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The emergence of phase asynchrony and frequency modulation in metacommunities

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Abstract

Spatial synchrony can summarize complex patterns of population abundance. Studies of phase synchrony predict that limited dispersal can drive either in-phase or out-of-phase synchrony, characterized by a constant phase difference among populations. We still lack an understanding of ecological processes leading to the loss of phase synchrony. Here, we study the role of limited dispersal as a cause of phase asynchrony defined as fluctuating phase differences among populations. We adopt a minimal predator-prey model allowing for dispersal-induced phase asynchrony, and show its dependence on species traits. We show that phase asynchrony in a homogeneous metacommunity requires a minimum of three communities and is characterized by the emergence of regional frequency modulation of population fluctuations. This frequency modulation results in spectral signatures in local time series that can be used to infer the causes and properties of metacommunity dynamics. Dispersal-induced phase asynchrony extends the application of ecological theories of synchrony to nonstationary time series, and is consistent with observed spatiotemporal patterns in marine metacommunities.

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  • 29 December 2018

    The original version of this article unfortunately contained a mistake. The name "Yuxian Zhang" should be corrected to "Yuxiang Zhang".

References

  • Abbott KC (2011) A dispersal-induced paradox: synchrony and stability in stochastic metapopulations. Ecol Lett 14(11):1158–1169

    PubMed  Google Scholar 

  • Allstadt AJ, Liebhold A, Johnson DM, Davis RE, Haynes KJ (2015) Temporal variation in the synchrony of weather and its consequences for spatiotemporal population dynamics. Ecology 96(11):2935–2946

    PubMed  Google Scholar 

  • Arumugam R, Dutta PS, Banerjee T (2015) Dispersal-induced synchrony, temporal stability, and clustering in a mean-field coupled Rosenzweig–MacArthur model. Chaos 25(10). https://doi.org/10.1063/1.4933300

    Google Scholar 

  • Ashwin P, King G, Swift JW (1990) Three identical oscillators with symmetric coupling. Nonlinearity 3(3):585–601

    Google Scholar 

  • Baesens C, Guckenheimer J, Kim S, MacKay R (1991) Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Physica D: Nonlinear Phenom 49(3):387–475

    Google Scholar 

  • Bjørnstad ON, Ims RA, Lambin X (1999) Spatial population dynamics: analyzing patterns and processes of population synchrony. Trends Ecol Evol 14(11):427–432

    PubMed  Google Scholar 

  • Blasius B, Huppert A, Stone L (1999) Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399:354–359

    CAS  PubMed  Google Scholar 

  • Boashash B (2016) Time-frequency signal analysis and processing, 2nd edn. Elsevier, New York

    Google Scholar 

  • Cavanaugh KC, Kendall BE, Siegel DA, Reed DC, Alberto F, Assis J (2013) Synchrony in dynamics of giant kelp forests is driven by both local recruitment and regional environmental controls. Ecology 94 (2):499–509

    PubMed  Google Scholar 

  • Cazelles B, Bottani S, Stone L (2001) Unexpected coherence and conservation. Proc R Soc B 268 (1485):2595–2602

    CAS  PubMed  PubMed Central  Google Scholar 

  • Cazelles B, Boudjema G (2001) The Moran effect and phase synchronization in complex spatial community dynamics. Am Nat 157(6):670–676

    CAS  PubMed  Google Scholar 

  • Cazelles B, Chavez M, Berteaux D, Ménard F, Vik JO, Jenouvrier S, Stenseth NC (2008) Wavelet analysis of ecological time series. Oecol 156(2):287–304

    Google Scholar 

  • Emelianova YP, Kuznetsov A, Sataev I, Turukina L (2013) Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D: Nonlinear Phenom 244(1):36–49

    Google Scholar 

  • Ermentrout B (2002) Simulating, analyzing and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM, Philadelphia

    Google Scholar 

  • Goldwyn EE, Hastings A (2007) When can dispersal synchronize populations? Theor Pop Biol 73:395–402

    Google Scholar 

  • Goldwyn EE, Hastings A (2009) Small heterogeneity has large effects on synchronization of ecological oscillators. Bull Math Biol 71(1):130–144

    PubMed  Google Scholar 

  • Goldwyn EE, Hastings A (2011) The roles of the Moran affect and dispersal in synchronizing oscillating populations. J Theor Biol 289:237–246

    PubMed  Google Scholar 

  • Gouhier TC, Guichard F (2014) Synchrony: quantifying variability in space and time. Meth Ecol Evol 5 (6):524–533

