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\begin{document}

\title{Functional diversity can facilitate the collapse of an undesirable
ecosystem state}



\author[1]{Romana Limberger}%
\author[1]{Uriah Daugaard}%
\author[1]{Anubhav Gupta}%
\author[1]{Rainer Krug}%
\author[1]{Kimberley Lemmen}%
\author[1]{Sofia van Moorsel}%
\author[1]{Marcel Suleiman}%
\author[1]{Debra Zuppinger-Dingley}%
\author[1]{Owen Petchey}%
\affil[1]{University of Zurich}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\selectlanguage{english}
\begin{abstract}
Biodiversity may increase ecosystem resilience. However, we have limited
understanding if this holds true for ecosystems that respond to gradual
environmental change with abrupt shifts to an alternative state. We used
a mathematical model of anoxic-oxic regime shifts and explored how trait
diversity in three groups of bacteria influences resilience. We found
that trait diversity did not always increase resilience: greater
diversity in two of the groups increased but in one group decreased
resilience of their preferred ecosystem state. We also found that
simultaneous trait diversity in multiple groups often led to reduced or
erased diversity effects. Overall, our results suggest that higher
diversity can increase resilience but can also promote collapse when
diversity occurs in a functional group that negatively influences the
state it occurs in. We propose this mechanism as a potential management
approach to facilitate the recovery of a desired ecosystem state.%
\end{abstract}%



\sloppy


\section*{Introduction}

{\label{introduction}}

In light of global change and biodiversity loss, it is a key challenge
to advance our understanding about ecosystem resilience and its
relationship with diversity. Resilience is defined as the capacity of
ecosystems to absorb changes in the environment without exhibiting large
changes in ecosystem state (Holling 1973). Biodiversity among and within
species may increase ecosystem resilience (Hughes \& Stachowicz 2004;
Isbell \emph{et al.} 2015), when species (or genotypes) perform similar
functions but respond differently to environmental change (Leary \&
Petchey 2009). Hence, a key aspect of diversity is variation in
functional traits, i.e. the traits that determine how organisms respond
to environmental change and affect ecosystem processes (Naeem \& Wright
2003; Suding \emph{et al.} 2008). Corresponding research has primarily
focused on ecosystems with single equilibria (Yachi \& Loreau 1999;
Loreau \& Mazancourt 2013), but was recently extended to ecosystems with
alternative stable states (Dakos \emph{et al.} 2019;
Chaparro-Pedraza\emph{et al.} 2021). When such ecosystems experience a
change in environmental conditions, higher biodiversity may prevent or
delay the shift to an alternative ecosystem state (Figure 1;
Chaparro-Pedraza\emph{et al.} 2021). Resilience is particularly crucial
in these systems because shifts to alternative states often occur
abruptly, are difficult to reverse, and can entail high costs when the
ecosystem transitions to a state that is undesired for ecological and/or
economic reasons (Scheffer \emph{et al.} 1993; Foley \emph{et al.} 2003;
Diaz \& Rosenberg 2008).

Ecosystems with alternative stable states are characterized by
displaying two or more different states at identical external conditions
(Fig. 1a). As an environmental driver increases gradually, the ecosystem
shifts abruptly to an alternative state once a threshold, the tipping
point, is passed, but recovers to its original state only at lower
threshold conditions (Fig. 1a; Scheffer \emph{et al.} 2001). Between
these two threshold values, the state of the ecosystem depends on its
history, a phenomenon termed hysteresis. A key mechanism behind
hysteresis is positive feedback between organisms and their environment
(K\selectlanguage{ngerman}éfi \emph{et al.} 2016). For example, in shallow lakes submerged
plants and phytoplankton can interact via mutual inhibition (a positive
feedback), where macrophytes inhibit phytoplankton by consuming
nutrients, while phytoplankton inhibit macrophytes by reducing water
clarity (Kéfi \emph{et al.} 2016). At intermediate nutrient
concentrations, shallow lakes can thus be in a clear state dominated by
submerged plants or in a turbid state dominated by phytoplankton,
depending on which of these two functional groups dominated in the past
(Scheffer \emph{et al.} 1993).

Effects of diversity on the resilience of systems with alternative
stable states were addressed conceptually (Dakos \emph{et al.} 2019) and
investigated with mathematical models of predator-prey systems
(Ceulemans \emph{et al.} 2019; Chaparro-Pedraza 2021; Wojcik \emph{et
al.} 2021) and shallow lakes (Chaparro-Pedraza \emph{et al.} 2021). Two
of these studies are particularly relevant here because they analyzed
how trait change influences the position of tipping points along an
environmental gradient (Chaparro-Pedraza 2021; Chaparro-Pedraza \emph{et
al.} 2021). Using three model systems (a population model, a
predator-prey model, and a model of a shallow lake ecosystem), the
authors report that trait change shifted the transition to an
alternative state to a higher level of environmental stress
(Chaparro-Pedraza 2021; Chaparro-Pedraza \emph{et al.} 2021). In the
shallow lake model system the shifts from a clear to turbid state and
back from turbid to clear both occurred at higher nutrient loading when
macrophytes varied in shade tolerance (Chaparro-Pedraza \emph{et
al.}2021). In this example, diversity was manipulated in one functional
group. Yet, multiple functional groups may simultaneously be present,
and the effect of diversity on ecosystem resilience may be different if
more than one functional group varied in traits.

