RESEARCH ARTICLE
Global s uc u al s abili y and he ole o
coope a ion in mu ualis ic sys ems
Jose
´R. Po illoID
1,4☯
*, Fe nando Sole -ToscanoID
2☯
, Jose
´A. LangaID
3,4☯
1Depa men o Applied Ma hema ics I, Uni e si y o Se ille, Se ille, Spain, 2Depa men o Philosophy,
Logic and Philosophy o Science, Uni e si y o Se ille, Se ille, Spain, 3Depa men o Di e en ial Equa ions
and Nume ical Analysis, Uni e si y o Se ille, Se ille, Spain, 4Ins i u o de Ma ema
´ icas de la Uni e sidad de
Se illa An onio de Cas o B zezicki, Se ille, Spain
☯These au ho s con ibu ed equally o his wo k.
*[email p o ec ed]
Abs ac
Dynamical sys ems on g aphs allow o desc ibe mul iple phenomena om di e en a eas o
Science. In pa icula , many complex sys ems in Ecology a e s udied by his app oach. In
his pape we analize he ma hema ical amewo k o he s udy o he s uc u al s abili y o
each s a iona y poin , easible o no , in oducing a gene aliza ion o his concep , de ined
as Global S uc u al S abili y. This app oach would i wi h he p ope ma hema ical concep
o s uc u al s abili y, in which we ind a ull desc ip ion o he complex dynamics on he
phase space due o nonlinea dynamics. This ac can be analyzed as an in o ma ional ield
g ounded in a global a ac o whose s uc u e can be comple ely cha ac e ized. These
a ac o s a e s able unde pe u ba ion and suppose he minimal s uc u ally s able se s.
We also s udy in de ail, ma hema ically and compu a ionally, he zones cha ac e izing di e -
en le els o biodi e si y in bipa i e g aphs desc ibing mu ualis ic an agonis ic sys ems o
popula ion dynamics. In pa icula , we in es iga e he dependence o he egion o maximal
biodi e si y o a sys em on i s connec i i y ma ix. On he o he hand, as he ne wo k opol-
ogy does no comple ely de e mine he obus ness o he dynamics o a complex ne wo k,
we s udy he co ela ion be ween s uc u al s abili y and se e al g aph measu es. A sys em-
a ic s udy on syn he ic and biological g aphs is p esen ed, including 10 mu ualis ic ne wo ks
o plan s and seed-dispe sal and 1000 andom syn he ic ne wo ks. We compa e he ole o
cen ali y measu es and modula i y, concluding he impo ance o jus coope a ion s eng h
among nodes when desc ibing a eas o maximal biodi e si y. Indeed, we show ha coope -
a ion pa ame e s a e he cen al ole o biodi e si y while o he measu es ac as seconda y
suppo ing unc ions.
In oduc ion
Phenomena om Na u al and Social Sciences a e usually modeled as complex ne wo ks o
which a dynamic is de ined among he nodes [1–4], some imes associa ed o dynamical g aphs
[3,5–9], and whe e he s udy o s abili y is equen ly a c ucial ac [10,11]. F om he keyno e
pape om S oga z [9], many s udies ha e ocused on possible scena ios o he long ime
PLOS ONE
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 1 / 21
a1111111111
a1111111111
a1111111111
a1111111111
a1111111111
OPEN ACCESS
Ci a ion: Po illo JR, Sole -Toscano F, Langa JA
(2022) Global s uc u al s abili y and he ole o
coope a ion in mu ualis ic sys ems. PLoS ONE
17(4): e0267404. h ps://doi.o g/10.1371/jou nal.
pone.0267404
Edi o : Pablo Ma in Rod iguez, Fede al Uni e si y
o Pe nambuco: Uni e sidade Fede al de
Pe nambuco, BRAZIL
Recei ed: No embe 16, 2021
Accep ed: Ap il 7, 2022
Published: Ap il 19, 2022
Pee Re iew His o y: PLOS ecognizes he
bene i s o anspa ency in he pee e iew
p ocess; he e o e, we enable he publica ion o
all o he con en o pee e iew and au ho
esponses alongside inal, published a icles. The
edi o ial his o y o his a icle is a ailable he e:
h ps://doi.o g/10.1371/jou nal.pone.0267404
Copy igh : ©2022 Po illo e al. This is an open
access a icle dis ibu ed unde he e ms o he
C ea i e Commons A ibu ion License, which
pe mi s un es ic ed use, dis ibu ion, and
ep oduc ion in any medium, p o ided he o iginal
au ho and sou ce a e c edi ed.
Da a A ailabili y S a emen : All eal wo ld seed-
dispe sal da abases iles a e a ailable om he
web-o -li e da abase (accession numbe (s).
M_SD_XX). h ps://www.web-o -li e.es/map.php?
dynamics o complex ne wo k wi h a gi en opology [12–15], being Popula ion Dynamics
[16–19], Economy [20–22] and Neu oscience [23–28] some o he a eas whe e his impo an
p oblem has been in ensi ely s udied. When he dynamics o he sys em is gi en by a se o di -
e en ial equa ions, i s beha iou gene ically depends on i s global a ac o [29–32], de ined as
in o ma ion s uc u e (IS) when i s geome ical cha ac e iza ion is a ailable [28,33]. An IS
includes no only he in o ma ion om he opology o he g aph (s uc u al ne wo k), bu
o he key componen s ha a e c ucial o unde s and all possible u u e scena ios. Indeed, an
IS is he skele on in he phase space desc ibing opological and geome ical s uc u al s abili y
in dynamical sys em [34]. No e ha , o an au onomous sys em, an IS is jus he de ailed s uc-
u e o he unique global a ac o . In g adien sys ems, his IS induces a whole de o ma ion o
he phase space, d awing an in o ma ional landscape whe e he ansien and asymp o ic
obse ed dynamics o he sys em hold [33]. This IS and in o ma ional landscape a e ixed and
a ac ing. As indica ed abo e, hey coincide wi h he global a ac o . Bu , in non-au onomous
sys ems, in which, o ins ance, pa ame e s depend on ime, his ixed s uc u e and associa ed
landscapes a e also changing in ime, loosing hei in a iance and a ac ing p ope ies, bu
s ill being c ucial o he desc ip ion o he dynamics. This ac has been used, o ins ance, in
Neu oscience o disc imina e in de ail subjec s wi h diso de s o consciousness [35]. Thus, and
IS could no coincide wi h he s anda d de ini ion o a global a ac o as he objec desc ibing
all he asymp o ic beha iou o he sys em. This is why, e en in an au onomous amewo k as
we use in his pape , o a ac o s and IS is be e i hey a e di e en ia ed.
