Mobility-dependent selection of competing strategy associations

Standard models of population dynamics focus on the interaction, survival, and extinction of the competing species individually. Real ecological systems, however, are characterized by an abundance of species (or strategies, in the terminology of evolutionary-game theory) that form intricate, complex interaction networks. The description of the ensuing dynamics may be aided by studying associations of certain strategies rather than individual ones. Here we show how such a higher-level description can bear fruitful insight. Motivated from different strains of colicinogenic Escherichia coli bacteria, we investigate a four-strategy system which contains a three-strategy cycle and a neutral alliance of two strategies. We find that the stochastic, spatial model exhibits a mobility-dependent selection of either the three-strategy cycle or of the neutral pair. We analyze this intriguing phenomenon numerically and analytically.

Figure 1

Four-strategy system. (a) Four strategies $A$, $B$, $C$, and $D$ compete in a cyclic manner. Because $B$ also dominates $D$, a three-strategy cycle emerges between $A$, $B$, and $D$. The two strategies $A$ and $C$ form a neutral alliance. Such interactions can arise for colicinogenic bacteria of types R, P, or S (text). (b) Reactions between individuals of the four strategies that produce the required interaction scheme.

Figure 2

Snapshots of a simulation of the spatially extended model on a lattice of $256\times 256$ sites at various times $t$. At $t=0$, each site is occupied randomly, with equal probability, with an individual of the four strategies $A$ (red or gray), $B$ (blue or dark gray), $C$ (yellow or light gray), or $D$ (green or medium gray). A low mixing rate then leads to the rapid extinction of strategy $C$ and the prevalence of the dynamic three-strategy cycle $A$, $B$, $D$. At a higher mixing rate, however, strategies $B$ and $D$ die out, leaving a frozen steady state of the neutral alliance $A$ and $C$.

Figure 3

Phase diagram for the spatially extended model. We show the survival probability $P_{\mathrm{surv}}\left(B\right)$ of strategy $B$, that informs which strategy association prevails, as a function of the initial densities and the mixing rate. Black hence corresponds to the $ABD$ phase, and white to the $AC$ phase. The initial densities are $\vec{x}=(\frac{\varphi }{2},\frac{1-\varphi }{2},\frac{\varphi }{2},\frac{1-\varphi }{2})$ (left) and $\vec{x}=(\frac{1-\psi }{3},\frac{1-\psi }{3},\psi ,\frac{1-\psi }{3})$ (right). Lattice simulations are from a grid of $N=256\times 256$ sites. Results for the critical value ${\epsilon }_c^{\mathrm{ana}}$ of the mixing rate obtained from a generalized pair approximation (red crosses) agree qualitatively, though not quantitatively, with the numerical results. The narrow $AC$ strip for $\psi >0.8$ in (b) is an artifact of finite system size, and becomes smaller as $N$ increases.

Figure 4

Competing domains. (a), (c) Initially, an $A$, $B$, $D$ domain occupies the lower half and an $A$, $C$ domain the upper half. The ratio $a/c$ within the $A$, $C$ domain is $95/5$ in (a) and $1/2$ in (c). (b), (d) The system's state after evolving for 25 (a) respectively 50 (b) Monte Carlo steps. The lattice size is $L=64$ and the mixing rate $\epsilon =0.03$.

Figure 5

Average domain-wall velocity. The numerical results have been obtained from a lattice of linear size $L=128$. The domain-wall velocity is positive for low mixing rates, crosses zero at a value ${\epsilon }_0$, and is negative for large mixing. The value of the zero ${\epsilon }_0$ depends on the initial conditions. We show results for different ratios of $a/c$ in the $A$, $C$ domain, namely for $a/c=25/75$ (blue triangles), for $a/c=50/50$ (green squares), and for $a/c=75/25$ (red diamonds). We also include the velocity as seen from the perspective of species $B$: $v_{\mathrm{B}}=(db/dt)/\left(2b_0\right)$ when starting with a $a/c=75/25$ (black circles). We observe that it is the inverse of the velocity $v$ as seen from species $C$.

Figure 6

Mean lifetime $T$ of an $A$, $C$ droplet (a) and an $A$, $B$, $D$ droplet (b). Results have been obtained from a lattice of linear size $L=128$ and for different values of the mixing rate $\epsilon$ (legends).