Studies on systems without alternative stable states have shown that
simultaneous changes of diversity in multiple functional groups may lead
to complex relationships between diversity and ecosystem processes
(Thébault \& Loreau 2003; Ceulemans \emph{et al.} 2021). Specifically,
diversity of one functional group can enhance (Eisenhauer \emph{et
al.}2012; Zhao \emph{et al.} 2019), dampen (Bruno \emph{et al.} 2008),
or reverse (Thébault \& Loreau 2003) the diversity effects of another
functional group. These previous studies focused on trophic and
facilitative interactions. It is unclear, however, how simultaneous
versus independent changes of diversity play out when functional groups
interact via mutual inhibition, an interaction that is characteristic of
many ecosystems with alternative stable states (Kéfi \emph{et al.} 2016;
Bush \emph{et al.} 2017).

Here we investigate how trait diversity in one, two, or three functional
groups influences the resilience of ecosystems that have alternative
stable states. We used a mathematical model developed by Bush \emph{et
al.} (2017) which describes abrupt oxic-anoxic regime shifts of aquatic
ecosystems in response to changes in oxygen diffusivity. In this system,
regime shifts are mediated by mutual inhibition of cyanobacteria and two
types of sulfur bacteria (sulfate-reducing bacteria and phototrophic
sulfur bacteria). The already published model (Bush \emph{et al.} 2017)
assumes no trait variation within each of the three functional groups,
which is equivalent to there being only one strain in each functional
group. We extended this model to include trait variation among multiple
strains within each of the three groups of bacteria, and tested if the
effect of diversity on resilience depends on (i) the amount of trait
variation, and (ii) the combination of functional groups that contain
variation. Increased resilience can be considered beneficial when it
concerns the desired ecosystem state (oxic), but detrimental when it
concerns the undesired state (anoxic). To describe our findings, we use
the visualization framework illustrated in Figure 1 and terminology
explained in Box 1.

We focused on a model that is simple enough to give generally applicable
insights about the biodiversity-resilience relationship, but that is of
sufficient complexity and realism to provide testable hypotheses for
experiments and suggest novel approaches in ecosystem management.
Moreover, by using a model of a specific ecosystem type, we may detect
mechanisms that might be overlooked with simpler and more abstract
models. We show that diversity can both enhance and reduce resilience
depending on which and how many functional groups vary in traits, which
illustrates the importance of considering multiple groups and their
complex interactions.

\section*{Methods and Materials}

{\label{methods-and-materials}}

\subsection*{The study ecosystem}

{\label{the-study-ecosystem}}

We studied a model of anoxic-oxic regime shifts in aquatic ecosystems in
response to changes in oxygen diffusivity. Depletion of oxygen has
detrimental effects on the survival of many aquatic organisms, leading
to changes in community composition and to a decline in biodiversity
(Vaquer-Sunyer \& Duarte 2008; Hughes \emph{et al.} 2015). Hypoxic
conditions can arise naturally when thermal stratification in summer
leads to reduced oxygen diffusivity to deep-water habitats (Wetzel
2001). Anthropogenic increases of nutrients and temperature have
enhanced the frequency and duration of low-oxygen episodes (Diaz \&
Rosenberg 2008; Scavia \emph{et al.} 2014; Jane \emph{et al.} 2021).
Moreover, severe hypoxia/anoxia can be difficult to reverse because of
feedbacks between organisms and their environment (Diaz \& Rosenberg
2008; Conley \emph{et al.} 2009; Bush\emph{et al.} 2017).

Using a mathematical model, Bush \emph{et al.} (2017) described
anoxic-oxic regime shifts resulting from interactions among microbial
organisms. The model ecosystem contains three functional groups of
bacteria (Figure 2; cyanobacteria (CB), sulfate-reducing bacteria (SB),
and phototrophic sulfur bacteria (PB)), four chemical substrates
(oxygen, phosphorous, reduced sulfur, and oxidized sulfur), and four
types of flows/interactions (production, consumption, inhibition, and
diffusion); there is also abiotic oxidization of the reduced sulfur.

Critical for the presence of alternative stable states in this system is
the mutual inhibition of cyanobacteria and sulfur bacteria due to their
intolerance of the chemical substrate that the other produces:
cyanobacteria inhibit sulfur bacteria by producing oxygen, whereas
sulfate-reducing bacteria inhibit cyanobacteria by producing reduced
sulfur, e.g. sulfide (Figure 2). Bush \emph{et al.} (2017) simulated
this ecosystem with a set of ordinary differential equations (ODEs) and
demonstrated that the state of the system depends on the rate at which
oxygen diffuses into (or out of) the system: the oxygen diffusivity. The
system is oxic and dominated by cyanobacteria when oxygen diffusivity is
high, but anoxic and dominated by the two groups of sulfur bacteria when
oxygen diffusivity is low. At intermediate oxygen diffusivity, the
system is bistable, i.e. either oxic or anoxic depending on whether
cyanobacteria or sulfur bacteria dominated in the past (Bush \emph{et
al.} 2017).