In his pape we ocus on N-dimensional Lo ka-Vol e a sys ems used in he s udy o popu-
la ion dynamics (see, o ins ance, [36,37]), bu , by he Fundamen al Theo em o Dynamical
Sys ems [38], he esul s o his esea ch can be ex ended o mo e gene al sys ems o di e en-
ial equa ions. We show he dependence o dynamics on he opology o he g aph, bu , in
addi ion, we claim ha his ac i is only pa o a mo e gene al p inciple: he dynamics on a
g aph is globally desc ibed by i s associa ed IS, which is di e en om he s uc u al base
g aph and whose na u e is essen ially in o ma ional. The IS o hese sys ems is desc ibed as an
hie a chical se o semi-s able s a iona y solu ions linked by associa ed s able and uns able
mani olds (see Fig 1), and in o ms no only on all he possible u u e scena ios o he sys em,
bu he way hey a e eached (me as abili y), he a e o con e gence, and he zones desc ibing
phase ansi ions be ween di e en s uc u es (bi u ca ion phenomena).
This mo e complex scena io leads o de ine a gene aliza ion o he concep o s uc u al s a-
bili y in oduced in [39], in line wi h [40,41], allowing o a mo e ine desc ip ion o in e nal
and ansien dynamics in ecological sys ems.
A ma hema ical model o di e en ial equa ions desc ibes he dynamics o nodes on a mu u-
alis ic sys em as ollows: suppose Pis he o al numbe o plan s and A he numbe o animals.
Plan s (and animals) a e in compe i ion among hem and coope a ion links a e se om plan s
o aminals and ice e sa. We in oduce he ollowing sys em o N=P+Adi e en ial equa-
ions o Spiand Saidesc ibing he popula ion densi y o he i- h species:
dSpi
d ¼SpiapiX
P
j¼1
bpij SpjþX
A
k¼1
gpik Sak
!
dSai
d ¼SaiaaiX
A
j¼1
baij SajþX
P
k¼1
gaik Spk
!
Spið0Þ ¼ Spi0
Saið0Þ ¼ Sai0
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
ð1Þ
o each p
i
o 1 �i�Pand a
i
wi h 1 �i�A.apiand aai(α
i
in sho ) a e he in insic g ow h
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 2 / 21
ype=6 All syn he ic compu e -gene a ed da abases
iles a e a ailable om h ps://gi hub.com/
DynamicG aphSys em/S uc u alS abili y Code is
also ull a ailable.
Funding: This wo k was pa ially suppo ed by
FEDER Minis e io de Economı
´a, Indus ia y
Compe i i idad g an PGC2018-096540-B-I00, and
P oyec os Fondo Eu opeo de Desa ollo Regional
(FEDER) and Conseje ı
´a de Economı
´a,
Conocimien o, Emp esas y Uni e sidad de la Jun a
de Andalucı
´a, by P og ama Ope a i o FEDER 2014-
2020 e e ences US-1254251 and P20-00592. The
unde s had no ole in s udy design, da a collec ion
and analysis, decision o publish, o p epa a ion o
he manusc ip .
Compe ing in e es s: The au ho s ha e decla ed
ha no compe ing in e es s exis .
a es in he absence o compe i ion and coope a ion o plan s and animals, espec i ely,
bpij �0,baij �0deno e he compe i i e in e ac ions and gpij �0and gaij �0 he mu ualis ic
s eng hs. S uc u al S abili y ocuses on he size o he egion o he in insic pa ame e s α
i
o
each op imal (maximal) biodi e si y. Obse e ha (1) can be w i en as a gene al Lo ka-Vol-
e a model o nspecies as:
_
ui¼uiaiþX
n
j¼1
aijuj
!;i¼1;. . . ;N;ð2Þ
o , equi alen ly,
_
u¼uðaþAuÞ;ð3Þ
wi h A= (α
ij
) he in e ac ion (o adjacency) ma ix gi en by
A¼B1G2
G1B2
" #ðPþAÞ�ðPþAÞ
:ð4Þ
S uc u al S abili y o his model is in oduced in [39] as a p ope concep uni ying he
in luences o ne wo k opology and pa ame e dependence in he sys em; i has been used in
Theo e ical Ecology o analyze obus ness o biodi e si y in hese complex ne wo ks [42–48].
Essen ially, s uc u al s abili y o sys em (1) measu es he egion o in insic pa ame e s o spe-
cies o which we ge maximal biodi e si y. No e ha a g ea e egion o s uc u al s abili y
allows o lowe alues o indi idual in insic g ow h pa ame e s bu p ese ing a high le el o
biodi e si y, poin ing o obus ness and esilience o species.
The s udy o he size o he egion o in insic g ow h pa ame e s (in ou case he α
i
pa am-
e e s) o which a sys em eaches i s op imal biodi e si y (all he species p esen ) is de ined as
S uc u al S abili y in [39]. This is a c ucial ac o he s udy o he obus ness o biodi e si y
in an ecosys em, as i cha ac e izes he bo de s o in insic g ow h o ge maximal
biodi e si y.
Fig 1. In o ma ion s uc u e. G aph wi h six nodes ( op le ) whe e a dynamics is de ined by means o a Lo ka-
Vol e a coope a i e sys em wi h α
i
and γ
ij
pa ame e s as shown in he ables below. The a ac o associa ed o he
sys em, he in o ma ion s uc u e (IS) is shown on he igh . I s eigh nodes co espond o non-nega i e s a iona y
poin s in he dynamics o he sys em. These s a iona y poin s a e cha ac e ised by he alue o each node in he sys em.
The nodes u
i
shown in whi e indica e ha u
i
= 0 a he co esponding poin o he IS. Black nodes indica e ha u
i
>0.
Links be ween nodes u
i
and u
j
a e hose in he sys em ( op le ) whe e bo h u
i
,u
j
>0. The blue a ows linking di e en
poin s o he IS ep esen ansi ions going om one s a iona y solu ion (limi wi h ime app oaching −1) o ano he
(when ime app oaches + 1). Fo cla i y, ansi i e a ows a e no shown.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g001
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 3 / 21
Following he Linea Complemen a y Theo y (LCP) associa ed o Lo ka-Vol e a sys ems
[36,37,49], we in oduce a pa i ion o he phase space [50] o which we can es ima e he a ea
in which each s a iona y solu ion is globally s able, by measu ing he in e sec ion o i s associ-
a ed cone o biodi e si y wi h he uni N-dimensional sphe e [45].
Howe e , specially in high dimensional sys ems, a s a iona y poin wi h all i s componen s
s ic ly posi i e ei he does no exis , o , i his is he case, he e also exis s a big se o semi-
s able s a iona y poin s. The p esence o hese s a iona y poin s is c ucial o he desc ip ion o
he ansien beha iou and me as abili y p ope ies o he sys em, so ha neglec ing i s s udy
could lead o w ong conclusions. Mo eo e , he ways o each a pa icula s a iona y solu ion
a e mul iple, depending o he di e en (in o ma ional) landscapes [33] desc ibed in de ail by
i s semis able solu ions (see Fig 2).
