Spatial social dilemmas promote diversity: Difference between revisions
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''B'' for populations with | ''B'' for populations with | ||
====Interactive labs, Figure 1 A==== | |||
{| class=wikitable align=center | {| class=wikitable align=center | ||
|colspan=" | |+ | ||
| colspan="4" | | |||
Dynamics in well-mixed populations with size | Dynamics in well-mixed populations with size | ||
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! well-mixed !! | |||
|- | |- | ||
! | |||
|{{EvoLudoTrigger| | |||
options="--module cSD --benefits 0 1 --costs 0 0.1 --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"}} | |||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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 1,10 --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| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0. | 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"}} | ||
|- | |- | ||
! | |||
|{{EvoLudoTrigger| | |||
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"}} | |||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0. | 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| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0. | 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==== | |||
{| class=wikitable align=center | |||
|+ | |||
| colspan="4" | | |||
Dynamics in on | |||
|- | |- | ||
! lattice, | |||
|- | |||
! | |||
|{{EvoLudoTrigger| | |||
options="--module cSD --benefits 0 1 --costs 0 0.1 --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"}} | |||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0.1 --delay 100 --fitnessmap exp 1, | 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"}} | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0. | 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"}} | ||
|- | |- | ||
! | |||
|{{EvoLudoTrigger| | |||
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"}} | |||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,10 --geometry | 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| | |{{EvoLudoTrigger| | ||
options="--module cSD --benefits 0 1 --costs 0 0.3 --delay 100 --fitnessmap exp 1,100 --geometry | 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, | Interestingly, these analytical predictions are not always borne out in individual-based models when the selection strength, | ||
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Interestingly, for strong selection in lattice populations not only lower investors can invade for | Interestingly, for strong selection in lattice populations not only lower investors can invade for | ||
====Interactive labs, Figure 2==== | |||
{| class=wikitable align=center | {| class=wikitable align=center | ||
|colspan="3"| | |colspan="3"| | ||
Dynamics on a square | Dynamics on a square | ||
|- | |||
! '''A-C:''' | |||
|- | |- | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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 0.1 --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"}} | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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 1,10 --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"}} | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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" | 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"}} | ||
|} | |} | ||
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=== Saturating benefits === | === Saturating benefits === | ||
The continuous prisoner's dilemma was first introduced in [[#References|Killingback et al. 1999]] with saturating benefits \(B(x)=b_0\left( | 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 | ||
\begin{align} | \begin{align} | ||
\label{eq:d:kdk:dB} | \label{eq:d:kdk:dB} | ||
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[[Image:Diversification - continuous PD, saturating benefits (dynamics).jpg|825px]] | [[Image:Diversification - continuous PD, saturating benefits (dynamics).jpg|825px]] | ||
'''Figure 3:''' Continuous prisoner's dilemma with saturating benefits | '''Figure 3:''' Continuous prisoner's dilemma with saturating benefits \(B(x)=b_0\left(1-\exp{(-b_1 x)}\right)\) and linear costs | ||
and strong selection, | and strong selection, | ||
The pairwise invasibility plots ( | The pairwise invasibility plots ( | ||
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''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'' for strong selection, | ''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'' for strong selection, | ||
====Interactive labs, Figure 3==== | |||
{| class=wikitable align=center | {| class=wikitable align=center | ||
|colspan="2"| | |colspan="2"| | ||
Dynamics on a square | |||
Dynamics on a square | |- | ||
! '''A-C:''' weak selection | |||
|- | |- | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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 1,1 --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"}} | ||
|{{EvoLudoTrigger| | |{{EvoLudoTrigger| | ||
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" | 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{kdk:dB}d) and reveals that | 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{kdk:dB}d) and reveals that | ||
== Evolution in the continuous snowdrift game == | == Evolution in the continuous snowdrift game == |
Revision as of 06:49, 10 September 2024
Cooperative investments in social dilemmas can spontaneously diversify into stably co-existing high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavourable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.
Social dilemmas with continuous traits
Social dilemmas are important mathematical metaphors for studying the problem of cooperation. The best studied models of social dilemmas are the prisoner's dilemma and the snowdrift game. Traditionally, such models are often restricted to the two distinct strategies of cooperate,
For example, in the donation game, which is the most prominent version of the prisoner's dilemma, cooperators confer a benefit
Continuous Donation game
A natural translation of the donation game to continuous traits is based on cost and benefit functions,
Continuous Snowdrift game
In a weaker form of a social dilemma, the snowdrift game, cooperators also provide benefits,
The gradual evolution of continuous traits can be described using the framework of adaptive dynamics. Below we extend this framework to spatial settings by amalgamating adaptive dynamics and pair approximation into spatial adaptive dynamics. This provides the toolbox to investigate the impact of spatial structures on the evolution of cooperation in the prisoner's dilemma as well as the snowdrift games.
