Interactive simulations complementing
Spatial social dilemmas promote diversity
PNAS 2021 Vol. 118 No. xx e2105252118, doi: 10.1073/pnas.2105252118

All figures below are copies of those in the above publication followed by a list of interactive simulations that illustrate the dynamics and reproduce the corresponding scenarios.

Contents:

Figure 1: Equilibria in linear continuous prisoner's dilemma
Figure 2: Spatial linear continuous prisoner's dilemma
Figure 3: Spatial continuous prisoner's dilemma with saturating benefits
Figure 4: Equilibria in the continuous snowdrift game
Figure 5: Spatial modes of diversification in the continuous snowdrift game
Figure S1: Spatial adaptive dynamics and modes of diversification in the continuous snowdrift game
Figure S2: Equilibria in the continuous snowdrift game with birth-death updating

Equilibria in linear continuous prisoner's dilemma:

Fig. 1 in https://doi.org/10.1073/pnas.2105252118 — **Figure 1:** Equilibrium investment levels (mean \(\pm\) standard deviation) in individual-based simulations of the linear continuous prisoner's dilemma (\(B(x)=x, C(x)=r x\)) as a function of the cost-to-benefit ratio, \(r\), for weak selection, \(w=1\) (red), moderate, \(w=10\) (black), and strong selection, \(w=100\) (blue). (A) in well-mixed populations with \(N=10^4\) individuals cooperative investments are close to zero regardless of \(r\) as predicted by adaptive dynamics (dashed line). The small variance further decreases for stronger selection emphasizing the disadvantage of mutants with higher investments. (B) for populations with \(k=4\) neighbours spatial adaptive dynamics predicts a threshold \(r^\ast=1/k\) (dash-dotted-line) below which investments reach the maximum but disappear above. Simulations on a \(100\times 100\) lattice confirm the trend but reveal an interesting susceptibility to noise: for weak selection (red) the maximum investment is not reached; intermediate selection (black) essentially follows pair approximation, while strong selection (blue) maintains non-zero investment levels beyond the predicted threshold. The large variation suggests co-existence of different traits and is confirmed through sample snapshots of the spatial configuration in Fig. 2I.

Interactive labs:

	Figure 1A: well-mixed population \(N=10'000\), favourable cost-to-benefit ratio (\(r=0.1\)) and moderate selection (\(w=10\)).
	Figure 1A: well-mixed population \(N=10'000\), harsher cost-to-benefit ratio (\(r=0.3\)) and moderate selection (\(w=10\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), favourable cost-to-benefit ratio (\(r=0.1\)) and weak selection (\(w=1\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), favourable cost-to-benefit ratio (\(r=0.1\)) and moderate selection (\(w=10\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), favourable cost-to-benefit ratio (\(r=0.1\)) and strong selection (\(w=100\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), harsher cost-to-benefit ratio (\(r=0.3\)) and weak selection (\(w=1\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), harsher cost-to-benefit ratio (\(r=0.3\)) and moderate selection (\(w=10\)).
	Figure 1B: lattice population \(N=100\times 100, k=4\), harsher cost-to-benefit ratio (\(r=0.3\)) and strong selection (\(w=100\)).

Spatial linear continuous prisoner's dilemma:

