From 0717fbede6b5490cc7a2373715a54129a4891ebc Mon Sep 17 00:00:00 2001 From: Silvia DE MONTE <demonte@evolbio.mpg.de> Date: Mon, 13 Mar 2023 17:34:07 +0100 Subject: [PATCH] intro0 --- manuscript/draft/chaotic_glv_draft.aux | 247 ++++++++------ manuscript/draft/chaotic_glv_draft.dvi | Bin 0 -> 182760 bytes manuscript/draft/chaotic_glv_draft.log | 305 ++++++++++++------ manuscript/draft/chaotic_glv_draft.pdf | Bin 2701742 -> 2725971 bytes manuscript/draft/chaotic_glv_draft.synctex.gz | Bin 217398 -> 239348 bytes manuscript/draft/chaotic_glv_draft.tex | 29 +- manuscript/draft/chaotic_glv_draft.toc | 46 +-- 7 files changed, 404 insertions(+), 223 deletions(-) create mode 100644 manuscript/draft/chaotic_glv_draft.dvi diff --git a/manuscript/draft/chaotic_glv_draft.aux b/manuscript/draft/chaotic_glv_draft.aux index f2be91e..851d350 100644 --- a/manuscript/draft/chaotic_glv_draft.aux +++ b/manuscript/draft/chaotic_glv_draft.aux @@ -16,6 +16,46 @@ \gdef\HyperFirstAtBeginDocument#1{#1} \providecommand\HyField@AuxAddToFields[1]{} \providecommand\HyField@AuxAddToCoFields[2]{} +\citation{deVargas} +\citation{review} +\citation{hoffman2007} +\citation{driscoll2016} +\citation{nemergut} +\citation{logares} +\citation{chesson?} +\citation{barabas_2018?} +\citation{lennon} +\citation{benincà } +\citation{huisman} +\citation{munch} +\citation{fuhrmann} +\citation{martinez-platero} +\citation{gilbert} +\citation{...} +\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction \& background}{1}{section.1}} +\citation{rosenzweig-mcarthur} +\citation{rinaldi} +\citation{huisman} +\citation{may} +\citation{bunin} +\citation{roy} +\citation{altieri} +\citation{pearce} +\citation{...} +\citation{barbieri} +\citation{bunin} +\citation{...} +\citation{pearce} +\citation{roy} +\citation{o'sullivan} +\citation{bunin?} +\citation{cazelles} +\citation{martinez-platero} +\citation{grilli} +\citation{doebeli2021} +\citation{ser-giacomi} +\citation{henderson&magurran} +\citation{ulrich2004} \citation{Hutchinson1961} \citation{Huisman1999} \citation{Scheffer2003} @@ -24,10 +64,9 @@ \citation{Medvinsky2015} \citation{MartinPlatero2018} \citation{SerGiacomi2018} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Why are some species rare while other are abundant?}{2}{subsection.1.1}} \citation{Rogers2022} \citation{Munch2022} -\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction \& background}{1}{section.1}} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Why are some species rare while other are abundant?}{1}{subsection.1.1}} \citation{Telesh2019} \citation{Hu2022} \citation{Mambuca2022} @@ -54,6 +93,11 @@ \citation{Bunin2017} \citation{Barbier2018} \citation{Hofbauer2002} +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{3}{subsection.1.2}} +\newlabel{sec:model}{{1.2}{3}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{subsection.1.2}{}} +\newlabel{sec:model@cref}{{[subsection][2][1]1.2}{[1][3][]3}} +\newlabel{eq:gLV}{{1}{3}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{equation.1.1}{}} +\newlabel{eq:gLV@cref}{{[equation][1][]1}{[1][3][]3}} \citation{Poley2022x} \citation{Mambuca2022} \citation{Akjouj2022} @@ -68,11 +112,6 @@ \citation{Lorenzana2022} \citation{Denk2022} \citation{Descheemaeker2021} -\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{2}{subsection.1.2}} -\newlabel{sec:model}{{1.2}{2}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{subsection.1.2}{}} -\newlabel{sec:model@cref}{{[subsection][2][1]1.2}{[1][2][]2}} -\newlabel{eq:gLV}{{1}{2}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{equation.1.1}{}} -\newlabel{eq:gLV@cref}{{[equation][1][]1}{[1][2][]2}} \citation{Bunin2017} \citation{Barbier2017x} \citation{Galla2018} @@ -93,15 +132,15 @@ \citation{Pearce2020} \citation{Denk2022} \citation{OSullivan2021} +\newlabel{eq:weak}{{2}{4}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{equation.1.2}{}} +\newlabel{eq:weak@cref}{{[equation][2][]2}{[1][4][]4}} +\newlabel{eq:ai}{{3}{4}{The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{equation.1.3}{}} +\newlabel{eq:ai@cref}{{[equation][3][]3}{[1][4][]4}} +\@writefile{toc}{\contentsline {section}{\numberline {2}Chaotic turnover from strong interactions}{4}{section.2}} +\newlabel{sec:chaos}{{2}{4}{Chaotic turnover from strong interactions}{section.2}{}} +\newlabel{sec:chaos@cref}{{[section][2][]2}{[1][4][]4}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}A chaotic turnover of rare and dominant species}{4}{subsection.2.1}} \citation{Xu2020} -\newlabel{eq:weak}{{2}{3}{The disordered Lotka-Volterra (dLV) model with weak or strong 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random colour) show a turnover of dominant species. \textbf {(B)} Bray-Curtis index of community composition similarity between the dominant component of the community at time $t$, and the equilibrium that would be attained if all rare species (here not visible on the linear abundance scale) were permanently removed at that time. It suggests that the dynamics moves close to few-species equilibria continually destabilized by invasion from the pool of rare species. Transient dynamics before $t=0$ has been discarded. \relax }}{4}{figure.caption.1}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces \textbf {Succession of dominant species.