Foraging at the Edge of Chaos: Internal Clock versus External Forcing

Activity rhythms in animal groups arise both from external changes in the environment, as well as from internal group dynamics. These cycles are reminiscent of physical and chemical systems with quasiperiodic and even chaotic behavior resulting from “autocatalytic” mechanisms. We use nonlinear differential equations to model how the coupling between the self-excitatory interactions of individuals and external forcing can produce four different types of activity rhythms: quasiperiodic, chaotic, phase locked, and displaying over or under shooting. At the transition between quasiperiodic and chaotic regimes, activity cycles are asymmetrical, with rapid activity increases and slower decreases and a phase shift between external forcing and activity. We find similar activity patterns in ant colonies in response to varying temperature during the day. Thus foraging ants operate in a region of quasiperiodicity close to a cascade of transitions leading to chaos. The model suggests that a wide range of temporal structures and irregularities seen in the activity of animal and human groups might be accounted for by the coupling between collectively generated internal clocks and external forcings.

Figure 1

Bifurcation diagram of the steady states of $A$ against the parameter $k$ (a). Time behavior of active individuals before the first bifurcation point $k_c$ ($k=0.0045$) (b), after the first bifurcation point ($k=0.007$) (c), and after the second bifurcation point ($k=0.025$) (d). Other parameter values are $\varphi =1$, $k^{\prime }=0.003$, and $K=1$.

Figure 2

Mean period of oscillations of $A$ as a function of the amplitude of the forcing $T(t)$. The vertical bars correspond to the standard deviation. Other parameter values are $k=0.007$, $k^{\prime }=0.003$, $K=1$, $\varphi =1$, and $\omega =0.005$.

Figure 3

Activity versus time, phase space plots and Poincaré surface section recorded at time intervals separated by the mean period for four different $ϵ$. (a) A quasiperiodic behavior ($ϵ=0.26$), (b) a chaotic behavior ($ϵ=0.36$), (c) a periodic behavior with period equal to the forcing ($ϵ=0.46$), and (d) a periodic behavior with a period twice the forcing ($ϵ=0.76$). Other parameter values as in Fig. 1c and $\omega =0.005$.

Figure 4

Temperature and activity time series on one nest for approximately 8 full days. Upper panel: Temperature time series. Lower panel: Activity time series (activity is measured as number of measured pulses every 36 seconds).

Figure 5

Monte Carlo simulation with a deterministic forcing fitted from the data with a sum of eight sinuses (a), autocorrelation of activity’s experimental and simulated data (b), probability distribution of the frequency of activity signal of the experimental and simulated data (c), and probability distribution of the variable $A\times (T-\overline{T})$ (d). Parameter values as in Fig. 1c but for a forcing equal to $\sum _{i=1}^8{ϵ}_i\sin ({\omega }_{i}t+{\phi }_i)$ with ${ϵ}_i$, ${\omega }_i$, and ${\phi }_i$ fitted to the data in Fig. 4: ${ϵ}_1=0.1359$, ${\omega }_1=0.004387$, ${\phi }_1=2.197$, ${ϵ}_2=0.01797$, ${\omega }_2=0.003825$, ${\phi }_2=-1.974$, ${ϵ}_3=0.03699$, ${\omega }_3=0.0005156$, ${\phi }_3=-1.599$, ${ϵ}_4=0.03763$, ${\omega }_4=0.001148$, ${\phi }_4=-0.2025$, ${ϵ}_5=0.04034$, ${\omega }_5=0.008723$, ${\phi }_5=-2.639$, ${ϵ}_6=0.6858$, ${\omega }_6=0.002471$, ${\phi }_6=1.167$, ${ϵ}_7=0.6865$, ${\omega }_7=0.00246$, ${\phi }_7=-1.925$, ${ϵ}_8=0.01727$, ${\omega }_8=0.005573$, ${\phi }_8=0.02816$.