of barrena females having cubs | 0.80 | 0.20 |
| Prop. of litters with 1 cub | 0.20 | 0.20 |
| Prop. of litters with 2 cubs | 0.60 | 0.60 |
| Prop. of litters with 3 cubs | 0.20 | 0.20 |
| Sex ratio at time of 2nd birthday | 50:50 | 50:50 |
a Barren means cubs born more than 1 year ago, or cubs lost during first year so that female actually breeds in 2 consecutive years.
Stochasticity in the model is in the form of binomial processes for births and deaths. For example, a 71% survival rate for adult males does not multiply the current population size by 0.71 to obtain the number of survivors. Rather, the N animals in the adult male population are assigned a fate (lived or died) according to the probability 0.71, using the RANBIN function of SAS. This binomial variation increases the reality of model simulations.
Two extensions to the grizzly bear model developed by Knight and Eberhardt (1985) were made. Beck (1991) showed that survival of cubs and yearlings and reproductive rates are affected by berry crop. Late spring frosts curtail mast and berry production. Therefore, some simulations incorporated 2 sets of values for birth and death rates, depending on whether the current year was a “good” berry crop or a “bad” berry crop. Bad berry crops occurred at random with a frequency of 1 year out of 10 using the RANUNI function (SAS 1985).
In addition to effects from bad berry years shown in Tables 1 and 2, all cubs born during bad years had reproductive rates as if it were a bad year until they were 6 years old. After a bad berry year, all barren adults had 0.90 reproduction instead of 0.80, compensation for the previous bad year (unless the second year was also a bad berry year).
The second modification of the model by Eberhardt and Knight was to include harvest. A major consideration of the various experimental harvests was the proportion of males versus females in the harvest. The proportion of females in the harvest can be affected by the timing of the season, plus the method of take (Gill and Beck 1990). Six harvest regimes were simulated: no harvest; spring seasons with the expected harvest consisting of 30 (labeled S30) and 35% (labeled S35) females; and fall seasons with the expected harvest consisting of 35 (labeled F35), 40 (labeled F40), and 45% (labeled F45) females. Because of the differential survival of males and females, an unhunted population would consist of 72% females for animals >2 years old. Therefore, experimental harvests incorporated differential vulnerability of females and males. Harvest vulnerability was determined based on the expected harvest and the relative proportion of the population consisting of males and females > 2 years old. For example, to obtain a harvest with 30% females, the proportion of females to harvest is equal to the expected number of females in the harvest divided by the number of females in the population, and similarly for males. The only exception to this procedure was when the proportion of the population segment to be harvested exceeded 90%, the maximum allowable harvest was set to 90%. Actual number of animals harvested was determined stochastically with the RANBIN function (SAS 1985). Cubs of any females harvested in a spring season (April-June) are all assumed to die, whereas cubs of females harvested in the fall (September-October) are assumed to survive at the rate of cubs without mothers. In addition, an increase in vulnerability to harvest of the entire population was included in the model using the RANUNI function (SAS 1985). On average, 1 year out of 10 had a double success rate (e.g., 20% harvest instead of 10%). This doubling of harvest was included in the model to represent the effects of variable weather on harvest (Gill and Beck 1990).
Initial population sizes were constructed by making 100 runs of the model for 30 years with no harvest, and using the resulting relative proportions of the various age and sex classes. Values for each of the age and sex classes are initialized the same in each simulation. A summary of these initial conditions are: cubs (both sexes) – 2499, yearlings (both sexes) – 1296, adult males – 1740, and adult females – 4465, for a total population at time 0 (N0) of 10,000. These initial conditions represent a large population, approximately the state-wide population of Colorado (Beck 1991).
All simulations were performed for 30 years, i.e., the population was projected forward for 30 years. This process was replicated 100 times to provide a mean and standard deviation for each of the parameters measured on the simulated population. Here, we report the population level at the end of the 30-year simulation (N30), plus annual rate of increase r, where r = (N30/N0)(1/30) – 1. All simulations were repeated for 100 times to estimate means and SDs.
Statistical tests between pairs of treatments were performed with a z-test using a pooled variance computed as s2 = [(n1 – 1)s12 + (n2 – 1)s22]/(n1 + n2 – 2).
