Func ional shi -induced degene a e ansc i ical
Neima k-Sacke bi u ca ion in a disc e e hype cycle
E nes Fon ich1,2, An oni Guillamon
*
3,4,2, J´ulia Pe ona3, and Josep Sa dany´es2
1Depa amen de Ma em`a iques i In o m`a ica, Uni e si a de Ba celona (UB),
G an Via de les Co s Ca alanes 585, 08007 Ba celona, Spain
2Cen e de Rece ca Ma em`a ica (CRM), Ce danyola del Vall`es 08193 Ba celona,
Ca alonia, Spain
3Depa amen de Ma em`a iques, Uni e si a Poli `ecnica de Ca alunya, EPSEB,
A . D . Ma a˜n´on 44-50, 08028 Ba celona, Ca alonia, Spain
4IMTech, Uni e si a Poli `ecnica de Ca alunya, 08028 Ba celona, Ca alonia, Spain
Abs ac
In his a icle we in es iga e he impac o unc ional shi s in a ime-disc e e c oss-ca aly ic
sys em. We use he hype cycle model conside ing ha one o he species shi s om a coop-
e a o o a deg ade . A he bi u ca ion caused by his unc ional shi , an in a ian cu e
collapses o a poin Pwhile, simul aneously, wo ixed poin s collide wi h Pin a ansc i ical
bi u ca ion. Mo eo e , all poin s o a line con aining Pbecome ixed poin s a he bi u ca-
ion and only a he bi u ca ion in a degene a e scena io. We p o ide a comple e analy ical
desc ip ion o his degene a e bi u ca ion. As a esul o ou s udy, we p o e he exis ence o
he in a ian cu e a ising om he ansi ion o coope a ion.
Keywo ds Disc e e dynamical sys em; Con inuous dynamical sys em; Neima k-Sacke bi u -
ca ion; T ansc i ical bi u ca ion; Nonlinea popula ion dynamics; Coope a ion; Func ional shi
In oduc ion
Hype cycles a e ca aly ic se s o sel - eplica ing mac omolecules, whe e each eplica o ca alyzes
he eplica ion o he nex species o he se . This concep and model was i s in oduced by [ES77]
and has played a pi o al ole in he s udy o p ebio ic e olu ion and he o e coming o he so-called
in o ma ion c isis [Eig71, ES82, SS95]. Resea ch in hype cycles p ima ily in es iga es coope a i e
*
Co esponding au ho .
ORCID: E nes Fon ich 0000-0002-2415-9310, An oni Guillamon 0000-0001-8268-4503, J´ulia Pe ona 0009-0005-4154-
5917, Josep Sa dany´es 0000-0001-7225-5158
1
in e ac ions among eplica o s [ES79]. Hype cycle heo y has been also applied o in es iga e dy-
namics o ecological sys ems [SS95, Sa 09] and has aided in modeling expe imen al sys ems unde -
going coope a ion in enginee ed bac e ia [AMDNS17]. Coope a ion has been p e iously desc ibed
in di e en expe imen al sys ems wi h coiled-coil pep ides [LSG97], yeas cell popula ions [SRV07],
and sel - eplica ing ibozymes [VMC+02]. Despi e coope a ion is a majo d i e in bo h molecula
and ecological in e ac ions, addi ional in e ac ions a e likely o eme ge. Fo ins ance, molecula
ca aly ic eplica o s expe ience mu a ional p ocesses ha could change unc ional p ope ies o
ca aly ic co es e.g., a shi om liga ion o clea age, while ecological species may unde go be-
ha io al shi s in esponse o en i onmen al and ecological changes e.g., a shi om acili a ion
o compe i ion. Gene ically, hese shi s may en ail ansi ions om coope a i e o an agonis ic
in e ac ions, which we e e he e o as unc ional shi s (see also [BFO+21]), and a e p one o
in ol e di e en local and global bi u ca ions. The dynamics a ising om hese unc ional shi s
in ca aly ic cycles ha e been sca cely in es iga ed. Few wo ks ha e add essed his subjec o
con inuous- [BH95, BFO+21] and disc e e- ime [PFS20] sys ems.
