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Appendix C—Population Dynamics and Harvest Management The population dynamics described above and the stability of the fixed-rate removal strategy depend on a few assumptions: first, that the environment remains constant on average (the mean r and mean K do not change); second, that changes in N can be monitored without bias and used to adjust the number of animals removed; third, that there are no Allee effects, or at least that the population never drops low enough that they are realized; and fourth, that eq. C1 adequately describes the dynamics. Note that although the model in eq. C1 looks deterministic, it can also serve as the central tendency of a stochastic model, and under reasonable assumptions about the nature of the annual variation, the fixedrate removal strategy will still robustly maintain a stochastic population near an equilibrium point, provided there is not also some change in the environment. The effect of environmental change on the yield curve What happens to the yield curve (and the harvestable surplus) if the environment changes? It’s not easy to see the answer to this question in the formulas for the discrete logistic model, because the underlying density-dependent processes are not explicitly written out in the formula. Instead, consider the following population model, NNNN tttt+1 = φφφφNNNN tttt (1 + RRRR tttt ) − hNNNN tttt (C4) where φ is the survival rate and R is the recruitment rate (the number of offspring produced per adult). Let’s assume that the survival rate, φ, is density independent (this can be a reasonable simplifying assumption for the adults of a large, long-lived mammal species). But let recruitment be density dependent and given by the linear function RRRR tttt = aaaa + bbbbNNNN tttt (C5) where a is the reproductive rate at very low densities (when there is no competition for resources), and b < 0 describes how much recruitment decreases for each unit increase in the population size. Then, substituting eq. C5 into eq. C4, NNNN tttt+1 = φφφφNNNN tttt (1 + aaaa + bbbbNNNN tttt ) − hNNNN tttt = φφφφ(1 + aaaa)NNNN tttt + φφφφbbbbNNNN tttt 2 − hNNNN tttt (C6) Now, by comparing eq. C6 to eq. C1 and making the following substitutions, φφφφ(1 + aaaa) = 1 + rrrr, and φφφφbbbb = −rrrr⁄ KKKK (C7) we calculate NNNN tttt+1 = (1 + rrrr)NNNN tttt − rrrrNNNN tttt 2 KKKK − hNNNN tttt = NNNN tttt + rrrrNNNN tttt 1 − NNNN tttt KKKK − hNNNN tttt (C8) which is identical to eq. C1. Thus, this new model, built from the underlying density-dependent relationships, is identical to the discrete logistic model. What’s helpful about this is that we can use the substitutions in eq. C7 to solve for the parameters of the discrete logistic (r and K) in terms of the parameters in the density-dependent formulation (a, b, and φ), rrrr = φφφφ(1 + aaaa) − 1 KKKK = 1 − φφφφ(1 + aaaa) φφφφbbbb (C9) A graphical depiction of the model in eq. C4 gives an intuitive sense of why it has the same behavior as the model in eq. C1 (Fig. C-2). In a population that is below its carrying capacity, the reproductive rate will exceed the level needed to offset mortality, so the population will grow. As the population grows, competition for resources increases, and the reproductive rate decreases. When the reproductive rate matches the mortality rate, there is a stable equilibrium point (K 0 ). Suppose now that there is a change in the environment such that the extent of habitat decreases, but the habitat that remains is the same quality as before. In this case, we could reason that b will decrease, because the competition for resources will be felt sooner as the population grows; but a will stay the same, because the reproductive rate at very low density would remain unchanged (Fig. C-2, top panel). The equilibrium point at which reproduction offsets mortality decreases (K 1 ). Another way in which the environment could change is that the extent of habitat doesn’t change, but the quality of it decreases. In this case, we might surmise that the reproductive rate decreases equally for all densities (Fig. C-2, bottom panel), thus a decreases, but b stays the same. Again, the equilibrium point at which reproduction offsets mortality decreases (K 1 ). Although the effect of these two types of environmental change on the carrying capacity looks similar, the effect on the yield curve is profoundly different (Fig. C-3). In the case of the effect of habitat quantity, b changes but not a; looking at eq. C9, this means that the carrying capacity (K) changes, but the intrinsic rate of growth (r) does not. Thus, the yield curve shrinks to the new carrying capacity, but does not change its proportions (Fig. 98 Polar Bear Conservation Management Plan