Thus, in his pape we s udy he s uc u al s abili y o e e y possible u u e scena io o he
sys em. We do i in wo di e en ways. Fi s ly, we conside he whole se o s a iona y poin s
(asymp o ically s able, semis able, o e en globally uns able), and no only he globally asymp-
o ically s able poin wi h all componen s posi i e (see [46,48] o a simila app oach). Fo
ins ance, he ansi ion o one globally asymp o ically s a iona y poin o ano he by a bi u ca-
ion pa ame e is usually desc ibed as a sudden phenomenon, bu , as we show in his pape , i
is o ally unde s andable by a ca e ul s udy o he pa ame e egion o s abili y o each s a ion-
a y poin and he way hey in e sec . Secondly, and maybe mo e impo an , we in oduce he
s udy o he in e nal dynamics o each le el o biodi e si y. The analy ic calcula ion o he ea-
sible egion o any dimension has been es ablished in Saa ed a e al. [42] and in Song e al
[51]. The egion o maximal biodi e si y is desc ibed by a cone [39,45,50] in he phase space
o he in insic g ow h pa ame e s αso ha , o e e y αin his cone, he sys em will end
asymp o ically o a s a iona y poin wi h all componen s s ic ly posi i e. Bu he e exis s many
ways o each his global a ac ing s a e, each one de ined by a di e en global a ac o whose
s uc u e de e mines he ansien beha iou . Indeed, in he in e io o he cone o maximal
biodi e si y holds a ich se o di e en dynamical scena ios desc ibing how species uses
di e se s a egies in o de o each he inal s a iona y poin . These dis inc scena ios a e
desc ibed by di e en global a ac o s o which a opological desc ip ion is a ailable.
On he o he hand, in Theo e ical Ecology he s udy o coope a i e in e ac ions be ween
g oups o plan s and pollina o s / seed-dispe sal / an s and how hey a ec o biodi e si y has
ecei ed an in ensi e esea ch in he las i een yea s [16–19,21]. Mo eo e , many s udies
conclude ha he unde lined opology o a complex ne wo k is somehow associa ed o he
obse ed dynamics. Indeed, he dependence o he o wa ds scena ios o a sys em on he opol-
ogy o he unde lying g aph is usually poin ed ou [13,14,17–19,21,22,26,52–57]. A ma he-
ma ical model by a sys em o di e en ial equa ions o mu ualis ic ne wo ks in Ecology was
in oduced in Bas olla e al. [22]. Since hen, many s udies ha e been ocused on his model
class, as hey p o ide a p ecise analysis o a global app oach o hese complex phenomena.
They a e ep esen ed by bipa i e g aphs ep esen ing wo kind o species (classi ied in o wo
se s, plan s and animals) and he coope a i e links be ween he g oups [17–19,39,53]. These
wo ks s udied how he a chi ec u e o he ne wo k ela es o biodi e si y. In pa icula , unde
some condi ions is obse ed ha he mo e nes edness o he ne wo k, he mo e p obabili y o
a iche biodi e si y [58]; on he o he hand, i also depends on o he p ope ies o he associ-
a ed g aph, and canno be conside ed as he only ma ke o a highe biodi e si y [59]. Mo e-
o e , se e al s udies sugges ha his index may no play he impo an ole in shaping he
ne wo k dynamics as i was p e iously belie ed. E.g., Pay a o
´e al. show ha nes edness is
ac ually an en opic consequence o he deg ee sequence o he mu ualis ic ne wo ks, and no
an i educibly mac oscopic ea u e [60].
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 4 / 21
We s udy in de ail he dependence o s uc u al s abili y on se e al g aph measu es cha ac-
e izing he unde lying ne wo k desc ibed by adjacency ma ix A(Resul s). Ou indings,
based on a deep compu a ional analysis o biological and syn he ic ne wo ks, conclude ha
coope a ion pa ame e s play he key ole in biodi e si y di e en o o he g aph measu es
such as modula i y.
Fig 2. The e olu ion o ou di e en scena ios de ined o e he same g aph in he same s a e. A h ee-nodes g aph
is conside ed. A sys em o di e en ial equa ions as (2) is de ined o he h ee nodes. He e, γ
ji
= 0.21 in all cases).
Below, he e olu ion in ime ( ed lines) om he s a e (0.2, 0.2, 0.2) o he sys em, depending on he alue o he α
i
pa ame e s which a ec he beha iou o he nodes o he g aph bu no o i s connec i i y. The s a ing poin o he
ed ajec o ies is always he same ini ial poin bu he ajec o ies a e qui e di e en . The changes in he ajec o ies
a e go e ned by he di e en in o ma ion s uc u es ( igu es delimi ed by he blue lines) in each o he dynamical
sys ems which de e mine he u u e scena ios o he sys em. In each case, he ajec o y goes o a special poin which is
he global s able solu ion in he phase space.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g002
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 5 / 21

Ma e ials and me hods
A ac o s as in o ma ion s uc u es
A ull ma hema ical s udy o sys ems like (1) is de eloped in [49]. In pa icula , su icien con-
di ions o exis ence and uniqueness o solu ions a e p o ided, so de ining a dynamical sys em
{T( )}
�0
o (1) which possesses a global a ac o A. The phase space X will be he space in
which he dynamics akes place; in ou case X¼IRN. We de ine adynamical sys em on Xas a
amily o non-linea ope a o s Sð Þg 2IRþ,
Sð Þ:X!X
u2X;Sð Þu2X;
which desc ibes he o wa ds dynamics o each u2X. In ou case, S( )u
0
=u( ;u
0
), he solu ion
ep esen s he solu ion o (1) a ime wi h ini ial condi ion u(0) = u
0
.
The global a ac o is he cen al concep in dynamical sys em heo y, since i desc ibes all
he u u e scena ios o he associa ed gi en phenomena. I is de ined as ollows [29–32,34,61,
62]: A se A�Xis a global a ac o o {S( ): �0} i i is
1. compac ,
2. in a ian unde {S( ): �0}, i.e. Sð ÞA¼A o all �0, and
3. a ac s bounded subse s o Xunde {S( ): �0}; ha is, o all B�Xbounded
dis HðSð ÞB;AÞ≔sup
b2B
in
a2AðSð Þb;aÞ !
!1 0:
Suppose Ain (3) belongs o class S
w
o is Lyapuno -s able [63], i.e., A2S
w
, in he sense ha
he e exis s a diagonal posi i e ma ix Wsuch ha WA +A
T
Wis nega i e de ini e. In his case
he whole s uc u e o he global a ac o o Lo ka-Vol e a sys ems can be cha ac e ized [49,
54]. Indeed, i is known ha he dynamics o (3) gene a es an a ac o , which is a s uc u ed
ini e se o s a iona y poin s (o equilib ia) o he sys em, o which he e exis s a globally s a-
ble s a iona y poin . The igh pa o Fig 1 ep esen s he a ac o co esponding o he g aph
on he le wi h he gi en α
i
and γ
ij
pa ame e s. Due o he in o ma ional na u e o a global
a ac o , his a ac o cha ac e iza ion has been de ined as in o ma ion s uc u e (IS) in [28].
The in o ma ion s uc u e o (1) no only in o ms on all he s a iona y poin s o he sys em,
bu he way hey a e connec ed, showing a p ecise hie alchical s uc u e by le els o in o ma-
ion (see Fig 1 and [28,33]).