Examples o unc ional shi s a e widesp ead in Ecology. In he ma ine en i onmen , la ge
pelagic p eda o s like una, sha ks, and dolphins a e known o collabo a e in loca ing and handling
small pelagic shoals [SCOH12]. Howe e , ins ances o p eda ion among hese species ha e also been
documen ed [Mal03, MSTG14]. Simila ly, seabi ds o m in e -speci ic locks o coope a e in inding
ood a sea, bu hey may also engage in p eda ion du ing b eeding in colonies [VFB+04, AO11].
Wa e bi ds, on he o he hand, o m mixed colonies o p o ec agains p eda o s, bu ce ain
species may oppo unis ically p ey on o he species, especially unde ad e se en i onmen al condi-
ions [Ha 70, And76]. These examples highligh he complexi y o ecological in e ac ions and how
coope a ion and an agonism can coexis wi hin di e se ecosys ems. Despi e he exis ence o such
obse a ions, he speci ic e ec s o unc ional shi s on he dynamics and s abili y o coope a ing
sys ems ha e no been ho oughly explo ed.
Dynamics in hype cycles ha e been ex ensi ely s udied by means o con inuous- ime ma he-
ma ical models, bo h in he limi o in ini e di usion [ES79, CFS00, SS06, SF08, PFG+18] and
in spa ially ex ended sys ems [BH95, C o95]. The use o disc e e models has been less explo ed
al hough hese sys ems a e o g ea impo ance o species wi h non-o e lapping gene a ions such
as insec s in bo eal clima es. The i s au ho o in es iga e a disc e e model was [Ho 84]. In
his model, he in e ac ions be ween species a e deno ed by ki, wi h xi he concen a ion o each
species. Since he wo k by Ho baue , e y ew esea ch has been pe o med in disc e e- ime hy-
pe cycles [Ho 84, HI84, PFS20]. Speci ically, in Re . [PFS20] we wen in dep h in o Ho baue ’s
disc e e model. We p o ed bo h he coope a ion in a n-dimensional sys em (i.e., no species goes
ex inc ) and he exis ence o a ixed poin in he in e io o he domain. This ixed poin is globally
asymp o ically s able in he h ee-dimensional sys em and uns able o n≥4. We also ca ied ou
a nume ical s udy o ind in a ian cu es o he ou -dimensional sys em. To compu e he in a i-
an cu e we buil a pseudo Poinca ´e map inspi ed by [GV19] aking ad an age ha he disc e e
sys em is close o a con inuous one. Ou main con ibu ion was he in es iga ion o dynamics and
he bi u ca ions when a species o he hype cycle shi s om coope a ion, ki>0, o deg ada ion
ki<0, in small hype cycles i.e., n= 2,3,4. We assumed ha he i s species was he one su -
e ing he shi , i.e., k1= 0. We analy ically p o ed ha he ajec o ies end asymp o ically o a
ixed poin in a co ne o he domain when he i s species shi s o deg ada ion, o any numbe
o species in he hype cycle. We also analy ically ob ained he a es o con e gence o he ixed
poin s o he s udied hype cycles. In he ou h-dimensional case, we showed nume ically ha he
disc e e- ime hype cycle is go e ned by an in a ian a ac ing cu e ha sh inks o a co ne o
he domain and disappea s h ough wha seems o be a Neima k-Sacke bi u ca ion when k1= 0.
2
Mo eo e , he in e io ixed poin collides wi h he co ne poin in a ansc i ical bi u ca ion and
a line o ixed poin s appea s, hus making he bi u ca ion mo e degene a e.
The main goal o his pape is o analy ically p o e he exis ence o he a o emen ioned in a i-
an cu es o he disc e e- ime hype cycle in oduced by [Ho 84] by inspec ing he coope a ion
pa ame e , k1, ha d i es he unc ional shi . Ou main con ibu ion is o disen angle he con-
luence o bi u ca ions by in oducing an ad hoc singula change o coo dina es ha b ings he
degene a e poin on he bounda y o a pa ame e -independen less degene a e in e io poin . This
change is essen ial o be able o w i e he sys em in such a way ha i is possible o apply a
heo em by [HI84] which gua an ees he exis ence o a amily o a ac ing in a ian cu es. This
las s ep also equi es a compu a ion o a no mal o m o he sys em.
