Fig 1.
Illustration of the update rules.
Each row represents one of the different evolutionary update mechanisms. The columns indicate the different steps of each evolutionary event. In column a) an individual is chosen from the whole population; it can be ‘selected’ through competition by fitness (red shading), or ‘picked’ at random, irrespective of its species (blue shading). This node is destined to either reproduce (pink shading), or to be replaced (brown shading), as shown in column b). Column c) indicates that one neighbour of this node is either selected (red), or picked (blue). This second node is destined to reproduce (pink), or to be replaced (brown), shown in column d). Column e) shows the result of the evolutionary event; the node chosen to reproduce places an offspring in place of the node chosen to die. Each row is composed of one box of each colour; the sequence of the colours distinguishes the different processes. From top to bottom, the rows correspond to: (i) global birth-death process (Bd): an individual is selected from the whole population to reproduce, and one of its neighbours is picked to be replaced by the first individual’s offspring; (ii) global death-birth process (Db): an individual is selected to die from the whole population, and one of its neighbours is picked to place an offspring in its place; (iii) local birth-death process (bD): an individual is picked from the whole population to reproduce, and one of its neighbours is selected to die; (iv) local death-birth process (dB): an individual is picked from the whole population to die, and one of its neighbours is selected to reproduce.
Fig 2.
Fixation probability as a function of the flow speed for unrestricted random initial positions (random geometric graphs, RGGs).
For the global death-birth process, increasing the flow speed increases the fixation probability. The reverse is found for the remaining three processes. Circle markers show fixation probabilities in the fast-flow limit; square markers are results for fixed connected random geometric graphs (CRGGs); see text for further details. The fixation probabilities on a complete graph are shown for reference.
Fig 3.
Fixed heterogeneous graphs amplify selection for birth-death processes and suppress it for death-birth processes.
The figure shows the fixation probability of an invading mutant (ϕ), averaged over static CRGGs. Data is shown relative to the corresponding fixation probability on a complete graph (ϕCG). Regardless of the population size, selection is amplified for Bd and bD processes, and suppressed for Db and dB processes.
Fig 4.
Significance of the degree of the initial mutant.
The upper panel shows the degree distribution, pk, of the ensemble of connected random geometric graphs (CRGGs), obtained by placing N = 100 individuals into the spatial domain 0 ≤ x, y ≤ 1 with uniform distribution, and using an interaction radius R = 0.11 and periodic boundary conditions. The lower panel shows the fixation probability obtained from simulating the evolutionary process on these graphs, as a function of the degree of the initial mutant. For the two death-birth processes the mutant’s success is below the one on a complete graph if its degree is low, and above ϕCG at high connectivity. The reverse is found for the two birth-death processes. Data points have been connected as a visual guide.
Fig 5.
Comparison of fixation probability for simulations started from unrestricted and connected random geometric graphs (RGGs and CRGGs, respectively).
The fixation probability as a function of the flow speed is shown as thick lines for simulations started on connected graphs; thin lines are for unrestricted initial positions (some of this data is also shown in Fig 2). Square markers indicate the fixation probabilities on static CRGGs; see text for details. The fixation probability on complete graphs is shown for reference. A minimum of ϕ is found for the Db process; maxima are discernible for Bd and bD when the dynamics are started from connected graphs. The effect of amplification/suppression of selection at slow flow speeds is more pronounced for simulations initialized from RGGs than from CRGGs.
Fig 6.
Fragmented initialization promotes the formation of clusters.
The main panel shows the average proportion of active links as the evolutionary dynamics proceed. Thick lines correspond to simulations started from connected graphs (CRGGs); thin dotted lines to simulations initialized from unrestricted random positions (RGGs). The fraction of active links is lower for RGGs, regardless of the evolutionary process. Inset: Fixation probability of the mutant species, once there are i mutants in the population. When mutants are a minority, a small increase in their frequency greatly increases their fixation probability. Conversely, reducing their numbers when they are a majority has only minor effects on their chances of success. Simulations in the inset are initialized from CRGGs.
Fig 7.
Fixation probability at different flow speeds for simulations started from a square lattice.
For the global death-birth process a minimum of fixation probability is found at intermediate flow speeds; conversely, the global birth-death process shows a maximum. For the local processes no extrema are found; instead varying the flow speed interpolates monotonously between the behaviour on fixed lattices and the limit of fast flows.