Appendix A. Maximizing fitness of conservative bet hedgers based on
normal fitness functions in the two-state variable environment
In this analysis, a conservative bet-hedger (CB) is a type of generalist that maximizes its fitness by adjusting the shape of its fitness function. Specifically, CB does this by widening its normally-distributed fitness function (i.e. standard deviation s →ks, where k > 1) at the expense of reducing the height of the curve at every point by the multiplicative factor 1/k, thus preserving a constant area under the curve. This fixed-area constraint, though arbitrary, enforces the width-height trade-off for the fitness curve in a reasonable way. (Note: CB incurs an additional cost in exchange for this width-height flexibility, the multiplicative factor 1-δC, with 0 < δC < 1, which has no effect on optimizing k.) Assuming that the relevant fitness function has the same shape as that of a specialist when k = 1, we have
(A-1)
Like all generalists here, CB is a variance-minimizing strategy in which R(c1) = R(c2), and thus both within-patch and between-patch variances are zero. CB optimizes k = to maximize fitness where dR/dk = 0 and d2R/dk2 < 0, unless < 1, in which case = 1. Differentiating,
. (A-2)
By inspection, d2R/dk2 < 0 for k > 1 where dR/dk = 0, because increasing (decreasing) k must decrease (increase) the positive term more than the negative term.
Now setting dR/dk = 0 and solving then yields
for , and
for .
Taking d2R/dc2 and setting the result to zero yields the inflection points, which are coincident with the ci in equation (A-3) as long as . In other words, for CB with > 1, the inflection points of the fitness curve correspond exactly to the environmental conditions and .
Appendix B. Mean, variance, and covariance calculations for the five strategies
The five strategies of interest here are specialists 1 and 2 (S1 and S2); fixed generalists (FG) and conservative bet hedgers CB; and diversified bet hedgers DB. Within-generation fitness R is a normally distributed function of environmental condition c:
, (B-1)
(just as in equation (A-1), except with c explicitly continuous), where δ is the cost factor for bet hedging (0 < δ < 1), k is the derived in Appendix A, is the environmental condition corresponding to the peak of the normal curve, and s is the standard deviation.
Means (M), variances (V), and covariances (C) for each strategy to be substituted into text equations (1) and (5) are indicated below. Fitnesses for these relationships are evaluated at each of the two environmental conditions, based on substituting particular magnitudes of c, , k, s, and δ into equation (B-1). Each of the strategies is assumed in some sense to be optimized to the conditions. For specialist 1 and specialist 2, this means having = ci for i = 1 or 2, so that for the relevant Rij, c = cj, = ci, k = 1, s = 1, and δ = 0. For the RGi of FG, c = ci, = (c1 + c2)/2, k = 1, s = 1, and δ = 0. For the RGi of CB, c = ci, = (c1 + c2)/2, k = max{1, (c2 – c1)/(2s)}, s = 1, and δ = δG > 0. CB may optimize k as described in Appendix A. Since diversified bet hedgers (DB) are a mix of specialists, their R-values are those of the specialists multiplied by the cost factor δD > 0. DB also optimizes the mix of specialists, where d1 is the proportion of specialist 1, and 1-d1 is the proportion of specialist 2. All strategies optimize individuals per patch m and the number of patches occupied n for a given total number of individuals I = mn.
The means in each case are arithmetic, based on expected fitnesses over the combinations of patterns and frequencies. The variances are determined over those combinations and measure the expected variation over time for individual randomly chosen locations or patches, depending on the variance being assessed. Similarly, covariances express expected covariation between pairs of randomly chosen locations or patches over time.
Subscripts in the equations refer to specialists 1 and 2 (1, 2), fixed generalists and conservative bet hedgers (both G), and diversified bet hedgers (D, W, and B). For DB, subscript D is the special case with m = 1 and n = I, W indicates a within-patch relationship with m > 1, and B is a between-patch relationship with m > 1. MW1 and MW2 are DB means within patches in condition 1 or 2, respectively. Recall that for fij, i is the condition and j is the type of year or annual pattern; for Rkl, k is the relevant strategy and l is the condition.
The relationships below are constructed as follows. Means are expected fitnesses for each strategy, with alternatives weighted by their frequencies. Variances are expected squared deviations from the relevant means weighted by their frequencies. For the variance of the DB within patches, the relevant means are those for patches in each environmental condition. For the variance of DB between patches, the relevant deviations are differences between means for patches in each condition and the overall fitness mean for DB. Covariances are frequency-weighted multiplicative products of deviations from means for pairs of individuals (or patches in the case of CB) that may or may not differ in fitness (or for CB, in mean within-patch fitness). Thus the squared terms result in the frequency-weighted combinations of multiplied deviations giving rise to the covariances.