The pape is s uc u ed as ollows. In Sec ion 1, we i s in oduce he model and, o he sake
o comple eness, we compu e he ixed poin s and hei s abili y as a unc ion o he pa ame e s.
In Sec ion 2, we ecall he Neima k-Sacke bi u ca ion o disc e e- ime dynamical sys ems and he
heo em om [HI84] ha p o es he exis ence o a amily o a ac ing in a ian cu es in a amily
o maps ha can be exp essed as a s ep o he Eule ’s in eg a ion me hod o a di e en ial equa ion.
Finally, Sec ion 3 con ains he p oo o he exis ence o he in a ian cu e o ou disc e e- ime
hype cycle wi h ou species, n= 4, o k1→0+. Fo his pu pose, we make a singula change o
coo dina es o make ou sys em less degene a e and we apply he abo emen ioned heo em.
1 Ho baue ’s disc e e- ime hype cycle model
In his sec ion we p esen he disc e e- ime model o he hype cycle, in oduced by [Ho 84], and
we ela e i o a con inuous- ime model. We also compu e he basic elemen s o he dynamics such
as he ixed poin s o he sys em and hei s abili y. This dynamical sys em consis s o a se o n
species si, 1 ≤i≤n, such ha he species si−1ca alyzes only he nex one si, in a cyclic manne ,
wi h a s eng h ki. Le xibe he concen a ion o he i- h species. Fo con enience o no a ion,
we w i e x0:= xnand xn+1 := x1and simila ly o k0, kn+1. The model assumes ha i he o al
popula ion is no malized o 1, i emains cons an . This is accomplished by he in oduc ion o a
lux φ(x), which also in oduces compe i ion be ween all he hype cycle species. This ac implies
ha he sys em will be de ined on he n-simplex
Sn=(x= (x1, . . . , xn)∈Rn|
n
X
i=1
xi= 1, xi≥0,1≤i≤n).(1)
We in oduce he hype plane
∆n=(x∈Rn|
n
X
i=1
xi= 1),
and he se e
∆n={x∈∆n|xi6= 0,1≤i≤n}.The sys em is de e mined by he map F=
(F1, . . . , Fn) : Sn→Sn, whe e
Fi(x) = C+kixi−1
C+φ(x)xi,1≤i≤n, (2)
3
C > 0 is a cons an o p opo ionali y and
φ(x) =
n
X
i=1
kixixi−1.(3)
In [Ho 84] he map (2) is ela ed o he co esponding con inuous- ime sys em
˙xi=xi(kixi−1−φ(x)),1≤i≤n, (4)
which also sa is ies ha i he ini ial o al popula ion is 1, hen i emains cons an . In pa icula
we can w i e
Fi(x)−xi
C−1=CC+kixi−1
C+φ(x)xi−xi=xi(kixi−1−φ(x)) C
C+φ(x).(5)
This exp ession allows us o compa e he map Fwi h he con inuous ime model, since C−1can
be in e p e ed as he ime in e al be ween wo gene a ions and F(x) can be seen as he Eule
s ep o leng h C−1o he con inuous- ime sys em (4) since
lim
C−1→0
xi( +C−1)−xi( )
C−1=xi( )(kixi−1( )−φ(x( ))),
iden i ying Fi(x)( ) wi h xi( +C−1) p o ided ha xi=xi( ). Then, o la ge alues o C, he
disc e e- ime model (2) app oxima es he di e en ial equa ion (4) and he e o e we expec ha ,
in such case, bo h models ha e simila p ope ies.
