Figures and data in Can Hamilton’s rule be violated? | eLife

Figures
Additional files

11 figures and 1 additional file

Figures

Figure 1

Download asset Open asset

The value of coefficients $β_{s e}$ and $β_{s i}$ may depend on the specification chosen.

If $x_{c o}$ is included (as in specification II), these values will be different from when $x_{c o}$ is not included (specification I). Including an interaction term (III) or a quadratic term (IV) will also make a difference for the value of $β_{s e}$ and $β_{s i}$ . All specifications that are linear, result in Hamilton’s rules, all of which agree with the direction of selection. Hamilton’s rule with specification I says that $r_{s i} b_{s i, I} - c_{I} > 0$ if and only if $Δ \bar{x} > 0$ – where $b_{s i, I}$ is the value of $β_{s i}$ , and $c_{I}$ is minus the value of $β_{s e}$ in this specification. Hamilton’s rule with specification II says that $r_{s i} b_{s i, II} + r_{c o} b_{c o, II} - c_{II} > 0$ if and only if $Δ \bar{x} > 0$ – where $b_{s i, II}$ is the value of $β_{s i}$ , $b_{c o, II}$ is the value of $β_{c o}$ , and $c_{II}$ is minus the value of $β_{s e}$ in this specification.

https://doi.org/10.7554/eLife.41901.002

Figure 2

Download asset Open asset

Within each panel, the fitness functions are depicted in the upper part.

In panels A, B and C the bottom part depicts population structure profiles of mutant co-operators at $p = 0$ (blue) and of mutant defectors at $p = 1$ (red). In panel D the bottom part depicts the distribution of what group types co-operators (blue) and defectors (red) find themselves in, both at the same intermediate equilibrium value for $p$ . No violations of Hamilton’s rule with equal gains from switching. In panels A and B, the fitness function is $π_{C} (i) = 0.6 + 2 (i / n)$ and $π_{D} (i) = 1 + 2 (i / n)$ . In panel A the difference in average fitness between co-operators and defectors is ${\bar{π}}_{C} - {\bar{π}}_{D} = - 0.128$ , both at $p = 0$ and at $p = 1$ . Cooperation therefore is selected against at both ends. Inclusive fitness is also $- 0.128$ at both ends. Panel B has a more assorted population structure, for which this difference, as well as inclusive fitness, is $+ 0.16$ at both ends, and cooperation is selected for. No violations in equilibrium with synergies. Panel C has the same population structure profiles as panel B, but a different fitness function: $π_{C} (i) = 0.5 + 2 {(i / n)}^{2}$ and $π_{D} (i) = 1 + 2 {(i / n)}^{2}$ . Here cooperation is selected against at $p = 0$ , where ${\bar{π}}_{C} - {\bar{π}}_{D} = - 0.33$ , and selected for at $p = 1$ , where ${\bar{π}}_{C} - {\bar{π}}_{D} = + 0.45$ . Inclusive fitness is $- 0.48$ at $p = 0$ and $+ 0.6$ at $p = 1$ . Violation in a mixed equilibrium. In panel D, the fitness function is $π_{C} (i) = 0.5 + 2 {(i / n)}^{0.5}$ and $π_{D} (i) = 1 + 2 {(i / n)}^{0.5}$ . Here, ${\bar{π}}_{C} - {\bar{π}}_{D} = 0$ at $p = 0.473$ – which makes it an equilibrium – while inclusive fitness is $0.113 \neq 0$ . Details are in Appendix 1, as are computations of inclusive fitness with costs and benefits according to the regression method instead of the counterfactual method.

https://doi.org/10.7554/eLife.41901.003

Figure 3

Download asset Open asset

Dynamics for two 2-player games.

Every point in the simplex represents a population state $(f_{0}, f_{1}, f_{2})$ . The left down corner is $(1, 0, 0)$ , which has only groups with $0$ co-operators; the right down corner is $(0, 0, 1)$ , which has only groups with $2$ co-operators; the top corner is $(0, 1, 0)$ , which has only groups with $1$ co-operator. The grey lines represent different population structures, all with constant relatedness. Any given grey line gives a population state for every overall frequency $p$ of co-operators. Dynamics make populations move along the line that represents the population structure it faces. All grey lines go through the left down corner, where $p = 0$ , and the right down corner, where $p = 1$ . The straight line on the bottom reflects a totally assorted population that has no mixed groups. The higher up, the more mixed groups there are, and the less assortment there is. The highest up grey line represents a well-mixed population. No violations of Hamilton’s rule in equilibrium with synergies. In panel A, $π_{C} (1) = 0.1$ , $π_{C} (2) = 3$ , $π_{D} (0) = 2$ and $π_{D} (1) = 3.1$ . The regions where cooperation is selected for (green), and where inclusive fitness is positive (blue) are not the same, but selection always takes populations out of the parts where they disagree. Violations with anti-synergies. In panel B $π_{C} (1) = 1.9$ , $π_{C} (2) = 3$ , $π_{D} (0) = 2$ and $π_{D} (1) = 4.9$ . Here populations can settle at mixed equilibria, while inclusive fitness is not $0$ . Violations at $p = 0$ and $p = 1$ are also possible for more extreme choices of $r$ .

https://doi.org/10.7554/eLife.41901.004

Figure 4

Download asset Open asset

A road map for empirical tests of Hamilton’s rule.