    Google Scholar 

  • Gouhier TC, Guichard F, Menge BA (2010) Ecological processes can synchronize marine population dynamics over continental scales. Proc Natl Acad Sci U S A 107(18):8281–8286

    CAS  PubMed  PubMed Central  Google Scholar 

  • Grainger TN, Gilbert B (2016) Dispersal and diversity in experimental metacommunities: linking theory and practice. Oikos 125(9):1213–1223

    Google Scholar 

  • Guichard F, Gouhier TC (2014) Non-equilibrium spatial dynamics of ecosystems. Math Biosci 255:1–10

    PubMed  Google Scholar 

  • Hastings A (2001) Transient dynamics and persistence of ecological systems. Ecol Lett 4:215–220

    Google Scholar 

  • Hastings A (2004) Transients: the key to long-term ecological understanding? Trends Ecol Evol 19(1):39–45

    PubMed  Google Scholar 

  • Haydon DT, Greenwood PE (2000) Spatial coupling in cyclic population dynamics: models and data. Theor Pop Biol 58(3):239–254

    CAS  Google Scholar 

  • Henden JA, Ims RA, Yoccoz NG (2009) Nonstationary spatio-temporal small rodent dynamics: evidence from long-term norwegian fox bounty data. J An Ecol 78(3):636–645

    Google Scholar 

  • Holland MD, Hastings A (2008) Strong effect of dispersal network structure on ecological dynamics. Nature 456(7223):792–794

    CAS  PubMed  Google Scholar 

  • Hoppensteadt FC, Izhikevech EM (1997) Weakly connected neural networks. Springer, New York

    Google Scholar 

  • Hoppensteadt FC, Izhikevich EM (1998) Thalamo-cortical interactions modeled by weakly connected oscillators: could the brain use fm radio principles? Biosystems 48:85–94

    CAS  PubMed  Google Scholar 

  • Jansen VA, de Roos A (2000) The role of space in reducing predator–prey cycles. In: The geometry of ecological interactions: simplifying spatial complexity, eds. Cambridge University Press, pp 183–201

  • Jassby AD, Powell TM (1990) Detecting changes in ecological time series. Ecology pp 2044–2052

  • Kim S, Kook H, Lee SG, Park MH (1998) Synchronization and clustering in a network of three globally coupled neural oscillators. Int J Bif Chaos 8(04):731–739

    Google Scholar 

  • Koelle K, Vandermeer J (2005) Dispersal-induced desynchronization: from metapopulations to metacommunities. Ecol Lett 8(2):167–175

    Google Scholar 

  • Lampert A, Hastings A (2016) Stability and distribution of predator–prey systems: local and regional mechanisms and patterns. Ecol Lett 19(3):279–288

    PubMed  Google Scholar 

  • Lande, Engen, Sæther (1999) Spatial scale of population synchrony: environmental correlation versus dispersal and density regulation. Am Nat 154(3):271–281

    PubMed  Google Scholar 

  • Liebhold A, Koenig WD, Bjørnstad ON (2004) Spatial synchrony in population dynamics. Ann Rev Ecol Evol Syst 35:467–490

    Google Scholar 

  • Louca S, Doebeli M (2014) Distinguishing intrinsic limit cycles from forced oscillations in ecological time series. Theor Ecol 7(4):381–390

    Google Scholar 

  • Marleau JN, Guichard F, Loreau M (1777) Meta-ecosystem dynamics and functioning on finite spatial networks. Proc R Soc B 281:20132094

    Google Scholar 

  • Montbrió E, Kurths J, Blasius B (2004) Synchronization of two interacting populations of oscillators. Phys Rev E 70:056125

    Google Scholar 

  • Morton ES (1975) Ecological sources of selection on avian sounds. Am Nat 109:17–34

    Google Scholar 

  • Ruxton G, Doebeli M (1996) Spatial self-organization and persistence of transients in a metapopulation model. Proc R Soc B 263(1374):1153–1158

    Google Scholar 

  • Truax B (2001) Handbook of acoustic ecology. Comput Mus J 25:93–94

    Google Scholar 

  • Turchin P (2003) Complex population dynamics: a theoretical/empirical synthesis, vol 35 of monographs in population biology. Princeton University Press, Princeton

    Google Scholar 

  • Wall E, Guichard F, Humphries AR (2013) Synchronization in ecological systems by weak dispersal coupling with time delay. Theor Ecol 6(4):405–418

    Google Scholar 

  • Zhang Y, Lutscher F, Guichard F (2015) How robust is dispersal-induced spatial synchrony? Chaos 25(036402). https://doi.org/10.1063/1.4906951

    Google Scholar 

Download references

Acknowledgments

F.G. and F.L. wish to thank the Natural Science and Engineering Research Council (NSERC) of Canada for their support through the Discovery Program.