I we keep he cons an s kiposi i e and bounded away om ze o, and we le one o hem, say
k`, go o ze o and hen become nega i e, he model can be in e p e ed biologically as he e has
been a unc ional shi meaning ha he ole o he species s`−1changes om coope a ion (k`>0)
o deg ada ion (k`<0). No e ha , due o he cyclic cha ac e o ou model, we can assume,
wi hou loss o gene ali y, ha he pa ame e ha ends o ze o is k1while all kiwi h i > 1 a e
bounded away om ze o.
In [PFS20] his bi u ca ion was s udied and i was ound nume ically ha o ou species he e
is an a ac ing in a ian cu e ha ends o he poin Q= (0,0,0,1) when k1→0+. Also, when
k1>0 he e is a unique ixed poin Pin he in e io o he simplex Sn∩e
∆n, al eady desc ibed
in [Ho 84]. The poin Pcollides wi h Qwhen k1= 0, and goes ou o Snin a ansc i ical-like
bi u ca ion. Mo eo e , in [PFS20] i is shown ha o any numbe o species, when k1≤0, he
basin o a ac ion o Qcon ains Sn∩e
∆n. This ac implies ha , in his case, he sys em has no
in a ian cu es in Sn.
As we s a ed abo e, he main goal o his con ibu ion is o p o e he exis ence o an in a ian
cu e when k1>0 gene a ed h ough a Neima k-Sacke bi u ca ion ha occu s a he same ime
ha bo h a ansc i ical bi u ca ion and he appea ance o a new line o ixed poin s.
We ema k ha in [Ho 84] i is p o ed he exis ence o an in a ian cu e o ampli ude O(1/√C)
when C→ ∞. He e, ins ead, we a e looking o an in a ian cu e in a di e en egion o he
space o pa ame e s, and ocus ou a en ion on he abo e-men ioned unc ional shi .
Since we assume C > 0, we can ew i e Fi(x) as 1+
e
kixi−1
1+
e
φ(x)xiwi h e
φ(x) = Pn
i=1 e
kixixi−1and
e
ki=ki/C. In his way we can ge id o C. We w i e kiagain ins ead o e
ki. No ice ha le ing C
4
go o ∞in (2) esul s in le ing he new pa ame e s ki end o ze o. He e, we will only le k1go o
ze o keeping he o he pa ame e s ixed. Conc e ely, we will ake kj>0, o 2 ≤j≤n, a bi a y
and k1 a iable such ha k1> k∗
1, wi h k∗
1=−(Pn
j=2 1
kj)−1<0.
1.1 Fixed poin s and s abili y
As a i s s ep o unde s and he dynamics and o he sake o comple eness, we gi e a b ie
desc ip ion o he ixed poin s o sys em (2) and hei s abili y. The unique ixed poin Pin he
in e io o he simplex Sn∩e
∆nwas s udied in [Ho 84]. In [PFS20] he ixed poin s in he bounda y
o he simplex we e also s udied.
Since he ixed poin s mus sa is y Fi(x) = xi o all i, o he poin s in e
∆n, om (2), we ge
he condi ion kixi−1=φ(x), 1 ≤i≤n, o equi alen ly,
k2x1=k3x2=··· =knxn−1=k1xn=φ(x).
Fo k1≤0, he e a e no ixed poin s in Sn∩e
∆n. I k1>0, he las se o equa ions gi es
xi=k1
ki+1 xn, 1 ≤i≤n−1. Using Pn
i=1 xi= 1, we ge ha he ixed poin is
P:= (p1, . . . , pn),wi h pi=1
ki+1M1
,1≤i≤n,
whe e M1=Pn
j=1 1
kj. When k1= 0, Pcoincides wi h Q= (0,0. . . , 0,1) and when k∗
1< k1<0,
M1<0 and he e o e pj<0, 2 ≤j≤n.
