# Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games

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## Abstract

**:**

## 1. Introduction

## 2. Model Description

#### 2.1. Demographical Dynamics

#### 2.2. Evolutionary Dynamics

## 3. Results

#### 3.1. Quadratic Benefit and Cost Functions

**Theorem**

**1.**

- Assume B is accelerating and that ${\mu}_{0}>{\mu}_{V}$, as is the case if C is decelerating. For $\mu <{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For ${\mu}_{V}<\mu <{\mu}_{0}$, ${r}^{*}\left(\mu \right)$ is decreasing and the evolutionary dynamics is bistable. For $\mu >{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.
- Assume B is accelerating and that ${\mu}_{0}<{\mu}_{V}$. For $\mu <{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For ${\mu}_{0}<\mu <{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor. For $\mu >{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.
- Assume B is decelerating and ${\mu}_{0}>{\mu}_{V}$, as is the case if C is accelerating. For $\mu <{\mu}_{0}$, we have ${r}^{*}\left(\mu \right)=0$ and selection is thus towards no cooperation. For $\mu >{\mu}_{V}$, ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor.
- Assume B is decelerating and ${\mu}_{0}<{\mu}_{V}$. For ${\mu}_{0}<\mu <{\mu}_{V}$, ${r}^{*}\left(\mu \right)$ is increasing and convergence stable, i.e., an evolutionary attractor. For $\mu >{\mu}_{V}$, we have ${r}^{*}\left(\mu \right)=1$ and selection is thus towards full cooperation.

**Proof.**

#### 3.2. General Cost and Benefit Functions

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Investigations

#### Appendix A.1. Condition for Evolutionary Stability

#### Appendix A.2. Impossible Region in the μ − σ^{2}-Plane

#### Appendix A.3. Implicit Differentiation

## Appendix B. Individual-Based Simulations

## References

- Dugatkin, L.A. Cooperation in animals: An evolutionary overview. Biol. Philos.
**2002**, 17, 459–476. [Google Scholar] [CrossRef] - Hölldobler, B.; Wilson, E.O. The Ants. 1990. Available online: http://www.hup.harvard.edu/catalog.php?isbn=9780674040755 (accessed on 31 October 2018).
- Hamilton, W.D. Altruism and related phenomena, mainly in social insects. Ann. Rev. Ecol. Syst.
**1972**, 3, 193–232. [Google Scholar] [CrossRef] - Cornforth, D.M.; Sumpter, D.J.T.; Brown, S.P.; Brännström, Å. Synergy and group size in microbial cooperation. Am. Nat.
**2012**, 180, 296–305. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sachs, J.L.; Mueller, U.G.; Wilcox, T.P.; Bull, J.J. The evolution of cooperation. Q. Rev. Biol.
**2004**, 79, 135–160. [Google Scholar] [CrossRef] [PubMed] - Nowak, M.A. Five rules for the evolution of cooperation. Science
**2006**, 314, 1560–1563. [Google Scholar] [CrossRef] [PubMed] - Hamilton, W.D. The evolution of altruistic behaviour. Am. Nat.
**1963**, 97, 354–356. [Google Scholar] [CrossRef] - Hofbauer, J.; Sigmund, K. Evolutionary Games Population Dynamics. 1998. Available online: https://www.cambridge.org/core/books/evolutionary-games-and-population-dynamics/A8D94EBE6A16837E7CB3CED24E1948F8 (accessed on 31 October 2018).
- Hofbauer, J.; Sigmund, K. Evolutionary game dynamics. Bull. Am. Math. Soc.
**2003**, 40, 479–519. [Google Scholar] [CrossRef] - Grafen, A. The hawk-dove game played between relatives. Anim. Behav.
**1979**, 27, 905–907. [Google Scholar] [CrossRef] - Eshel, I.; Cavalli-Sforza, L.L. Assortment of encounters and evolution of cooperativeness. Proc. Natl. Acad. Sci. USA
**1982**, 79, 1331–1335. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tao, Y.; Lessard, S. Frequency-dependent selection in sexual family-structured populations. J. Theor. Biol.
**2002**, 217, 525–534. [Google Scholar] [CrossRef] [PubMed] - Bergstrom, T.C. The algebra of assortative encounters and the evolution of cooperation. Int. Game Theor. Rev.
**2003**, 5, 211–228. [Google Scholar] [CrossRef] - Taylor, C.; Nowak, M.A. Evolutionary game dynamics with non-uniform interaction rates. Theor. Popul. Biol.
**2006**, 69, 243–252. [Google Scholar] [CrossRef] [PubMed][Green Version] - Allen, B.; Nowak, M.A. Games among relatives revisited. J. Theor. Biol.
**2015**, 378, 103–116. [Google Scholar] [CrossRef] [PubMed] - Cooney, D.; Allen, B.; Veller, C. Assortment and the evolution of cooperation in a Moran process with exponential fitness. J. Theor. Biol.
**2016**, 409, 38–46. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brush, E.; Brännström, Å.; Dieckmann, U. Indirect reciprocity with negative assortment and limited information can promote cooperation. J. Theor. Biol.
**2018**, 443, 56–65. [Google Scholar] [CrossRef] [PubMed] - Doebeli, M.; Hauert, C.; Killingback, T. Evolutionary origin of cooperators and defectors. Science
**2004**, 306, 859–862. [Google Scholar] [CrossRef] [PubMed] - Deng, K.; Chu, T. Adaptive evolution of cooperation through Darwinian dynamics in public goods games. PLoS ONE
**2011**, 6, e25496. [Google Scholar] [CrossRef] [PubMed] - Brännström, Å.; Gross, T.; Blasius, B.; Dieckmann, U. Consequences of fluctuating group size for the evolution of cooperation. J. Math. Biol.
**2011**, 63, 263–281. [Google Scholar] [CrossRef] [PubMed] - Ito, K.; Ohtsuki, H.; Yamauchi, A. Relationship between aggregation of rewards and the possibility of polymorphism in continuous snowdrift games. J. Theor. Biol.
**2015**, 372, 47–53. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sasaki, T.; Okada, I. Cheating is evolutionarily assimilated with cooperation in the continuous snowdrift game. Biosystems
**2015**, 131, 51–59. [Google Scholar] [CrossRef] [PubMed][Green Version] - Molina, C.; Earn, D.J. Evolutionary stability in continuous nonlinear public goods games. J. Math. Biol.
**2017**, 74, 499–529. [Google Scholar] [CrossRef] [PubMed] - Geritz, S.A.H.; Kisdi, É.; Meszéna, G.; Metz, J.A.J. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evolut. Ecol.
**1998**, 12, 35–57. [Google Scholar] [CrossRef][Green Version] - Metz, J.A.J.; Geritz, S.A.H.; Meszéna, G.; Jacobs, F.J.A.; Van Heerwaarden, J.S. Adaptive dynamics: A geometrical study of the consequences of nearly faithful reproduction. In Stochastic and Spatial Structures of Dynamical Systems; van Strien, S.J., Verduyn, L.S.M., Eds.; Amsterdam: New York, NY, USA, 1995; pp. 183–231. [Google Scholar]
- Dercole, F.; Rinaldi, S. Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications. 2008. Available online: https://press.princeton.edu/titles/8703.html (accessed on 31 October 2018).
- Brännström, Å.; Johansson, J.; von Festenberg, N. The hitchhiker’s guide to adaptive dynamics. Games
**2013**, 4, 304–328. [Google Scholar] [CrossRef][Green Version] - Gilbert, O.M.; Foster, K.R.; Mehdiabadi, N.J.; Strassmann, J.E.; Queller, D.C. High relatedness maintains multicellular cooperation in a social amoeba by controlling cheater mutants. Proc. Natl. Acad. Sci. USA
**2007**, 104, 8913–8917. [Google Scholar] [CrossRef] [PubMed][Green Version] - West, S.A.; Pen, I.; Griffin, A.S. Conflict and cooperation—Cooperation and competition between relatives. Science
**2002**, 296, 72–75. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Examples of cost and benefit functions. Parameters used are for decelerating benefit ${b}_{2}$ = −1, ${b}_{1}=20$; for accelerating benefit ${b}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{b}_{1}=1$; for decelerating cost ${c}_{2}=-2.75,\phantom{\rule{4pt}{0ex}}{c}_{1}=6$; and for accelerating cost ${c}_{2}=2,\phantom{\rule{4pt}{0ex}}{c}_{1}=1$. In choosing the horizontal range for the plots of the benefit functions, we have assumed $N=10$.