Global s uc u al s abili y
Unde he hypo heses o Ain (3) o be Lyapuno -s able, i is known ha he e exis s a unique
global asymp o ically s able s a iona y poin [36]. Bu he e also exis s a huge se (a mos 2
N
)
o ac ual s a iona y poin s which a e de e mining he ansien dynamics, desc ibing he close-
ness o phase ansi ions be ween di e en scena ios o biodi e si y. This in o ma ion is con-
ained in he IS desc ibed abo e.
Con ex cone pa i ion o IRN
Le us in oduce he p ecise de ini ions ela ed o global s uc u al s abili y: le D= {1, . . .,n}, I
�Dand J=DnI. Le Aa Lypauno -s able ma ix and B
.j
=−A
.j
o j2Jand B
.j
=−I.
j
(nega i e
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 6 / 21
o iden i y ma ix) o j2I, whe e B
.j
he j-column o ma ix B. De ine he con ex cone as
posðB:1;. . . ;B:NÞ ¼ a2IRN:a¼ 1B:1þ. . . þ nB:N; i�0g:ð5Þ
Then, o any α2pos(B.
1
,. . .,B.
N
), he unique globally s able s a iona y solu ion o (3)
u�
J¼ u�ð1Þ;...;u�ðNÞgsa is ies [36]:
j2D=u�ðjÞ>0g ¼ J;and
i2D=u�ðiÞ ¼ 0g ¼ I:
This is an impo an esul as, gi en any possible s a iona y poin o he sys em, he e exis s
an associa ed con ex cone as desc ibed in pos(B.
1
,. . .,B.
N
) such ha , when α2pos(B.
1
,. . .,
B.
N
), his s a iona y poin is globally asymp o ically s able [36].
Fo ins ance, i we conside a 4D Lo ka-Vol e a sys em, and J= {1, 2} (so ha I= {3, 4},)
he po ion (i.e., he cone C
J
) o he IR4space o pa ame e αassu ing ha he global asymp-
o ic s a iona y poin is o he o m u�
Jis gi en by
CJ¼ a2IR4:a¼ 1ð A:1Þþ 2ð A:2Þþ 3ð I:3Þþ 4ð I:4Þ; i>0g:
E en mo e in e es ing, he six een possible cones (2
4
)C
J
, o all possible J−choices, o m a
pa i ion o IR4, i.e., he union o cones ills all he space and he e is no in e sec ion be ween
hei in e io s.
Fig 3 shows an example o he calcula ion o a g aph o i e nodes (n
1
,n
2
a he le and n
3
,
n
4
,n
5
a he igh ). Compe i ion (dashed lines) is assumed be ween e e y pai o nodes on he
same side o he g aph and coope a ion (solid lines) exis s when he e is a link n
i
$n
j
joining
nodes o di e en sides. No e ha when conside ing coope a ion ela ionships is a bipa i e
Fig 3. The cone o maximal biodi e si y and s uc u al s abili y. Le : Example (2 + 3)-bipa i e g aph wi h wo
nodes on he le and h ee nodes on he igh . Compe i ion (dashed lines) is be ween all elemen s on he same side,
while coope a ion occu s be ween elemen s on di e en sides o which he e is an edge ep esen ed by a solid line.
Coope a i e ela ionships o m a bipa i e g aph wi h wo g oups o nodes (le and igh ). Top igh : Connec i i y
ma ix −Mo he sys em. Compe i ion pa ame e s β
ij
>0 a e se o all pai s o nodes n
i
,n
j
in he same g oup (bo h a
he le o he igh ). Coope a ion pa ame e s γ
ij
>0 exis o nodes n
i
,n
j
in di e en g oups (one in he le and he
o he in he igh ) only when he e is an a ow n
i
$n
j
in he g aph. In ou expe imen s β
ij
=β
ji
and γ
ij
=γ
ji
. Bo om
igh : Equa ion o de e mine i a gi en poin a
�¼ ða1;...;a5Þ 2 IR5is in he maximal biodi e si y cone. I he e exis
some
i
�0, 1 �i�5, which e i y (7), hen a
�is in he maximal biodi e si y cone o M. To ensu e he exis ence o a
solu ion, he sum o he absolu e alues o each ow o column o M(including he 1 in he diagonal) mus be always
lowe han 2. This is equi alen o bound o 1 he sum o weigh s o all edges adjacen o each node o he g aph (node
deg ee bounded o 1). Gi en M,s uc u al s abili y is de ined as he p opo ion o poin s a
�2IR5 o which (7), has a
solu ion. Since he cones a e cen ed a 0, conside ed poin s can be limi ed o hose on he su ace o he IR5sphe e o
adius 1 cen ed a 0.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g003
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 7 / 21
g aph. In gene al, β
ij
is no necessa ily equal o β
ji
and he same o γ
ij
and γ
ji
. Ma ix Min (6)
(no e ha , in ou case M=−A) con ains all connec i i y pa ame e s. Obse e ha he diagonal
is 1 and 0 ep esen s no in e ac ion be ween nodes. Gi en M, a ce ain a
�2IRNis in he maxi-
mal biodi e si y cone when he e exis
1
,
2
,. . .,
N
�0 e i ying (7). S uc u al s abili y o M
is de ined as he p opo ion o a
�2IRNin he maximal biodi e si y cone; i.e., he s uc u al s a-
bili y o Mis equal o he p opo ion o poin s o he IRNsphe e o adius 1 cen ed a 0 in he
maximal biodi e si y cone.
Resul s
Global s uc u al s abili y
In his sec ion, we s udy he s uc u al s abili y o each s a iona y poin in he sys em, indepen-
den ly o hei s abili y p ope ies. This is e e ed as Global S uc u al S abili y. Fo (3), he
ze o solu ion is globally uns able, each s a iona y u�
jpoin belongs o an in o ma ional le el E
i
and possesses s able and uns able di ec ions, and he e exis s jus one s a iona y easible poin
u�in he lowe le el which is globally s able (see Fig 1).
To illus a e he desc ip ion o global s uc u al s abili y, conside a wo-dimensional coop-
e a i e sys em gi en by
_
u1¼u1ða1u1þau2Þ
_
u2¼u2ða2u2þbu1Þ
(ð8Þ
wi h ai2IR and a,b>0. Fo a ixed ne wo k o connec ions in he sys em (gi en by alues o
he aand bpa ame e s), he in insic g ow h a e o each species plays a c ucial ole. Indeed, a
con ex cone o αpa ame e s in (8) is associa ed o hese s a iona y poin s, and all o hese con-
ex cones o m a pa i ion o IR2[50], i.e., each cone has a non oid in e io , he union o all
cones is IR2and each pai o he in e io o cones is disjoin (see Fig 4).
This means ha a gi en ec o αo (8) belongs ei he o jus he in e io o one cone (de e -
mining he easible s a iona y poin , and so he u u e biodi e si y o he sys em) o o he
in e sec ion o cones, made by ich mani olds showing a phase ansi ion and a high sensibili y
o bi u ca ion scena ios in biodi e si y.
