Multiple phase transitions shape biodiversity of a migrating population

In a wide variety of natural systems, closely related microbial strains coexist stably, resulting in high levels of fine-scale biodiversity. However, the mechanisms that stabilize this coexistence are not fully understood. Spatial heterogeneity is one common stabilizing mechanism, but the rate at which organisms disperse throughout the heterogeneous environment may strongly impact the stabilizing effect that heterogeneity can provide. An intriguing example is the gut microbiome, where active mechanisms affect the movement of microbes and potentially maintain diversity. We investigate how biodiversity is affected by migration rate using a simple evolutionary model with heterogeneous selection pressure. We find that the biodiversity-migration rate relationship is shaped by multiple phase transitions, including a reentrant phase transition to coexistence. At each transition, an ecotype goes extinct and dynamics exhibit critical slowing down (CSD). CSD is encoded in the statistics of fluctuations due to demographic noise—this may provide an experimental means for detecting and altering impending extinction.

Figure 1

Coexistence of genetically distinct ecotypes leads to phase transitions. (a) Schematic of the two-habitat system. (b) In case 1 (left), native ecotypes ${\vec{\psi }}_1$ (gray diamonds) and ${\vec{\psi }}_2$ (blue squares) share common genotypes, but in case 2 (right) they are genetically distinct. Lower panels: fitness landscapes ${\varphi }_{1,i}$ (gray diamonds) and ${\varphi }_{2,i}$ (blue squares). (c) Steady-state population in habitat 2 with $R=R_c/2\approx 0.41$ ($k\approx 0.10$) and $\widehat{R}={\vec{\psi }}_1$ in two cases. In case 1 (left), habitat 2 supports a single ecotype. The genotypes shared by ${\vec{\psi }}_1$ and ${\vec{\psi }}_2$ have the highest abundance. In case 2 (right), habitat 2 supports coexistence of ecotype 1 (genotypes 1 to 11) and ecotype 2 (genotypes 25 to 31). (d) Habitat 2's total population in case 1 (black) and case 2 (green, gray in grayscale). In case (2), habitat 2's total population remains constant at the carrying capacity $K_2$ until $R$ reaches $R_c$. (e) Relaxation time $\tau$. $\tau \to \infty$ at $R=R_c$ in case (2), but not in case (1). Phases of coexistence ($R<R_c$) and ecotype 1 ($R>R_c$) are indicated. ${\rho }_1=0.2, {\rho }_2=1$, and $\gamma =0.01$.

Figure 2

Phase transitions shape the relationship between biodiversity and migration rate. (a) Migrant flux $R\left(k\right)$ as a function of $k$ for a system with $R_{\text{max}}<R_c$ (left), $R_{\text{max}}=R_c$ (middle), and $R_{\text{max}}>R_c$ (right). (b) Steady state total population of ecotype 1, $E_1$ (dark blue) and ecotype 2, $E_2$ (light orange) in habitat 2, and habitat 2's total population $n_2^{*\text{tot}}$ (thin black). (c) Biodiversity of habitat 2, $H\left[{\vec{n}}_2\right]$, defined by the second-order Renyi entropy (solid black) and Shannon entropy (dotted red). (d) Phases in different regimes. Diagrams indicate the coexistence phase and phases with only ecotype 1 or ecotype 2 (E. 2). The regime of $R_{\text{max}}>R_c$ is characterized by the presence of ecotype 1 phase and the reentrance of coexistence phase at intermediate $k$ values, both of which are absent in other regimes. All panels show the same setting as in Fig. 1 case (2), except with ${\rho }_1=0.4$ for $R_{\text{max}}<R_c$ (left), ${\rho }_1=0.304$ for $R_{\text{max}}=R_c$ (middle), and ${\rho }_1=0.2$ for $R_{\text{max}}>R_c$ (right).

Figure 3

Resource competition between ecotypes causes critical slowing down. (a) Relaxation time $\tau =-1/w_1$ (thick teal, gray in grayscale), ecological relaxation time ${\tau }_E$ (thin black), and time constant of autocorrelation decay ${\tau }_A$ (red crosses). Vertical dashed lines indicate critical migration rates. Same setting as Fig. 1, case (2). (b) Closeup of the peak in $\tau$ at $k=k_{-}$. Similar peaks occur at $k=k_{+}$ and $k={\lambda }_1$. (c) The zero mode in habitat 2, ${\vec{\zeta }}_2$. The two terms ${\vec{\psi }}_2$ and $-a{({\lambda }_2-{\widehat{V}}_2)}^{+}\vec{J}$ (indicated in the figure) comprise the same genotypes as ecotype 2 and ecotype 1, respectively. Due to the opposite sign, they describe a reallocation of population from ecotype 2 to ecotype 1, as a result of resource competition.