Mo eo e , in P oposi ion 1 o [PFS20] i was p o ed ha x∈∆n∩e
∆nis a ixed poin i and
only i kixixi−1= 0 ∀i. In he ou species case, o k16= 0 in he bounda y o he simplex we
ha e he segmen s o ixed poin s {(α, 0,1−α, 0) |α∈[0,1] }and {(0, α, 0,1−α)|α∈[0,1] }. I
k1= 0 we ha e he addi ional segmen o ixed poin s {(α, 0,0,1−α)|α∈[0,1] }. In pa icula ,
he e ices q(m):= (δm,1, . . . , δm,4), 1 ≤m≤4, o he simplex S4a e always ixed poin s. He e
δk,l is he K onecke del a. When k1→0+ he inne ixed poin P ends o he ixed poin
Q=q(4) = (0,0,0,1).
Again, in he ou species case, we ha e ha he eigen alues o he inne ixed poin Pa e
λj= 1 + 1
M1+ 1eiθj, θj=e2πij
4, j = 1,2,3,(6)
oge he wi h λ0= 1 + 1
M1+1 which has he eigen ec o (1,1,1,1) o hogonal o S4. The e o e,
conce ning he dynamics in S4 he ele an eigen alues a e λ1, λ2and λ3(see [Ho 84, PFS20]).
We also ha e ha
|λ1|2=|λ3|2= 1 + 1
M1+ 12>1,
|λ2|2=1−1
M1+ 12<1.
The e o e, Pis uns able.
5
The eigen alues o q(m)a e gi en in [PFS20]. They a e 1 (double) and 1 + km+1. In pa icula
Q=q(4) has one eigen alue ha goes om bigge han 1 o less han 1 when k1goes om posi i e
o nega i e.
2 A non-gene ic Neima k-Sacke bi u ca ion heo em
In his sec ion, we ecall a esul by [HI84], ha s udies a Neima k-Sacke bi u ca ion o di e ence
equa ions. In ha pape he au ho s call i Hop bi u ca ion bu we p e e o e e o i as Neima k-
Sacke since i seems o us ha nowadays his e m is mo e used o maps, see o ins ance [Kuz13].
The esul deals wi h a disc e iza ion o a di e en ial equa ion nea a ixed poin wi h wo pu ely
imagina y eigen alues and he emaining ones wi h nega i e eal pa . The inal goal is o p o e
ha an in a ian cu e appea s a ound he ixed poin o he disc e e ime sys em. In ou case
we canno apply he s anda d no mal o m o he Neima k-Sacke bi u ca ion as i appea s, o
ins ance in [Kuz13] since, among o he hings, ou amily o maps is close o he iden i y and he
eigen alues a he bi u ca ion a e all equal o one.
Fi s , we conside an au onomous di e en ial equa ion
˙x= (x) (7)
de ined in an open se o Rnand we assume ha he o igin is an equilib ium poin , i.e. (0) = 0.
As in he Eule ’s me hod o in eg a ion, we conside he ollowing amily o maps
Tε:x7→ x+ε (x), ε > 0 small.(8)
Since (0) = 0 we can w i e
(x) = Ax +O(kxk2),(9)
whe e A=D (0). We immedia ely ha e ha he maps Tεha e he o m
Tε(x) = (Id + εA)x+εO(kxk2).
I is clea ha λis an eigen alue o Ai and only i 1 + ελ is an eigen alue o DTε(0). I Ahas
a pai o pu ely imagina y eigen alues, hen we ha e ha he ixed poin is uns able o he map
(8) o e e y ε > 0, al hough x= 0 could be asymp o ically s able o sys em (7).
De ini ion 2.1 Assume ha he o igin is an equilib ium poin o sys em (7) and i has a pai
o pu ely imagina y eigen alues ±i ω. Suppose is su icien ly di e en iable and ha (7) can be
ans o med, a ound he o igin, by a change o coo dina es, in o he o m
˙z=iωz +
2k
X
j=1
αjz|z|2j+O(|z|+| |)4k+2,
˙ =A +O(|z|2+| |2),
(10)
whe e |z|2=zz, and αj=aj+ibj. We say ha he o igin is a weakly s able equilib ium poin o
o de ki he e exis s k≥1such ha a1=··· =ak−1= 0 and ak<0.
6
In [HI84] he ollowing heo em is p o ed.
