Appendix B—Specific Conservation and Recovery Actions Considered 4. Evaluate the effects of concentrated attractants (e.g., dumps) and supplemental feeding (e.g., remains of subsistence-harvested whales) on polar bear distribution, habitat use, nutritional status, and human-bear interactions. 5. Expand non-invasive genetic sampling around seasonally abundant, concentrated food sources (e.g., bone piles). 96 <strong>Polar</strong> <strong>Bear</strong> Conservation Management Plan
Appendix C—Population Dynamics and Harvest Management APPENDIX C—POPULATION DYNAMICS AND HARVEST MANAGEMENT The harvest strategy described in the Conservation Management Plan is founded on an extensive literature on harvest theory (Wade 1998, Runge et al. 2009) and a detailed population model for polar bears (Regehr et al. 2015, Regehr et al. in press), and is customized to reflect the cultural practices of Alaska Native people and the principles of the Marine Mammal Protection Act. This Appendix describes the underlying harvest theory and technical details of the harvest strategy. Harvest Theory Sustained removal of animals from a population, whether for subsistence harvest, sport harvest, incidental take, or population control, is possible because of density-dependent feedback mechanisms. A reduction in the population size via removals can—through any of a number of processes—free up resources (food, space, breeding territory, etc.) for the remaining individuals, increasing their survival rates, reproductive rates, or both. The increase in the demographic rates provides a surplus of individuals relative to what is needed to maintain the population at a constant size. This surplus can be sustainably removed, as long as a number of conditions are met. The simplest model that can capture these population dynamics is the discrete logistic model, which describes the trajectory of a population using the formula, NNNN tttt+1 = NNNN tttt + rrrrNNNN tttt 1 − NNNN tttt KKKK − hNNNN tttt (C1) where N t is the population size at time t, r is the intrinsic rate of growth of the population, K is the carrying capacity, and h is the rate of removal (the harvest rate). If such a population is subjected to a fixed rate of removal for some period of time, it will eventually settle to an equilibrium population size NNNN eeeeeeee = KKKK 1 − h rrrr (C2) that allows a sustained annual removal (annual harvest) of a number of individuals as calculated by the formula: HHHH eeeeeeee = hNNNN eeeeeeee = hKKKK 1 − h rrrr = hKKKK − h2 KKKK rrrr . Plotting H eq against N eq depicts a “yield curve” (Fig. C-1). For a given harvest rate, h, there is a (C3) corresponding equilibrium population size and annual harvest. The yield curve traces the combinations of N eq and H eq : when h = 0, N eq = K and the annual harvest is, of course, 0. As h increases, N eq decreases and H eq increases until a maximum sustainable annual harvest is reached. (For the discrete logistic model, this maximum occurs at h = r/2, N eq = K/2, and H eq = rK/4.) The harvest rate can continue to increase, pushing the equilibrium point over to the left side of the yield curve; now N eq continues to decrease, but so does H eq . When h ≥ r, N eq = 0, that is, the harvest rate is greater than the fastest-possible population growth rate, and the only resulting equilibrium condition is extirpation of the population. Annual Harvest (H) 0 0 N eq K Population Size (N) Figure C-1. The yield curve for a population that is described by a discrete logistic population model. If the annual harvest is a fixed fraction, h, of the population size (black line), the population size will converge to a stable equilibrium point. When the harvest rate is fixed, the equilibrium point is a stable attractor for the population dynamics: if the population size is lower than N eq , the surplus production (as indicated by the yield curve) will be greater than the annual harvest (which occurs on the thin black line), and the population will increase back toward N eq ; if the population size is greater than N eq , the annual harvest (on the black line) will exceed the surplus production, and the population will decrease back toward N eq . The stability of the equilibrium point is the reason that using a sustainable, fixed harvest rate is a robust strategy—the population dynamics are self-correcting in the face of stochastic fluctuations. The critical thing to note, however, is that this stability works for a fixed harvest rate (i.e., percentage of current population size), not a fixed harvest quota (i.e., a fixed number of individuals removed each year, regardless of changes in population size). To achieve a fixed harvest rate, the harvest quota needs to be able to change in response to changes in the population size on a regular basis. h <strong>Polar</strong> <strong>Bear</strong> Conservation Management Plan 97
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