movie.gif from Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and contain a similar percentage of the human population. Theoretical models and palaeoclimatic data predict a grim future, with rapid shifts towards a desert state, with accelerated diversity losses and ecological collapses. These shifts are a consequence of the special nonlinearities resulting from ecological facilitation. Here, we investigate a simple model of semiarid ecosystems identifying the so-called ghost, which appears after a catastrophic transition from a vegetated to a desert state once a critical rate of soil degradation is overcome. The ghost which involves a slowdown of transients towards the desert state, making the ecosystem to seem stable even though vegetation extinction is inevitable. We use this model to show how to exploit the ecological ghosts to avoid collapse. Doing so involves the restoration of small fractions of desert areas with vegetation capable of maintaining a stable community once the catastrophic shift condition has been achieved. This intervention method is successfully tested under the presence of demographic stochastic fluctuations

Supporting Information from Trans-heteroclinic bifurcation: a novel type of catastrophic shift

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as <i>trans-heteroclinic bifurcation</i>, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication–mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication–mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems

movie.gif from Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and contain a similar percentage of the human population. Theoretical models and palaeoclimatic data predict a grim future, with rapid shifts towards a desert state, with accelerated diversity losses and ecological collapses. These shifts are a consequence of the special nonlinearities resulting from ecological facilitation. Here, we investigate a simple model of semiarid ecosystems identifying the so-called ghost, which appears after a catastrophic transition from a vegetated to a desert state once a critical rate of soil degradation is overcome. The ghost involves a slowdown of transients towards the desert state, making the ecosystem seem stable even though vegetation extinction is inevitable. We use this model to show how to exploit the ecological ghosts to avoid collapse. Doing so involves the restoration of small fractions of desert areas with vegetation capable of maintaining a stable community once the catastrophic shift condition has been achieved. This intervention method is successfully tested under the presence of demographic stochastic fluctuations

Vidiella_et_al_SI_R2.pdf from Exploiting delayed transitions to sustain semiarid ecosystems after catastrophic shifts

Semiarid ecosystems (including arid, semiarid and dry-subhumid ecosystems) span more than 40% of extant habitats and contain a similar percentage of the human population. Theoretical models and palaeoclimatic data predict a grim future, with rapid shifts towards a desert state, with accelerated diversity losses and ecological collapses. These shifts are a consequence of the special nonlinearities resulting from ecological facilitation. Here, we investigate a simple model of semiarid ecosystems identifying the so-called ghost, which appears after a catastrophic transition from a vegetated to a desert state once a critical rate of soil degradation is overcome. The ghost which involves a slowdown of transients towards the desert state, making the ecosystem to seem stable even though vegetation extinction is inevitable. We use this model to show how to exploit the ecological ghosts to avoid collapse. Doing so involves the restoration of small fractions of desert areas with vegetation capable of maintaining a stable community once the catastrophic shift condition has been achieved. This intervention method is successfully tested under the presence of demographic stochastic fluctuations

Scheme of the experimental evolution procedure and virulence and <i>ID</i>₅₀ (infectivity) estimation.

<p>Seven dpi, virus accumulation (titer) was evaluated by local-lesion assays on <i>C. quinoa</i> and then concentrations were made equal so each newly infected plant received 20 LFU per evolutionary passages or to 30 LFU/µL for <i>ID</i>₅₀ determination.</p

Phase portraits obtained numerically from <b>Eqs. (1)</b>–<b>(2)</b> displaying the dynamics in the phase plane (<i>x</i>₁, <i>x</i>₂), with <i>x</i>₁+<i>x</i>₂ = 1, and the stability of the fixed points: <i>P</i>₁^*, <i>P</i>₂^*, <i>P</i>₃^*, and <i>P</i>₄^* (stable and unstable equilibria are shown, respectively, in black and white circles).

<p>In (a) we use the experimental values used in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0017917#pone-0017917-g004" target="_blank">Fig. 4 (a)</a> right. In (b) we use the same values of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0017917#pone-0017917-g004" target="_blank">Fig. 4(b)</a> left. In both cases the origin is a repeller; <i>P</i>₃^* is a saddle; <i>P</i>₂^* (outcompetition of <i>x</i>₂ by <i>x</i>₁) is stable and the equilibrium <i>P</i>₄^* is outside the phase plane. In (c) we show the asymptotic coexistence scenario, where <i>P</i>₂^* becomes a saddle and the fixed point <i>P</i>₄^*, which is stable, is inside the phase plane (here we use <i>r_max</i>_,1 = 1.07, <i>r_max</i>_,2 = 0.621, <i>K</i>₁ = 0.8 and <i>K</i>₂ = 0.2). The arrows in all the plots indicate the directions of the flows.</p

Percentage of infected plants infected by one or the two viral strains as determined by the RT-PCR/restriction analysis.

<p>Percentage of infected plants infected by one or the two viral strains as determined by the RT-PCR/restriction analysis.</p

Supplementary info from Noise-induced bistability in the quasi-neutral coexistence of viral RNAs under different replication modes. 21 February 2018 8 May 2018

Evolutionary and dynamical investigations into real viral populations indicate that RNA replication can range between the two extremes represented by the so-called ‘stamping machine replication’ (SMR) and ‘geometric replication’ (GR). The impact of asymmetries in replication for single stranded, (+) sense RNA viruses has been mainly studied with deterministic models. However, viral replication should be better described by including stochasticity, as the cell infection process is typically initiated with a very small number of RNA macromolecules, and thus largely influenced by intrinsic noise. Under appropriate conditions, deterministic theoretical descriptions of viral RNA replication predict a quasi-neutral coexistence scenario, with a line of fixed points involving different strands' equilibrium ratios depending on the initial conditions. Recent research into the quasi-neutral coexistence in two competing populations reveals that stochastic fluctuations fundamentally alter the mean-field scenario, and one of the two species outcompetes the other one. In this manuscript, we study this phenomenon for viral RNAs replication modes by means of stochastic simulations and a diffusion approximation. Our results reveal that noise has a strong impact on the amplification of viral RNAs, also causing the emergence of noise-induced bistability. We provide analytical criteria for the dominance of (+) sense strands depending on the initial populations on the line of equilibria, which are in agreement with direct stochastic simulation results. The biological implications of this noise-driven mechanism are discussed within the framework of the evolutionary dynamics of RNA viruses with different modes of replication

Supplementary Information from Population dynamics of synthetic terraformation motifs

Ecosystems are complex systems, currently experiencing several threats associated with global warming, intensive exploitation and human-driven habitat degradation. Because of a general presence of multiple stable states, including states involving population extinction, and due to the intrinsic nonlinearities associated with feedback loops, collapse in ecosystems could occur in a catastrophic manner. It has been recently suggested that a potential path to prevent or modify the outcome of these transitions would involve designing synthetic organisms and synthetic ecological interactions that could push these endangered systems out of the critical boundaries. In this paper, we investigate the dynamics of the simplest mathematical models associated with four classes of ecological engineering designs, named <i>Terraformation motifs</i> (TMs). These TMs put in a nutshell different ecological strategies. In this context, some fundamental types of bifurcations pervade the systems; dynamics. Mutualistic interactions can enhance persistence of the systems by means of saddle-node bifurcations. The models without cooperative interactions show that ecosystems achieve restoration through transcritical bifurcations. Thus, our analysis of the models allows us to define the stability conditions and parameter domains where these TMs must work.