Fig 4. Desc ip ion o cones o alpha pa ame e s associa ed o 8wi h 2 ×2 ma ices as indica ed. A. Compe i i e
case. B. Coope a i e case. No e ha he e a e ou egions C
ij
,i,j= 0, 1, each o he ou possible s a iona y poin s. I α
2C
ij
, he globally asymp o ically s able poin u� o (8) has he posi i e componen s poin ed by ij, i.e., α2C
11
means
ha in u�
1;u�
2>0:No e ha bo de s o each cone a e bi u ca ion lines, in he sense ha a sudden a ac o bi u ca ion
occu s when passing h ough his bo de . Mo eo e , inside each cone, he e exis in e es ing zones ma king di e en
a ac o s uc u es wi h he same globally asymp o ically s able solu ion, which is also c ucial o he s udy o he
s uc u al s abili y.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g004
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 8 / 21
The global s uc u al s abili y o a sys em allows o s udy bi u ca ion and ansi ions
be ween di e en biodi e si y scena ios. Indeed, he bo de s, which a e now ma hema ically
well de ined, o each cone a e c i ical zones o he sudden ansi ion o one biodi e si y sce-
na io o a di e en one. Mo eo e , we can also obse e in a global way he dependence o he
cone pa i ion on he pa ame e s o he sys em (see Fig 5).
In Fig 6 we show he di e en cones o a h ee dimensional Lo ka-Vol e a sys em. No e,
once mo e, ha he union o cones o ms a pa i ion o IR3.
Fig 5. E olu ion o global s uc u al s abili y on pa ame e s on a 3D LV sys em. A. We obse e he e olu ion o he
size o he cones when changing pa ame e γ
23
om compe i ion (0.4) o coope a ion alues (−0.4). We obse e ha
cones o maximal biodi e si y (u
111
) and he cone associa ed o s a iona y poin u
011
beha es mono onically inc easing
wi h γ
23
. B. The same esul , now simul aneously changing γ
23
and γ
32
. No e ha he cones o u
011
and u
111
now g ow
as e , while o he cones wi h cons an size now dec eases (as hose associa ed o u
010
and u
110
). C I is shown he
Global S uc u al S abili y om a compe i i e sys em o a coope a i e one, by changing all he pa ame e s in ma ix A.
No e ha , among all, i is he cone wi h maximal biodi e si y he only one inc easing wi h an income o coope a ion in
he sys em.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g005
Fig 6. Two ep esen a ions o he eigh cones desc ibing global s uc u al s abili y. A. a 3D compe i i e LV sys em,
wi h he cone o maximal biodi e si y (da k blue). B. a 3D coope a i e LV sys em, wi h a bigge cone o maximal
biodi e si y, poin ing ou he key ole o coope a ion in biodi e si y.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g006
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 9 / 21
way hey a e joined. All his in o ma ion is usually beli led in many esea ch con ibu ions.
ISs a e desc ibed in ou Lo ka-Vol e a sys em wi h p ecision, so allowing o u he s udies
on obus ness, bi u ca ion phenomena o me as abili y o solu ions, in which he ole o he
associa ed in o ma ional ield may be c ucial [28,33,62,72,73].
The concep o s uc u al s abili y is used in [22], and de ined as in he p esen pape in [39,
74]. In his wo k we ha e in oduced a global amewo k o s udy he s uc u al s abili y o
e e y possible s a iona y poin o a mu ualis ic sys em. I is e y impo an o de e mine he
obus ness o each asymp o ic egime, measu ed by he egion ha associa ed pa ame e s
each. In his sense, we ha e in oduced an IRN-pa i ion o he α-pa ame e s desc ibing he
di e en con ex egions o each s a iona y solu ion, which is mo eo e globally asymp o i-
cally s able in hese egions. To ou knowledge, his is he i s ime s uc u al s abili y is used
o s udy all he possible u u e scena ios, and no only o de e mine maximal biodi e si y. We
a e awa e we ha e no aken all he impo an in o ma ion om he exis ence o an in o ma-
ion s uc u e. Indeed, he e a e many possible con igu a ions possessing he same global
asymp o ic s able s a iona y solu ion (see o ins ance Fig 1 in which he se o semis able s a-
iona y poin s abo e he las asymp o ically s able one could be e y di e en ), and would
dese e u he esea ch.
Fig 11. Op imal modula i y and s uc u al s abili y. Le ( op o bo om): Compa ison o M_SD_20 wi h GB,GC
and GD. Righ ( op o bo om): Compa ison o M_SD_50 wi h EB,EC and ED. Spea man co ela ion coe icien is
lowe han −0.057 in all hese cases excep M_SD_20 (Table 2). I sugges ha he o ma ion o s ongly in e ela ed
communi ies wi hin a biological sys em has a nega i e in luence on maximal biological di e si y.
h ps://doi.o g/10.1371/jou nal.pone.0267404.g011
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 16 / 21

The le el o he in e dependence be ween s uc u e and dynamics on complex ne wo ks is
no always clea ; some imes i seems ha his ela ion induces de e mina ion, and usually jus
co ela ion in mos cases. In his pape we conclude ha dynamics, al hough closely ela ed, is
mainly de e mined by he signed sum o he deg ees o he ne and no o o he pa ame e s o
he opology o he ne wo k.
To s udy he dependence o he opology and i s associa ed dynamics we ha e in oduced
an N-dimensional Lo ka-Vol e a sys em o di e en ial equa ions.
Ou esul s also sugges ha op imal modula i y has a nega i e impac on biological di e -
si y (s uc u al s abili y), bu in a smoo he way ha he posi i e in luence o he sum o coop-
e a ion γ
ij
alues. Tha is, he o ma ion o s ongly in e ela ed communi ies wi hin a
biological sys em has a nega i e in luence on maximal biological di e si y. Mo e conclusi e
esul s on modula i y may equi e mo e ex ensi e s udies, maybe no malizing he connec i i y
ma ix o he ne wo ks o a cons an cen ali y deg ee o cons an sum o coope a ion coe i-
cien s. Tha way, he in luence o modula i y could be be e es ima ed. Simila s udies ha e
been done in [39,43,60], and hey ha e shown ha some ne wo k p ope ies, such as nes ed-
ness, a e also a seconda y p ocess (no a co e p ocess) shaping coexis ence.
We conjec u e ha compe i ion/an agonism β
ij
coe icien s ha e nega i e in luence on
maximal biological di e si y, bu mo e ex ensi e expe imen s could also be equi ed o con-
i m his claim.
Acknowledgmen s
Au ho s hank P o . F ancisco J. Es eban, a he Facul y o Biology a Jaen Uni e si y (Spain)
o hei use ul sugges ions o imp o e a p e ious e sion o his pape . We also wan o hank
he Compu a ional Cen e a he Compu e Enginee ing High Technical School a Se ille Uni-
e si y and Jeśus Cano o echnical asis ence.