} \textbf {(A)} The stacked abundances of all species (each with a distinct random colour) show a turnover of dominant species. \textbf {(B)} Bray-Curtis index of community composition similarity between the dominant component of the community at time $t$, and the equilibrium that would be attained if all rare species (here not visible on the linear abundance scale) were permanently removed at that time. It suggests that the dynamics moves close to few-species equilibria continually destabilized by invasion from the pool of rare species. Transient dynamics before $t=0$ has been discarded. \relax }}{5}{figure.caption.1}} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} -\newlabel{fig:turnover}{{1}{4}{\textbf {Succession of dominant species.} \textbf {(A)} The stacked abundances of all species (each with a distinct random colour) show a turnover of dominant species. \textbf {(B)} Bray-Curtis index of community composition similarity between the dominant component of the community at time $t$, and the equilibrium that would be attained if all rare species (here not visible on the linear abundance scale) were permanently removed at that time. It suggests that the dynamics moves close to few-species equilibria continually destabilized by invasion from the pool of rare species. 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Transient dynamics before $t=0$ has been discarded. \relax }{figure.caption.1}{}} +\newlabel{fig:turnover@cref}{{[figure][1][]1}{[1][4][]5}} +\newlabel{eq:Neff}{{4}{5}{A chaotic turnover of rare and dominant species}{equation.2.4}{}} +\newlabel{eq:Neff@cref}{{[equation][4][]4}{[1][5][]5}} \citation{Mezard1986} \citation{Roy2019} \citation{Grilli2020} \citation{Descheemaeker2020} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces \textbf {Abundance fluctuation distribution (AFD) and snapshot species abundance distribution (SAD).} The black line shows the distribution of abundances observed across all times and species (avarage AFD); shaded in grey is $\pm $ one std across species. The histogram is a `snapshot' distribution of species abundances at one arbitrary point in time (SAD). 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The histogram is a `snapshot' distribution of species abundances at one arbitrary point in time (SAD). The few missing bars are not systematic, but due to high-abundance brackets only ever containing a couple of species for $S=500$.\relax }}{6}{figure.caption.2}} +\newlabel{fig:dist}{{2}{6}{\textbf {Abundance fluctuation distribution (AFD) and snapshot species abundance distribution (SAD).} The black line shows the distribution of abundances observed across all times and species (avarage AFD); shaded in grey is $\pm $ one std across species. The histogram is a `snapshot' distribution of species abundances at one arbitrary point in time (SAD). The few missing bars are not systematic, but due to high-abundance brackets only ever containing a couple of species for $S=500$.\relax }{figure.caption.2}{}} +\newlabel{fig:dist@cref}{{[figure][2][]2}{[1][5][]6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Species' abundance fluctuations follow a power-law}{6}{subsection.2.2}} +\newlabel{sec:power}{{2.2}{6}{Species' abundance fluctuations follow a power-law}{subsection.2.2}{}} +\newlabel{sec:power@cref}{{[subsection][2][2]2.2}{[1][5][]6}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {Comparison of stochastic to chaotic model}. \textbf {(A)} Time series of one species in the deterministic model, chaotic phase (blue), and a realization of the stochastic focal-species model with parameters $u^*,k^*,\tau ^*$ (see main text). {(\textbf {B})} Comparison of resulting abundance fluctuation distribution $P^*(x)$; dashed pink corresponds to universal coloured noise approximation of the stochastic model.\relax }}{6}{figure.caption.3}} +\newlabel{fig:eff}{{3}{6}{\textbf {Comparison of stochastic to chaotic model}. \textbf {(A)} Time series of one species in the deterministic model, chaotic phase (blue), and a realization of the stochastic focal-species model with parameters $u^*,k^*,\tau ^*$ (see main text). {(\textbf {B})} Comparison of resulting abundance fluctuation distribution $P^*(x)$; dashed pink corresponds to universal coloured noise approximation of the stochastic model.