Hostility prevents the tragedy of the commons in metapopulation with asymmetric migration: A lesson from queenless ants

TL;DR

The model validates that there is evolutionary benefit behind the queenless ants' behavior of showing more hostility towards the immigrants from nearby colonies than those from the far-off ones.

Abstract

A colony of the queenless ant species, \emph{Pristomyrmex punctatus}, can broadly be seen as consisting of small-body sized worker ants and relatively larger body-sized cheater ants. Hence, in the presence of inter-colony migration, a set of constituent colonies act as a metapopulation exclusively composed of cooperators and defectors. Such a set-up facilitates an evolutionary game-theoretic replication-selection model of population dynamics of the ants in a metapopulation. Using the model, we analytically probe the effects of territoriality induced hostility. Such hostility in the ant-metapopulation proves to be crucial in preventing the tragedy of the commons, specifically, the workforce, a social good formed by cooperation. This mechanism applies to any metapopulation -- not necessarily the ants -- composed of cooperators and defectors where inter-population migration occurs asymmetrically, i.e., cooperators and defectors migrate at different rates. Furthermore, our model validates that there is evolutionary benefit behind the queenless ants' behavior of showing more hostility towards the immigrants from nearby colonies than those from the far-off ones. In order to calibrate our model's parameters, we have extensively used the data available on the queenless ant species, \emph{Pristomyrmex punctatus}

Figures (7)

• Figure 1: Numerics validate the stability diagram: We illustrate the validation of the analytically obtained stability diagram in Fig. \ref{['fig1_migration']}(d). Extinction, bistability and heterogeneous existence of the metapopulation are respectively marked by the white markers $*$, $+$ and $\times$ which represent the direct numerical time evolution of Eq. (\ref{['evolution_Cooperator_with_migration']}) and Eq. (\ref{['evolution_Defector_with_migration']}). For the numerics, we uniformly divide $\mu^C$--$\mu^D$ plane in $8 \times 8$ grid. At each grid point, we evolve ten randomly chosen initial conditions (for a time $t=15000$ by the fourth order Runge--Kutta method with discrete time step $dt=10^{-3}$) whose final states match with the corresponding analytical results: The red, the yellow, and the green patches---which respectively indicate extinction, bistability and heterogeneous existence---exclusively contain $*$, $+$ and $\times$ markers respectively.
• Figure 2: Hostility averts extinction: This figure depicts the stability diagram of Eq. (\ref{['evolution_Cooperator_with_migration']}) and Eq. (\ref{['evolution_Defector_with_migration']}) with PD ($T=1.5$ and $S=-0.01$) as the underlying game structure. We also fix $\delta=10^{-4}$ and $M=2$. By keeping the hostility ($h$) fixed, we vary $\mu^C$ along $x$-axis and $\mu^D$ along $y$-axis for generating every subplot. The red, yellow, green, and blue colors respectively depict the parameter region for which the metapopulation goes extinct, sustains bistability between extinction and non-homogeneous extant state, sustains non-homogeneous extant state, and sustains homogeneous extant state. Subplot $(a)$ is for the non-hostile case while subplot $(b)$ is for the hostile case with $h=0.5$. The lower row of subplots $(c)$ to $(d)$ is zoomed in plots about the origin and the value of hostility increases from $(a)$ to $(d)$ as marked in the figure; here the ranges of $\mu^C$ and $\mu^D$ are close to what was reported for Pristomyrmex punctatusDobata2010Dobata2013.
• Figure 3: Hostility averts the TOC: The first row represents the realization and the prevention of the TOC in metapopulation, whereas the second row represents the same for the constituent populations/colonies separately. We find that the presentation is conspicuous if we plot the logarithm of the total number of cooperators; for the extinction state we replace $N_1^C=N_2^C=0$ with a small number, $10^{-20}$. Fixed parameters are same as in Fig. \ref{['fig1_migration']}. The red and the darkest green regions respectively depict the complete realization and the prevention of TOC, whereas the all other colors represent the partial TOC. We can see (in subplots $(a)$, $(d)$ and $(e)$) that the TOC is an unavoidable fate without hostility; however, (as shown in subplots $(b)$, $(f)$ and $(g)$) with hostility $(h=0.5)$, the prevention of TOC always possible for suitable parameter values. Subfigure (c) shows the increase in size of metapopulation with the hostility: The solid line corresponds to homogeneous stable fixed point, while the dashed line represents the non-homogeneous fixed point. Here $\mu^C=0.015$ and $\mu^D=0.5$.
• Figure 4: Diagrammatic depiction of multi-colony metapopulation: Subfigure $(a)$ shows nearest neighbor interaction, and subfigure $(b)$ depicts the random rewiring at a time step. The directed arrow heads show the directed migrations from one colony to the other. A nearest-neighbor migration is indicated by solid green arrow, a broken connection (dashed arrow with scissors) indicates no nearest-neighbor migration at that time step and a new connection (violet arrow) indicates migration from far-off node.
• Figure 5: The propensity of preventing the extinction is higher with the nasty-neighbor effect than with the dear-enemy effect. Subplots $(a)$ and $(b)$ are the stability diagram for the dear-enemy ($h_d=0.75>0.5=h_n$) and nasty-neighbor ($h_n=0.75>0.5=h_d$) effects, respectively. Rewiring probability has been fixed to $0.1$. Colors bears the same meaning and all other fixed parameters' values are same as in Fig. \ref{['fig1_migration']}.