Funding

This study is financially supported by the NSF of China (No. 11601386). We also wish to acknowledge financial support from the Centre de Recherches Mathématiques (CRM) and from the Canadian Healthy Ocean Network (CHONe).

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Correspondence to Frederic Guichard.

Additional information

The original version of this article was revised: The name “Yuxian Zhang” should be corrected to “Yuxiang Zhang”.

Appendix: Model analysis

Appendix: Model analysis

Here, we provide details on the qualitative analysis of the planar dynamical system given by the phase-difference equations (6), i.e.,

$$\begin{array}{@{}rcl@{}} \frac{\text{d} \psi_{1}}{\text{d}t}\!&=&\!\frac{1}{2}[H(-\psi_{1}) + H(\psi_{2}) - \!H(\psi_{1}) - \!H(\psi_{1}+\psi_{2})],\\ \frac{\text{d} \psi_{2}}{\text{d}t}\!&=&\!\frac{1}{2}[ H(-\psi_{1} - \psi_{2}) + H(-\psi_{2})-H(-\psi_{1})- H(\psi_{2})]. \end{array} $$
(A.1)

The most important ingredient of these equations is the \(2\pi \)-periodic function H. It is related to the so-called infinitesimal phase-response curve, \(\hat \gamma ,\) along a periodic orbit (Goldwyn and Hastings 2007). The infinitesimal phase-response curve is the solution of the differential equation

$$ \frac{\text{d}\hat\gamma}{\text{d}t}=-\text{DF}(\gamma(t))\hat\gamma $$
(A.2)

with the normalization condition \(\hat \gamma (t)\cdot \gamma ^{\prime }(t)= 1\). In other words, \(\hat \gamma \) is given by the linearization equation of the vector field F around the stable periodic orbit \(\gamma \). With this notation, function H is given as the average:

$$ H(x)=\frac 1 T{{\int}^{T}_{0}}\hat{\gamma}(t)\cdot (\gamma(t+x/{\Omega})-\gamma(t))\text{d}t, $$
(A.3)

where T is the period of the periodic orbit \(\gamma (t)\) of the Rosenzweig-MacArthur model and \(\Omega = 2\pi /T\) is the frequency.

Even though function H is not explicitly given, we can still proceed with the qualitative analysis of the phase-difference system by finding steady states and calculating their stability conditions. Periodicity of the function H is an important aspect in almost all the considerations to follow.

Zero phase differences

The “all-in-phase” state \((\psi _{1}, \psi _{2})=(0, 0)\) is a steady-state solution, independent of parameter values. Hence, all-in-phase synchrony is always possible. Moreover, this state is always locally asymptotically stable: if a metacommunity is initially close to in-phase synchrony, it will converge to in-phase synchrony. To see that (0,0) is locally stable, we linearize (A.1) and get the Jacobi matrix

$$ J(0,0)=\frac{H^{\prime}(0)}2\left( \begin{array}{cc} -3 & 0 \\ 0 & -3 \end{array} \right). $$

Thus, the local stability of (0,0) is determined by the sign of \(H^{\prime }(0)\). If \(H^{\prime }(0)>0\) (\(H^{\prime }(0)<0\)), the all-in-phase steady state is linearly stable (unstable). This condition is exactly the same as for the two-patch model studied in Goldwyn and Hastings (2007) and Zhang et al. (2015). Based on those results, the all-in-phase state will be locally stable for all parameter sets that we study here.

Symmetry

As we look for further steady states, we note that the phase-difference equations are \(2\pi \)-periodic and possess several symmetries. In particular, if \((\psi _{1}^{*}, \psi _{2}^{*})\) is a steady state of system (A.1), then the following are also steady states: \((2\pi +\psi _{1}^{*}, \psi _{2}^{*}), (\psi _{1}^{*}, 2\pi +\psi _{2}^{*}),\)\((2\pi +\psi _{1}^{*}, 2\pi +\psi _{2}^{*})\) and \((2\pi -\psi _{2}^{*}, 2\pi -\psi ^{*}_{1})\). In fact, the vector field in Eq. A.1 on the square \([0,2\pi )\times [0,2\pi )\) is symmetric with respect to the diagonal \(y = 2\pi -x\). It can therefore be completely represented by its value on the triangle \(0\leq \psi _{1}+\psi _{2}\leq 2\pi \). The schematic representations in Fig. 7 only show one such triangle for each case. The computationally generated phase-plane plots in Fig. 8 show the entire square \([0,2\pi )\times [0,2\pi )\) and illustrate the symmetry along the diagonal.

