Globally coupled chaotic maps and demographic stochasticity

Abstract

The affect of demographic stochasticity of a system of globally coupled chaotic maps is considered. A two-step model is studied, where the intra-patch chaotic dynamics is followed by a migration step that coupled all patches; the equilibrium number of agents on each site, N , controls the strength of the discreteness-induced fluctuations. For small N (large fluctuations) a period-doubling cascade appears as the coupling (migration) increases. As N grows an extremely slow dynamic emerges, leading to a flow along a one-dimensional family of almost period 2 solutions. This manifold become a true solutions in the deterministic limit. The degeneracy between different attractors that characterizes the clustering phase of the deterministic system is thus the N → ∞ limit of the slow dynamics manifold.PACS numbers: 87.23. Cc , 64.70.qj, 05.45.Xt, The dynamics of coupled chaotic maps have attracted a lot of interest in the last decades, following the pioneering works of Kaneko [1,2]. A substantial part of the study is focused around the paradigmatic model of globally coupled maps, where many fundamental results like mutual synchronization, dynamical clustering and glassy behavior were demonstrated [3]. The universal character of the chaotic dynamics makes the coupled maps model relevant to many phenomena, ranging from neural systems and human body rhythms to coupled lasers and cryptography [4].Here we consider the effect of demographic stochasticity (shot noise) on various phases of a globally coupled system. This problem emerges naturally while applying the theory to spatially extended ecologies.Many old [5] and recent [6] experiments suggest that the well-mixed dynamics of simple ecosystems (single species or victim-exploiter system) are extinction-prone, and that the system acquires stability only due to its spatial structure, a result supported also by numerical simulations of many models [7,8]. The extended (spatial) system survives due to the possibility of migration among spatial patches. This migration should be large enough to allow for recolonization of empty patches be emigrants. On the other hand [9], too much migration is also dangerous, as it leads to global synchronization, in which case the system acts essentially as a single, wellmixed patch, with its vulnerability to extinction.The globally coupled system, which obeys,is a natural and popular framework to discuss the dynamics of so-called meta-populations [10] on various patches with migration between the patches [9]. Here, s i is the population density on the i-th site and F is the chaotic map. ν is the migration parameter (the chance of an individual agent to leave its site) and 1 ≤ i ≤ L, where L is the number of patches. To address the problem of extinction properly, however, it is necessary to account for the discrete nature of the population and the absorbing character of the zero population state. This is achieved by the addition of demographic (shot) noise to the system. Clearly, if the local populations are all large, this effect is tiny. Howeve...

Consistent scaling of persistence time in metapopulations

“…At D 1 a wellunderstood phase transition occurs (Snyder 2000, Kessler and Shnerb 2010 and the metapopulation forms a single domain, allowing recolonization of an extinct patch even when the nearest occupied patch is many dispersal steps away. At D 1 a wellunderstood phase transition occurs (Snyder 2000, Kessler and Shnerb 2010 and the metapopulation forms a single domain, allowing recolonization of an extinct patch even when the nearest occupied patch is many dispersal steps away.…”

“…Even when local dynamics are extinction prone, the metapopulation as a whole can exhibit a stable equilibrium (Hastings 1993, Ben-Zion et al 2010, Kessler and Shnerb 2010, causing the persistence time to scale exponentially with the number of patches L. Fig. Accordingly, this is the most extinction robust scenario.…”

“…As D increases from zero towards the critical value D 1 , the size and lifetime of the domains increases, but the same considerations still hold. At D 1 a wellunderstood phase transition occurs (Snyder 2000, Kessler and Shnerb 2010 and the metapopulation forms a single domain, allowing recolonization of an extinct patch even when the nearest occupied patch is many dispersal steps away. This marks the transition to the intermediate-dispersal regime.…”

Synchronization-induced persistence versus selection for habitats in spatially coupled ecosystems

“…Therefore, the dynamics are reduced to metapopulation dynamics, and we may assume that each site is either empty or populated by N 0 individuals. The extinction rate equals a single site's (MTE) 21 , which was derived by Assaf & Meerson [16] using the Wentzel-Kramers-Brillouin (WKB) approach, and colonization rate equals lN 0 c times the fixation probability of a single individual. Because the population can stably persist if and only if the average number of established migrants that are being sent from a single site before its population becomes extinct is greater than one, it follows that…”

Sustainability without coexistence state in Durrett–Levin hawk–dove model

“…Some deterministic models that do not support a coexistence fixed point still admit other attractive manifolds, like a limit cycle or chaotic attractor (Wilson and Abrams 2005;Abrams and Holt 2002;Armstrong and McGehee 1980). The effect of demographic stochasticity on the extinction rate in these cases is also exponentially small, where now N should be taken as the minimal number of agents along the deterministic trajectory (Kessler and Shnerb 2010;McKane and Newman 2005). This holds unless the stable manifold is "excitable" and the system undergoes long excursions under weak perturbations (Ben Zion et al 2010).…”