Fig 1.
A schematic view of switching landscapes and adaptive walks demonstrating possible attractor types.
Cycling distinct yet correlated rugged landscapes can drive population escape from specialists—genotypes fit in a particular environment (orange/gray star in environment A/B) and open new paths (arrows) to generalists—genotypes that remain fit in different environments (blue star). While generalists constitute fixed point attractors in changing environments, specialists located in each other’s basin of attraction on alternating landscapes form limit cycle attractors.
Fig 2.
The onset, abundance and fitness of generalists.
Expected number (A) and average fitness (C) of shared peaks are shown as a function of the number np of conserved fitness contributions at color-coded values of K. In (C), dashed lines indicate average fitness of all local optima at K = 3 (grey) and K = 11 (blue), respectively, which correspond to values at np = N = 12, whereby all peaks are shared. Above (marked by arrows) generalists are on average fitter than specialists; this requires a higher level of fitness conservation as ruggedness decreases (e.g.
for K = 11 whereas
for K = 3). (B) The fraction of generalists among all fitness peaks increases with np;
indicates the minimum conservation level for generalists to exist. Curves correspond to K = 3, 7, 11, increasing in the direction of the arrow. Inset:
deceases with increasing K. Each data point is an average over 1000 landscape pairs.
Fig 3.
Dynamic basin enlargement via ratchet effects under environmental cycling.
(A) Schematically, in a static landscape, only a single basin of attraction (shaded orange/gray in landscape A/B) leads to a generalist (blue peak). Under environmental switches (ABAB or BABA, indicated with dotted arrows), a population initialized in any genotype inside the linked basins (six amorphous shapes with orange or gray borders) can reach the generalist. This chain of basins contains
and
hinged at the generalist peak,
and
as two ends, along with
and
as intermediate links. Thus, the total coverage of linked basins defines the effective accessibility of a generalist in switching landscapes. (B) The fraction of generalists that gain basin size under landscape switching, where K increases from top to bottom. Each data point is an average over 1000 realizations of landscape pairs. (C) Correlation between fitness and basin size of local optima in a static landscape (orange filled symbols) and in switching landscapes (blue open symbols).
Fig 4.
Accessibility of generalists under environmental cycling.
(A) The fraction of genotypes that can reach a generalist via an adaptive walk. (B) The ratio of the total basin size of fit generalists (within the top 30% of maximum fitness) to that of all generalists. Both heatmaps are obtained by averaging over 1000 pairs of landscapes at each combination of np and K. In both diagrams, the gray area corresponds to phase I in which all fitness peaks are specific to one environment; in this no-generalist phase, landscape switching leads to oscillations (panel B, upper inset) or limit cycles (lower inset) among specialists. The colored region represents phase II, where generalists act like hubs into which evolutionary trajectories enclosed by linked basins converge, following multiple landscape switches.