RESULTS
Annual rate of increase for simulations with constant annual survival and reproduction, berry crop failures 1 year in 10, double the normal harvest 1 year in 10, and both berry crop failures 1 year in 10 and double harvest 1 year in 10 are presented in Fig. 1. Harvest rate is 10%. Effect of berry crop failures on the model population was to increase variability of population size after 30 years and to lower expected N30. As an example, 35% females in the spring harvest under no berry failures had a 3.7% annual increase, whereas with berry failures, the annual growth rate was 2.8% (P < 0.001). Under the fall harvest scenario with 35% females, no berry failures generated a population growth rate of 4.2%, berry failures 3.2% (P < 0.001). SD’s of r for the no annual variation simulations were 0.085, 0.071, 0.083, 0.089, and 0.095, for S30, S35, F35, F40, and F45 seasons, respectively. Corresponding SD’s for the berry failure simulations were 0.540, 0.571, 0.664, 0.592, and 0.614.
A 10% chance of double harvest reduces (P < 0.001) the rate of population growth compared to where harvest is always the same from year to year. Another impact of the occasional double harvest is an increase in variability of the final populations. For example, SD’s of r for the occasional double harvest simulations were 0.188, 0.236, 0.220, 0.245, and 0.274 for S30, S35, F35, F40, and F45 seasons, respectively. These values are intermediate between the no annual variation and occasional berry failure simulations presented above.
Impact of both berry crop failures and occasional double harvest is to further increase variation, and to further lower the average annual rate of increase. However, impacts of increased annual variation does not change the relative impact of the 5 seasons (Fig. 1). The ordering of the annual rate of increase for S30 through F45 is the same for all 4 scenarios shown in Fig. 1, and absolute differences appear to be constant across the 4 scenarios.
For reproductive and survival rates measured in south-central Colorado, approximately 15% of the population can be harvested annually if no density-dependent compensation is assumed (Fig. 2). Occasional double harvest and berry crop failures are included in the simulations in Fig. 2. This annual variation would decrease the proportion of the population that can be harvested annually. Under constant annual conditions, the proportion of the population that could be harvested would be >15%.
DISCUSSION
Additional stochasticity included in the model by incorporating survival and reproduction as binomial processes makes the results more realistic for small populations. Typically black bear populations are small in a numerical sense, even though large areas may be inhabited. Chance variation should be included in a realistic model. Additional variation from berry crop failures and occasional double harvest also makes the model’s predictions more realistic. Also, this annual variation makes the model more conservative in its predictions of the permissible annual harvest.
Responses of black bear populations to hunting removals are not well known. Some studies suggest that black bear populations respond to hunting removals with increased survival of subadults and juveniles (??? 19??). Others suggest that increased survival occurs only among subadult and juvenile males (??? 19??). Still others suggest that populations do not respond with increased survival at all, rather dispersing subadults colonize home ranges vacated by the harvest removals of resident bears (??? 19??). This controversy if more than academic because allowable harvests depend upon the degree of compensation.
Simulations reported here portray harvest as additive mortality, with no compensation assumed to result from removal of harvested animals. Thus, our results magnify effects of harvest — any compensation would reduce differences between no harvest and harvest scenarios. We chose to not simulate a compensatory mechanism because adequate data are lacking to perform a realistic simulation. We know a priori that any compensation occurring in the population dynamics of black bears means that a greater harvest can be exerted, and less effect observed in the resulting population levels. Hence, the question of compensation is of interest only to the degree that compensation results. However, no reliable data exist to justify a level of compensation for our simulations. Thus, any results pertaining to compensation would only reflect our potentially unrealistic assumptions, and not provide insight into bear population dynamics. Our results provide the maximum effect expected from the harvest scenarios we simulated. If compensation occurs, then the effect of the harvest will be less than portrayed here.
MANAGEMENT IMPLICATIONS
The 15% harvest rate estimated with this model is too liberal if we consider that some illegal kill takes place, plus loss of depredating bears from the population. This harvest rate represents the total allowable removals from the population, because our simulation model did not include illegal harvest or removal of problem bears. Thus, to maintain existing population levels, we recommend that black bear populations similar to the one simulated should be harvested at a rate <15% to permit additional removals from illegal harvest and problem bears.
LITERATURE CITED
Beck, T. D. I. 1991. Black bears in west-central Colorado. Tech. Publ. 39, DOW-R-T-39-91. Colorado Div. of Wildl., Fort Collins.
Gill, R. B., and T. D. I. Beck. 1990. Black bear management plan. Colorado Div. Wildl. Rep. No. 14, Fort Collins. 44pp.
Knight, R. R. and L. L. Eberhardt. 1985. Population dynamics of Yellowstone grizzly bears. Ecology 66:323-334.
SAS Institute Inc. 1985. SAS Language Guide for Personal Computers, Version 6 Edition. SAS Institute Inc., Cary, NC. 429pp.