The Bush \emph{et al.} model (2017) assumes zero diversity within each
of the three functional groups. There is, however, empirical evidence of
intra- and inter-specific variation in tolerance (Rolfe \emph{et
al.}1978; Knoll \& Bauld 1989; Miller \& Bebout 2004; Ramel \emph{et
al.}2015). Furthermore, experimental evolution studies demonstrated
considerable and rapid increases in tolerance of bacteria
(Martín-Clemente \emph{et al.} 2019; Schoeffler \emph{et al.} 2019).

\subsection*{The model and our extension of
it}

{\label{the-model-and-our-extension-of-it}}

Bush \emph{et al.} (2017) has an accessible and complete description of
the model of the ecosystem, including rate equations, parameter values,
and initial conditions. Our implementation of the model is documented in
the \emph{microxanox} R package (R Core Team 2021; Krug \& Petchey
2022), which has two vignettes (a user guide and a partial reproduction
of the analyses in Bush \emph{et al.} (2017)). The R code to run,
analyze, and visualize the numerical experiments of our study is
available in a repository of supplementary content (Petchey \emph{et
al.} 2022), which also includes all variables and parameter values.

We extended the Bush \emph{et al.} model (2017) by modelling multiple
strains per functional group. Each of the strains of a functional group
has its own parameter set and state variable. For example, a system with
nine strains per functional group has 27 state variables for the
strains, and four for the chemical substrates. Other than having
potentially different parameter values for maximum growth rate and
tolerance, the dynamics of the strains in a functional group were
described by the same equations. We adjusted the chemical substrate ODEs
to correctly account for the number of strains, by summing their
substrate consumption and production.

\subsection*{Creating and varying within functional group
diversity}

{\label{creating-and-varying-within-functional-group-diversity}}

We generated diversity within functional groups by creating multiple
strains that differed in maximum growth rate and in tolerance to the
respective inhibiting substrate (sulfide tolerance of cyanobacteria, and
oxygen tolerance of sulfate-reducing bacteria and phototrophic sulfur
bacteria). We assumed a trade-off between the two traits, in line with
empirical studies that report a negative correlation between maximum
growth rate and resistance to environmental stress in microbial
organisms (Ferenci 2016). For simplicity, we used a linear trade-off
between log\textsubscript{10}-transformed trait values, with among
strain variation distributed evenly on a log-scale (see Supplementary
Report Section 3 for more information). In the main text, we report
results for nine strains in each functional group. Results were
relatively robust to changes in the number of strains, although some
effects of strain number were observed at high levels of trait diversity
(see Supplementary Report Section 7).

Functional diversity has multiple components, and we manipulated only
one of these, namely the range of trait space represented, which has
been termed functional richness (Villéger \emph{et al.} 2008). We
compared 20 different amounts of trait diversity ranging from no
variation, i.e. all nine strains had the reference trait value (Table
1), to greatest trait variation with the range of trait values according
to Table 1. The maximum trait range was selected to ensure that the
region of bistability was mostly within the range of oxygen diffusivity
values used in Bush \emph{et al.} (2017) when all functional groups
contained the maximum amount of variation. The other 18 trait diversity
levels were evenly distributed between the no diversity scenario and the
greatest amount of diversity. Twenty diversity levels gave sufficient
resolution and required reasonable computation time to simulate. A
detailed account is in Supplementary Report Section 3.

We also manipulated the combination of functional groups that contained
diversity. Specifically, we examined the effect of adding diversity to
only one group (either CB, or SB, or PB), simultaneously to two
functional groups (CB-SB, CB-PB, and SB-PB), and simultaneously to all
three functional groups (CB-SB-PB). This led to 7 combinations * 20
diversity levels, but with seven of these being the same zero diversity
in all functional groups, there were 134 diversity scenarios in total.

We have confirmed that when there is no within functional group strain
variation, the dynamics of the model are not affected by the number of
strains in each of the functional groups. This is a necessary condition
for such a model (Moisset de Espanés \emph{et al.} 2021), and gives
confidence that our methods and their implementation are correct and
robust.

\subsection*{Stable state finding}

{\label{stable-state-finding}}

To investigate the effect of trait diversity on tipping point positions,
we parameterized the system according to each of the diversity scenarios
described above, and then ran two simulations, one with a stepwise
increase and one with a stepwise decrease of oxygen diffusivity. This
technique was used to find stable states, rather than to simulate the
effects of environmental change per se. At each of 300 values of oxygen
diffusivity, ranging from 0 to -8
(log\textsubscript{10}h\textsuperscript{-1}), we simulated the dynamics
for 1,000,000 hours, which was sufficient for the position of the
tipping points to stabilize, even though some strain dynamics had not
yet completely stabilized (Supplementary Report Section 6.2). At the
start of the two simulations, total initial abundance of each functional
group was 10\textsuperscript{5} cells, equally divided among the strains
of each functional group. To avoid very low strain densities which would
cause computational problems, we added 1 cell L\textsuperscript{-1} to
every biological state variable every 1,000 hours (an alternative
approach produced the same results, see Supplementary Report Section
4.3). At the end of each period of constant oxygen diffusivity, the
system state was recorded and used as the initial conditions for the
simulation at the next step of oxygen diffusivity. This approach for
finding stable states differs from the method used by Bush \emph{et al.}
(2017). See the Supplementary Report (Section 4) for a detailed
description of our rationale for using our approach, and also for
evidence of qualitatively similar results produced by both approaches.