**Figure 2.**Bifurcation diagrams showing how the evolutionary dynamics depend on average relatedness for all four possible combinations of the accelerating and decelerating benefit and cost functions in Figure 1. Green thin line represents convergence stability without evolutionary stability, blue thick line represents convergence stability with evolutionary stability, black dotted line represents neither convergence stability nor evolutionary stability, and black thick line represents evolutionary stability without convergence stability. We see that the four cases are qualitatively different. We also see that, when B and C are both decelerating, a certain range of relatedness leads to evolutionary branching. The parameters used were in all cases $N=10$, and ${b}_{1},{b}_{2},{c}_{1},{c}_{2}$ as in Figure 1.

**Figure 3.**Plots of variance in relatedness against average relatedness. The grey region represents combinations of mean and variance that are not logically possible (see Appendix A.2). We see that for decreasing cost and benefit, the region with evolutionary branching shrinks with increased variance in relatedness. We also see that. for increasing cost and benefit, the region with neither convergence stability nor evolutionary stability shrinks with increased variance in relatedness. In the latter case, with increased average relatedness or variance in relatedness, the singular strategy moves from evolutionary stability (ESS) without convergence stability to neither evolutionary stability nor convergence stability. The parameters used are in both cases $N=10$, and ${b}_{1},{b}_{2},{c}_{1},{c}_{2}$ as in Figure 1.

**Figure 4.**Simulations using group size $N=10$, and B and C both decelerating or both accelerating. As predicted by the analytical investigation, when B and C are decelerating, the population undergoes evolutionary branching when $\mu =0.4$, and it reaches evolutionary stability when $\mu =0.8$. When B and C are accelerating, bistability occurs when $\mu =0.5$. In all three cases, ${\sigma}^{2}=0$.

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**MDPI and ACS Style**

Coder Gylling, K.; Brännström, Å. Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games. *Games* **2018**, *9*, 87.
https://doi.org/10.3390/g9040087

**AMA Style**

Coder Gylling K, Brännström Å. Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games. *Games*. 2018; 9(4):87.
https://doi.org/10.3390/g9040087

**Chicago/Turabian Style**

Coder Gylling, Kira, and Åke Brännström. 2018. "Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games" *Games* 9, no. 4: 87.
https://doi.org/10.3390/g9040087