Spatial social dilemmas promote diversity: Difference between revisions

A natural translation of the donation game to continuous traits is based on cost and benefit functions, \(C(x)\) and \(B(x)\), where the trait \(x\in[0,x_\text{max}]\) represents a level of cooperative investment that can vary continuously. In the spirit of the donation game, an individual with strategy \(x\) interacting with an \(y\)-strategist obtains a payoff \(P(x,y)=B(y)-C(x)\). Assuming that  

# zero investments incur no costs and provide no benefits, \(B(0)=C(0)=0\),  

# benefits exceed costs, \(B(x)>C(x)\geq0\), and  

# are increasing functions, \(B^\prime(x), C^\prime(x)\geq0\),  

recovers the social dilemma of the donation game for continuous investment levels: the level of investment invariably evolves to zero, which corresponds to pure defection, despite the fact that both players would be better off at non-zero investment levels. The reason is that an actor can only influence the cost \(C(x)\) of an interaction, but not the received benefit \(B(y)\), and hence selection can only act to minimize costs.  

In the continuous prisoner's dilemma the payoff to an individual with strategy \(y\) interacting with an \(x\)-strategist is \(P(y,x)=B(x)-C(y)\). This implies that the selection gradient in well-mixed populations is proportional to \(-C'(x)\), and hence, assuming monotonously increasing costs, no singular strategy exists apart from the pure defection state \(x=0\), and the population always evolves to that state, regardless of how large the benefits of cooperation are. In contrast, for death-birth updating in structured populations the selection gradient for [[spatial adaptive dynamics]] becomes

Thus, in structured populations a singular strategy, \(x^\ast\), may exist as a solution to \(D_\text{db}(x^\ast)=0\). If \(x^\ast\) exists and is convergence stable then it is also evolutionarily stable because the two stability conditions are identical ([[spatial adaptive dynamics#Selection gradient, convergent and evolutionary stability|the mixed derivatives are zero]]). In particular, cooperation can be maintained in spatially structured populations, a result that is of course in line with classical theory. More specifically, cooperative investments can increase if the marginal benefits, \(B^\prime(x)\), exceed the \(k\)-fold marginal costs, \(C^\prime(x)\), which is reminiscent of the \(b>ck\)-rule in the traditional donation game (see [[Spatial social dilemmas promote diversity#References|Ohtsuki et al. 2006]]).

The evolutionary analysis becomes particularly simple for linear benefit and cost functions \(B(x)=x\) and \(C(x)=r x\), where \(r\) denotes the (marginal) cost-to-benefit ratio. Again, population structures are capable of supporting cooperation for sufficiently small \(r\). More precisely, the selection gradient \(D_\text{db}(x)=w\frac{k-2}{k(k-1)}(1-k r)\) is reduced to a constant (c.f. Eq. \eqref{eq:d:pd:dB}), and hence no singular strategy exists. However, the gradient changes sign for \(r^\ast=1/k\). Thus, for favourable cost-to-benefit ratios, \(r<r^\ast\), investments increase to their maximum, whereas for harsher conditions, \(r>r^\ast\) cooperative investments cannot be sustained, see Fig. 1.

''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.

| colspan="4" |

Dynamics in well-mixed populations with size \(N=10^4\) for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).

options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1,10 --geometry M --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1,100 --geometry M --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,10 --geometry M --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,100 --geometry M --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

|}

 

====Interactive labs, Figure 1 B====

Dynamics in on \(N=100\times100\) square lattices with \(k=4\) neighbours for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1,10 --geometry m --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1,100 --geometry m --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,1 --geometry m --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,10 --geometry m --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

|{{EvoLudoTrigger|

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,100 --geometry m --init gaussian 0.5,0.01 --interactions r1 --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

Interestingly, these analytical predictions are not always borne out in individual-based models when the selection strength, \(w\), is large. The reasons for the differences can be appreciated by looking at pairwise invasibility plots (\(PIP\)), which show the regions in \((x,y)\)-space in which a mutant \(y\) can invade a resident \(x\), i.e., regions for which the invasion fitness \(f(x,y)\) is positive. To construct these plots, we first solve the dynamical equation \(\dot q_{m|m}=0\) (see [[spatial adaptive dynamics]]) to find the equilibrium \(q^\ast_{m|m}\) for given resident and mutant trait values \(x, y\) and then determine the sign of the corresponding invasion fitness. An example is shown in Fig. 2g with \(r>1/k\) so that adaptive dynamics predicts evolution to pure defection.  