Fig. 2 in https://doi.org/10.1073/pnas.2105252118 — **Figure 2:** Linear continuous prisoner's dilemma in \(100\times 100\) lattice populations with \(k=4\) neighbours. Evolutionary dynamics is illustrated for favourable cost-to-benefit ratios, \(r=0.1<1/k\), *A-C*, and for harsher conditions, \(r=0.3>1/k\), with moderate selection, \(w=10\) *D-F*, as well as strong selection, \(w=100\) *G-I* The left column shows the pairwise invasibility plots (*PIP*), which indicate whether mutant traits are capable of invading a particular resident population (white regions) or not (black regions). The middle column shows the evolutionary trajectory of the distribution of investments over time in individual-based simulations (darker shades indicate higher trait densities in the population with the highest densities in yellow). The right column depicts snapshots of the population configuration at the end of the simulation runs. The colour hue indicates the investment levels ranging from low (red) to intermediate (green) and high (blue). For \(r<1/k\) higher investors can always invade and eventually the maximum investment is reached *A-C*, regardless of selection strength. The situation is reversed for \(r>1/k\) and weak to moderate selection where only lower investors can invade and investments dwindle to zero *D-F*. Interestingly, for strong selection in lattice populations not only lower investors can invade for \(r>1/k\) but also those that invest significantly more than the resident *G-I*. Note that the width of the region of unfavourable mutants decreases with increasing selection strength \(w\), i.e. the gap becomes easier to bridge (but does not depend on the resident trait \(x\); grey area for \(w=50\) and black area for \(w=100\)). Nevertheless adaptive dynamics predicts that investments evolve to zero because of the assumption that mutations are small, which restrict the dynamics to the diagonal of the *PIP*. However, individual-based simulations show that rare larger mutations or a sequence of smaller ones can give rise to the co-existence of high and low investors. The outcome does not depend on the initial configuration of the population.

Interactive labs:

	Figure 2B: favourable cost-to-benefit ratio (\(r=0.1\)) and moderate selection (\(w=10\)).
	Figure 2E: harsher cost-to-benefit ratio (\(r=0.3\)) and moderate selection (\(w=10\)).
	Figure 2H: harsher cost-to-benefit ratio (\(r=0.1\)) and strong selection (\(w=100\)).

Spatial continuous prisoner's dilemma with saturating benefits:

Fig. 3 in https://doi.org/10.1073/pnas.2105252118 — **Figure 3:** Continuous prisoner's dilemma with saturating benefits \(B(x)=b_0\left(1-\exp(-b_1 x)\right)\) and linear costs \(C(x)=c_0 x\) for \(b_0=8, b_1=1, c_0=0.7\) in \(100\times 100\) lattice populations for weak selection, \(w=1\) (top row) and strong selection, \(w=100\) (bottom row). The pairwise invasibility plots (*PIP*, left column) show that higher investing mutants can invade for low resident investments and lower investing mutants can invaded high investing residents. However, near the singular investment level, \(x^\ast=\log\left(\frac{b_0 b_1}{k c_0}\right)/b_1\approx1.050\), selection strength gives rise to interesting differences in the dynamics. A for weaker selection, \(w=1\), \(x^\ast\) is evolutionarily stable. This is confirmed through individual-based simulations. B depicts the investment distribution over time (darker shades indicate higher trait densities in the population with the highest densities in yellow). C shows a snapshot of the spatial configuration at the end of the simulation. The colour hue indicates the investment level ranging from low (red) to intermediate (green) to high (blue). In contrast, *D-F* for strong selection, \(w=100\), \(x^\ast\) can be invaded but only by higher investing mutants. As a consequence, a degenerate form of evolutionary branching may occur. Individual-based simulations confirm branching already for \(w=10\) (*E, F*).

Interactive labs:

	Figure 3A-C: weak selection (\(w=1\)).
	Figure 3D-F: moderate selection (\(w=10\)).
	Figure 3D-F: strong selection (\(w=100\)).

Equilibria in continuous snowdrift game with quadratic benefit and cost functions:

Fig. 4 in https://doi.org/10.1073/pnas.2105252118 — **Figure 4:** Continuous snowdrift game with quadratic benefit and cost functions, \(B(x)=b_1 x+b_2 x^2\) and \(C(x)=c_1 x+c_2 x^2\). Evolutionary outcomes are shown as a function of the benefit parameter \(b_1\) and cost parameter \(c_2\) with \(b_2=-1/4\) and \(c_1 = 2\). Note that \(b_1<2\) violates the assumption \(B(x)>C(x)\) at least for small \(x\) and hence effectively mimics the characteristics of the prisoner's dilemma. A analytical predictions based on adaptive dynamics in well-mixed populations and B results from individual-based simulations for populations with \(N=10^4\) individuals. C analytical predictions based on spatial adaptive dynamics and complementing individual-based simulations on \(100\times 100\) lattices for \textbf{\textsf{d}} moderate selection, \(w=10\), and E strong selection, \(w=100\). In lattice populations the parameter region admitting singular strategies is shifted to both smaller values of \(b_1\) and of \(c_2\) and the size of the region admitting evolutionary branching is markedly smaller than in well-mixed populations (*A, C*). Interestingly, spatial adaptive dynamics predicts branching only for parameters where defection dominates in well-mixed populations, \(b_1<c_1\), mimicking the continuous prisoner's dilemma. For weak to moderate selection predictions by adaptive dynamics (*A, C*) are in good agreement with results from individual-based simulations (*B, D*), where equilibrium investment levels range from the minimum (black) to intermediate (grey) and the maximum (white) augmented by convergence stability (red) and evolutionary instability (blue) with the overlapping region indicating evolutionary branching (maroon) in adaptive dynamics and diversification in simulations. For strong selection (E) striking differences arise with a much increased region of diversification. The points labelled *a-d* indicate the parameter settings for the invasion analysis in Fig. 5. Note that the automated classification of investment distributions becomes more difficult whenever the singular investment \(x^\ast\) is close to zero or one (for details see SI Text S2).

Interactive labs:

	Figure 4D: evolutionary branching (\(b_1=1.65, c_2=-0.5\)) and weak selection (\(w=1\)).
	Figure 4D: evolutionary branching (\(b_1=1.65, c_2=-0.5\)) and moderate selection (\(w=10\)).
	Figure 4E: evolutionary branching (\(b_1=1.65, c_2=-0.5\)) and strong selection (\(w=100\)).

Spatial modes of diversification in the continuous snowdrift game:

Fig. 5 in https://doi.org/10.1073/pnas.2105252118 — **Figure 5:** Spatial modes of diversification in the continuous snowdrift game with quadratic benefit and cost functions. *A-D* depict pairwise invasibility plots (*PIP*, top row) for the four scenarios illustrating increased spatial diversification due to strong selection (c.f. parameter combinations *a-d* marked in Fig. 4). In all cases the width of the region of disadvantageous mutants decreases with selection strength (grey for \(w=1\); black for \(w=100\)). A the *PIP* suggests gradual evolution towards minimal investments, except for smaller resident traits, where not only lower investing mutants can invade but also those making markedly higher investing. B higher investing mutants can always invade, but so can traits investing markedly less. C selection strength distorts the *PIP* in the vicinity of the convergence and evolutionarily stable \(x^\ast\) resulting in a degenerate form of branching (c.f. Fig. 3). D the *PIP* indicates that \(x^\ast\) is a repellor such that residents with \(x<x^\ast\) are invaded by lower investors while those with \(x>x^\ast\) by higher investors. However, as a consequence of strong selection, mutants with markedly higher (lower) investments can also invade. *E-H* depict corresponding plots of regions of mutual invasibility (*PIP\(^\text{2}\)*, white regions). Regions where mutants or residents are unable to invade (grey) are marked with \((+,-)\) and \((-,+)\), respectively. The vector field shows the divergence (see SI Text S2 for details) and indicates the direction of selection for two co-existing residents based on analytical approximations of the spatial invasion dynamics. In all cases divergence drives the traits away from the diagonal and hence preserves diversity.
Parameters: \(b_2=-1/4, c_1=2\), A \(b_1=1.55\), \(c_2=-0.6\); B \(b_1=1.65\), \(c_2=-0.625\); C \(b_1=1.9\), \(c_2=-0.3\); D \(b_1=1.5\), \(c_2=-0.72\).

Interactive labs:

	Figure 5A, E: evolutionary diversification (\(b_1=1.55, c_2=-0.6\)) and strong selection (\(w=100\)).
	Figure 5B, F: evolutionary diversification (\(b_1=1.65, c_2=-0.625\)) and strong selection (\(w=100\)).
	Figure 5C, G: evolutionary diversification (\(b_1=1.9, c_2=-0.3\)) and strong selection (\(w=100\)).
	Figure 5D, H: evolutionary diversification (\(b_1=1.5, c_2=-0.72\)) and strong selection (\(w=100\)).