We calculated the tipping point position of the oxic to anoxic
(anoxic-oxic) transition as the minimum (maximum) level of oxygen
diffusivity at which the total density of sulfate-reducing bacteria was
more than 1000 abundance units different between the increasing and
decreasing oxygen diffusivity simulations. We did not calculate tipping
point positions when coexistence patterns were atypical of the classical
pattern shown in Figure 1a (e.g. when cyanobacteria coexisted with
sulfur bacteria at low oxygen diffusivities; see Results and
Supplementary Report Section 10.2 for details). We calculated the effect
size of trait variation on tipping point position as the difference in
oxygen diffusivity between a tipping point without trait variation and
with a given level of trait variation (Fig. 1c-d). A positive effect
size indicates increased resilience.

\section*{Results}

{\label{results}}

\subsection*{Resilience of the oxic
state}

{\label{resilience-of-the-oxic-state}}

The trait diversity effect on the resilience of the oxic state depended
on which functional group varied in traits (Fig. 3 a-f, red lines).
Resilience increased with increasing trait variation in the group that
dominated the oxic state (cyanobacteria; Fig. 3a, b). In contrast,
increasing diversity in the two suppressed groups either reduced
resilience of the oxic state (sulfate-reducing bacteria; Fig. 3c, d) or
had no effect on resilience (phototrophic sulfur bacteria; Fig. 3e, f).

Simultaneous variation in more than one functional group reduced or did
not change the diversity effect of individual functional groups (Fig. 3
g-o; red lines). Simultaneous trait variation in cyanobacteria and
sulfate-reducing bacteria (Fig. 3g) led to smaller diversity effects
than variation in only one of these two groups; the negative diversity
effect of sulfate-reducing bacteria outweighed the positive diversity
effect of cyanobacteria when trait variation was low, but the effect of
cyanobacteria diversity prevailed at high trait variation (Fig. 3b, d,
h). Diversity in the phototrophic sulfur bacteria did not change the
diversity effects of other functional groups (Fig. 3 i-o).

\subsection*{Resilience of the anoxic
state}

{\label{resilience-of-the-anoxic-state}}

The diversity effect on the resilience of the anoxic state also differed
among groups (Fig. 3 a-f, blue lines). The two groups that dominated the
anoxic state had contrasting diversity effects on its resilience:
increasing diversity in the sulfate-reducing bacteria increased the
resilience of the anoxic state (Fig. 3 c, d), whereas increasing
diversity in the phototrophic sulfur bacteria decreased resilience (Fig.
3e, f). Diversity in the suppressed group (cyanobacteria) had no effect
on resilience when diversity was low but slightly reduced resilience
when diversity was high (Fig. 3 a, b).

When more than one functional group varied in traits, their effects were
either additive (cyanobacteria and sulfate-reducing bacteria) or one
group erased the diversity effect of another group (Fig. 3 g-o,
Supplementary Report Section 12). Specifically, variation in
sulfate-reducing bacteria erased the diversity effect of phototrophic
sulfur bacteria (Fig. 3 l, m), and variation in phototrophic sulfur
bacteria erased the (small) diversity effect of cyanobacteria (Fig. 3 i,
k).

\subsection*{Dynamics of strains and
substrates}

{\label{dynamics-of-strains-and-substrates}}

Trait variation led to compositional turnover along the oxygen
diffusivity gradient when a functional group was on the collapse
trajectory (see Box 1 for explanation of terms). Less tolerant strains
were replaced by more tolerant strains as the concentration of the
inhibiting substrate increased (Fig. 4). On the trajectory of decreasing
oxygen diffusivity, the ecosystem was initially dominated by the
cyanobacteria strain with lowest sulfide tolerance and highest maximum
growth rate (Fig. 4a). As oxygen diffusivity declined, more tolerant
strains of cyanobacteria replaced the fastest growing strain until the
strain with highest sulfide tolerance dominated. Subsequently, the
cyanobacteria collapsed, probably due to their reduced capacity to
suppress the sulfur bacteria, and simultaneously the ecosystem shifted
from oxic to anoxic (Fig. 4 d, e). Prior to the tipping point, the most
tolerant strains of the two sulfur bacteria groups slightly increased in
abundance, in particular in the sulfate-reducing bacteria (Fig. 4b).
However, once the system shifted to anoxic, the sulfur bacteria strains
with lowest tolerance and highest maximum growth rate dominated (Fig. 4
b, c).

Strain dynamics were similar on the trajectory of increasing oxygen
diffusivity (Fig. 4). Replacement of less tolerant by more tolerant
strains in both groups of sulfur bacteria was followed by the collapse
of the phototrophic sulfur bacteria. Then sulfate-reducing bacteria
collapsed, simultaneously with the shift from anoxic to oxic and the
rise of the least tolerant cyanobacteria strain. In contrast to the
trajectory of decreasing oxygen diffusivity, the switch from least to
most tolerant strains occurred over a comparatively broad range of
oxygen diffusivity.