\relax }{figure.caption.3}{}} +\newlabel{fig:eff@cref}{{[figure][3][]3}{[1][6][]6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}A stochastic focal-species model reproduces rarity--dominance patterns}{6}{subsection.2.3}} +\newlabel{sec:focal}{{2.3}{6}{A stochastic focal-species model reproduces rarity--dominance patterns}{subsection.2.3}{}} +\newlabel{sec:focal@cref}{{[subsection][3][2]2.3}{[1][6][]6}} +\newlabel{eq:eff}{{6}{6}{A stochastic focal-species model reproduces rarity--dominance patterns}{equation.2.6}{}} +\newlabel{eq:eff@cref}{{[equation][6][]6}{[1][6][]6}} +\newlabel{eq:g}{{7}{6}{A stochastic focal-species model reproduces rarity--dominance patterns}{equation.2.7}{}} +\newlabel{eq:g@cref}{{[equation][7][]7}{[1][6][]6}} \citation{Xu2020} -\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Species with lower overall interactions are more often dominant}{6}{subsection.2.4}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces \textbf {Species differences in dominance.} \textbf {(A)} A long abundance time series of the three species who are ranked first, in the middle, and last, with respect to time spent in the dominant component. \textbf {(B)} Shown for $S \in \{320,640,1280,2560,5120\}$: the dominance-time-\textit {vs}-rank distribution, referring to fraction of time spent in the dominant component (as defined in main text), normalized by the median value. Ranks have been normalized by the total number of species. The inset demonstrates that the median time scales as $1/S$. \textbf {(C)} A scatter plot of dominance time versus each species' shifted-and-rescaled sum of interaction coefficients ($z_i$ in \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:zi}\unskip \@@italiccorr )}}). Colour indicates the `typicality' $\theta _i$ of a species AFD, as defined in \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:typ}\unskip \@@italiccorr )}}. Here, $S=1280$. {\color {WildStrawberry}[Fix rank by mean; missing $S$ point]} \relax }}{6}{figure.caption.4}} -\newlabel{fig:diff}{{4}{6}{\textbf {Species differences in dominance.} \textbf {(A)} A long abundance time series of the three species who are ranked first, in the middle, and last, with respect to time spent in the dominant component. \textbf {(B)} Shown for $S \in \{320,640,1280,2560,5120\}$: the dominance-time-\textit {vs}-rank distribution, referring to fraction of time spent in the dominant component (as defined in main text), normalized by the median value. Ranks have been normalized by the total number of species. The inset demonstrates that the median time scales as $1/S$. \textbf {(C)} A scatter plot of dominance time versus each species' shifted-and-rescaled sum of interaction coefficients ($z_i$ in \eqref {eq:zi}). Colour indicates the `typicality' $\theta _i$ of a species AFD, as defined in \eqref {eq:typ}. Here, $S=1280$. \inpink {[Fix rank by mean; missing $S$ point]} \relax }{figure.caption.4}{}} -\newlabel{fig:diff@cref}{{[figure][4][]4}{[1][6][]6}} -\newlabel{eq:zi}{{10}{6}{Species with lower overall interactions are more often dominant}{equation.2.10}{}} -\newlabel{eq:zi@cref}{{[equation][10][]10}{[1][6][]6}} -\newlabel{eq:typ}{{11}{6}{Species with lower overall interactions are more often dominant}{equation.2.11}{}} -\newlabel{eq:typ@cref}{{[equation][11][]11}{[1][6][]6}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Species with lower overall interactions are more often dominant}{7}{subsection.2.4}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces \textbf {Species differences in dominance.} \textbf {(A)} A long abundance time series of the three species who are ranked first, in the middle, and last, with respect to time spent in the dominant component. \textbf {(B)} Shown for $S \in \{320,640,1280,2560,5120\}$: the dominance-time-\textit {vs}-rank distribution, referring to fraction of time spent in the dominant component (as defined in main text), normalized by the median value. Ranks have been normalized by the total number of species. The inset demonstrates that the median time scales as $1/S$. \textbf {(C)} A scatter plot of dominance time versus each species' shifted-and-rescaled sum of interaction coefficients ($z_i$ in \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:zi}\unskip \@@italiccorr )}}). Colour indicates the `typicality' $\theta _i$ of a species AFD, as defined in \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:typ}\unskip \@@italiccorr )}}. Here, $S=1280$. {\color {WildStrawberry}[Fix rank by mean; missing $S$ point]} \relax }}{7}{figure.caption.4}} +\newlabel{fig:diff}{{4}{7}{\textbf {Species differences in dominance.} \textbf {(A)} A long abundance time series of the three species who are ranked first, in the middle, and last, with respect to time spent in the dominant component. \textbf {(B)} Shown for $S \in \{320,640,1280,2560,5120\}$: the dominance-time-\textit {vs}-rank distribution, referring to fraction of time spent in the dominant component (as defined in main text), normalized by the median value. Ranks have been normalized by the total number of species. The inset demonstrates that the median time scales as $1/S$. \textbf {(C)} A scatter plot of dominance time versus each species' shifted-and-rescaled sum of interaction coefficients ($z_i$ in \eqref {eq:zi}). Colour indicates the `typicality' $\theta _i$ of a species AFD, as defined in \eqref {eq:typ}. Here, $S=1280$. \inpink {[Fix rank by mean; missing $S$ point]} \relax }{figure.caption.4}{}} +\newlabel{fig:diff@cref}{{[figure][4][]4}{[1][7][]7}} +\newlabel{eq:zi}{{10}{7}{Species with lower overall interactions are more often dominant}{equation.2.10}{}} +\newlabel{eq:zi@cref}{{[equation][10][]10}{[1][7][]7}} +\newlabel{eq:typ}{{11}{7}{Species with lower overall interactions are more often dominant}{equation.2.11}{}} +\newlabel{eq:typ@cref}{{[equation][11][]11}{[1][7][]7}} \citation{Bunin2017} \citation{Hardin1960} \citation{Kessler2015} \citation{Baron2019} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf {Dynamical phase diagram}. Probability of persistent chaos in long-time simulations. For each $\mu $ and $\sigma $ (with a 0.01 resolution), {\color {WildStrawberry}N} simulations were made from different, random abundance initial conditions and realizations of the interaction matrix. Parameters yielding divergence every time are marked with grey. The identification of the different phases is discussed in the main text.\relax }}{7}{figure.caption.5}} -\newlabel{fig:chpr}{{5}{7}{\textbf {Dynamical phase diagram}. Probability of persistent chaos in long-time simulations. For each $\mu $ and $\sigma $ (with a 0.01 resolution), \inpink {N} simulations were made from different, random abundance initial conditions and realizations of the interaction matrix. Parameters yielding divergence every time are marked with grey. 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A single interaction matrix is shifted and rescaled to given $\mu ,\sigma $, and simulations are run in horizontal lines from right to left in order for the abundance dynamics to track a moving attractor until it vanishes. \textbf {(B)} Stacked abundances after transient simulation time between each $\Delta \mu $ step. Simulated sequences is $\mu = 1.4 \to 0.6 \to 1.4$. {\color {WildStrawberry}[update so that data comes from exactly same rnd seed as above]} \relax }}{7}{figure.caption.6}} -\newlabel{fig:adiab}{{6}{7}{\textbf {Adiabatic simulation outcomes}. \textbf {(A)} Attractor type converged to in an adiabatic left $\to $ right simulation approach (colour quality), and the effective community size \eqref {eq:Neff} (colour gradation). A single interaction matrix is shifted and rescaled to given $\mu ,\sigma $, and simulations are run in horizontal lines from right to left in order for the abundance dynamics to track a moving attractor until it vanishes. \textbf {(B)} Stacked abundances after transient simulation time between each $\Delta \mu $ step. Simulated sequences is $\mu = 1.4 \to 0.6 \to 1.4$. \inpink {[update so that data comes from exactly same rnd seed as above]} \relax }{figure.caption.6}{}} -\newlabel{fig:adiab@cref}{{[figure][6][]6}{[1][7][]7}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf {Dynamical phase diagram}. Probability of persistent chaos in long-time simulations. For each $\mu $ and $\sigma $ (with a 0.01 resolution), {\color {WildStrawberry}N} simulations were made from different, random abundance initial conditions and realizations of the interaction matrix. Parameters yielding divergence every time are marked with grey. The identification of the different phases is discussed in the main text.\relax }}{8}{figure.caption.5}} +\newlabel{fig:chpr}{{5}{8}{\textbf {Dynamical phase diagram}. Probability of persistent chaos in long-time simulations. For each $\mu $ and $\sigma $ (with a 0.01 resolution), \inpink {N} simulations were made from different, random abundance initial conditions and realizations of the interaction matrix. Parameters yielding divergence every time are marked with grey. The identification of the different phases is discussed in the main text.\relax }{figure.caption.5}{}} +\newlabel{fig:chpr@cref}{{[figure][5][]5}{[1][8][]8}} +\@writefile{toc}{\contentsline {section}{\numberline {3}When chaos?---varying the interaction statistics}{8}{section.3}} +\newlabel{sec:phase}{{3}{8}{When chaos?---varying the interaction statistics}{section.3}{}} +\newlabel{sec:phase@cref}{{[section][3][]3}{[1][8][]8}} +\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Dynamical phase diagram}{8}{subsection.3.1}} +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf {Adiabatic simulation outcomes}. \textbf {(A)} Attractor type converged to in an adiabatic left $\to $ right simulation approach (colour quality), and the effective community size \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:Neff}\unskip \@@italiccorr )}} (colour gradation). A single interaction matrix is shifted and rescaled to given $\mu ,\sigma $, and simulations are run in horizontal lines from right to left in order for the abundance dynamics to track a moving attractor until it vanishes. \textbf {(B)} Stacked abundances after transient simulation time between each $\Delta \mu $ step. Simulated sequences is $\mu = 1.4 \to 0.6 \to 1.4$. {\color {WildStrawberry}[update so that data comes from exactly same rnd seed as above]} \relax }}{8}{figure.caption.6}} +\newlabel{fig:adiab}{{6}{8}{\textbf {Adiabatic simulation outcomes}. \textbf {(A)} Attractor type converged to in an adiabatic left $\to $ right simulation approach (colour quality), and the effective community size \eqref {eq:Neff} (colour gradation). A single interaction matrix is shifted and rescaled to given $\mu ,\sigma $, and simulations are run in horizontal lines from right to left in order for the abundance dynamics to track a moving attractor until it vanishes. \textbf {(B)} Stacked abundances after transient simulation time between each $\Delta \mu $ step. Simulated sequences is $\mu = 1.4 \to 0.6 \to 1.4$. \inpink {[update so that data comes from exactly same rnd seed as above]} \relax }{figure.caption.6}{}} +\newlabel{fig:adiab@cref}{{[figure][6][]6}{[1][8][]8}} \citation{Descheemaeker2021} \citation{Afraimovich2008} \citation{Pearce2020} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}From multiple attractors to chaos: a transcritical bifurcation}{8}{subsection.3.2}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Relationships between community-level observables}{8}{subsection.3.3}} -\newlabel{eq:Xbar}{{13}{8}{Relationships between community-level observables}{equation.3.13}{}} -\newlabel{eq:Xbar@cref}{{[equation][13][]13}{[1][8][]8}} -\newlabel{eq:rho_crit}{{14}{8}{Relationships between community-level observables}{equation.3.14}{}} -\newlabel{eq:rho_crit@cref}{{[equation][14][]14}{[1][8][]8}} -\@writefile{toc}{\contentsline {section}{\numberline {4}Discussion}{8}{section.4}} 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of stochastic focal-species model from dynamical mean-field arguments}{10}{subsection.Alph0.7}} -\newlabel{app:noise}{{Appendix G}{10}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{subsection.Alph0.7}{}} -\newlabel{app:noise@cref}{{[subappendix][7][2147483647,0]Appendix G}{[1][10][]10}} -\newlabel{eq:acf}{{35}{10}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{equation.Alph0.35}{}} -\newlabel{eq:acf@cref}{{[equation][35][2147483647]35}{[1][10][]10}} +\newlabel{eq:ninv}{{25}{11}{Invasion analysis of few-species equilibria}{equation.Alph0.25}{}} +\newlabel{eq:ninv@cref}{{[equation][25][2147483647]25}{[1][11][]11}} +\@writefile{toc}{\contentsline {subsection}{\numberline {G}Derivation of stochastic focal-species model from dynamical mean-field arguments}{11}{subsection.Alph0.7}} +\newlabel{app:noise}{{Appendix G}{11}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{subsection.Alph0.7}{}} +\newlabel{app:noise@cref}{{[subappendix][7][2147483647,0]Appendix G}{[1][11][]11}} +\newlabel{eq:acf}{{35}{11}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{equation.Alph0.35}{}} +\newlabel{eq:acf@cref}{{[equation][35][2147483647]35}{[1][11][]11}} \citation{Jung1987} \citation{Fox1986a} \citation{Fox1986b} @@ -207,21 +246,21 @@ \bibcite{Scheffer2003}{3} \bibcite{Beninca2008}{4} \bibcite{Telesh2019}{5} -\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces \textbf {Statistical properties of the effective noise}. \textbf {(A,B)} Time series and distribution of $a_\text {rel} = a/\overline {a} -1$, \emph {etc.}. \textbf {(C,D)} Histograms of $\eta _i(t)$ across all species and time (grey), over just species for one random time (green), over all time for the first- (blue), mid- (pink), and last-ranked (yellow) species with respect to average abundance, with $\mathcal {N}(0,1)$ (black, dashed) for reference. \textbf {(E)} Autocorrelation functions . Left inset, \relax }}{11}{figure.caption.8}} -\newlabel{fig:noise}{{8}{11}{\textbf {Statistical properties of the effective noise}. \textbf {(A,B)} Time series and distribution of $a_\text {rel} = a/\overline {a} -1$, \etc . \textbf {(C,D)} Histograms of $\eta _i(t)$ across all species and time (grey), over just species for one random time (green), over all time for the first- (blue), mid- (pink), and last-ranked (yellow) species with respect to average abundance, with $\mathcal {N}(0,1)$ (black, dashed) for reference. \textbf {(E)} Autocorrelation functions . 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Left inset, \relax }}{12}{figure.caption.8}} +\newlabel{fig:noise}{{8}{12}{\textbf {Statistical properties of the effective noise}. \textbf {(A,B)} Time series and distribution of $a_\text {rel} = a/\overline {a} -1$, \etc . \textbf {(C,D)} Histograms of $\eta _i(t)$ across all species and time (grey), over just species for one random time (green), over all time for the first- (blue), mid- (pink), and last-ranked (yellow) species with respect to average abundance, with $\mathcal {N}(0,1)$ (black, dashed) for reference. \textbf {(E)} Autocorrelation functions . Left inset, \relax }{figure.caption.8}{}} +\newlabel{fig:noise@cref}{{[figure][8][2147483647]8}{[1][11][]12}} +\newlabel{eq:eta-oup}{{38}{12}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{equation.Alph0.38}{}} +\newlabel{eq:eta-oup@cref}{{[equation][38][2147483647]38}{[1][12][]12}} +\newlabel{eq:g=a+bn}{{39}{12}{Derivation of stochastic focal-species model from dynamical mean-field