Au ho Con ibu ions
Concep ualiza ion: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
Da a cu a ion: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
Fo mal analysis: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
Funding acquisi ion: Jose
´A. Langa.
In es iga ion: Jose
´R. Po illo, Fe nando Sole -Toscano.
Me hodology: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
P ojec adminis a ion: Jose
´A. Langa.
Resou ces: Jose
´A. Langa.
So wa e: Jose
´R. Po illo, Fe nando Sole -Toscano.
Supe ision: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
Valida ion: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
Visualiza ion: Jose
´R. Po illo, Fe nando Sole -Toscano.
W i ing – o iginal d a : Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
W i ing – e iew & edi ing: Jose
´R. Po illo, Fe nando Sole -Toscano, Jose
´A. Langa.
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 17 / 21
Re e ences
1. G eene D, Doyle D, Cunningham P. T acking he E olu ion o Communi ies in Dynamic Social Ne -
wo ks. In: P oceedings o he 2010 In e na ional Con e ence on Ad ances in Social Ne wo ks Analysis
and Mining. ASONAM’10. Washing on, DC, USA: IEEE Compu e Socie y; 2010. p. 176–183. A ailable
om: h p://dx.doi.o g/10.1109/ASONAM.2010.17.
2. Ma ko i ch O, K asnogo N. P edic ing species eme gence in simula ed complex p e-bio ic ne wo ks.
PLOS ONE. 2018; 13(2):1–19. h ps://doi.o g/10.1371/jou nal.pone.0192871 PMID: 29447212
3. Palla G, Ba abasi AL, Vicsek T. Quan i ying social g oup e olu ion. Na u e. 2007; 446(7136):664–667.
h ps://doi.o g/10.1038/na u e05670 PMID: 17410175
4. Xie T, F ance-Lano d A, Wang Y, Shao-Ho n Y, G ossman J. G aph dynamical ne wo ks o unsupe -
ised lea ning o a omic scale dynamics in ma e ials. Na u e Communica ions. 2019; 10:2667. h ps://
doi.o g/10.1038/s41467-019-10663-6 PMID: 31209223
5. A aimo ich V, Dmi iche A, Shchapin D, Neko kin V. Complexi y unc ions o ne wo ks: Dynamical
hubs and complexi y clus e s. Communica ions in Nonlinea Science and Nume ical Simula ion. 2018;
55:166–173. h ps://doi.o g/10.1016/j.cnsns.2017.07.005
6. Dellni z M, Hessel- on Molo M, Me zne P, P eis R, Schu¨ e C. G aph Algo i hms o Dynamical Sys-
ems. In: Mielke A, edi o . Analysis, Modeling and Simula ion o Mul iscale P oblems. Be lin, Heidel-
be g: Sp inge Be lin Heidelbe g; 2006. p. 619–645.
7. Jos J. Dynamical Ne wo ks. In: Feng J, Jos J, Qian M, edi o s. Ne wo ks: F om Biology o Theo y.
London: Sp inge London; 2007. p. 35–62. A ailable om: h ps://doi.o g/10.1007/978-1-84628-780-0_
3.
8. Le ellie C, Sendiña Nadal I, Agui e LA. Nonlinea g aph-based heo y o dynamical ne wo k obse -
abili y. Phys Re E. 2018; 98:020303. h ps://doi.o g/10.1103/PhysRe E.98.020303 PMID: 30253528
9. S oga z SH. Explo ing complex ne wo ks. Na u e. 2001; 410(3):268–276. h ps://doi.o g/10.1038/
35065725 PMID: 11258382
10. Pi ani M, Cos a T, Sunda am S. S abili y o dynamical sys ems on a g aph. 53 d IEEE Con e ence on
Decision and Con ol. 2014; p. 613–618.
11. Deco G, Senden M, Ji sa V. How ana omy shapes dynamics: a semi-analy ical s udy o he b ain a es
by a simple spin model. F on ie s in Compu a ional Neu oscience. 2012; 6:68. h ps://doi.o g/10.3389/
ncom.2012.00068 PMID: 23024632
12. Boccale i S, La o a V, Mo eno Y, Cha ez M, Hwang DU. Complex ne wo ks: S uc u e and dynamics.
Physics Repo s. 2006; 424(4):175–308. h ps://doi.o g/10.1016/j.phys ep.2005.10.009
13. Cse mely P, London A, Wu LY, Uzzi B. S uc u e and dynamics o co e/pe iphe y ne wo ks. Jou nal o
Complex Ne wo ks. 2013; 1(2):93–123. h ps://doi.o g/10.1093/comne /cn 016
14. Danzige MM, Bonamassa I, Boccale i S, Ha lin S. Dynamic in e dependence and compe i ion in mul i-
laye ne wo ks. Na u e Physics. 2018. h ps://doi.o g/10.1038/s41567-018-0343-1.
15. Boccale i S, Bianconi G, C iado R, del Genio CI, Gomez-Ga deñes J, Romance M, e al. The s uc u e
and dynamics o mul ilaye ne wo ks. Physics Repo s. 2014; 544(1):1–122. h ps://doi.o g/10.1016/j.
phys ep.2014.07.001 PMID: 32834429
16. Guima ães PR J , Pi es MM, Jo dano P, Bascomp e J, Thompson JN. Indi ec e ec s d i e coe olu ion
in mu ualis ic ne wo ks. Na u e. 2017; 550:511–514. h ps://doi.o g/10.1038/na u e24273
17. Bascomp e J, Jo dano P, Olesen JM. Asymme ic Coe olu iona y Ne wo ks Facili a e Biodi e si y
Main enance. Science. 2006; 312(5772):431–433. h ps://doi.o g/10.1126/science.1123412 PMID:
16627742
18. Bascomp e J, Jo dano P. The S uc u e o Plan -Animal Mu ualis ic Ne wo ks. In: Ecological ne wo ks:
linking s uc u e o dynamics in ood webs / edi o s, Pascual Me cedes, Dunne Jenni e A. San a Fe
Ins i u e s udies in he sciences o complexi y. Ox o d, UK: Ox o d Uni e si y P ess; 2006. p. 143–159.
19. Bascomp e J, Jo dano P. Plan -Animal Mu ualis ic Ne wo ks: The A chi ec u e o Biodi e si y. Annual
Re iew o Ecology, E olu ion, and Sys ema ics. 2007; 38(1):567–593. h ps://doi.o g/10.1146/annu e .
ecolsys.38.091206.095818
20. Naimzada AK, S e ani S, To ie o Ae. Ne wo ks, Topology and Dynamics. Theo y and Applica ions o
Economics and Social Sys ems. Lec u e No es in Economics and Ma hema ical Sys ems. Sp inge -
Ve lag Be lin Heidelbe g; 2009. A ailable om: h ps://www.sp inge .com/gp/book/9783540684077.
21. Saa ed a S, S ou e DB, Uzzi B, Bascomp e J. S ong con ibu o s o ne wo k pe sis ence a e he
mos ulne able o ex inc ion. Na u e. 2014;.
