Polar Bear
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Appendix C—Population Dynamics and Harvest Management<br />
The population dynamics described above and the<br />
stability of the fixed-rate removal strategy depend<br />
on a few assumptions: first, that the environment<br />
remains constant on average (the mean r and mean<br />
K do not change); second, that changes in N can<br />
be monitored without bias and used to adjust the<br />
number of animals removed; third, that there are no<br />
Allee effects, or at least that the population never<br />
drops low enough that they are realized; and fourth,<br />
that eq. C1 adequately describes the dynamics. Note<br />
that although the model in eq. C1 looks deterministic,<br />
it can also serve as the central tendency of a<br />
stochastic model, and under reasonable assumptions<br />
about the nature of the annual variation, the fixedrate<br />
removal strategy will still robustly maintain<br />
a stochastic population near an equilibrium point,<br />
provided there is not also some change in the<br />
environment.<br />
The effect of environmental change on the yield<br />
curve<br />
What happens to the yield curve (and the harvestable<br />
surplus) if the environment changes? It’s<br />
not easy to see the answer to this question in the<br />
formulas for the discrete logistic model, because<br />
the underlying density-dependent processes are<br />
not explicitly written out in the formula. Instead,<br />
consider the following population model,<br />
NNNN tttt+1 = φφφφNNNN tttt (1 + RRRR tttt ) − hNNNN tttt<br />
(C4)<br />
where φ is the survival rate and R is the recruitment<br />
rate (the number of offspring produced per adult).<br />
Let’s assume that the survival rate, φ, is density<br />
independent (this can be a reasonable simplifying<br />
assumption for the adults of a large, long-lived<br />
mammal species). But let recruitment be density<br />
dependent and given by the linear function<br />
RRRR tttt = aaaa + bbbbNNNN tttt<br />
(C5)<br />
where a is the reproductive rate at very low densities<br />
(when there is no competition for resources),<br />
and b < 0 describes how much recruitment<br />
decreases for each unit increase in the population<br />
size. Then, substituting eq. C5 into eq. C4,<br />
NNNN tttt+1 = φφφφNNNN tttt (1 + aaaa + bbbbNNNN tttt ) − hNNNN tttt =<br />
φφφφ(1 + aaaa)NNNN tttt + φφφφbbbbNNNN tttt 2 − hNNNN tttt<br />
(C6)<br />
Now, by comparing eq. C6 to eq. C1 and making the<br />
following substitutions,<br />
φφφφ(1 + aaaa) = 1 + rrrr, and φφφφbbbb = −rrrr⁄ KKKK (C7)<br />
we calculate<br />
NNNN tttt+1 = (1 + rrrr)NNNN tttt − rrrrNNNN tttt 2<br />
KKKK − hNNNN tttt =<br />
NNNN tttt + rrrrNNNN tttt 1 − NNNN tttt<br />
KKKK − hNNNN tttt<br />
(C8)<br />
which is identical to eq. C1. Thus, this new model,<br />
built from the underlying density-dependent<br />
relationships, is identical to the discrete logistic<br />
model. What’s helpful about this is that we can use<br />
the substitutions in eq. C7 to solve for the parameters<br />
of the discrete logistic (r and K) in terms of the<br />
parameters in the density-dependent formulation (a,<br />
b, and φ),<br />
rrrr = φφφφ(1 + aaaa) − 1<br />
KKKK =<br />
1 − φφφφ(1 + aaaa)<br />
φφφφbbbb (C9)<br />
A graphical depiction of the model in eq. C4 gives<br />
an intuitive sense of why it has the same behavior as<br />
the model in eq. C1 (Fig. C-2). In a population that<br />
is below its carrying capacity, the reproductive rate<br />
will exceed the level needed to offset mortality, so<br />
the population will grow. As the population grows,<br />
competition for resources increases, and the reproductive<br />
rate decreases. When the reproductive rate<br />
matches the mortality rate, there is a stable equilibrium<br />
point (K 0 ). Suppose now that there is a change<br />
in the environment such that the extent of habitat<br />
decreases, but the habitat that remains is the same<br />
quality as before. In this case, we could reason that b<br />
will decrease, because the competition for resources<br />
will be felt sooner as the population grows; but a will<br />
stay the same, because the reproductive rate at very<br />
low density would remain unchanged (Fig. C-2, top<br />
panel). The equilibrium point at which reproduction<br />
offsets mortality decreases (K 1 ).<br />
Another way in which the environment could change<br />
is that the extent of habitat doesn’t change, but<br />
the quality of it decreases. In this case, we might<br />
surmise that the reproductive rate decreases equally<br />
for all densities (Fig. C-2, bottom panel), thus a<br />
decreases, but b stays the same. Again, the equilibrium<br />
point at which reproduction offsets mortality<br />
decreases (K 1 ).<br />
Although the effect of these two types of environmental<br />
change on the carrying capacity looks<br />
similar, the effect on the yield curve is profoundly<br />
different (Fig. C-3). In the case of the effect of<br />
habitat quantity, b changes but not a; looking at<br />
eq. C9, this means that the carrying capacity (K)<br />
changes, but the intrinsic rate of growth (r) does not.<br />
Thus, the yield curve shrinks to the new carrying<br />
capacity, but does not change its proportions (Fig.<br />
98 <strong>Polar</strong> <strong>Bear</strong> Conservation Management Plan