Equal phase differences

The next particular steady-state solution that we investigate has equal, non-zero phase differences between all three oscillators. Such a state is known as a “traveling-wave state” (Goldwyn and Hastings 2011) a “rotating wave” (Ashwin et al. 1990) or “splay state”. If we assume that \((\psi _{1}^{*},\psi _{2}^{*})=(x^{*}, x^{*})\) is such a state, we have the equations:

$$\begin{array}{@{}rcl@{}} 0&=&H(-x^{*})+ H(x^{*})- H(x^{*})- H(2x^{*}),\\ 0&=& H(-2x^{*})+ H(-x^{*})-H(-x^{*})- H(x^{*}). \end{array} $$
(A.4)

Hence, if \(2x^{*}=-x^{*}\) modulo \(2\pi \) then these equations are satisfied. Therefore, traveling-wave states with phase difference \(x^{*}= 2\pi /3\) or \(x^{*}={4\pi }/3\) exist independently of parameter values in the system.

Knowing the stability behavior of traveling-wave states will turn out crucial to understanding the dynamics of the system. The Jacobi matrix at the state (x,x) = (2π/3, 2π/3) has the form

$$ J=\frac{1}{2}\begin{pmatrix} -2H^{\prime}(2x^{*})-H^{\prime}(x^{*}) & H^{\prime}(x^{*})-H^{\prime}(2x^{*}) \\ H^{\prime}(2x^{*})-H^{\prime}(x^{*}) & -H^{\prime}(2x^{*})-2H^{\prime}(x^{*}) \end{pmatrix}. $$
(A.5)

To get this form, we used the fact that for \(x^{*}= 2\pi /3\) we have \(2x^{*}=-x^{*}\) mod \(2\pi \) and that H is \(2\pi \)-periodic. We calculate the trace and determinant of this matrix as follow:

$$\begin{array}{@{}rcl@{}} \text{tr}J&=&-3(H^{\prime}(x^{*})+H^{\prime}(2x^{*})),\\ \det J &=& 3 [H^{\prime}(x^{*})^{2}+H^{\prime}(2x^{*})^{2}+ H^{\prime}(x^{*})H^{\prime}(2x^{*})]. \end{array} $$

To evaluate the eigenvalues of J, we calculate the discriminant as following:

$$(\text{tr}J)^{2} - 4\det J = -3(H^{\prime}(x^{*})-H^{\prime}(2x^{*}))^{2}\leq 0. $$

Hence, the eigenvalues are not real (unless \(H^{\prime }(x^{*})=H^{\prime }(2x^{*})\)) and therefore, the point is a focus or spiral. This observation also follows from general symmetry considerations (Ashwin et al. 1990). The stability is then given by the sign of the trace of J. In particular, the traveling-wave state is stable if \(\text {tr}J<0\) or

$$ H^{\prime}(x^{*})+H^{\prime}(2x^{*})>0. $$
(A.6)

We evaluate this quantity numerically for our parameter set. The plot in Fig. 6 shows that the traveling-wave state is stable for intermediate values \(\eta \) but unstable for small and for large values.

Fig. 6
figure 6

Numerical evaluation of the stability condition (A.6) for our default parameter set. The traveling-wave state is stable when the quantity in the figure is positive

Two in-phase states

Finally, we will see steady-state solutions where one phase difference is zero, so-called “two in-phase states” (Ashwin et al. 1990), for example, \((\psi _{1}^{*},\psi _{2}^{*})=(y^{*},0)\). These arise when \(y^{*}\) satisfies

$$ 2H(y^{*})=H(-y^{*}). $$
(A.7)

These states are located in an invariant set. In fact, if \(\psi _{2}= 0\) then \(\text {d}\psi _{2}/\text {d}t = 0\). Hence, if the phase difference \(\psi _{2}\) is zero initially, then it will be zero for all times. In that case, the dynamics reduce to the line \((\psi _{1},0),\) with \(0\leq \psi _{1}\leq 2\pi \). The two end-points are locally stable, as we have seen above when studying the stability of (0,0). Since the dynamics are one-dimensional, there has to be at least one steady state on the line with \(0<\psi _{1}<2\pi ,\) and if there is only one it has to be unstable. Hence, the two in-phase state always exists. By symmetry, the same reasoning holds for \(\psi _{1}= 0\) and there have to be at least three such states, namely \((y^{*},0),\)\((2\pi -y^{*},y^{*}),\) and \((0,2\pi -y^{*})\).