Theo em 1 Conside equa ion (7) in an open neighbou hood o x= 0 in Rnwi h su icien ly
di e en iable. Assume ha
(1) (0) = 0 and D (0) has wo pu ely imagina y eigen alues ±iω, and he es o he eigen alues
ha e nega i e eal pa , and
(2) he equilib ium poin x= 0 is a weakly s able equilib ium poin o o de k, wi h 4k+ 2 ≤ .
Then, o any amily o maps Tεo class C , o he o m
Tε(x) = x+ε (x) + O(ε2kxk2), ε > 0,(11)
he e exis s an ε-dependen amily o in a ian and a ac ing closed cu es a ound he ixed poin
x= 0 o adius O(ε1/2k).
Rema k 2.1 In he n-species case, he eigen alues co esponding o he inne ixed poin a e λ0
and λjas in (6) wi h j= 1, . . . , n −1. We see ha Ahas a pai o (conjuga e) pu e imagina y
eigen alues only when nis a mul iple o 4 and, i nis bigge han 4, Ahas eigen alues bo h wi h
posi i e and nega i e eal pa . Then Theo em 1 would no be applicable.
3 A degene a e ansc i ical Neima k-Sacke bi u ca ion
In his sec ion, ou goal is o p o e analy ically he exis ence o an in a ian cu e applying Theo em
1 o ou disc e e- ime sys em (see Fig. 1 o a nume ical ep esen a ion o he in a ian cu e and
he bi u ca ion). In o he wo ds, we will p o e ha an in a ian cu e is bo n when k1= 0 pe sis s
o k1posi i e and su icien ly small. Since he bi u ca ion is e y degene a e we will uncouple
he Neima k-Sacke and he ansc i ical bi u ca ions. Fo his pu pose, we o ce he inne ixed
poin o be loca ed a he “cen e ” o he simplex o all alues o k1. This is accomplished using
ba ycen ic coo dina es. Le
yi=ki+1xi
P4
j=1 kj+1xj
,1≤i≤4.
Indeed, his change allows o sepa a e he inne ixed poin P om he e ex Q= (0,0,0,1)
ans o ming he ixed poin Pin o he “cen e poin ” o he simplex S4:
p=1
4,1
4,1
4,1
4.
This ans o ma ion is singula a k1= 0, bu i acili a es he s udy o he sys em nea o he
ixed poin p, since i b ings i a om he o he ixed poin s.
7
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
0 100 200 300
0
0.2
0.4
0.6
0.8
a
b
c
050 100 150 200 250 300
0
0.2
0.4
0.6
0.8
0 200 400 600 800 1000
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
b
ac
a
b
c
Figu e 1: E olu ion o he in a ian cu e shown in he space (k1, x3, x4). The in a ian cu e
sh inks as k1dec eases, inally collapsing a he bi u ca ion alue k1= 0. Panels show ime se ies
o all hype cycle species ( igh ) and a p ojec ion o he cu e in he phase space (x1, x2) (bo om).
He e we use k2,3,4= 1 and: (a) k1= 1, (b) k1= 0.5, and (c) k1= 0.1. Ini ial condi ions in all
panels a e x1(0) = 0.9, x2,3,4(0) = 0.01.
8
Now, we a e going o compu e he new sys em in ba ycen ic coo dina es. Fi s , we no ice ha
xi=yiP4
j=1 kj+1xj
ki+1
.
Since his change o coo dina es sends ∆n o ∆nwe can w i e
4
X
i=1
xi= 1 ⇐⇒
4
X
i=1
yiP4
j=1 kj+1xj
ki+1
= 1 ⇐⇒
4
X
j=1
kj+1xj=1
P4
i=1
yi
ki+1
,
and we ob ain
xi=yi
N(y)ki+1
,whe e N(y) =
4
X
j=1
yj
kj+1
.
We can exp ess ou map Fin (2) in he new a iables yias
Fi(y) = ki+1Fi(x)
P4
j=1 kj+1Fj(x)=
ki+11+kixi−1
1+φ(x)xi
P4
j=1 1+kjxj−1
1+φ(x)xjkj+1
=
ki+11 + kiyi−1
kiN(y)yi
ki+1N(y)
P4
j=1 1 + kjyj−1
kjN(y) yj
kj+1N(y)kj+1
=1 + yi−1
N(y)
P4
j=1 1 + yj−1
N(y)yj
yi.
