A spatially explicit model of synchronization in fiddler crab waving displays

Abstract

Fiddler crabs (Uca spp., Decapoda: Ocypodidae) are commonly found forming large aggregations in intertidal zones, where they perform rhythmic waving displays with their greatly enlarged claws. While performing these displays, fiddler crabs often synchronize their behavior with neighboring males, forming the only known synchronized visual courtship displays involving reflected light and moving body parts. Despite being one of the most conspicuous aspects of fiddler crab behavior, little is known about the mechanisms underlying synchronization of male displays. In this study we develop a spatially explicit model of fiddler crab waving displays using coupled logistic map equations. We explored two alternative models in which males either direct their attention at random angles or preferentially toward neighbors. Our results indicate that synchronization is possible over a fairly large region of parameter space. Moreover, our model was capable of generating local synchronization neighborhoods, as commonly observed in fiddler crabs under natural conditions.

Figure 1. Waving-display in male of Uca leptodactylus illustrating fiddler crab waving behavior.

(a) Initial upward movement; (b) waving apex; (c) final downward movement. Photo: Ana C. Rorato.

Figure 2. Logistic map.

Left: Bifurcation diagram shoes the asymptotic value of xn as function µ. Right: Temporal evolution for five different values of µ: µ = 2.5 (stable fixed point), µ = 3.2 (period two), µ = 3.6 (chaos) and µ = 4.0 (chaos). In all cases the initial conditions were set on x₀ = 0.1.

Figure 3. Illustration of male spatial distribution with random visual attention orientations.

The intersections of the grid represent all possible home positions and the black dots represent all occupied sites. Filled gray area shows the field of attention, defined by R and θ.

Figure 4. Analytical restriction for two individuals, obtained by Eq. (9) and Eq. (13).

The contour plot shows how fast the system becomes synchronized: the darker is the plot, the faster it occurs. The upper contour plot refers to the follower-leader coupling, and the contour line values correspond to ν = (1−D)µ. The lower contour plot refers to the case where the coupling is bidirectional, and the contour lines values correspond toν = |(1−2D)| µ.

Figure 5. (Color online) Spatial distribution at time n = 800 for M1 and M2.

Each dot represents one individual, whose claw displacement value,, is identified by the color scale on right of the graphs. Links shows the presence of interactions. Narrow lines means that the interaction occurs in just one direction (the interaction direction is not plotted) and wide lines means bidirectional interaction. The imposed density values, ρ = {0.1, 0.4}, and the logistic map values, µ = {3.2, 4.0} are shown on left of each graph. For all graphs the coupling constant value was set on D = 0.5. In order to contrast the spatial interaction network predicted by M1 and M2, in these graphs the spatial positions of the crabs are the same (for a given density), the unique difference is the orientation.

Figure 6. (Color online) Bifurcation diagram for M1 and M2 considering ρ = 0.4 and D = 0.5.

Gray dots show all possible claw displacement values assumed by all individuals. Colored dots show one possible claw asymptotic displacement values (assumed by one individual) for 800<n<1000. Colored scale refers to the global correlation value, <r>. For µ<3 both models predict a stable fix point (gray dots are behind to colored dots). Vertical black dot lines highlight the cases where µ = {3.2, 3.74, 3.84, 4}. The first and the last values are the ones set in Figs. 5 and 7. The other two values have, respectively, five and three period orbits only for M1 model (as predict by uncoupling logistic map).

Figure 7. (Color online) Phase diagrams.

The graphs show the global correlation <r>, Eq. (5), for M1 and M2 models and two values of logistic map constant, as function of coupling constant, D, and the density of individuals, ρ. The global correlation value corresponds to the color scale on the right of the graphs. The open black squares highlight the parameters present in Fig. 5.

Figure 8. Consequence of spatial organization for M1 (black dots) and M2 (gray dots) obtained by the average of 5000 initial conditions for each ρ value.

Above: average number of leader per follower. Observe that, for M1 and 0.15≤ ρ≤0.45, there is more than one leader per follower, causing the asynchronicity observed in Fig. 6. As in M2 the individuals rearrange themselves in order to maximize the number of individuals in their field of attention, there is not any leader present. Below: average number of reciprocal couplings by individuals that makes at least one reciprocal coupling. Observe that, for M2 and 0.45≤ ρ≤0.85, there are more than two reciprocal couplings per individual, causing a decrease of synchrony, as observed in Fig. 7 for µ = 3.2.