Stability changes

In the corresponding two-patch system, the antiphase-locked solution changes stability, and additional out-of-phase-locked solutions appeared, when parameters are varied in such a way that the relative temporal scales between different processes differed (Goldwyn and Hastings 2007; Zhang et al. 2015; Wall et al. 2013). We focus on how the dynamics of Eq. 6 change as parameter \(\eta \) decreases. We describe several scenarios in Figs. 7 (schematic) and 8 (actual) and summarize the results in a schematic bifurcation diagram (Fig. 9, see also Fig. 2 in the main text). For large values of \(\eta \geq 0.32\), the traveling-wave state is an unstable focus; all nonconstant solutions converge to the all-in-phase-locked state. As \(\eta \) decreases, the traveling-wave state becomes stable through a subcritical Hopf bifurcation (SH) and an unstable limit cycle emerges. Continuing to decrease \(\eta ,\) two additional limit cycles appear in a saddle-node bifurcation of limit cycles (SNL) so that there are now three limit cycles. The middle one is stable (e.g., η = 0.19). The two inner limit cycles eventually collide and disappear in another saddle-node bifurcation of limit cycles (near \(\eta = 0.175\)). The traveling-wave state undergoes another Hopf bifurcation (H), this time supercritical, so that a stable limit cycle emerges (η = 0.17). The stable limit cycles correspond to solutions that are not phase-locked, and hence show true asynchrony.

Fig. 7
figure 7

Schematic of the phase plane of system (6) for 0.15 < η < 0.4. Blue dots correspond to the all-in-phase-locked state and are locally stable for all values. Red dots correspond to two in-phase states and are saddles. The green dot stands for the traveling-wave state. The latter state is an unstable spiral for η ≥ 0.35 (top left). After a subcritical Hopf bifurcation, it becomes stable with a surrounding unstable limit cycle (η = 0.31, top right). After the first saddle-node bifurcation, there are three limit cycles of which the inner and outer are unstable (η = 0.19, bottom left). After the second saddle-node bifurcation, there is only a large unstable limit cycle (no plot). Then, a supercritical Hopf bifurcation generates a stable limit cycle and the traveling-wave state becomes unstable again (η = 0.17, bottom right). The other parameter values are 𝜖 = 0.1, and α = 0.4

Fig. 8
figure 8

Phase planes of system (6) for different values of η, corresponding to Fig. 7. Plot a shows two complete trajectories, connecting the unstable traveling-wave state to the stable all-in-phase state. Plot b also shows two complete orbits, one located inside the unstable limit cycle (upper right triangle), one outside (lower left triangle). The upper right triangle in plot c shows how two forward trajectories approach two stable objects: the traveling-wave state and the stable periodic orbit of asynchronous phase. The lower left triangle in plot c shows how two backwards orbits approach the two unstable limit cycles. Plot d highlights the single stable limit cycle as it is approached from the exterior (top right triangle) and the interior (bottom left triangle). The large unstable limit cycle is extremely difficult to capture because it is so close to the axes that numerical accuracy becomes an issue. These plots were obtained using Matlab (Mathworks), after generating an expression of the function H from XPPAUT (Ermentrout 2002)

Fig. 9
figure 9

Schematic summary of the bifurcation behavior of the phase-difference dynamics of Eq. 6 as η varies in (0.15,0.4). The straight horizontal lines correspond to the tall-in-phase state at (0,0) and the traveling-wave state at (2π/3,2π/3); the curves represent amplitudes of limit cycles. Thicker lines are stable objects, thin lines unstable. Letters H and SH indicate the Hopf and the subcritical Hopf bifurcation. SNL stands for the saddle-node bifurcation of limit cycles. The vertical lines correspond to the values of η for which the phase plane schematic is given in Fig. 7

The situation is similar to the generic bifurcation diagram for three identical, weakly coupled oscillators in Ashwin et al. (1990). Those authors mention that the branch that emerges from the (subcritical) Hopf bifurcation can “fold back on itself and create saddle-node bifurcations of tori.” What we do not observe in our parameter range is the change of stability of the all-in-phase-locked state that is required for the global bifurcation that these authors observed.

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Guichard, F., Zhang, Y. & Lutscher, F. The emergence of phase asynchrony and frequency modulation in metacommunities. Theor Ecol 12, 329–343 (2019). https://doi.org/10.1007/s12080-018-0398-8

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