Small Heterogeneity Has Large Effects on Synchronization of Ecological Oscillators

Abstract

Heterogeneity in habitat plays a crucial role in the dynamics of spatially extended populations and is often ignored by both empiricists and theoreticians. A common assumption made is that spatially homogeneous systems and those with slight heterogeneity will behave similarly and, therefore, the results and data from studies of the former can be applied to the latter. Here, we test this assumption by deriving a phase model from two weakly coupled predator-prey oscillators and analyze the effect of spatial heterogeneity on the phase dynamics of this system. We find that even small heterogeneity between the two patches causes substantial changes in the phase dynamics of the system which can have dramatic effects on both population dynamics and persistence. Additionally, if the prey and predator time scales are similar, the effect of heterogeneity is much greater.

The Paradox of Enrichment in Metaecosystems

“…Spatial synchrony usually has negative effects on ecosystem stability by increasing the global extinction risk or reducing the effect of dispersal (Earn et al 2000). In homogeneous metaecosystems, we observe perfect in-phase synchrony even for very low diffusion rates of living compartments (phase locking; Jansen 1999; Goldwyn and Hastings 2008), which cancels any potential stabilization via source-sink dynamics (Vogwill et al 2009). However, if heterogeneity prevents perfect synchrony, diffusion can synchronize the dynamics while they are stabilized by other mechanisms (e.g., Abbott 2011).…”

Dispersal and noise: Various modes of synchrony in ecological oscillators

“…This allows us to define a stochastic phase variable for each oscillator according to Θ i = Π(X i ) with X i evolving according to the Langevin equation (5). Moreover, the standard deterministic phase reduction method used in [11,12] can be extended to the Langevin equation (5), provided that the noise is treated in the sense of Stratonovich [9]. That is, the corresponding Wiener processes dW h (t) = η h (t)dt and dW p (t) = η p (t)dt are considered the zero correlation limits of a coloured noise process so that the normal rules of calculus can be applied.…”

“…The phase-reduction method can then be used to analyze synchronization of an ensemble of oscillators by approximating the high-dimensional limit cycle dynamics as a closed system of equations for the corresponding phase variables [19,4]. Within the context of ecology, Goldwyn and Hastings have used the theory of weakly coupled phase oscillators to investigate various modes of synchronous and asynchronous phase-locking in predator-prey systems weakly coupled by dispersal [11,12]. They have also examined the joint effects of dispersal and environmental fluctuations, by simulating ensembles of predator-prey oscillators with diffusive or global coupling and spatially correlated Poisson inputs [13]; each predator-prey patch is described by the Rosenzweig and MacArthur (RM) model [35,17] Recently, a complementary approach to analyzing the effects of external fluctuations on the synchronization of predator-prey populations has been developed by Lai et.…”

Different types of synchrony in chaotic and cyclic communities

“…Beside in-phase synchronization between spatially separated populations, where populations cycle in tune, a more subtle form, phase locking, has been suggested by ecological models for periodic and chaotically oscillating populations 7,11,[25][26][27] . Phase locking occurs when one (or more) oscillator influences a second oscillator in such a way that their phases oscillate in tune but that there is a constant difference in the phase of two oscillators 1,2 .…”