options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1,10 --geometry n --init gaussian 0.1,0.01 --interactions all --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,10 --geometry n --init gaussian 0.1,0.01 --interactions all --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,100 --geometry n --init gaussian 0.1,0.01 --interactions all --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,1 --view Strategies_-_Distribution"}}

For mutants \(y\) sufficiently close to the resident \(x\), \(f(x,y)>0\) only holds for \(y<x\), so that evolution by (infinitesimally) small mutations, as envisaged in adaptive dynamics, indeed leads to ever smaller investment. However, the \(PIP\) also shows positive invasion fitness for mutants \(y>x\) if \(y\) is sufficiently larger than \(x\) (white area above the diagonal in Fig. 2g). In individual-based models, mutations are normally distributed around the parental trait and of finite size such that stochastically appearing larger mutations or sequences of smaller mutations in the right direction can give rise to the co-existence of high and low investors (see Fig. 2i). Thus, evolutionary diversification is possible in individual-based models despite the absence of evolutionary branching points in the adaptive dynamics, and even if the analytical prediction is evolution of pure defection. Note that this mode of evolutionary diversification is a feature of spatial structure that does not occur in the corresponding well-mixed models.

The continuous prisoner's dilemma was first introduced in [[#References|Killingback et al. 1999]] with saturating benefits \(B(x)=b_0\left( \exp{(-b_1 x)} \right)\) and linear costs \(C(x)=c_0 x\) (\(b_0, b_1, c_0>0\)). Using individual-based simulations these authors showed that structured populations are capable of supporting cooperation, whereas well-mixed populations are not. Spatial adaptive dynamics provides an analytical perspective on these results. For the given cost and benefit functions, we have:

The singular point is always convergence stable as well as evolutionarily stable (see Fig. 3b).  

'''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)  

Dynamics on a square \(100\times100\) lattice with \(k=4\) neighbours for saturating benefits, \(B(x)=b_0\left(1-\exp{(-b_1 x)}\right)\) and linear costs \(C(x)=c_0 x\) for \(b_0=8, b_1=1, c_0=0.7\) with \(b_0=8, b_1=1, c_0=0.7\) with weak, \(w=1\) and stronger \(w=10\) selection.

options="--module cSD --benefits 4 8,1 --costs 0 0.7 --delay 100 --fitnessmap exp 1,1 --geometry n --init gaussian 0.2,0.02 --interactions all --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,2 --view Strategies_-_Distribution"}}

options="--module cSD --benefits 4 8,1 --costs 0 0.7 --delay 100 --fitnessmap exp 1,10 --geometry n --init gaussian 0.2,0.02 --interactions all --mutation 0.01 gaussian 0.01 --popsize 100x --popupdate d --timestep 20 --traitrange 0,2 --view Strategies_-_Distribution"}}

Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A pairwise invasibility plot for strong selection is shown in Fig. 3D) and reveals that \(x^\ast\) is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns \(x^\ast\) into a (degenerate) branching point, i.e., a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in [[#References|Killingback et al. 1999]], but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best performing neighbour, including itself, but this update rule essentially corresponds to death-birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in [[#References|Killingback et al. 1999]] is of the same type as the one seen here for strong selection.

Following [[spatial adaptive dynamics]], the selection gradient for the continuous snowdrift game in structured populations with death-birth updating is given by

The numerator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if the condition for evolution away from zero, \(b_1>\frac{k}{k+1}c_1\), is satisfied. Similarly, the denominator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if \(2b_2(k+1)<c_2k\) (recall \(b_2, c_2<0\)). It is clear from Eq. \eqref{eq:css:csd:dB} that this is also the condition for convergence stability. Thus, if a  singular point, \(x^\ast\), exists, cooperative traits either evolve away from zero and \(x^\ast\) is convergence stable, or the trait cannot evolve away from zero and \(x^\ast\) is a repellor.

 

 

 

 

The results for well-mixed populations summarized in Fig. 4A & B can be replicated and verified in the tutorial [[Origin of Cooperators and Defectors]]. The following labs illustrate the dynamical regimes in spatially structured populations.  

Dynamical regimes in the spatial continuous snowdrift game for moderate selection, \(w=10\), on \(N=100\times100\) square lattices with \(k=4\) neighbours.

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! branching, \(c_2=-0.5,b_1=1.65\)

! attractor, \(c_2=-0.3,b_1=1.9\)

|{{EvoLudoTrigger|

! repellor, \(c_2=-0.75,b_1=1.5\)

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.75 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

! full investments, \(c_2=-0.5,b_1=2\)

 

 

[[Image:Diversification - continuous SD (dB modes).jpg|825px]]

 

<strong>A-D</strong>  depict pairwise invasibility plots (<em>PIP</em>, top row) for the four scenarios illustrating increased spatial diversification due to strong selection (c.f. parameter combinations <strong>a-d</strong> 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 \(w=100\)).