The strain dynamics deviated from this pattern when there was medium to
high variation in only the phototrophic sulfur bacteria. In this case,
all three functional groups coexisted at low levels of oxygen
diffusivity likely because high maximum growth rates in the phototrophic
sulfur bacteria led to lower sulfide concentrations (Supplementary
Report Section 10.2). The same pattern occurred for simultaneous
variation in phototrophic sulfur bacteria and cyanobacteria, albeit only
on the trajectory of increasing oxygen diffusivity (Supplementary Report
Section 10.2).

\subsection*{Mechanisms of the functional diversity
effects}

{\label{mechanisms-of-the-functional-diversity-effects}}

At low levels of trait variation, the diversity effects were driven
entirely by the most tolerant strains. At higher levels of trait
variation, however, strains with low tolerance (and high maximum growth
rate) also contributed to the diversity effects in five of the seven
combinations (Supplementary Report Section 11). For example, in the
sulfate-reducing bacteria, absence of strains with high maximum growth
rates led to reduced production of sulfide and therefore to coexistence
with cyanobacteria at low oxygen diffusivity. That is, high maximum
growth rates often had ecosystem-level consequences.

\section*{Discussion}

{\label{discussion}}

Using a model of anoxic-oxic regime shifts, we found three key results:
first, not all functional groups had positive diversity effects on the
resilience of their preferred ecosystem state (Fig. 3a-f). Second, the
effect of diversity was often smaller or absent when a functional group
was on the recovery trajectory rather than on the collapse trajectory
(Figure 3a-f). And third, diversity effects were often reduced or erased
when multiple groups varied in traits, except when a functional group
had no individual effect on resilience (Figure 3g-o). From a management
perspective, our results suggest that higher biodiversity in a
functional group that dominates a desired ecosystem state may prevent or
delay the shift to an undesired state but does not necessarily promote
recovery. A more promising approach to facilitate recovery may be to
identify functional groups in the undesired state (here: phototrophic
sulfur bacteria) that promote their own collapse and therefore the
recovery of the desired state.

\subsection*{Diversity did not always increase
resilience}

{\label{diversity-did-not-always-increase-resilience}}

Greater diversity in cyanobacteria or sulfate-reducing bacteria
increased the resilience of the state they dominated, whereas greater
diversity in phototrophic sulfur bacteria reduced resilience of their
preferred state (Figure 3a-f). These contrasting diversity effects
reflect differences in interactions: whereas cyanobacteria and
sulfate-reducing bacteria interact in a strictly antagonistic way,
phototrophic sulfur bacteria have both positive and negative effects on
the other two functional groups (Figure 2). By oxidizing sulfide,
phototrophic sulfur bacteria indirectly increase the level of oxygen
because the lower sulfide concentration (i) promotes cyanobacteria and
therefore oxygen production, and (ii) leads to less oxygen consumption
during abiotic oxidation of sulfide (Supplementary Report Section 10.1).
Via these two pathways, phototrophic sulfur bacteria deteriorate their
own environment and promote the shift to the oxic state. High diversity
in this group even resulted in failure to exclude cyanobacteria likely
because the fastest-growing strain lowered the sulfide concentration to
such an extent that cyanobacteria were able to grow despite low oxygen
diffusivity (Supplementary Report Section 10.2).

Stimulating the growth of groups that have negative diversity effects
may be used as a management tool to promote the recovery of a desired
ecosystem state. Such an approach would require identifying functional
groups that co-dominate an undesired ecosystem state and negatively
affect it. Microbes may be particularly useful for this approach because
(i) they often dominate undesired ecosystem states (e.g. turbid lake
state, anoxic state), and (ii) they often interact via modification of
the chemical environment (Ratzke \emph{et al.} 2018; Ratzke \& Gore
2018), including even microbes that drive themselves extinct by
deteriorating their environment (Ratzke \emph{et al.} 2018).

\subsection*{Diversity effects were often smaller or absent on recovery
trajectories}

{\label{diversity-effects-were-often-smaller-or-absent-on-recovery-trajectories}}

In two functional groups, the effect of trait diversity on resilience
was very small (cyanobacteria; Figure 3b) or absent (phototrophic sulfur
bacteria; Figure 3 f) when the group was on the recovery trajectory
rather than on the collapse trajectory. For example, except for very
high diversity levels, cyanobacteria diversity had no effect on the
shift from the anoxic to the oxic state likely due to the abrupt change
of the inhibiting substrate: sulfide was maintained by sulfate-reducing
bacteria at a constantly high level such that not even the most tolerant
cyanobacteria strain was able to grow (Figure 4a, e). At the tipping
point to the oxic state, sulfide dropped abruptly to a level at which
all cyanobacteria strains would be able to grow, resulting in immediate
dominance of the strain with lowest sulfide tolerance and highest
maximum growth rate (Figure 4a). Previous work showed that evolutionary
trait change can even delay recovery when organisms adapt to the
alternative state and then have reduced performance once environmental
conditions return to original (Chaparro-Pedraza \emph{et al.} 2021). Our
study adds a further mechanism for why diversity might not aid recovery
of a suppressed group: the greater abruptness of change of the
inhibiting substrate on recovery trajectories might hinder attempts to
restore ecosystems by introducing tolerant strains.

