Minimum conditions for bet-hedging to evolve in a finite population.

*R* =

*p̂*_{fix} (0,

*m̂*_{opt},

*N*,

*θ*,

*s*) /

*p̂*_{fix} (

*m̂*,0,

*N*,

*θ*,

*s*) denotes the estimated advantage provided by using the optimal bet-hedging strategy over using no bet-hedging.

**A** and

**B**: When

(from Equation 6),

*s* = ∞, and

*m*−1−

*e*^{−}^{θ} ≈

*m̂*_{opt}, the ratio

*p̂*_{fix} (0,

*m̂*_{opt},

*N*,

*θ*,

*s*) /

*p̂*_{fix}(

*m̂*_{opt}, 0,

*N*,

*θ*,

*s*) , shown on the vertical axis, is approximately

*R*. Curves in

**A** are shown for

*R* <

*N*^{2}/100, and in

**B** for

*R* > 1 + 10 log(

*N*)

^{2}/

*N*, roughly the range for which the approximation is accurate. (Note that

**B** covers the range from

*R* = 1.001 to

*R* = 2.)

**C**: Rectangular regions in which

*s* and

*θ* must fall to allow for a given advantage

*R*; these are necessary but not sufficient conditions. For smaller

*R* these rectangles are encroached on by the approximate equation for

*N*_{min} Computed estimates are shown for

*θ* ≤

*s*.

**D**: Estimated parameters

*θ* and

*s* for which

*R* = 100, given as hyperbolas in log (

*s*) and log(

*θ*). Our computed estimates become less accurate for small

*sN* and large

*θ*, and are shown only for

*sN* > 10. The curve for

*N* = 10

^{8} is extrapolated using Equation 7.

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