Synchronization and chaos in complex ecological communities with delayed interactions

Abstract

Explaining the wide range of dynamics observed in ecological communities is challenging due to the large number of species involved, the complex network of interactions among them, and the influence of multiple environmental variables. Here, we consider a general framework to model the dynamics of species-rich communities under the effects of external environmental factors, showing that it naturally leads to delayed interactions between species, and analyze the impact of such memory effects on population dynamics. Employing the generalized Lotka-Volterra equations with time delays and random interactions, we characterize the resulting dynamical phases in terms of the statistical properties of community interactions. Our findings reveal that memory effects can generate persistent and synchronized oscillations in species abundances in sufficiently competitive communities. This provides an additional explanation for synchronization in large communities, complementing known mechanisms such as predator-prey cycles and environmental periodic variability. Furthermore, we show that when reciprocal interactions are negatively correlated, time delays alone can induce chaotic behavior. This suggests that ecological complexity is not a prerequisite for unpredictable population dynamics, as intrinsic memory effects are sufficient to generate long-term fluctuations in species abundances. The techniques developed in this work are applicable to any high-dimensional random dynamical system with time delays.

Figures (11)

• Figure 1: Schematic illustration of the projection mechanism for the dynamics of a community embedded in a broader ecosystem. The state of the system is characterized by the abundances of the community species $x_i$ and additional ecological variables $y_\mu$. These external variables represent environmental factors such as abiotic resources or environmental conditions. The full dynamical system is described by $\dot{x}_i = x_i g_i(\bm{x}, \bm{y})$ and $\dot{y}_\mu = f_\mu(\bm{x}, \bm{y})$. By integrating out the $y_\mu$ variables, an effective dynamical framework for the species abundances is derived, yielding a reduced description in the $x$-subspace given by $\dot{x}_i = x_i G_i[\{x(t'): t' \leq t\}]$. This effective dynamics encapsulates the influence of the environmental variables as memory effects, where species interactions are mediated through time-delayed kernels.
• Figure 2: Phases of generalized Lotka-Volterra equations with discrete delay. Panel (A): Phase diagram of the generalized Lotka-Volterra equations with a discrete delay $\tau = 3$. Along the unique fixed-point (UFP) and multiple attractors (MA) phases, also observed for $\tau = 0$, two new oscillatory phases emerge if competition in the community is high enough, that is, when $\mu$ is sufficiently negative. The line separating the oscillatory phases and the non-oscillatory ones is obtained numerically. The unbounded growth (UG) phase is instead unaffected by the delay. Panel (B): Typical dynamics in the Oscillatory UFP phase. All the species fluctuate around a unique fixed point. Only a subset of the species composing the community is shown. Panels (C-D): Phase shift $\phi$ and relative amplitude $A$ of oscillations of species abundances in the Oscillatory UFP phase. Triangles are obtained by numerical simulations, while the black line is given by Eq. (\ref{['eq:discrete-phi(z)-A(z)']}). Panel (E): Typical dynamics in the Oscillatory MA phase. At difference with Oscillatory UFP, there is no unique fixed-point structure around which these oscillations occur. Only a subset of the species composing the community is shown. Panel (F): Comparison between the power spectral densities in the Oscillatory MA phase in the case of instantaneous and delayed interactions. Upon turning on a sufficiently large delay, equally-spaced peaks appear due to the oscillations in species abundances.
• Figure 3: Phase diagram of generalized Lotka-Volterra equations with distributed delay and homogeneous interactions. Main: Phase diagram as a function of the mean delay and the coefficient of variation of a Gamma-distributed kernel. When $\textrm{CV} \geq 1$, sustained oscillations do not appear. In contrast, when the memory effects are sufficiently localized in the past, precisely if $\textrm{CV} < 1$, oscillations can emerge if the mean delay memory exceeds a certain threshold and if the community is sufficiently competitive (shaded blue region). Inset: The case of an exponentially decaying memory kernel, which can display overdamped or underdamped oscillations during the relaxation to equilibrium.
• Figure 4: Phase diagram of generalized Lotka-Volterra equations with delayed intraspecific interactions. Panel (A): Phase diagram of the generalized Lotka-Volterra equations with a discrete delay $\tau = 3$ and a fraction of delayed intraspecific interactions $u = -0.5$. As discussed in the main text, introducing delayed intraspecific interactions is qualitatively similar to considering a correlation $\gamma$ between species pair interactions. In addition to the phases observed without delayed intraspecific interactions ($u=0$), an additional phase of delay-induced chaos emerges. Panel (B): An example of the dynamics in the phase of delay-induced chaos. Only a subset of the species composing the community is shown. Species abundances in this phase exhibit irregular and unpredictable fluctuations. Panel (C): Power spectrum in the phase of delay-induced chaos. Multiple and nested frequency peaks can be seen, which are characteristic of the emergent chaotic behavior.
• Figure S1: Exact and approximated critical line for GLV with discrete delay Phase diagram comparing the critical lines for various values of $\tau$ obtained through the approximated DMFT (solid lines) and from numerical simulations (dashed lines). The error bars indicate numerical uncertainty due to the finite size $S=1000$ of the community. The dashed line is obtained as a polynomial fourth-order fit of the numerical points.