The three synergy conditions are that $π_{C} (i) \geq π_{C} (1) + (i - 1) [π_{D} (1) - π_{D} (0)]$ for all $i = 1, ..., n$ , that $π_{D} (i) \geq π_{D} (n - 1) - (n - i - 1) [π_{C} (n) - π_{C} (n - 1)]$ for all $i = 0, ..., n - 1$ , and that ${\bar{π}}_{C} (p) - {\bar{π}}_{D} (p)$ increases with $p$ .

https://doi.org/10.7554/eLife.41901.005

Appendix 1—figure 1

Download asset Open asset

Population structure profiles $\underline{u}$ (blue) for co-operator invaders at $p ↓ 0$ , and $\bar{u}$ (red) for defector invaders at $p ↑ 1$ .

Defectors at $p ↓ 0$ only find themselves in groups with $0$ defectors, co-operators at $p ↑ 1$ only find themselves in groups with $n$ co-operators (not depicted in the figures). In the lower panel mutants assort more than in the upper one, and random matching would result in a population structure profile $\bar{u}$ with single spike at $1$ , and a single spike at $n - 1$ for $\bar{u}$ . Section IX describes the population structures that result in these population structure profiles. For this population structure ${\underline{u}}_{i} = {\bar{u}}_{n - i}$ is satisfied, which means that the red bars are the mirror image of the blue ones.

https://doi.org/10.7554/eLife.41901.008

Appendix 1—figure 2

Download asset Open asset

The population structure profiles are depicted in the lower half of the figure, the payoffs in the upper half.
https://doi.org/10.7554/eLife.41901.009

Appendix 1—figure 3

Download asset Open asset

The population structure profiles are depicted in the lower half of the figure, the payoffs in the upper half.
https://doi.org/10.7554/eLife.41901.010

Appendix 1—figure 4

Download asset Open asset

The population structure profiles are depicted in the lower half of the figure, the payoffs in the upper half.
https://doi.org/10.7554/eLife.41901.011

Appendix 1—figure 5

Download asset Open asset

The previous figures depicted $\underline{u}$ , which is the population structure profile at $p = 0$ , and $\bar{u}$ , which is the population structure profile at $p = 1$ .

The payoffs co-operators get, when they find themselves in groups according to $\underline{u}$ would be compared to the payoffs defectors get when they are in a group of defectors only in order to determine whether co-operators can invade defectors, at $p = 0$ , and the mirror image of that in order to determine whether defectors can invade co-operators, at $p = 1$ . The $u_{i, C} (p^{*})$ and $u_{i, D} (p^{*})$ depicted here all pertain to one and the same equilibrium $p^{*}$ , at which ${\bar{π}}_{C} (p^{*}) - {\bar{π}}_{D} (p^{*}) = 0$ . With the same payoff function as at Appendix 1—figure 4, we find an equilibrium frequency of co-operators of $p^{*} = 0.473$ , at which inclusive fitness is $\frac{1}{4} (2.103) - (0.412) \approx 0.113 \neq 0$ .

https://doi.org/10.7554/eLife.41901.012

Appendix 1—figure 6

Download asset Open asset

The $π_{D}$ according to the regression method coincides the true $π_{D}$ at $p ↓ 0$ (top panel, red), but the $π_{C}$ according to the regression method (dotted blue) differs from the true $π_{C}$ (solid blue).

The $π_{D}$ and $π_{C}$ according to the regression method coincide with each other at $p = 0.5$ (middle panel, dotted lines) but do not coincide with the true $π_{D}$ (solid red) and $π_{C}$ (solid blue). The $π_{C}$ according to the regression method coincides the true $π_{C}$ at $p ↑ 1$ (bottom panel, blue), but the $π_{D}$ according to the regression method (dotted red) differs from the true $π_{D}$ (solid red).

https://doi.org/10.7554/eLife.41901.013

Appendix 1—figure 7

Download asset Open asset

With parameters $α = \frac{1}{2}$ , $β = 2$ and $γ = 0.4$ , the equilibrium frequency of co-operators, at which ${\bar{π}}_{C} (p^{*}) = {\bar{π}}_{D} (p^{*})$ , is $p^{*} = 0.465$ .

Inclusive fitness there is $\frac{1}{4} (1.82) - (0.4) \approx 0.055 \neq 0$ . Section X specifies the $f_{p} (x)$ for this example.

https://doi.org/10.7554/eLife.41901.014

Additional files

Transparent reporting form: https://doi.org/10.7554/eLife.41901.006
Download elife-41901-transrepform-v2.pdf

Download links

A two-part list of links to download the article, or parts of the article, in various formats.

Downloads (link to download the article as PDF)

Article PDF

Open citations (links to open the citations from this article in various online reference manager services)

Mendeley

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Matthijs van Veelen

(2018)

Can Hamilton’s rule be violated?

eLife 7:e41901.

https://doi.org/10.7554/eLife.41901

Sign up for email alerts

Privacy notice