arguments}{equation.Alph0.39}{}} +\newlabel{eq:g=a+bn@cref}{{[equation][39][2147483647]39}{[1][12][]12}} +\@writefile{toc}{\contentsline {subsection}{\numberline {H}Solution of intermittency model under unified coloured noise approximation}{12}{subsection.Alph0.8}} +\newlabel{app:ucna}{{Appendix H}{12}{Solution of intermittency model under unified coloured noise approximation}{subsection.Alph0.8}{}} +\newlabel{app:ucna@cref}{{[subappendix][8][2147483647,0]Appendix H}{[1][12][]12}} +\newlabel{eq:P*-app}{{40}{12}{Solution of intermittency model under unified coloured noise approximation}{equation.Alph0.40}{}} +\newlabel{eq:P*-app@cref}{{[equation][40][2147483647]40}{[1][12][]12}} +\newlabel{eq:x-tilde-P*}{{45}{12}{Solution of intermittency model under unified coloured noise approximation}{equation.Alph0.45}{}} +\newlabel{eq:x-tilde-P*@cref}{{[equation][45][2147483647]45}{[1][12][]12}} +\@writefile{toc}{\contentsline {section}{Acknowledgments}{12}{equation.Alph0.46}} \bibcite{Medvinsky2015}{6} \bibcite{MartinPlatero2018}{7} \bibcite{SerGiacomi2018}{8} @@ -273,5 +312,5 @@ \bibcite{Jung1987}{54} \bibcite{Fox1986a}{55} \bibcite{Fox1986b}{56} -\@writefile{toc}{\contentsline {section}{References}{13}{equation.Alph0.46}} +\@writefile{toc}{\contentsline {section}{References}{14}{equation.Alph0.46}} \ttl@finishall diff --git a/manuscript/draft/chaotic_glv_draft.dvi b/manuscript/draft/chaotic_glv_draft.dvi new file mode 100644 index 0000000000000000000000000000000000000000..a7e4f371d511a17d0ffc446658abb3cce622a9db GIT binary patch literal 182760 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allowed for them to be stable despite competition for shared resources and space. The coexistence of a large number of different taxa reaches its paroxism in microbial communities, where severeal thousands of Operational Taxonomic Units (referred to as 'species' in the following) can be detected when sequencing samples of soil or of water \cite{deVargas, review}. A general feature of microbial communties is that abundance varies widely among the species that co-occur at the time of observation. Generally, just a handful of {\it abundant species} makes up most of the total community biomass, whereas the large majority of {\it rare species} are present in such low numbers that they are only detectable by genomic methods. + +Multiple explanations (not mutually exclusive) have been put forward to explain the nature and origin of such a 'rare biosphere'. One possibility is that distinct taxa belong preferentialy to the abundant or to the rare subcommunities, either because of their traits, or because of evolution driving a continual shift in the most adapted phenotype \cite{hoffman2007,driscoll2016}, a concept supported by the observation that they display different macroecological patterns \cite{nemergut,logares}. On the other hand, what determines the abundance or rareness of species may only be the moment time of observation, so that any species can be either rare or abundant. In this view, diversity in community composition is maintained by variations in the environment, that determines what species are most adapted to a given spatiotemporal niche \cite{chesson?,barabas_2018?}. The concept of 'microbial seed bank' encapsulates the idea that a small number of dominant taxa are maintained by environmental filtering, while most taxa remain dormant until they meet conditions favourable to their growth \cite{lennon}. +A third hypothesis is that the cycling of a givin species between the rare and abundant component of the community is driven by intrinsic ecological fluctuations, that self-sustain even in the absence of environmental variation. Such oscillations have been evidenced in controlled settings \cite{benincà }, but have been argued to be relevant also for natural communities \cite{huisman, munch}. For instance, planktonic communities display a fast turnover of OTUs within seasons, when thus the abiotic considtions do not vary massively \cite{fuhrmann,martinez-platero,gilbert,...}, supporting the notion that species interactions are key and perhaps the most important players in determining what species are abundant and when. + +Ecology-driven, chaotic fluctuations are indeed expected due to the nonlinearity of ecological interactions, whenever competition among species for a given niche destabilises abundances. Even though chaotic fluctuations are possible with just a handful of species \cite{rosenzweig-mcarthur, rinaldi, huisman}, they have been argued to be unavoidble when their number gets large \cite{may}. + +The conditions under which species-rich ecosystems loose stability, and what regimes they display as a result, have been addressed through simple but general models representing intra-specific and inter-specific interactions as determinants of the populations' dynamic. +The disordered Lotka-Volterra equations, for instance, where interaction coefficients among a large number of species are randomly assigned, have been broadly used to explore the collective regimes of species-rich communities \cite{bunin,roy,altieri,pearce,...