Spatial adaptive dynamics and modes of diversification in the continuous snowdrift game:

Fig. S1 in https://doi.org/10.1073/pnas.2105252118 — **Figure S1:** Trait distributions in the continuous snowdrift game with quadratic benefit and cost functions for different modes of spatial diversification (c.f. Fig. 5 in the main text). Smaller mutational steps \(\sigma_\mu\) recover predictions of spatial adaptive dynamics: the trait distribution arising from parameter combinations marked *a-d* in Fig. 4 in the main text are shown for \(\sigma_\mu=10^{-2}\) (dark grey) after \(3\cdot10^4\) generations and \(\sigma_\mu=10^{-4}\) (red) after \(3\cdot10^5\) generations with strong selection, \(w=100\). To speed simulations up, the probability of mutations was set to one (instead of \(0.01\) for \(\sigma_\mu=10^{-2}\)). In *A, B*, the smaller mutational steps, \(\sigma_\mu=10^{-4}\), prevent diversification and the adaptive dynamics predictions of zero and maximal investments, respectively, are recovered (dashed line). In D the singular strategy (dashed line) is a repellor. For larger \(\sigma_\mu\) the distribution is bi-modal whereas for \(\sigma_\mu=10^{-4}\) the result depends on the starting point (two unimodal distributions are shown in red from simulations with initial traits above and below the singular strategy). Only in C branching cannot be suppressed even for \(\sigma_\mu=10^{-6}\) (orange, with \(w=800\) after \(3\cdot10^6\) generations for an initial distribution with a sharp Gaussian peak at the singular point, \(\sigma_0=10^{-6}\)) but the process is much slower.

Interactive labs:

	Figure S1 A: same as Fig. 5A, E but smaller mutations, \(\sigma_\mu=10^{-4}\).
	Figure S1 B: same as Fig. 5B, F but smaller mutations, \(\sigma_\mu=10^{-4}\).
	Figure S1 C: same as Fig. 5C, G but smaller mutations, \(\sigma_\mu=10^{-4}\).
	Figure S1 C: same as Fig. 5C, G but even smaller mutations, \(\sigma_\mu=10^{-6}\), and stronger selection, \(w=800\).
	Figure S1 A: same as Fig. 5D, H but smaller mutations, \(\sigma_\mu=10^{-4}\).

Equilibria in the continuous snowdrift game with birth-death updating:

Fig. S2 in https://doi.org/10.1073/pnas.2105252118 — **Figure S2:** Continuous snowdrift game with quadratic benefit and cost functions, \(B(x)=b_1 x+b_2 x^2\) and \(C(x)=c_1 x+c_2 x^2\). Evolutionary outcomes are shown as a function of the benefit parameter \(b_1\) and cost parameter \(c_2\) with \(b_2=-1/4\) and \(c_1 = 2\). Note that for \(b_1<2\), \(B(x)<C(x)\) holds at least for small \(x\), which means that lower investors invariably dominate and effectively recover the characteristics of the prisoner's dilemma. A The predictions by adaptive dynamics differ from well-mixed populations only in that the parameter region of evolutionary branching is smaller (c.f. Fig. 4A in the main text) while all other thresholds are unchanged. B In contrast, individual-based simulations with \(w=10\), where equilibrium investment levels range from the minimum (black) to intermediate (grey) and the maximum (white) augmented by convergence stability (red), bi-stability (blue), and diversification (maroon), indicate much larger regions of parameter space that result in diversification than predicted by evolutionary branching.

Interactive labs:

	Figure S2 B: evolutionary branching (\(b_1=2.1, c_2=-0.42\)).
	Figure S2 B: evolutionary diversification through degenrate evolutionary branching (\(b_1=2.1, c_2=-0.3\)).
	Figure S2 B: evolutionary diversification through (almost) frozen regions (\(b_1=2.1, c_2=-0.52\)).