Evolution in the continuous prisoner's dilemma
In the continuous prisoner's dilemma the payoff to an individual with strategy
Linear costs and benefits
The evolutionary analysis becomes particularly simple for linear benefit and cost functions
Figure 1: Equilibrium investment levels (mean
Interactive labs, Figure 1 A
Dynamics in well-mixed populations with size | |||
well-mixed | |||
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Interactive labs, Figure 1 B
Dynamics in on | |||
lattice, |
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Interestingly, these analytical predictions are not always borne out in individual-based models when the selection strength,
Figure 2: Linear continuous prisoner's dilemma in
Interactive labs, Figure 2
Dynamics on a square | ||
A-C: |
D-F: |
G-I: |
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For mutants
Saturating benefits
The continuous prisoner's dilemma was first introduced in Killingback et al. 1999 with saturating benefits
Figure 3: Continuous prisoner's dilemma with saturating benefits
Interactive labs, Figure 3
Dynamics on a square | |
A-C: weak selection |
D-F: stronger selection |
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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{kdk:dB}d) and reveals that
Evolution in the continuous snowdrift game
In the continuous snowdrift game the payoff to an individual with strategy
The selection gradient for the continuous snowdrift game in structured populations, \eq{d:dB}, is given by
Quadratic costs and benefits
For suitable cost and benefit functions the (spatial) adaptive dynamics is analytically accessible. Here we focus on the quadratic cost and benefit functions used in Doebeli et al. 2004 for the well-mixed case:
For small resident values
For the above quadratic costs and benefits the singular strategy is given by
We note that if the singular point exists,
Conclusions & Discussion
In all scenarios considered here we find that population structures can promote and facilitate spontaneous diversification in social dilemmas into high and low investors, especially when selection is strong. However, at the same time, classical evolutionary branching tends to be inhibited, but compensated for by other modes of diversification. We derive an extension of adaptive dynamics for continuous games in (spatially) structured populations based on pair approximation, which tracks the frequencies of mutant-resident pairs during invasion. It turns out that predictions derived from this spatial adaptive dynamics framework are independent of selection strength. More precisely, selection strength only scales the magnitude of the selection gradient as well as that of convergence and evolutionary stability but neither affects the location of singular strategies nor their stability. Nevertheless, from the invasion analysis of mutant traits
Structured populations offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Trait variation is more easily maintained in structured populations due to the slower spreading of advantageous traits as compared to well-mixed populations. Spatial adaptive dynamics is unable to capture these new modes of diversification because of the underlying assumption that the resident population is composed of discrete traits (monomorphic before branching and dimorphic or polymorphic after branching). Nevertheless, invasion analysis and pairwise invasibility plots,
Previous attempts at amalgamating adaptive dynamics and spatial structure have not observed spontaneous diversification or evolutionary branching. In particular, Allen et al. (2013) augment adaptive dynamics by structure coefficients (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
Interestingly, in well-mixed populations evolutionary branching is only observed for the continuous snowdrift game, where two distinct traits of high and low investors can co-exist and essentially engage in a classical (discrete) snowdrift game. In contrast, in structured populations with death-birth updating, evolutionary branching is only observed for prisoner's dilemma type interactions where lower investments invariably dominate higher ones, which applies both in the continuous prisoner's dilemma as well as the continuous snowdrift game with sufficiently high costs. The reason for this surprising difference can be understood intuitively by considering the preferred spatial configurations in the two classical (discrete) games: in the prisoner's dilemma cooperators form compact clusters to reduce exploitation by defectors (minimize surface), while in the snowdrift game filament like clusters form because it is advantageous to adopt a strategy that is different from that of the interaction partners (maximize surface). In the continuous variants of those games it is naturally much harder to maintain and spread distinct traits in fragmented filament-like structures because they are more prone to effects of noise than compact clusters. Effectively this fragmentation inhibits evolutionary branching because diverging traits tend to trigger further fragmentation and as a consequence do not survive long enough to get established and form their own branch. In contrast, the compact clusters promoted by the prisoner's dilemma provide structural protection for higher investors and thus help drive diversification.
Because of global competition the spatial dynamics for birth-death updating is (unsurprisingly) much closer to results for well-mixed populations. For example, evolutionary branching was again only observed for continuous snowdrift game. Also because of global competition, structured populations are updated in a non-uniform manner. In particular, regions of high payoffs experience a much higher turnover than regions of low payoffs. For strong selection this can result in almost frozen parts of the population. As a consequence unsuccessful traits are able to stay around for long times and, in some cases, those traits turn out to be advantageous again at later times when the surroundings have sufficiently changed, so that the stragglers then contribute to diversification. This mode of diversification, however, introduces historical contingencies where the evolutionary end state can sensitively depend on the initial configuration.
Overall, we find that evolutionary diversification is a robust feature of continuous spatial games, and that spatial structure can sometimes hinder, but generally promotes diversification through modes of diversification that complement traditional evolutionary branching.
Publications
- Hauert, C. & Doebeli, M. (2021) Spatial social dilemmas promote diversity, Proc. Natl. Acad. Sci. USA 118 42 e2105252118 doi: 10.1073/pnas.2105252118
References
- Allen, B., Nowak, M.A., & Dieckmann, U. (2013) Adaptive dynamics with interaction structure. Am. Nat. 181 (6) E139–E163.
- Doebeli M, Hauert C, Killingback T (2004) The evolutionary origin of cooperators and defectors. Science 306 (5697) 859–62.
- Doebeli M, Hauert C, Killingback T (2013) A comment on "Towards a rigorous framework for studying 2-player continuous games" by Shade T. Shutters, Journal of Theoretical Biology 321, 40--43, 2013. J. Theor. Biol. 336 240–241.
- Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428 643–646.
- Killingback, T., Doebeli, M., & Knowlton, N. (1999) Variable investment, the continuous prisoner's dilemma, and the origin of cooperation. Proc. R. Soc. B 266 1723–1728.
- Ohtsuki, H., Hauert, C., Lieberman, E., & Nowak, M.A. (2006) A simple rule for the evolution of cooperation on graphs. Nature 441 502–505.
- Parvinen, K., Ohtsuki, H., & Wakano, J.Y. (2017) The effect of fecundity derivatives on the condition of evolutionary branching in spatial models. J. Theor. Biol. 416 129–143.
- Tarnita, C.E., Wage, N., & Nowak, M.A. (2011) Multiple strategies in structured populations. Proc. Natl. Acad. Sci. USA 108 2334–2337.