\subsection*{Diversity effects were often reduced or erased when multiple
groups varied in
traits}

{\label{diversity-effects-were-often-reduced-or-erased-when-multiple-groups-varied-in-traits}}

Antagonistic diversity effects were to be expected in a system where
organisms interact via mutual inhibition. Surprisingly, however,
opposing diversity effects were not necessarily caused by groups that
dominated alternative ecosystem states but also by two groups that
dominated the same state and interacted via facilitation. Contrary to
other examples where functional groups that interact via facilitation
enhance each other's diversity effects (Eisenhauer \emph{et al.} 2012),
variation in sulfate-reducing bacteria erased the diversity effect of
phototrophic sulfur bacteria. Apparently, when both groups varied in
traits, the positive effect of sulfate-reducing bacteria on sulfide
outweighed the negative effect of phototrophic sulfur bacteria,
precluding the earlier shift to the oxic state. Simultaneous variation
did usually not alter diversity effects when a functional group had no
diversity effect on resilience, which was often the case on recovery
trajectories. If it is a general phenomenon that diversity plays out
less on recovery trajectories, the positive diversity effect on
resilience of the dominant group would not be dampened by diversity in
the suppressed group. Collectively, our results illustrate that
predicting the outcome of simultaneous diversity effects of multiple
functional groups can be difficult in systems where organisms interact
via a multitude of complex interactions.

In natural systems, the alternative states of an ecosystem are often
dominated by functional groups that strongly differ in intra- and
interspecific diversity, and in how quickly new variation arises. In
some of the most prominent examples of ecosystems with alternative
states, macro-organisms dominate the desired state and micro-organisms
the undesired state. For example, benthic macro-organisms and bacteria
dominate the oxic and anoxic state of coastal ecosystems, respectively
(Diaz \& Rosenberg 2008), and macrophytes and phytoplankton dominate the
clear and turbid state of shallow lakes, respectively (Scheffer et al.
1993). Prokaryotic and eukaryotic micro-organisms are both characterized
by tremendous diversity (de Vargas et al. 2015; Locey \& Lennon 2016),
including diversity in functional traits (Litchman \& Klausmeier 2008;
Escalas et al. 2019). Furthermore, microorganisms might adapt more
rapidly to new environmental conditions than macro-organisms due to
their short generation times, high population sizes, and low complexity
(Orr 2000; Barraclough 2015). Such greater standing variation and higher
ability to adapt to stressful conditions of micro-organisms may pose an
additional challenge to reversing shifts to an undesired ecosystem state
dominated by micro-organisms.

\subsection*{Ecology, evolution, or
both?}

{\label{ecology-evolution-or-both}}

The observed replacement of strains along the gradient of oxygen
diffusivity may be interpreted as changes in the abundances of species
(an ecological process) or of genotypes (an evolutionary process).
Furthermore, models such as ours are interpreted in both of two ways
(Govaert \emph{et al.} 2019): as representing ecological processes
(Ceulemans \emph{et al.} 2019; Wojcik \emph{et al.} 2021) or
eco-evolutionary feedbacks (Jones \emph{et al.} 2009). To be clear, this
modelling approach only includes sorting of standing heritable
variation. Including mutation and recombination could be a useful next
step to investigate how these evolutionary processes influence regime
shifts. Chaparro-Pedraza et al. (2021) used a quantitative genetics
approach to show that changes in macrophyte shade tolerance can shift
the tipping point between the clear and turbid state of a lake to higher
levels of stress. Despite its different approach, our study yielded
similar results for two of the three functional groups. This gives some
credence to interpreting models such as ours as usefully informing about
the effects of evolution on ecosystem resilience.

\subsection*{Limitations and future
directions}

{\label{limitations-and-future-directions}}

Naturally, our model is a simplification of natural ecosystems with
assumptions that might have influenced our results. The model system
focuses on three groups of bacteria; however, other functional groups
(e.g. eukaryotic phytoplankton, colorless sulfur bacteria) might also
have relevant effects on oxic-anoxic regime shifts (Lavik \emph{et
al.}2009). We demonstrated that including more than two functional
groups can reveal counter-intuitive diversity effects; expanding models
such as ours to include even more functional groups may therefore give
further useful insights into the response of ecosystems to environmental
change. Furthermore, we did not vary the values of parameters other than
those of maximum growth rate and environmental tolerance. For the
remaining parameters we used the same values as Bush \emph{et al.}
(2017) because the authors found high consistency of their modelling
results and empirical observations. Also, we did not vary the shape of
the trade-off between environmental tolerance and maximum growth rate.
We speculate that such changes to the trade-off would have altered the
pattern of strain replacement but not the position of tipping points.