Nex , we pe o m a ansla ion o ha e he ixed poin a he o igin:
zi=yi−1
4,1≤i≤4.
In hese new coo dina es,
4
X
j=1
zj= 0 (12)
and he alue o N(z) is gi en by
N(z) =
4
X
j=1
(zj+1
4)
kj+1
=
4
X
j=1
zj
kj+1
+1
4
4
X
j=1
1
kj+1
.
Mo eo e , using (12), he componen s o Fbecome:
Fi(z) = Fi(y)−1
4=
1 + yi−1
N(y)
P4
j=1 1 + yj−1
N(y)yj
yi−1
4=N(y) + yi−1
P4
j=1 N(y) + yj−1yj
yi−1
4
=N(z) + zi−1+1
4
W(z)(zi+1
4)−1
4=zi+N(z) + zi−1+1
4
W(z)−1(zi+1
4)
=zi+N(z) + zi−1+1
4−W(z)
W(z)(zi+1
4),
9
[Mal03] D. Maldini. E idence o p eda ion by a ige sha k (Galeoce do cu ie ) on a spo ed
dolphin (S enella a enua a) o O’ahu, Hawai’i. Aqua . Mamm., 29:84–87, 2003.
[MSTG14] K. Melillo-Swee ing, S. D. Tu nbull, and T. L. Gu idge. E idence o sha k a acks
on A lan ic spo ed dolphins (S enella on alis) o Bimini, The Bahamas. Ma .
Mammal Sci., 30:1158–1164, 2014.
[PFG+18] J. Puig, G. Fa ´e, A. Guillamon, E. Fon ich, and J. Sa dany´es. Bi u ca ion gaps
in asymme ic and high-dimensional hype cycles. In . J. Bi u ca ion and Chaos,
28(1):1830001, 2018.
[PFS20] J. Pe ona, E. Fon ich, and J. Sa dany´es. Dynamical e ec s o loss o coope a ion in
disc e e- ime hype cycles. Physica D, 406:132425, 2020.
[Sa 09] J. Sa dany´es. The hype cycle: F om molecula o ecosys ems dynamics. in Landscape
Ecology Resea ch T ends. No a Publishe s: Hauppauge. NY, USA, 2009.
[SCOH12] M. D. Sco , S. J. Chi e s, R. J. Olson, and K. Holland. Pelagic p eda o associa ions:
Tuna and dolphins in he eas e n opical Paci ic Ocean. Ma . Ecol. P og. Se .,
458:283–302, 2012.
[SF08] D. A. M. M. Sil es e and J. F. Fon ana i. The in o ma ion capaci y o hype cycles.
J. Theo . Biol., 4:804–806, 2008.
[SRV07] W. Shou, S. Ram, and J. M. G. Vila . Syn he ic coope a ion in enginee ed yeas
popula ions. P oc. Na l. Acad. Sci. USA, 104:1877–1882, 2007.
[SS95] J. M. Smi h and E. Sza hm´a y. The Majo T ansi ions in E olu ion. Ox o d Uni e -
si y P ess: Ox o d, UK, 1995.
[SS06] J. Sa dany´es and R. V. Sol´e. Bi u ca ions and phase ansi ions in spa ially ex ended
wo-membe hype cycles. J. Theo . Biol., 243:468–482, 2006.
[VFB+04] S. C. Vo ie , R. W. Fu ness, S. Bea hop, J. E. C ane, R. W. G. Caldow, and e al.
Changes in ishe ies disca d a es and seabi d communi ies. Na u e, 427:727–730,
2004.
[VMC+02] N. Vaidya, M. L. Manapa , I. A. Chen, R. Xul i-B une , E. J. Hayden, and
N. Lehman. Spon aneous ne wo k o ma ion among coope a i e RNA eplica o s.
Na u e, 491:72–77, 2002.
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