Calculating the covariances CD and CW for finite broods size I is complicated by a negative bias that arises from the concept of comparing randomly drawn pairs of individuals. Whichever type of specialist is drawn first (S1 with probability d1 or S2 with probability 1 – d1) is thus less likely to be drawn second because drawing the first depleted the pool of that type by one individual. This effect is more important for smaller brood sizes but must in any case be accounted for in the calculations. For CD, all individuals are assumed to occupy the n separate patches, and so this depletion effect depends on d1 and n; for CW, within-patch covariance and the depletion effect depends on d1 and the m individuals per patch. In these cases, when I ∞, these two equations simplify to those shown at the bottom of the set. Note that CB is not subject to this bias, because patches are assigned conditions independently of each other: drawing one type first does not alter the chance of drawing either type next.
Here we summarize how these relationships are used to determine lineage fitness W for each strategy. The mean values M1, M2, MG, and MD are substituted into text equation (2) for M in each case. (An important caveat not explicitly indicated in the equations is that all R values when used in bet hedging calculations are first multiplied by 1- δC or1-δD to incorporate costs, both = 0.05 here.) VT for each strategy, constructed from the other components, is substituted for V in that equation. Variance components V1, V2, and VG are substituted into equation (5) for VL and for CL (since VL = CL and thus ρL = 1 within patches for those strategies). Covariances C1, C2, and CG are substituted into equation (5) for CP. For DB, when m = 1, VD is substituted for VL, and when m > 1, VW is substituted for VL; CD (= CB) is substituted for CP, and CW for CL. Moreover, VD = VW + VB and CD = CW+ CB, so that the within and between components partition the magnitudes for m = 1, where, as in the S&K analysis, both sources of variation and of covariation are combined.
When I = mn = ∞, CDand CW simplify as follows:
ε = 1.5 / ε = 3.0m = I,
n =1 / m = n
= / m = 1,
n = I / m = I,
n =1 / m = n
= / m = 1,
n = I
I =4 / f11 = 1,
f12 = 0 / SP / 0.5816* / 0.5816* / 0.5816* / 0.3134* / 0.3134* / 0.3134*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.75,
f12 = 0.25 / SP / 0.5816 / 0.6107 / 0.6257* / 0.3134 / 0.3749 / 0.4101*
DB / 0.6292* / 0.6292 / 0.6141 / 0.4803* / 0.4803 / 0.4391
f11 = 0.5,
f12 = 0.5 / SP / 0.5816 / 0.6207 / 0.6412* / 0.3134 / 0.3980 / 0.4486*
DB / 0.6292* / 0.6292 / 0.6091 / 0.4803* / 0.4803 / 0.4262
I =16 / f11 = 1,
f12 = 0 / SP / 0.5816* / 0.5816* / 0.5816* / 0.3134* / 0.3134* / 0.3134*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.75,
f12 = 0.25 / SP / 0.5816 / 0.6257 / 0.6373* / 0.3134 / 0.4101 / 0.4387*
DB / 0.6292* / 0.6292 / 0.6254 / 0.4803* / 0.4803 / 0.4696
f11 = 0.5,
f12 = 0.5 / SP / 0.5816 / 0.6412 / 0.6570* / 0.3134 / 0.4486 / 0.4907*
DB / 0.6292* / 0.6292 / 0.6241 / 0.4803* / 0.4803 / 0.4661
I =64 / f11 = 1,
f12 = 0 / SP / 0.5816* / 0.5816* / 0.5816* / 0.3134* / 0.3134* / 0.3134*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.75,
f12 = 0.25 / SP / 0.5816 / 0.6334 / 0.6402* / 0.3134 / 0.4289 / 0.4461*
DB / 0.6292* / 0.6292 / 0.6282 / 0.4803* / 0.4803 / 0.4776
f11 = 0.5,
f12 = 0.5 / SP / 0.5816 / 0.6517 / 0.6610* / 0.3134 / 0.4762 / 0.5018*
DB / 0.6292* / 0.6292 / 0.6279 / 0.4803* / 0.4803 / 0.4767
I =144 / f11 = 1,
f12 = 0 / SP / 0.5816* / 0.5816* / 0.5816* / 0.3134* / 0.3134* / 0.3134*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.75,
f12 = 0.25 / SP / 0.5816 / 0.6360 / 0.6407* / 0.3134 / 0.4354 / 0.4475*
DB / 0.6292* / 0.6292 / 0.6289 / 0.4803* / 0.4803 / 0.4791
f11 = 0.5,
f12 = 0.5 / SP / 0.5816 / 0.6552 / 0.6617* / 0.3134 / 0.4858 / 0.5039*
DB / 0.6292* / 0.6292 / 0.6286 / 0.4803* / 0.4803 / 0.4787
I =∞ / f11 = 1,
f12 = 0 / SP / 0.5816* / 0.5816* / 0.5816* / 0.3134* / 0.3134* / 0.3134*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.75,
f12 = 0.25 / SP / 0.5816 / 0.6412* / 0.6412* / 0.3134 / 0.4486 / 0.4486*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
f11 = 0.5,
f12 = 0.5 / SP / 0.5816 / 0.6623* / 0.6623* / 0.3134 / 0.5056 / 0.5056*
DB / 0.6292* / 0.6292* / 0.6292* / 0.4803* / 0.4803* / 0.4803*
1