An interesting avenue for future studies would be the analysis of
transient dynamics, that is, the response of the system to temporal
environmental change. Because we addressed questions about alternative
stable states, we focused on responses of stable states, and therefore
ran simulations for 1,000,000 hours at each level of oxygen diffusivity.
However, in natural systems environmental conditions would not remain
constant over such long timescales, and new trait variation could arise
by mutation and recombination. Most of our results remained
qualitatively similar when we reduced simulation length to 10,000 hours,
but a further reduction in simulation length considerably changed the
results (Supplementary Report Section 6). Investigation of transient
dynamics in a model of shallow lakes showed that the capacity of
evolution to prevent a regime shift depended on the relative rates of
environmental change and trait change (Chaparro-Pedraza \emph{et
al.}2021). It would be interesting to test with our system if the
capacity of functional diversity to increase resilience is contingent on
a slow rate of environmental change. Possibly, however, the rate of
change is less influential in systems such as ours because sorting of
standing variation might allow a faster response to environmental change
than if relevant variation has yet to arise.

\subsection*{Conclusions}

{\label{conclusions}}

Our results illustrate that greater trait diversity of the functional
group that dominates a desired ecosystem state may increase the ability
of the system to absorb environmental change before tipping to the
alternative, undesired state. However, once the system has shifted to an
undesired state, recovery of a desired ecosystem state might be more
easily achieved by managing functional groups that dominate the
undesired state than by managing groups that dominate the desired state.
Trying to facilitate recovery by bringing in more stress tolerant
strains of the functional group that dominates the desired state might
be unsuccessful because of the abrupt change of the inhibiting substrate
on the recovery trajectory. A more promising management approach could
be to stimulate the growth of functional groups that co-dominate the
undesired state but have negative effects on it. To identify such
groups, it is useful to consider more than two functional groups in
models of systems with alternative stable states. Collectively, our
results highlight the importance of considering multiple interacting
groups when predicting the response of ecosystems with alternative
states to environmental change.

\section*{Acknowledgements}

{\label{acknowledgements}}

Statement is on the title page, or one after that.

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\textbf{Table 1:} Range of growth rates and tolerances. Without trait
variation, all strains of a functional group had the reference trait
value. With trait variation, the trait values of the strains varied
around the reference trait value. Here we give the range of growth rates
and tolerances when trait variation was at its maximum and for the
results displayed in Figure 4. The maximum amount of variation was set
to a range so that the region of bistability was usually within the
range of oxygen diffusivity values used in Bush \emph{et al} (2017).
Tolerance is given as the concentration of the inhibiting substrate
where growth rate is reduced by 50\%.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Trait\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Functional group\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Reference trait value\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Range for maximum trait variation (min, max)\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Range for results in Figure 4 (min, max)\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Maximum growth rate (h\textsuperscript{-1})\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Cyanobacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.05\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0489, 0.0511\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0490, 0.0510\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Maximum growth rate (h\textsuperscript{-1})\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Sulfate-reducing bacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0936, 0.1068\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0943, 0.1061\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Maximum growth rate (h\textsuperscript{-1})\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Phototrophic sulfur bacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.07\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0656, 0.0748\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
0.0660, 0.0742\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Sulfide tolerance (\selectlanguage{greek}µ\selectlanguage{english}M)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Cyanobacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
300\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
335.2, 268.5\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
331.3, 271.7\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Oxygen tolerance (\selectlanguage{greek}µ\selectlanguage{english}M)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Sulfate-reducing bacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
100\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
383.2, 26.1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
332.6, 30.1\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Oxygen tolerance (\selectlanguage{greek}µ\selectlanguage{english}M)\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Phototrophic sulfur bacteria\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
100\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
383.2, 26.1\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
332.6, 30.1\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\textbf{Box 1:} Glossary of terms.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\begin{minipage}[b]{0.47\columnwidth}\raggedright\strut
\textbf{Term}\strut
\end{minipage} & \begin{minipage}[b]{0.47\columnwidth}\raggedright\strut
\textbf{Definition}\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Resilience\strut
\end{minipage} & \begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Ability of the ecosystem to absorb a change in oxygen diffusivity
without tipping to the alternative ecosystem state.\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Resilience of the oxic state\strut
\end{minipage} & \begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Ability of the ecosystem to absorb a decrease in oxygen diffusivity
without shifting from oxic to anoxic. Resilience is therefore higher
when the \emph{oxic-anoxic} tipping point is at lower oxygen
diffusivity.\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Resilience of the anoxic state\strut
\end{minipage} & \begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Ability of the ecosystem to absorb an increase in oxygen diffusivity
without shifting from anoxic to oxic. Resilience is therefore higher
when the anoxic-oxic tipping point is at higher oxygen
diffusivity.\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Dominant group\strut
\end{minipage} & \begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Functional group that dominates a given ecosystem state (oxic state:
cyanobacteria; anoxic state: sulfate-reducing bacteria and phototrophic
sulfur bacteria).\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Suppressed group\strut
\end{minipage} & \begin{minipage}[t]{0.47\columnwidth}\raggedright\strut
Functional group that is inhibited in a given ecosystem state and
therefore has zero or low abundance (oxic state: sulfate-reducing
bacteria and phototrophic sulfur bacteria; anoxic state:
cyanobacteria).\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.48\columnwidth}\raggedright\strut
Collapse trajectory\strut
\end{minipage} & \begin{minipage}[t]{0.48\columnwidth}\raggedright\strut
Trajectory with increasingly unfavourable conditions for a given
functional group, leading to its collapse. (cyanobacteria: trajectory of
decreasing oxygen diffusivity; sulfur bacteria: trajectory of increasing
oxygen diffusivity)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.48\columnwidth}\raggedright\strut
Recovery trajectory\strut
\end{minipage} & \begin{minipage}[t]{0.48\columnwidth}\raggedright\strut
Trajectory with increasingly favourable conditions for a given
functional group, leading to its recovery. (cyanobacteria: trajectory of
increasing oxygen diffusivity; sulfur bacteria: trajectory of decreasing
oxygen diffusivity)\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}

