Evolutionary game dynamics with environmental feedback in a network with two communities

TL;DR

The study analyzes eco‑evolutionary game dynamics on a two‑community network where edge‑dependent environmental feedback modulates payoffs, formulating a five‑dimensional model for $(x,y,n_1,n_2,n_{12})$ and reducing to a three‑dimensional system under symmetry. It reveals that for $\theta_1\neq\theta_{12}$ there are no interior equilibria, with stable states migrating between faces and edges through transcritical bifurcations as the inter‑community rate $\delta$ changes; the full model exhibits multistability and anti‑synchronous oscillations between the two communities. The results highlight the critical role of community structure and edge‑level environmental feedback in shaping cooperation dynamics in structured populations, with implications for ecological and social systems. The framework supports extensions to other payoff structures and larger networks, paving the way for broader applications in complex adaptive systems.

Abstract

Recent developments of eco-evolutionary models have shown that evolving feedbacks between behavioral strategies and the environment of game interactions, leading to changes in the underlying payoff matrix, can impact the underlying population dynamics in various manners. We propose and analyze an eco-evolutionary game dynamics model on a network with two communities such that players interact with other players in the same community and those in the opposite community at different rates. In our model, we consider two-person matrix games with pairwise interactions occurring on individual edges and assume that the environmental state depends on edges rather than on nodes or being globally shared in the population. We analytically determine the equilibria and their stability under a symmetric population structure assumption, and we also numerically study the replicator dynamics of the general model. The model shows rich dynamical behavior, such as multiple transcritical bifurcations, multistability, and anti-synchronous oscillations. Our work offers insights into understanding how the presence of community structure impacts the eco-evolutionary dynamics within and between niches.

Figures (8)