22. Bas olla U, Fo una MA, Pascual-Ga cı
´a A, Fe e a A, Luque B, Bascomp e J. The a chi ec u e o mu u-
alis ic ne wo ks minimizes compe i ion and inc eases biodi e si y. Na u e. 2009; 458:1018–1020.
h ps://doi.o g/10.1038/na u e07950 PMID: 19396144
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 18 / 21
23. Bullmo e E, Spo ns O. Complex b ain ne wo ks: g aph heo e ical analysis o s uc u al and unc ional
sys ems. Na u e Re iews Neu oscience. 2009; 10. h ps://doi.o g/10.1038/n n2575 PMID: 19190637
24. Deco G, Ji sa VK. Ongoing co ical ac i i y a es : c i icali y, mul is abili y, and ghos a ac o s. J Neu-
osci. 2012; 32:3366–3375. h ps://doi.o g/10.1523/JNEUROSCI.2523-11.2012 PMID: 22399758
25. del Valle Rod ı
´guez A, Ce a M, Po illo JR. A ne wo k app oach o analyze neu onal lineage and laye
inne a ion in he D osophila op ic lobes. PLOS ONE. 2020. h ps://doi.o g/10.1371/jou nal.pone.
0227897 PMID: 32023281
26. Pa k HJ, F is on K. S uc u al and Func ional B ain Ne wo ks: F om Connec ions o Cogni ion. Science.
2013; 342 (6158). h ps://doi.o g/10.1126/science.1238411
27. Oizumi M, Alban akis L, Tononi G. F om he Phenomenology o he Mechanisms o Consciousness:
In eg a ed In o ma ion Theo y 3.0. PLOS Compu a ional Biology. 2014; 10(5):1–25. h ps://doi.o g/10.
1371/jou nal.pcbi.1003588
28. Es eban FJ, Galadı´JA, Langa JA, Po illo JR, Sole -Toscano F. In o ma ional s uc u es: A dynamical
sys em app oach o in eg a ed in o ma ion. PLOS Compu a ional Biology. 2018; 14(9):1–33. h ps://
doi.o g/10.1371/jou nal.pcbi.1006154 PMID: 30212467
29. Hale JK. Asymp o ic Beha io o Dissipa i e Sys ems. No. 25 in Ma hema ical Su eys and Mono-
g aphs. P o idence: Ame ican Ma hema ical Socie y; 1988.
30. Temam R. In ini e dimensional dynamical sys ems in mechanics and physics. No. 68 in Applied Ma he-
ma ical Sciences. Sp inge ; 1997.
31. Babin AV, Vishik MI. Regula a ac o s o semig oups and e olu ion equa ions. Ma h Pu es e Appl.
1938; 62:441–491.
32. Ladyzhenskaya OA. A ac o s o semig oups and e olu ion equa ions. Camb idge Uni e si y P ess;
1991.
33. Kali a P, Langa JA, Sole -Toscano F. In o ma ional S uc u es and In o ma ional Fields as a P o o ype
o he Desc ip ion o Pos ula es o he In eg a ed In o ma ion Theo y. En opy. 2019; 21(5). h ps://doi.
o g/10.3390/e21050493 PMID: 33267207
34. Bo olan MC, Ca alho AN, Langa JA. A ac o s unde au onomous and non-au onomous pe u ba ion.
ol. 246 o Ma hema ical Su eys and Monog aphs. Ame ican Ma hema ical Socie y P o idence RI;
2020. A ailable om: h ps://books o e.ams.o g/su -246/.
35. Galadi JA, Sil a-Pe ei a S, Sanz Pe l Y, K ingelbach ML, I G, Lau s H, e al. Cap u ing he non-s a io-
na i y o whole-b ain dynamics unde lying human b ain s a es. Neu oimage. 2021;(244). h ps://doi.o g/
10.1016/j.neu oimage.2021.118551 PMID: 34506913
36. Takeuchi Y. Global Dynamical P ope ies o Lo ka-Vol e a Sys ems. Wo ld Scien i ic; 1996. A ailable
om: h ps://books.google.es/books?id=HM dmO8aSycC.
37. Takeuchi Y, Adachi N. The exis ence o globally s able equilib ia o ecosys ems o he gene alized Vol-
e a ype. Jou nal o Ma hema ical Biology. 1980; 10(4):401–415. h ps://doi.o g/10.1007/BF00276098
38. No on DE. The undamen al heo em o dynamical sys ems. Commen a iones Ma hema icae Uni e si-
a is Ca olinae. 1995; 36(3):585–597.
39. Roh RP, Saa ed a S, Bascomp e J. On he s uc u al s abili y o mu ualis ic sys ems. Science. 2014;
345(6195):1253497. h ps://doi.o g/10.1126/science.1253497 PMID: 25061214
40. Thom R. S uc u al S abili y and Mo phogenesis. An ou line o a gene al heo y o models. W.A. Benja-
min, INC; 1975.
41. And ono AA, Pon yagin LS. Coa se sys ems. Doklady Akademii Nauk SSSR. 1937; 14 (5):247–250.
42. Saa ed a S, Roh R, Olesen J, Bascomp e J. Nes ed species in e ac ions p omo e easibili y o e s a-
bili y du ing he assembly o a pollina o communi y. Ecol E ol. 2016; 6:997–1007. h ps://doi.o g/10.
1002/ece3.1930 PMID: 26941941
43. G illi J, Ado isio M, Suweis S, Ba aba
´s G, Bana a JR, Allesina S, e al. Feasibili y and coexis ence o
la ge ecological communi ies. Na u e Communica ions. 2017; 8. h ps://doi.o g/10.1038/ncomms14389
PMID: 28233768
44. Cenci S, Song C, Saa ed a S. Re hinking he impo ance o he s uc u e o ecological ne wo ks unde
an en i onmen ?dependen amewo k. Ecology and E olu ion. 2018; 8:6852–6859. h ps://doi.o g/10.
1002/ece3.4252 PMID: 30073049
45. Song C, Saa ed a S. S uc u al s abili y as a consis en p edic o o phenological e en s. P oceedings
o he Royal Socie y B: Biological Sciences. 2018; 285(1880):20180767. h ps://doi.o g/10.1098/ spb.
2018.0767 PMID: 29899073
46. Saa ed a S, Medei os LP, Aladwani M. S uc u al o ecas ing o species pe sis ence unde changing
en i onmen s. Ecology le e s. 2020;. h ps://doi.o g/10.1111/ele.13582 PMID: 32776667
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 19 / 21
47. Song C, Roh RP, Vasseu D, Saa ed a S. Disen angling he e ec s o ex e nal pe u ba ions on coex-
is ence and p io i y e ec s. Jou nal o Ecology. 2020; 108(4):1677–1689. h ps://doi.o g/10.1111/1365-
2745.13349
48. Medei os LP, Boege K, del Val E, Zaldi a -Ri e o
´n A, Saa ed a S. Obse ed ecological communi ies
a e o med by species combina ions ha a e among he mos likely o pe sis unde changing en i on-
men s. The Ame ican Na u alis . 2021; 197:E17–E29. h ps://doi.o g/10.1086/711663 PMID: 33417517
49. Gue e o G, Langa JA, Sua
´ ez A. A chi ec u e o a ac o de e mines dynamics on mu ualis ic complex
ne wo ks. Nonlinea Anal Real Wo ld Appl. 2017; 34:17–40. h ps://doi.o g/10.1016/j.non wa.2016.07.
