Pattern Formation in a Spatial Public Goods Dilemma due to Diffusive or Directed Motion

Abstract

The costly provision of public goods serves as a model problem for the evolution of cooperative behavior, presenting a social dilemma between the collective benefits of shared resources and the individual incentive to free-ride in resource production. The spatial structure of populations can also impact cooperation over public goods, as diffusion of public goods and intentional motion of individuals towards regions with greater resources can interact with population and public goods dynamics to produce heterogeneous patterns in the spatial distribution of strategies and resources. In this paper, we build off a model introduced by Young and Belmonte for the reaction dynamics of interacting individuals and explicit public good, deriving a system of PDEs that describes the spatial profiles of strategies and the public good in the presence of both diffusive motion of individuals and resources and chemotaxis-like directed motion of individuals in response to gradients in the concentration of public goods. Through linear stability analysis, we show that spatial patterns in strategic and public goods profiles can emerge due to either Turing instability with high defector diffusivity or a directed-motion instability through strong sensitivity of cooperators towards increasing resource concentration. We further explore the emergent spatial patterns with a mix of weakly nonlinear stability analysis and numerical simulation, showing that diffusion-driven instability appears to increase cooperation and public goods across the spatial domain, while directed motion of cooperators towards regions with great public goods provision tends to decrease cooperation and environmental quality across the environment.

Figures (15)

• Figure 1: Plots of the threshold value $D_v^\ast(k)$ as a function of $k$ with $L=8$, both for the range of integer wavenumbers between $1$ and $12$. The right panel shows a zoomed-in view of the left panel near the minimal threshold $D_v^\ast$ is attained as $k=8$, with $D_v^\ast\approx 0.04861$. The orange squares represent the thresholds obtained from the condition $Q_3(k) = -\det(M(k))<0$, the horizontal dashed line in the left panel indicates the limit of zero diffusivity $D_v = 0$, and the vertical dotted lines in both panels highlights the threshold wavenumber for pattern formation given by $k^* = 8$. The parameters for the reaction dynamics are taken from Table \ref{['table 1']}, and the PDE parameters are chosen as $D_u=D_\phi=0.01$.
• Figure 2: Dispersion relation and instability condition for the one-dimensional background domain $[0, L]$ with $L=8$ and wavenumber $k^\ast =8$. The parameters for the reaction dynamics are taken from Table \ref{['table 1']}. In panel (a), we plot the dispersion relation describing the largest real part of the lineriazation matrix $M(k)$ as a function of the wavenumber $k$ for the choices of defector diffusivity of $D_v=0.03$, $0.04861$, and $0.08$. In panel (b), we plot the threshold $D_v^*$ for pattern-forming instability in the $(D_u, D_v)$-plane, with the orange region describing the pattern-formation regime and the white region describing the regime of a locally stable uniform coexistence equilibrium. The black curve represents the boundary determined by evaluating the maximum value of the threshold $D_v^*(k)$ from Equation \ref{['eq:D_v-threshold']} over all feasible integer wavenumbers. In particular, when $D_u=D_\phi=0.01$, the threshold value is $D_v^{\ast}\approx 0.04861$.
• Figure 3: Threshold value $D_v^\ast(k)$ as a function of $k$ with $L=8$. The ODE parameters are taken from Table \ref{['table 1']}, and the PDE parameters are chosen as $D_u=D_v=D_\phi=0.03$. The orange squares represent the thresholds obtained from the condition $P_3<0$, the blue triangles represent those obtained from $P_3>P_1P_2$, the horizontal black dashed line corresponds to a cooperator movement sensitivity of $w_u = 0$, and the vertical black dotted line corresponds to the critical wavenumber $k^* = 8$. The right panel shows a zoomed-in view of the left panel near the minimal $w_u^\ast(k)$. The minimal threshold is attained at $k=8$, with $w_u^\ast \approx 6.4603$.
• Figure 4: Dispersion relation and stability condition for the one-dimensional background domain $[0, L]$ with $L=8$ and wavenumber $k^\ast =8$. The ODE parameters are taken from Table \ref{['table 1']}. The diffusion coefficients are set to be $D_u=D_v=D_\phi=0.03$. In part (a), we choose $w_u=4$, $6.4603$, and $8$. For the stability condition in the $(w_v, w_u)$-plane, the orange region corresponds to instability, while the white region corresponds to stability. The blue curve represents the boundary determined by Equation \ref{['eq:w_u-threshold']}. In particular, when $w_v=1$, the threshold value is $w_u^{\ast}\approx 6.4603$.
• Figure 5: $\eta$ - $D_u$ graph and $\beta$ - $D_u$ graph on the one-dimensional domain $[0,L]$ with $L=8$. Wavenumbers $k^\ast$ are chosen based on Equations \ref{['eq:D_v-threshold']} and \ref{['eq:D_v-threshold-min']}. The parameters are $w_u=0$, $w_v=0$, and the ODE parameters are taken from Table \ref{['table 1']}. The horizontal axis corresponds to the values of $D_u$. In particular, when $D_u = D_\phi = 0.01$, the critical wavenumber is $k^\ast=8$. The horizontal dashed lines represent $\eta = 0$ (left) and $\beta = 0$ (right), separating regimes in which we expect a supercritical pitchfork bifurcation *$\eta > 0$ and $\beta < 0$) and in which subcritical bifurcations may be possible.

Theorems & Definitions (8)

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• Lemma 1