The emergence of phase asynchrony and frequency modulation in metacommunities

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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Change history

  • 29 December 2018

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

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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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Corresponding author

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
figure6

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
figure7

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
figure8

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
figure9

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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Keywords

  • Phase synchrony
  • Metacommunities
  • Spatial dynamics
  • Predator-prey dynamics
  • Self-organization
  • Weakly coupled oscillators