009
50. Mu y KG. Linea Complemen a i y, Linea and Non Linea P og amming. Sigma se ies in applied ma h-
ema ics. Helde mann Ve lag; 1988. A ailable om: h ps://books.google.es/books?id=
ERhRAAAAMAAJ.
51. Song C, Roh R, Saa ed a S. A guideline o s udy he easibili y domain o mul i- ophic and changing
ecological communi ies. Jou nal o Theo e ical Biology. 2018; 450. h ps://doi.o g/10.1016/j.j bi.2018.
04.030 PMID: 29702110
52. Zenil H, Sole -Toscano F, Dingle K, Louis AA. Co ela ion o au omo phism g oup size and opological
p ope ies wi h p og am-size complexi y e alua ions o g aphs and complex ne wo ks. Physica A: S a-
is ical Mechanics and i s Applica ions. 2014; 404:341–358. h ps://doi.o g/10.1016/j.physa.2014.02.
060
53. Bascomp e J, Jo dano P, Melia
´n CJ, Olesen JM. The nes ed assembly o plan –animal mu ualis ic ne -
wo ks. P oceedings o he Na ional Academy o Sciences. 2003; 100(16):9383–9387. h ps://doi.o g/
10.1073/pnas.1633576100 PMID: 12881488
54. Gue e o G, Langa JA, Sua
´ ez A. A ac ing complex ne wo ks. In: Complex ne wo ks and dynamics.
ol. 683 o Lec u e No es in Econom. and Ma h. Sys ems. Sp inge , [ Cham]; 2016. p. 309–327.
55. Jalili M, Pe c M. In o ma ion cascades in complex ne wo ks. Jou nal o Complex Ne wo ks. 2017; 5
(5):665–693. h ps://doi.o g/10.1093/comne /cnx019
56. Muezzinoglu MK, T is an I, Hue a R, A aimo ich VS, Rabino ich MI. T ansien s e sus a ac o s in
complex ne wo ks. In e na J Bi u Chaos Appl Sci Eng g. 2010; 20(6):1653–1675. h ps://doi.o g/10.
1142/S0218127410026745
57. Molken hin N, Reh eld K, Ma wan N, Ku hs J. Ne wo ks om Flows—F om Dynamics o Topology. Sci-
en i ic Repo s. 2014; 4. h ps://doi.o g/10.1038/s ep04119 PMID: 24535026
58. Bana a JR, Suweis S, Ma i an A. Eme gence o s uc u al and dynamical p ope ies o ecological
mu ualis ic ne wo ks. Na u e. 2013; 500:449–452. h ps://doi.o g/10.1038/na u e12438 PMID:
23969462
59. Bascomp e J, Fe e a A. A s uc u al heo y o mu ualis ic ne wo ks. In: Theo e ical Ecology: concep s
and applica ions. Ox o d Uni e si y P ess; 2020. p. 93–115.
60. Pay a o
´-Bo às C, He na
´ndez L, Mo eno Y. B eaking he Spell o Nes edness: The En opic O igin o
Nes edness in Mu ualis ic Sys ems. Phys Re X. 2019; 9:031024. h ps://doi.o g/10.1103/PhysRe X.9.
031024
61. Hen y DB. Geome ic heo y o semilinea pa abolic equa ions. Be lin: Sp inge -Ve lag; 1981.
62. Ca alho A, Langa JA, Robinson J. A ac o s o in ini e-dimensional non-au onomous dynamical sys-
ems. Applied Ma hema ical Sciences. Sp inge New Yo k; 2012. A ailable om: h ps://link.sp inge .
com/book/10.1007/978-1-4614-4581-4.
63. C oss GW. Th ee ypes o ma ix s abili y. Linea Algeb a and i s Applica ions. 1978; 20(3):253–263.
h ps://doi.o g/10.1016/0024-3795(78)90021-6
64. Fo una MA, S ou e DB, Olesen JM, Jo dano P, Mouillo D, K asno BR, e al. Nes edness e sus
modula i y in ecological ne wo ks: wo sides o he same coin? Jou nal o Animal Ecology. 2010; 79
(4):811–817. h ps://doi.o g/10.1111/j.1365-2656.2010.01688.x PMID: 20374411
65. Spea man C. The P oo and Measu emen o Associa ion be ween Two Things. The Ame ican Jou nal
o Psychology. 1904; 15(1):72–101. h ps://doi.o g/10.2307/1412159
66. O ega R, Fo una MA, Bascomp e J. Web o Li e; 2021. h p://www.web-o -li e.es.
67. Ba elas A. A Ma hema ical Model o G oup S uc u es. Human O ganiza ion. 1948; 7(3):16–30.
h ps://doi.o g/10.17730/humo.7.3. 4033344851gl053
68. F eeman LC. Cen ali y in social ne wo ks concep ual cla i ica ion. Social Ne wo ks. 1978; 1(3):215–
239. h ps://doi.o g/10.1016/0378-8733(78)90021-7
69. Sun J, Tang J. A Su ey o Models and Algo i hms o Social In luence Analysis. In: Agga wal CC, edi-
o . Social Ne wo k Da a Analy ics. Bos on, MA: Sp inge US; 2011. p. 177–214. A ailable om:
h ps://doi.o g/10.1007/978-1-4419-8462-3_7.
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 20 / 21
70. Newman MEJ. Modula i y and communi y s uc u e in ne wo ks. P oceedings o he Na ional Academy
o Sciences. 2006; 103(23):8577–8582. h ps://doi.o g/10.1073/pnas.0601602103
71. Csa di G, Nepusz T. The ig aph so wa e package o complex ne wo k esea ch. In e Jou nal. 2006;
Complex Sys ems:1695.
72. A agão-Cos a ER, Ca aballo T, Ca alho AN, Langa JA. S abili y o g adien semig oups unde pe u -
ba ions. Nonlinea i y. 2011; 24(7):2099. h ps://doi.o g/10.1088/0951-7715/24/7/010
73. A agão-Cos a ER, Ca aballo T, Ca alho AN, Langa JA. Con inui y o Lyapuno unc ions and o ene gy
le el o a gene alized g adien semig oup. Topol Me hods Nonlinea Anal. 2012; 39(1):57–82.
74. S one L. The Google ma ix con ols he s abili y o s uc u ed ecological and biological ne wo ks.
Na u e Communica ions. 2016; 7. h ps://doi.o g/10.1038/ncomms12857 PMID: 27687986
PLOS ONE
Global s uc u al s abili y and he ole o coope a ion in mu ualis ic sys ems
PLOS ONE | h ps://doi.o g/10.1371/jou nal.pone.0267404 Ap il 19, 2022 21 / 21