Specialization and Bet Hedging in Heterogeneous Populations

Phenotypic heterogeneity is a strategy commonly used by bacteria to rapidly adapt to changing environmental conditions. Here, we study the interplay between phenotypic heterogeneity and genetic diversity in spatially extended populations. By analyzing the spatiotemporal dynamics, we show that the level of mobility and the type of competition qualitatively influence the persistence of phenotypic heterogeneity. While direct competition generally promotes persistence of phenotypic heterogeneity, specialization dominates in models with indirect competition irrespective of the degree of mobility.

Figure 1

Illustration of competition in the heterogeneous ecological model. When individuals with genotypes $\alpha$ and $\beta$ engage, their phenotypic states $s_m$ and $s_n$ are chosen randomly according to a probability distribution. The outcome of the reaction is specified by the interaction matrix $A_{s_m{s}_n}$.

Figure 2

(a) Time evolution of genetic diversity for the ML model. (b) A typical configuration of the ML model for large times [$t=660$, dotted line in (a)] and a low value of the diffusion constant ($D=10^{-4}$). Different colors (gray scales) signify the probability to be in any of three phenotypic states: red (light gray), green (medium gray), or blue (dark gray) denotes a high probability to be in the phenotypic states $s_1$, $s_2$, or $s_3$, respectively. (c) Time evolution of genetic diversity for the LV model. (d) A typical configuration of the LV model for large times [$t=10^3$, dotted line in (c), $L=100$].

Figure 3

Asymptotic genotypes for the heterogeneous LV model. (a) Probability density ${\mathcal{P}}_{\infty }(\overrightarrow{\pi })$ of the asymptotic genotypes $\overrightarrow{\pi }$ for different values of the diffusion coefficient $D$. Color (gray scale) denotes the value of ${\mathcal{P}}_{\infty }$, such that white signifies a high value and black a low value. The maxima of ${\mathcal{P}}_{\infty }$ identify successful genotypes. (b) Marginal probability distribution to be in any of the components of $\overrightarrow{\pi }$. We identify two threshold values, $D_1(N)$ and $D_2$, separating distinct outcomes of the evolutionary dynamics as indicated in the graph ($L=80$). (c) Scaling of the threshold values with system size $L=\sqrt{N}$: $D_1\propto 1/N$ and $D_2=\text{const}$.