Bet Hedging against Demographic Fluctuations

Biological organisms have to cope with stochastic variations in both the external environment and the internal population dynamics. Theoretical studies and laboratory experiments suggest that population diversification could be an effective bet-hedging strategy for adaptation to varying environments. Here we show that bet hedging can also be effective against demographic fluctuations that pose a trade-off between growth and survival for populations even in a constant environment. A species can maximize its overall abundance in the long term by diversifying into coexisting subpopulations of both “fast-growing” and “better-surviving” individuals. Our model generalizes statistical physics models of birth-death processes to incorporate dispersal, during which new populations are founded, and can further incorporate variations of local environments. In this way, we unify different bet-hedging strategies against demographic and environmental variations as a general means of adaptation to both types of uncertainties in population growth.

Figure 1

Schematic illustration of the birth, death, and dispersal processes. Each patch is represented by a grid and has a finite capacity represented by the number of cells in the grid; an empty cell represents a vacant site, and a colored cell represents an individual, whose phenotype is indicated by its color. The cells highlighted by thick borders are being updated: A cell with a “$+$” sign means an individual appears, either being born to another individual (sharing a thick border) or having immigrated from another patch (dashed arrow); a cell with a “$-$” sign means an individual disappears, either due to death (isolated thick border) or emigration (dashed arrow). Individuals can move freely within a patch or disperse to any other patch.

Figure 2

Asymptotic expansion rate $W$ as a function of phenotype distribution $\rho$. The birth, death, and dispersal rates are ${\beta }_A=2$, ${\delta }_A=1$, ${\beta }_B=0.5$, ${\delta }_B=0.1$, and $\mu =0.002$, and the carrying capacity is $K=100$. Inset: Time course of the total population size $N$ and the number of occupied patches $M$, simulated using the Gillespie algorithm. The slope of the curves determines the value of $W$ for a given $\rho$.

Figure 3

Optimal phenotype distribution ${\rho }^{*}$ as a function of dispersal rate $\mu$, where one phenotype has a faster growth rate and the other has a lower extinction risk. The birth and death rates of each phenotype are ${\beta }_A=2$, ${\delta }_A=1$, ${\beta }_B=0.5$, and ${\delta }_B=0.1$; the carrying capacity of each patch is $K=100$. For dispersal rates between ${\mu }_L$ and ${\mu }_R$, a mixed strategy offers the maximum asymptotic expansion rate for the species.

Figure 4

Optimal phenotype distribution ${\rho }^{*}$ for different values of the dispersal rate $\mu$ and the environment distribution $\epsilon$. The shaded region marks a $0<{\rho }^{*}<1$ phase in which a mixed strategy offers the maximum asymptotic expansion rate. The birth and death rates are ${\beta }_A^{(X)}=2$, ${\delta }_A^{(X)}=1$, ${\beta }_A^{(Y)}=0$, ${\delta }_A^{(Y)}=10$, ${\beta }_B^{(X)}={\beta }_B^{(Y)}=0.5$, and ${\delta }_B^{(X)}={\delta }_B^{(Y)}=0.1$. Each patch has carrying capacity $K=100$ and environment switching rates $\alpha =0.1$.