} and their emergent patterns \cite{barbieri,bunin,hu,...}. +High-amplitude abundance variations have been numerically observed under different settings, however it has been argued that self-sustained oscillations that mimic the turnover of natural communities require a metacommunity structure \cite{pearce,roy,o'sullivan,bunin?}. In this setting, where a number of communities are connected, species that are locally driven to extinction get replenished by migration of individuals from neighbouring patches, where they were not outcompeted. +Among the key features of natural ecosystems that such models for species-rich metacommunities reproduce are a large number of coexisting taxa, out-of equilibrium fluctuations, and very skewed species abundance distributions - with prominent power-law trends. However, given the complexity of the models, it seems inevitable that analytical discriptions are only possible under specific assumptions, for instance on the scaling or symmetry of interactions. + +On the other hand, much simpler models, that assume from the start the existence of intermittent population dynamics also predict highly diverse communities where species alternate between rareness and abundance \cite{cazelles,martinez-platero}. Similarly, single-species stochastic equations seem to account for many features of empirical abundance distributions \cite{grilli}. When species are also allowed to evolve, then booms and busts allow to maintain a larger number of species than expected by purely ecological dynamics \cite{doebeli2021}. + +Here, we show that under the hypotesis that ecological interactions are strong, single-patch disordered Lotka-Volterra equations with constant immigration display intermittent dynamics, where only a handful of species dominate the community at any moment. Over time, all species alternate between rareness and abundance, albeit not in a completely disorganised fashion: communities approach 'quasi-equilibria' where a few species coexist, before being overturned by immigration of different, non-random, groups of species. The turnover time scale depends on the satistical properties of the interactions, and becomes shorter, the stronger the interactions and the more pronounced the fluctuations of the total community abundance. +All species undergo a boom and bust dynamics and their abundance distribution closely matches that of neutral models \cite{azaele}, suggesting an emergent equivalence among different taxa - as previously proposed also for asymmetrically interacting viruses and hosts \cite{hoffman,pearce}. Such equivalence allows us to propose an approximate, effective model for the ecological dynamics of a 'typical' focal species. With the help of such model, we characterize the region where the ecological dynamics is chaotic and point to some scaling relations that may explain the weak geographical variation of plankton protist communities' abundance distributions \cite{ser-giacomi}. +The emergent equivalence among species is nonetheless not perfect. Species that bloom more frequently have a smaller average interaction strength with all other species, indicating the importance, in any community composed of a finite number of species, of relative, besides absolute, interaction statistics. Moreover, we establish a relation between different facets of rareness and dominance, where species are compared either at a fixed time, or based on their frequency of observation in time \cite{henderson&magurran,ulrich2004}. +Our results suggest statistical relationships that could be tested on empirical time series of communities displaying different dynamical regimes. + +In the following Section 2, we introduce the disordered Lotka-Volterra equations and the relation between the regimes we address here with the collective dynamics previously characterized in the limit of weak interactions. +Section 3 is dedicated to the exam of the chaotic dynamics by numerical simulation and through the introduction of an approximate effective model. We describe first the time series of the whole community and of different species, and the associated abundance distributions. We then show that their typical features are reproduced by a single-species stochastic equation, and finally examine the species-specific variations around such average behaviour. +In Section 4, we explore where in parameter space occurs the intermittent chaotic dynamics, and what other regimes are observed when interspecies interactions are strong. Finally, we use insight gained from the reduced model to reveal surprising regularities in community-level observables. +In the Discussion, we link our results to previous studies of Lotka-Volterra systems and to empirical observations of species-rich comunities. + + \subsection{Why are some species rare while other are abundant?