Altruism in populations at the extinction transition

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Altruism in populations at the extinction transition

Konstantin Klemm and Nagi Khalil

Phys. Rev. Research 2, 013374 – Published 30 March 2020

Abstract

We study the evolution of cooperation as a birth-death process in spatially extended populations. The benefit from the altruistic behavior of a cooperator is implemented by decreasing the death rate of its direct neighbors. The cost of cooperation is the increase of a cooperator's death rate proportional to the number of its neighbors. When cooperation has higher cost than benefit, cooperators disappear. Then, the dynamics reduces to the contact process with defectors as the single-particle type. Increasing the benefit-cost ratio above 1, the extinction transition of the contact process splits into a set of nonequilibrium transitions between four regimes when increasing the baseline death rate $p$ as a control parameter: (i) defection only, (ii) coexistence, (iii) cooperation only, (iv) extinction. We investigate the transitions between these regimes. As the main result, we find that full cooperation is established at the extinction transition as long as benefit is strictly larger than cost. Qualitatively identical phase diagrams are obtained for populations on square lattices and in pair approximation. Spatial correlations with nearest neighbors only are thus sufficient for sustained cooperation.

Received 20 June 2019
Accepted 10 March 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013374

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Population dynamics

Social dynamics

Nonlinear Dynamics

Authors & Affiliations

Konstantin Klemm ^1,2,* and Nagi Khalil ^3,†

¹Institute for Cross-Disciplinary Physics and Complex Systems IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain
²Bioinformatics Group, Department of Computer Science, Universität Leipzig, Härtelstr. 16-18, 04107 Leipzig, Germany
³Escuela Superior de Ciencias Experimentales y Tecnología (ESCET) & GISC, Universidad Rey Juan Carlos, Móstoles 28933, Madrid, Spain

^*klemm@ifisc.uib-csic.es
^†nagi.khalil@urjc.es

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Vol. 2, Iss. 1 — March - May 2020

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Images

Figure 1
Average densities of agents on a square lattice of $50 \times 50$ sites as a function of parameters $p$ and $ε$ (large center panel). For each combination of parameters, the agents' concentrations $〈c〉$ and $〈d〉$ are encoded by color. Red indicates high concentration of cooperators; blue indicates high concentration of defectors; green is for coexistence of the two types, and white for a low overall concentration of agents. Arrows labeled with letters (a)–(e) indicate parameter combinations further analyzed in Fig. 3. The panels in the top row are snapshots of typical system states encountered for $p \in {0.2, 0.4, 0.7}$ (panels left to right) with $ε = 0.75$ .
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Figure 2
Total concentration of agents (a) on square lattices with $N = 50 \times 50$ sites and (b) from the numerical solution of the pair approximation, Eqs. (47, 48, 49, 50, 51). In both (a) and (b), the three curves are for parameter values $ε = 0.99, 0.75, 0.10$ (top to bottom). The insets zoom in on the curves for $ε = 0.10$ . The inset of (a) shows these curves for different system sizes $N = 30 \times 30$ (dotted curve), $N = 50 \times 50$ (solid curve), and $N = 100 \times 100$ (dashed curve).
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Figure 3
Distributions of the number of agents on a square lattice with $50 \times 50$ sites. In the lower row, (c), (d), and (e), $ε = 0.75$ . Each panel describes a transition between presence and absence of a type of agent. The transitions are also marked in Fig. 1 with the panel identifiers (a)–(e).
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Figure 4
(a) Stationary mean concentrations for dynamics on a square lattice where parameter $p$ of the model varies with the horizontal location $x \in {1, 2, \dots, L}$ according to Eq. (11). Lattice size is $L \times L$ with $L = 100$ . Cooperators' survival rate is $ε = 0.75$ constant in space. Showing the concentration dependence on $x$ , each plotted value is, for a given $x$ , a uniform average over the $y$ coordinate of the lattice and over time $t \in [0, 10^{6}]$ . (b) Snapshot of a state in the simulation as described for (a).
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Figure 5
Phases in pair approximation. The extinction transition between the regime of only cooperators (red area) to the empty system (white area) is given by the expression in $p$ and $ε$ in Eq. (55). The transition between coexistence (green area) and the regime of only cooperators is described by Eq. (56) using expressions (57, 58, 59). The transition between coexistence and the regime of only defectors (blue area) has an approximate description (dashed curve) in Eq. (60) using expressions (61)–(63). The exact solution (boundary between blue and green area) has been obtained as well, details given elsewhere.
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