<strong>A</strong> the <em>PIP</em> 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.

<strong>B</strong>  higher investing mutants can always invade, but so can traits investing markedly less.

<strong>C</strong>  selection strength distorts the <em>PIP</em> in the vicinity of the convergence and evolutionarily stable \(x^\ast\) resulting in a degenerate form of branching (c.f. Fig. 3).

<strong>D</strong> the <em>PIP</em> indicates that \(x^\ast\) is a repellor such that residents with \(x&lt;x^\ast\) are invaded by lower investors while those with \(x&gt;x^\ast\) by higher investors. However, as a consequence of strong selection, mutants with markedly higher (lower) investments can also invade.

<strong>E-H</strong> depict corresponding plots of regions of mutual invasibility (<em>PIP\(^\text{2}\)</em>, white regions). Regions where mutants or residents are unable to invade (grey) are marked with \((+,-)\) and \((-,+)\), respectively. The vector field shows the [[Spatial adaptive dynamics#Evolutionary divergence of coexisting traits|evolutionary divergence]] 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.<br>

<em>Parameters:</em> \(b_2=-1/4, c_1=2\), <strong>A</strong> \(b_1=1.55\), \(c_2=-0.6\); <strong>B</strong> \(b_1=1.65\), \(c_2=-0.625\); <strong>C</strong> \(b_1=1.9\), \(c_2=-0.3\); <strong>D</strong> \(b_1=1.5\), \(c_2=-0.72\).

 

====Interactive labs, Figure 5====

Modes of diversification in the spatial continuous snowdrift game. The parameter settings refer to points <strong>a,b,c,d</strong> in Fig. 4E.

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

! Figure 5A & E, Figure 4E a, \(c_2=-0.6,b_1=1.55\)

|{{EvoLudoTrigger|

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options="--game cSD --benefitfcn 11 --benefitparams 1.65,-0.25 --costfcn 1 --costparams 2,-0.625 --delay 100 --fitnessmap exp --geometry n --initmean 0.2 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

! Figure 5C & G, Figure 4E c, \(c_2=-0.3,b_1=1.9\)

! Figure 5D & H, Figure 4E d, \(c_2=-0.72,b_1=1.5\)

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.72 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

Previous attempts at amalgamating adaptive dynamics and spatial structure have not observed spontaneous diversification or evolutionary branching. In particular, [[Spatial social dilemmas promote diversity#References|Allen et al. (2013)]] augment adaptive dynamics by structure coefficients ([[Spatial social dilemmas promote diversity#References|Tarnita et al. 2011]]), which restrict the analysis to weak selection. Moreover, their framework is fundamentally different from ours because it is based on fixation probabilities rather than invasion fitness. More specifically, their analysis is based on the fixation probabilities \(p_x\) and \(p_y\) of two types \(x\) and \(y\) in a population consisting of \(x\) and \(y\): \(x\) is favoured over \(y\) if \(p_x>p_y\) (see  Eq. 4 and formulas following Eq. 7 in [[Spatial social dilemmas promote diversity#References|Allen et al. 2013]]). Clearly, it is then impossible to have both types be favoured in an \((x,y)\)-population, but this is exactly the requirement for mutual coexistence. In fact, Allen et al. acknowledge that fixation probabilities are "less appropriate" for studying evolutionary branching and that instead "establishment probabilities" should be considered for studying diversification. As a consequence their adaptive dynamics derivation precludes its most intriguing feature, that of evolutionary branching. However, in the limit of weak selection the two approaches are compatible in some aspects. For example, the selection gradient and the condition for convergence stability reported for the continuous prisoner's dilemma are identical (c.f. Eqs. \eqref{eq:d:csd:dB}, \eqref{eq:css:csd:dB} and their Eqs. (11), (13)). In our framework, the use of invasion fitness allows to determine evolutionary stability, and the corresponding \(PIP\)'s reveal the potential for degenerate forms of branching – but only for stronger selection (see Fig. 3). Another attempt at spatial adaptive dynamics considers spatial structure in the form of demes ([[Spatial social dilemmas promote diversity#References|Parvinen et al. 2017]]). However, quite naturally such deme structures suppress branching because in the long run it is impossible to maintain multiple co-existing traits in small demes, which quickly become homogeneous. Instead, branching is only possible if demes are large enough to support branching in every single one.