• Figure 1: Schematic of the two-community network. A filled circle represents a player. Two players from the same community interact at rate $1-\delta$. Two players from the opposite communities interact at rate $\delta$. Without loss of generality, we normalize the rate parameter $0< \delta < 1$. We only show some edges for visualization purposes.
• Figure 2: Convergence to face equilibria. Shown are numerically obtained trajectories of the three-dimensional system given by Eqs. \ref{['3x']}, \ref{['3n1']}, and \ref{['3n12']}. The green dots represent the face equilibria given in Table \ref{['3dimface1']}. We use the payoff matrices given by Eq. \ref{['nummatri']} and initial conditions $(x,n_1,n_{12})=(0.5, 0.4, 0.1)$ and $(0.6, 0.5, 0.8)$, of which the corresponding trajectories are shown in blue and orange, respectively. (a) $\theta_1=8$, $\theta_{12}=5$, and $\delta=0.6$. (b) $\theta_1=8$, $\theta_{12}=5$, and $\delta=0.8$. (c) $\theta_1=5$, $\theta_{12}=8$, and $\delta=0.2$. (d) $\theta_1=5$, $\theta_{12}=8$, and $\delta=0.4$.
• Figure 3: Impact of the inter-community interaction rate $\delta$ on stability. Stable edge and face equilibria when $\theta_1 \neq \theta_{12}$ are shown as a function of $\delta$. In both (a) and (b), we use the payoff values given by Eq. \ref{['nummatri']}. (a) $\theta_1 < \theta_{12}$. The face equilibrium with $n_{12}^\ast=1$ is stable for $\delta<\delta_{\rm c,1}$. The edge equilibrium $(x_1^\ast, n_1^\ast, n_{12}^\ast) = \left(\frac{P_0-S_0-\delta(P_0-P_1-S_0+S_1)}{R_0-T_0-S_0+P_0-\delta\gamma},0,1\right)$ is stable for $\delta_{\rm c,1}<\delta<\delta_{\rm c,2}$. The face equilibrium with $n_1^\ast=0$ is stable for $\delta>\delta_{\rm c,2}$. (b) $\theta_1 > \theta_{12}$. The face equilibrium with $n_{12}^\ast=0$ is stable for $\delta<\delta_{\rm c,3}$. The edge equilibrium $(x_1^\ast, n_1^\ast, n_{12}^\ast) = \left(\frac{P_1-S_1+\delta(P_0-P_1-S_0+S_1)}{R_1-T_1-S_1+P_1+\delta\gamma},1,0\right)$ is stable for $\delta_{\rm c,3}<\delta<\delta_{\rm c,4}$. The face equilibrium with $n_1^\ast=1$ is stable for $\delta>\delta_{\rm c,4}$. In (a), we set $\theta_1 = 5$ and $\theta_{12} = 8$, yielding $\delta_{\rm c,1}=7/22$ and $\delta_{\rm c,2}=10/21$. In (b), we set $\theta_1 = 8$ and $\theta_{12}=5$, yielding $\delta_{\rm c,3}=21/31$ and $\delta_{\rm c,4} = 15/22$.
• Figure 4: Visualization of the transcritical bifurcations as $\delta$ varies. We use the payoff matrices given by Eq. \ref{['nummatri']}. The solid and dashed lines indicate stable and unstable equilibria, respectively, both disregarding the $0$ eigenvalues along the direction of $L$ in the case of $\theta_1 = \theta_{12}$. (a) Movement of three equilibria in the full state space as $\delta$ varies when $\theta_1 = 5$ and $\theta_{12} = 8$. A transcritical bifurcation occurs involving the face equilibrium on $n_{12}=1$ and the edge equilibrium $\left(x^\ast,0,1\right)$, where $x^\ast = 1/6$, at $\delta=7/22$. The second transcritical bifurcation occurs involving the face equilibrium on $n_1=0$ and the edge equilibirium $\left(x^\ast,0,1\right)$, where $x^\ast = 1/9$, at $\delta=10/31$. (b) Positions of all the same three edge and face equilibria as a function of $\delta$. The $\theta_1$ and $\theta_{12}$ values are the same as those used in (a). In (b), the three curves do not meet at a single point, as shown in the inset, which is a magnification of the main panel. (c) Same as (a) but when $\theta_1=\theta_{12} = 5$. A transcritical bifurcation occurs involving the face equilibrium on $n_{12}=1$ and that on $n_1=0$ at $\left(\frac{1}{6},0,1\right)$ when $\delta=7/22$. Edge equilibrium $\left(x^\ast, 0, 1\right)$ also collides with the two face equilibria at this value of $\delta$. (d) Same as (b) but when $\theta_1=\theta_{12} = 5$. There is another triplet of equilibria in addition to the triplet of equilibria shown in (c). For this second set of triplet of equilibria, a transcritical bifurcation occurs involving the face equilibrium on $n_{12}=0$ and that on $n_1=1$, and edge equilibrium $\left(x^\ast,1,0\right)$ collides with the bifurcation point, at $\delta=15/22$. Note that $x^\ast$ is not constant along the trajectories in (b), whereas it is in (d).
• Figure 5: Real part of the eigenvalues of the Jacobian near transcritical bifurcations as a function of $\delta$. We use the payoff matrices given by Eq. \ref{['nummatri']}. It should be noted that the third eigenvalue in (a) and (b) is always negative and thus is not shown, and that the third eigenvalue in (c) and (d) is always $0$. (a) $\theta_1=5$ and $\theta_{12}=8$. Each color represents a face or edge equilibrium. Two eigenvalues become $0$ at $\delta = \delta_{\rm c, 1} = 7/22 \approx 0.31818$, and another two eigenvalues become $0$ at $\delta = \delta_{\rm c, 2} = 10/31 \approx 0.32258$. Each of these $\delta$ values marks a transcritical bifurcation. At $\delta \approx 0.31822$ and $0.32321$, the eigenvalues of the stable face equilibrium turns from real to imaginary and vice versa. (b) $\theta_1=8$ and $\theta_{12}=5$. Two eigenvalues become $0$ at $\delta = \delta_{\rm c, 3} = 21/31 \approx 0.67742$, and another two eigenvalues become $0$ at $\delta = \delta_{\rm c, 4} = 15/22 \approx 0.68182$. Each of these $\delta$ values marks a transcritical bifurcation. At $\delta \approx 0.67732$ and $0.68194$, the eigenvalues of the stable face equilibrium turns from real to imaginary and vice versa. (c) $\theta_1 = \theta_{12} = 5$ and near the first transcritical bifurcation at $\delta = \delta_{\rm c, 1} = 7/22 \approx 0.31818$. At $\delta \approx 0.31806$ and $0.31830$, the eigenvalues of the stable face equilibrium turns from real to imaginary and vice versa. (d) $\theta_1 = \theta_{12} = 5$ and near the second transcritical bifurcation at $\delta = \delta_{\rm c, 3} = 15/22 \approx 0.68182$. At $\delta \approx 0.68170$ and $0.68194$, the eigenvalues of the stable face equilibrium turns from real to imaginary and vice versa.