} diff --git a/manuscript/draft/chaotic_glv_draft.toc b/manuscript/draft/chaotic_glv_draft.toc index fe0162d..fc7a9b3 100644 --- a/manuscript/draft/chaotic_glv_draft.toc +++ b/manuscript/draft/chaotic_glv_draft.toc @@ -1,25 +1,25 @@ \contentsline {section}{\numberline {1}Introduction \& background}{1}{section.1} -\contentsline {subsection}{\numberline {1.1}Why are some species rare while other are abundant?}{1}{subsection.1.1} -\contentsline {subsection}{\numberline {1.2}The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{2}{subsection.1.2} -\contentsline {section}{\numberline {2}Chaotic turnover from strong interactions}{3}{section.2} -\contentsline {subsection}{\numberline {2.1}A chaotic turnover of rare and dominant species}{3}{subsection.2.1} -\contentsline {subsection}{\numberline {2.2}Species' abundance fluctuations follow a power-law}{5}{subsection.2.2} -\contentsline {subsection}{\numberline {2.3}A stochastic focal-species model reproduces rarity--dominance patterns}{5}{subsection.2.3} -\contentsline {subsection}{\numberline {2.4}Species with lower overall interactions are more often dominant}{6}{subsection.2.4} -\contentsline {section}{\numberline {3}When chaos?---varying the interaction statistics}{7}{section.3} -\contentsline {subsection}{\numberline {3.1}Dynamical phase diagram}{7}{subsection.3.1} -\contentsline {subsection}{\numberline {3.2}From multiple attractors to chaos: a transcritical bifurcation}{8}{subsection.3.2} -\contentsline {subsection}{\numberline {3.3}Relationships between community-level observables}{8}{subsection.3.3} -\contentsline {section}{\numberline {4}Discussion}{8}{section.4} -\contentsline {section}{Appendices}{8}{section.4} -\contentsline {subsection}{\numberline {A}Integration scheme for dLV}{8}{subsection.Alph0.1} -\contentsline {subsection}{\numberline {B}Simulation of dynamical phase diagram}{8}{subsection.Alph0.2} -\contentsline {subsection}{\numberline {C}Metrics}{8}{subsection.Alph0.3} -\contentsline {subsection}{\numberline {D}Derivation of mean total abundance}{8}{subsection.Alph0.4} -\contentsline {subsection}{\numberline {E}Selective advantage}{9}{subsection.Alph0.5} -\contentsline {subsection}{\numberline {F}Invasion analysis of few-species equilibria}{9}{subsection.Alph0.6} -\contentsline {subsection}{\numberline {G}Derivation of stochastic focal-species model from dynamical mean-field arguments}{10}{subsection.Alph0.7} -\contentsline {subsection}{\numberline {H}Solution of intermittency model under unified coloured noise approximation}{11}{subsection.Alph0.8} -\contentsline {section}{Acknowledgments}{11}{equation.Alph0.46} -\contentsline {section}{References}{13}{equation.Alph0.46} +\contentsline {subsection}{\numberline {1.1}Why are some species rare while other are abundant?}{2}{subsection.1.1} +\contentsline {subsection}{\numberline {1.2}The disordered Lotka-Volterra (dLV) model with weak or strong interactions}{3}{subsection.1.2} +\contentsline {section}{\numberline {2}Chaotic turnover from strong interactions}{4}{section.2} +\contentsline {subsection}{\numberline {2.1}A chaotic turnover of rare and dominant species}{4}{subsection.2.1} +\contentsline {subsection}{\numberline {2.2}Species' abundance fluctuations follow a power-law}{6}{subsection.2.2} +\contentsline {subsection}{\numberline {2.3}A stochastic focal-species model reproduces rarity--dominance patterns}{6}{subsection.2.3} +\contentsline {subsection}{\numberline {2.4}Species with lower overall interactions are more often dominant}{7}{subsection.2.4} +\contentsline {section}{\numberline {3}When chaos?---varying the interaction statistics}{8}{section.3} +\contentsline {subsection}{\numberline {3.1}Dynamical phase diagram}{8}{subsection.3.1} +\contentsline {subsection}{\numberline {3.2}From multiple attractors to chaos: a transcritical bifurcation}{9}{subsection.3.2} +\contentsline {subsection}{\numberline {3.3}Relationships between community-level observables}{9}{subsection.3.3} +\contentsline {section}{\numberline {4}Discussion}{9}{section.4} +\contentsline {section}{Appendices}{9}{section.4} +\contentsline {subsection}{\numberline {A}Integration scheme for dLV}{9}{subsection.Alph0.1} +\contentsline {subsection}{\numberline {B}Simulation of dynamical phase diagram}{9}{subsection.Alph0.2} +\contentsline {subsection}{\numberline {C}Metrics}{9}{subsection.Alph0.3} +\contentsline {subsection}{\numberline {D}Derivation of mean total abundance}{9}{subsection.Alph0.4} +\contentsline {subsection}{\numberline {E}Selective advantage}{10}{subsection.Alph0.5} +\contentsline {subsection}{\numberline {F}Invasion analysis of few-species equilibria}{10}{subsection.Alph0.6} +\contentsline {subsection}{\numberline {G}Derivation of stochastic focal-species model from dynamical mean-field arguments}{11}{subsection.Alph0.7} +\contentsline {subsection}{\numberline {H}Solution of intermittency model under unified coloured noise approximation}{12}{subsection.Alph0.8} +\contentsline {section}{Acknowledgments}{12}{equation.Alph0.46} +\contentsline {section}{References}{14}{equation.Alph0.46} \contentsfinish -- GitLab