Fig 1.
A) General model: individuals can adopt N different phenotypes with probabilities αj (j = 1, ⋯, N) and experience M different environmental conditions with probabilities pi (i = 1, ⋯, M). The fitness of an individual with phenotype j in an environment i is given by sij. B) Two-phenotypes model: Individuals can adopt either a “risky” or a “safe” phenotype with probabilities α, and 1 − α respectively. The safe phenotype is characterized by an environment-independent growth rate ss. The growth rate of the risky phenotype is sa or sb, depending on whether the current environment is “adverse” (a) or “favorable” (b). C) and D) Sketch of range expansion in a population having 0 ≤ α ≤ 1 for temporally varying C) and spatially varying D) environments, respectively.
Fig 2.
Bet-hedging region in temporally varying environments.
Optimal strategy α* as a function of growth rates and
for range expansions in temporally varying environments under the limits of environmental change rate (A) k → 0, see Eq (7), and (B) k → ∞. In all panels, lines delimit the bet-hedging region 0 ≤ α* ≤ 1. Two dots in the panels mark parameter values chosen for the analysis of Figs 3, 4 and 5.
Fig 3.
The asymptotic mean velocity increases with k in temporally varying environments.
(A) Velocities obtained by numerical integration of Eq 5 for sa = 0.75, ss = 1, sb = 3 (yellow dot of Fig 2) for different switching rates k shown in the figure legend. (B) The same for sa = 0.25, ss = 1, sb = 2 (blue dot of Fig 2). In (A), the optimal strategy is α = 1 for all k values. In (B), bet-hedging optimal strategies appear depending on the value of k. The continuous red and yellow lines (both in A and B) illustrate analytical predictions under the two limits vM(k → 0) = (va(α) + vb(α))/2 and , respectively.
Fig 4.
The bet-hedging region is expanded for range expansions in spatially varying environments compared to temporally varying environments.
A) Optimal strategy α* as a function of the parameters for spatially varying environments in the limit ks → 0, Eq (9). White lines mark the limits of the bet-hedging region. The limit for which the strategy α = 1 is optimal in temporally fluctuating environments for k → 0 is also shown (gray line) for comparison. B) The velocity obtained by numerical integration of Eq (5) for sa = 0.25, ss = 1, sb = 2 (corresponding to the blue dot of panel A) and different values of kS shown in the figure legend. Light and dark gray lines correspond to the analytical limits for temporally varying environments, vM(k → 0) = (va(α) + vb(α))/2, and , respectively. The red curve is the analytical solution for a spatially fluctuating environment with kS → 0, see Eq (9). Note that in this case, the asymptotic mean velocity does not increase monotonically with kS but is maximal at kS ≈ 0.1.
Fig 5.
The optimal strategy is robust with respect to noise induced by finite population size in temporally varying environments.
(A) Asymptotic mean velocities obtained by numerical integration of the stochastic Fisher Eq (10) for , ss = 0.01,
(yellow dot of Fig 2) and different population sizes. (B) The same for
, ss = 1,
(blue dot of Fig 2). In both panels, the temporal switching rate of the environment is k = 0.001. Green dots corresponds to the results of Fig 3A and 3B for k = 0.001. Insets show a collapse of the curves according to Eq (11), with a fitted value of C = 3.