Appendix C—Population Dynamics and Harvest Management Demographic Rate 0.4 0.3 0.2 0.1 Original reproductive rate Loss of habitat quantity Loss of habitat quality Mortality rate Annual Harvest Original reproductive rate Loss of habitat quantity Loss of habitat quality 0.0 0 K 1 K 0 Population Size 0.4 0 0 K 1 K 0 Population Size Demographic Rate 0.3 0.2 0.1 0.0 0 K 1 K 0 Population Size Figure C-2. Density-dependent demographic rates, carrying capacity, and environmental change. As the population size increases, the density-dependent reproductive rate decreases until is just matches the mortality rate; at this point, additions and subtractions from the population are equal and the population size is stable (K 0 ). Changes in the environment that affect the reproductive rate (here we assume a negative effect) can shift the carrying capacity to a new level (K 1 ), either through an effect on habitat quantity (top panel) or through an effect on habitat quality (bottom panel). C-3, aqua curve). On the other hand, in the case of the effect of habitat quality, the change in a affects both K and r; the yield curve becomes flatter as well as smaller (Fig. C-3, purple curve). These two changes will affect harvest management differently. In the case of the habitat quantity change (change in b only), the fixed harvest rate strategy (using the desired harvest rate from before the change) will still work, and will maintain the population at the same proportion of its carrying capacity as it did previously, because only K (but not r) is affected. In the case of the habitat quality change (change in a), the fixed harvest rate strategy that worked before the environmental change will no longer hold the population at the same proportion of K and might not even be sustainable. Thus, the demographic mechanism of environmental change matters to the management of harvest. Figure C-3. The effect of environmental change on the yield curve, for a discrete logistic population model. A note about the intrinsic rate of growth In the models described above, the intrinsic rate of growth (r) is the growth rate the population would have if its density were close to 0; that is, it is a descriptor of the underlying dynamics of a particular population in a habitat of a particular quality. Further, eq. C9 shows that the intrinsic rate of growth for a particular population could change, if the survival rate or the reproductive rate at low density changed as a result of changes in the environment. Thus, the way we are using the term, the intrinsic rate of growth is not a property of the species as a whole (i.e., it is not the theoretical maximum growth rate that the species could experience under the best possible conditions). Maximum Net Productivity Level The phrase “maximum net productivity level” arises from language in the MMPA, but invokes the population theory described by yield curves. The maximum net productivity is “the greatest net annual increment in population numbers or biomass resulting from additions to the population due to reproduction and/or growth less losses due to natural mortality” (50 CFR 403.02); this annual increment corresponds to the surplus production that allows annual harvest, thus, the maximum net productivity is the value on the y-axis that corresponds to the peak of the yield curve (Fig. C-1). The maximum net productivity level, then, is the population size (the value on the x-axis) that corresponds to this peak. The harvest theory derivations shown above demonstrate that the equilibrium population size that produces the greatest sustainable annual harvest will change if the underlying demographic dynamics change. The policy distinction between MNPL (referenced to a historic value K 0 that could potentially be reduced by habitat effects) and mnpl Polar Bear Conservation Management Plan 99
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U.S. Fish and Wildlife Service Pola
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Polar Bear Conservation Management
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Contents Executive Summary.........
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Executive Summary Today, polar bear
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and connectivity in each recovery u
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I. Background I. BACKGROUND Polar b
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I. Background the extent and charac
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II. Conservation Strategy II. CONSE
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III. Management Goals and Criteria
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B. Conservation Criteria under the
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IV. Conservation Management Strateg
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Page 55 and 56: V. Literature Cited V. LITERATURE C
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Page 59 and 60: V. Literature Cited Wade, P.R. 1998
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Page 63 and 64: Appendix A—Background APPENDIX A
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