General scaling law in the saddle–node bifurcation: a complex phase space study

Fontich, Ernest; Sardanyés, Josep

doi:10.1088/1751-8113/41/1/015102

Cited by 32 publications

(60 citation statements)

References 26 publications

(46 reference statements)

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“…Here, all hypercycle members oscillate for a long period of time before collapsing, in what is known as a delayed transition. Such a dynamics, which arises near saddle-node bifurcations, have been characterized in hypercycles [6,7,9,10,15,19], in autocatalytic replicators [10,11], as well as in singlespecies dynamics under Allee effects [20]. Interestingly, the same dynamics is found below the saddle-node bifurcation of fixed points and below the saddle-node of periodic orbits (Fig.…”

Section: B Bifurcation Analysismentioning

confidence: 62%

“…Saddle-node bifurcations are typically responsible of asymptotic extinction in catalytic networks. These bifurcations, which have been identified in small hypercycles [7,9,15] as well as in autocatalytic replicators [10,11], are known to separate the survival from the extinction phases.…”

Section: Discussionmentioning

confidence: 98%

“…On the contrary, low initial conditions involve the asymptotic extinction of the hyperycle since the dynamics are captured by a stable attractor found in the origin of the phase space. The dynamics of autocatalytic replicators [10,11] and two-member hypercycles [7,10] have been widely characterized. Hypercycles with three and four species display the same asymptotic behavior than smaller catalytic networks but the coexistence attractor is approached by means of damped oscillations [1,6].…”

Section: Introductionmentioning

confidence: 99%

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Bifurcations analysis of oscillating hypercycles

Guillamón

Fontich

Sardanyés

2015

Journal of Theoretical Biology

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We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n > 4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named Q SS ). It has also been reported that, for values below Q SS , the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system, which is used to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q, undergoes a critical fidelity threshold, Q P O . The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q < Q P O . Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of Q P O is slightly higher than Q SS , thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q = 1.

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Section: B Bifurcation Analysismentioning

confidence: 62%

Section: Discussionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Bifurcations analysis of oscillating hypercycles

Guillamón

Fontich

Sardanyés

2015

Journal of Theoretical Biology

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“…We establish a procedure to analytically derive the scaling law for saddle-node bifurcations applied to one-dimensional unimodal maps using complex variable techniques. These calculations enable us to analytically provide the number of iterates in the bottleneck region (see Fontich and Sardanyés [21] for equivalent results in autonomous, one-dimensional continuous systems). The scaling law measures the number of iterates required to pass through x c = (−b + 1)/(2c), as a function of the difference of the parameter β to its bifurcation value β c .…”

Section: Scaling Law In a General One-dimensional Map Quadratic Formmentioning

confidence: 99%

“…A similar approach was recently applied to study the scaling properties for saddle-node bifurcations in one-dimensional, time continuous dynamical systems [21]. Here, we first develop our calculations for a general one-dimensional quadratic map.…”

Section: Introductionmentioning

confidence: 99%

Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

Duarte¹,

Januário²,

Martins³

et al. 2011

Nonlinear Dyn

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The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ , scales according to an inverse square-root power law, τ ∼ (μ − μ c ) −1/2 , as the bifurcation parameter μ, is driven further away from its critical value, μ c . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of onedimensional discrete dynamical systems with saddlenode bifurcations.

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Habitat loss‐induced tipping points in metapopulations with facilitation

2019

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Habitat loss is known to pervade extinction thresholds in metapopulations. Such thresholds result from a loss of stability that can eventually lead to collapse. Several models have been developed to understand the nature of these transitions and how they are affected by the locality of interactions, fluctuations or external drivers. Most models consider the impact of grazing or aridity as a control parameter that can trigger sudden shifts, once critical values are reached. Others explore instead the role played by habitat loss and fragmentation. Here we consider a minimal model incorporating facilitation between the individuals of the same species along with habitat destruction, with the aim of understanding how local cooperation and habitat loss interact with each other. A mathematical model incorporating facilitation and habitat destruction is derived, along with a spatially explicit simulation model. It is found that a catastrophic shift is expected for increasing levels of habitat loss, but the bifurcation becomes continuous when dispersal is local. Under these conditions, spatial patchiness is found and the qualitative change from discontinuous to continuous results are in agreement with previous studies on ecological systems. Our results suggest that species exhibiting facilitation and displaying short‐range dispersal will be markedly more capable of avoiding catastrophic tipping points.

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General scaling law in the saddle–node bifurcation: a complex phase space study

Cited by 32 publications

References 26 publications

Bifurcations analysis of oscillating hypercycles

Bifurcations analysis of oscillating hypercycles

Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

Habitat loss‐induced tipping points in metapopulations with facilitation

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