Amplitude-Dependent Frequency, Desynchronization, and Stabilization in Noisy Metapopulation Dynamics

The enigmatic stability of population oscillations within ecological systems is analyzed. The underlying mechanism is presented in the framework of two interacting species free to migrate between two spatial patches. It is shown that the combined effects of migration and noise cannot account for the stabilization. The missing ingredient is the dependence of the oscillations’ frequency upon their amplitude. A simple model of diffusively coupled oscillators allows the derivation of quantitative results, like the functional dependence of the desynchronization upon diffusion strength and frequency differences. The oscillations’ amplitude is shown to be (almost) noise independent.

Figure 1

The LV phase space (left panel) admits a marginally stable fixed point surrounded by close trajectories (three of these are plotted). Each trajectory corresponds to single $H$ defined in Eq. (2), where $H$ increases monotonically along the (dashed) line connecting the center with the $a=0$ wall, as shown in the lower right panel. In the upper right panel, the period of a cycle $T$ is plotted against $H$, and is shown to increase almost linearly.

Figure 2

The survival probability $Q(t)$ is plotted versus time for a single patch noisy LV system. Equations (1) (with the symmetric parameters) were integrated numerically (Euler integration with time step 0.001), where the initial conditions are at the fixed point $a=b=1$. At each time step, a small random number $\eta (t)\Delta t$ was added to each population density, where $\eta (t)\in [-\Delta ,\Delta ]$. A typical phase space trajectory, for $\Delta =0.5$, is shown in the inset. Using 300 different noise histories, the survival probability is shown here for $\Delta =0.5$ (full line), $\Delta =0.3$ (dotted line) and $\Delta =0.25$ (dashed line). Clearly, $Q(t)$ decays exponentially at long times, $Q(t)\sim \exp (-t/\tau )$, where $1/\tau$ scales with ${\Delta }^2$.

Figure 3

The typical persistence time as a function of the diffusion rate for different levels of noise. The values of $\tau$ were gathered from survival probability plots (like those in Fig. 1) and are displayed here for the two-patch system. $\tau$ grows very rapidly with the migration rate for small diffusion values, and decays with $D$ for large diffusivities. Data is shown for different noise intensities $\Delta =0.3$ (triangles), 0.5 (squares), and 1.0 (circles).

Figure 4

Histograms showing the probability to be at a distance $R$ from the origin as a function of $R$, for two coupled noisy oscillators, where $\omega =1+\alpha r$ with $D=0.01$, and various values of noise strength $\Delta$, and angular velocity gradient $\alpha$. As expected, the phase space confinement is proportional to $\alpha$, from $\alpha =1$ (triangles) to $\alpha =0.5$ (circles) to $\alpha =0.1$ (solid line), all for the same level of noise $\Delta =0.1$. On the other hand, as predicted by the linear analysis close to the invariant manifold, the confinement is noise independent, and the three solid lines corresponding to different levels of noise ($\Delta =0.1$, 0.5, 1 with the same $\alpha =0.1$) almost coincide. The inset shows the flow lines on the $r=0$ plane. The invariant manifold $\varphi =0$ is marginally stable, but there is some flow towards the center for any finite $\varphi$.

Figure 5

The average $\Delta H$ at an elementary time step (0.001 of a unit time) as a function of the angle $\varphi$ between the patches. While a simple phase space random walk yields on average positive $\Delta H$, this property is shown here to hold only for small $\varphi$. At larger angles, the diffusion between patches forces the system toward the center and the average $\Delta H$ becomes negative. Results are shown for $\Delta =0.1$ (full line) and $\Delta =1$ (dashed line). The inset shows the probability distribution function for $H$ at these two noise levels.