\section*{Figure legends:}

{\label{figure-legends}}

\textbf{Figure 1.} Various complementary visualizations of how diversity
could influence the position of tipping points in systems with
alternative stable states. \textbf{(a, b)} Illustrations of the possible
dependence of ecosystem state on an environmental driver
\textbf{(a)}with no diversity and \textbf{(b)} with diversity. The
ecosystem illustrated can exist in either an anoxic state with high
density of sulfate-reducing bacteria or an oxic state with low density
of sulfate-reducing bacteria (other functional groups are omitted for
clarity). In the region of bistability (green) the state of the system
depends on its history. At tipping points, the ecosystem shifts abruptly
(grey arrows) to the alternative state. \textbf{(b)} illustrates
replacement of less tolerant by more tolerant strains as the
environmental driver increases (strain dynamics caused by a decreasing
environmental driver are omitted for clarity.) \textbf{(c, d)} Two
complementary visualizations of the effect of trait variation on
resilience. \textbf{(c)} The position of the two tipping points at
different levels of trait variation. \textbf{(d)} The effect size of
trait variation on resilience, i.e. on the position of the tipping
points. Effect size is the difference in oxygen diffusivity between a
tipping point without trait variation and with a given level of trait
variation. A positive effect size indicates increased resilience. The
depicted example shows a scenario where increasing trait variation
increases the resilience of both the oxic and the anoxic state. A
non-linear effect and linear effect are shown only to aid understanding.
Please see Box 1 for further explanation of terms.

\textbf{Figure 2.} The model ecosystem. We simulated interactions among
three functional groups of bacteria. Cyanobacteria inhibit phototrophic
sulfur bacteria and sulfate-reducing bacteria by producing oxygen, while
sulfate-reducing bacteria inhibit cyanobacteria by producing reduced
sulfur. All three groups consume phosphorus. In addition, the two sulfur
bacteria groups interact by producing the substrate that the other group
consumes: sulfate-reducing bacteria reduce oxidized sulfur to reduced
sulfur, while phototrophic sulfur bacteria oxidize reduced sulfur to
oxidized sulfur. Reduced sulfur is also oxidized abiotically. Finally,
all four chemical substrates diffuse into or out of the system depending
on their concentrations. Modified version of Figure 1 of Bush et al
(2017).

\textbf{Figure 3.} Effect of trait variation on resilience. As in Figure
1c-d, we here display the effects of trait variation on resilience with
two types of visualization which together provide two complementary
views of the same data and patterns. The left column of panels shows the
position of the tipping points at different levels of trait variation
(red: oxic-anoxic tipping point; blue: anoxic-oxic tipping point). Data
is missing in panels e, and i, when patterns of coexistence were
atypical of the classical pattern shown in Figure 1a (e.g. when
cyanobacteria coexisted with sulfur bacteria at low oxygen
diffusivities). The amount of trait variation was standardized to range
from 0 to 1 by dividing each value of trait variation by the maximum
amount of trait variation of the respective functional group. The green
area depicts the extent of the region of bistability, the thick grey
vertical bars indicate the bounds of oxygen diffusivity analyzed in our
simulations. The right column of panels shows the effect size of trait
variation on resilience. Effect size is displayed only when the tipping
point was within the investigated range of oxygen diffusivity. Figure
labels indicate which functional group(s) varied in traits (CB:
cyanobacteria, SB: sulfate-reducing bacteria, PB: phototrophic sulfur
bacteria). See Figure 1c-d and Box 1 for further explanation of terms.

\textbf{Figure 4.} Relationship of ecosystem state variables and oxygen
diffusivity when there is functional diversity among nine strains within
each functional group. The range of trait values is given in Table 1.
The different strains are shaded differently, so that in each functional
group (a-c) the darkest strain is the most tolerant, the lightest the
least tolerant. The left column of panels shows the trajectory of
decreasing oxygen diffusivity, the right column of panels shows the
trajectory of increasing oxygen diffusivity. Points (which often are so
close as to appear as thick lines) show the stable state at the end of
each period of constant oxygen diffusivity. Thin lines are for
visualization only and join the points. Grey arrows show the oxygen
diffusivity at which the system shifts, and the direction of the shift.
The length of the grey arrows is without meaning.\selectlanguage{english}
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