The “Destabilizing” Effect of Cannibalism in a Spatially Explicit Three-Species Age Structured ...

16 Pages·2017·2.82 MB·English

Hindawi
Complexity
Volume 2017, Article ID 3896412, 15 pages
https://doi.org/10.1155/2017/3896412
Research Article
( )
The Destabilizing Effect of Cannibalism in a Spatially Explicit
Three-Species Age Structured Predator-Prey Model
AladeenAlBasheer,JingjingLyu,AdomGiffin,andRanaD.Parshad
DepartmentofMathematics,ClarksonUniversity,Potsdam,NY13699,USA
CorrespondenceshouldbeaddressedtoRanaD.Parshad;[email protected]
Received1July2016;Revised6October2016;Accepted7November2016;Published11January2017
AcademicEditor:DimitriVolchenkov
Copyright©2017AladeenAlBasheeretal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
Cannibalism,theactofkillingandconsumptionofconspecifics,isgenerallyconsideredtobeastabilisingprocessinODEmodels
ofpredator-preysystems.Ontheotherhand,Sunetal.werethefirsttoshowthatcannibalismcancauseTuringinstability,inthe
classicalRosenzweig-McArthurtwo-speciesPDEmodel,whichisanimpossibilitywithoutcannibalism.Magnu´sson’sclassicwork
isthefirsttoshowthatcannibalisminastructuredthree-speciespredator-preyODEmodelcanactuallybedestabilising.Inthe
currentmanuscriptweconsiderthePDEformofthethree-speciesmodelproposedinMagnu´sson’sclassicwork.Weprovethat,in
theabsenceofcannibalism,Turinginstabilityisanimpossibilityinthismodel,foranyrangeofparameters.However,theinclusion
ofcannibalismcancauseTuringinstability.Thus,tothebestofourknowledge,wereportthefirstcannibalisminducedTuring
instabilityresult,inspatiallyexplicitthree-speciesagestructuredpredator-preysystems.Wealsoshowthat,intheclassicalODE
modelproposedbyMagnu´sson,cannibalismcanactasalifeboatmechanism,fortheprey.
1.Introduction predator-preymodelshavebeeninvestigated[6,10–16].Most
worksintheliteratureconsidercannibalisminthepredator
Background.Cannibalismisdefinedastheactofkillingand
species. However, there are a few works that involve canni-
atleastpartialconsumptionofconspecifics.Itisubiquitous
balismamongpreyaswell[17].Inmostmodels,cannibalism
innaturalpredator-preycommunities[1]andisobservedin
is seen to have a strong stabilising influence [11, 14]. But
morethan1300speciesinnature[2].Itoccursasasocioan-
some works do report that it can destabilize as well [18].
thropological as well as ecological phenomenon [3, 4], in
Since species disperse in space in search of food, shelter,
manyvariedforms.Forexample,asectofasceticsinNorth-
and mates and to avoid predators, spatially dispersing pop-
ern India, called “Aghoris,” consume corpses in the belief
ulations are often modeled via partial differential equations
thatsuchritualisticpractice,orendocannibalism,willenable
themtoattainimmortality[5].Cannibalismisalsorampant (PDE)/spatiallyexplicitmodelsofinteractingspecies[19–22].
amongst certain insects. Many arachnid species practice OfparticularinteresttomanyisthephenomenonofTuring
sexualcannibalism,wherethefemalecannibaliseshermate, instability,firstintroducedbyTuring[23],whichshowsthat
before, during, or after copulation. In population models, diffusioninasystemcanleadtopatternforminginstabilities.
theinclusionofcannibalismcanbringaboutinterestingand Inthecontextofcannibalism,thereareveryfewworksinthe
sometimes counter intuitive dynamics, such as the life boat literaturethatconsiderspatiallyexplicitmodelsandinvesti-
mechanism.Thereintheactofcannibalismcausespersistence gateTuringinstabilitytherein.Sunetal.[24]werethefirstto
in a population, otherwise doomed to go extinct [6]. The showthatcannibalismintheclassicaldiffusiveRosenzweig-
effectsofclimatechangeinducedcannibalismhavealsobeen McArthurmodelcanbringaboutTuringpatterns.Thisisan
intenselyinvestigatedrecently[7–9]. impossibilityinthesystemwithoutcannibalism.Fasaniand
A detailed survey of the current and past mathemat- Rinaldi[25]generalisedthecannibalismtermfrom[24]and
ical literature on cannibalism shows that various types of foundthatcannibalismcanstillbringaboutTuringpatterns.
2 Complexity
Theabovemodelsconsidercannibalismamongstthepreda- (i)Thisscalingdoesnotpermitonetoconsiderthelim-
tors.In[17]wefoundthatcannibalismamongstpreycanalso iting case of no cannibalism (𝑆 = 0), in a tractable
bringaboutTuringpatterns,inthespatiallyexplicitHolling- way.
Tannermodel,withratiodependentfunctionalresponse.
(ii)There is a prey free equilibrium or no “cost” to the
One approach in the literature, when considering ODE
adultashecannibalisesthejuvenile.
models, has been to structure the predator class into adults
andjuvenilesandallowcannibalismofthejuvenilepredator
RescaledModelofTang.Partoftheseissueswasfirstaddressed
by the adult predator. This essentially yields a three-species
in[26],whereamorerefinedformofscalingisintroduced.
age structured ODE model. The reader is referred to [12,
14, 27], and again therein cannibalism is seen to provide a 𝑉
𝑢1 = 𝑋,
stabilisinginfluence,dependingonparameterregime.How- 𝑀
1
ever,dependingonmodelstructureandparameterregimes,
𝑉𝐴
cannibalismcanalsodestabilize[1,17,18].Tothebestofour V1 = 𝑌,
knowledge, there is a complete lack of investigation of the 𝑀2
1
effects of cannibalism, on the stability of spatially explicit/ (2)
𝐶
PDEthree-speciespredator-preymodels,withagestructure 𝑟1 = 𝑍,
𝑀
(orwithout). 1
𝑀
ClassicalModelofMagnu´sson.Tothisendweinvestigatethe 𝜏= 1𝑡,
𝐶
spatially explicit version of the model first proposed in [18]
by Magnu´sson and further analyzed by Kaewmanee et al. where𝑡1 =𝑇/𝑀1,𝑚=(𝐴+𝑀2)/𝑀1,𝑟=𝑅𝐴/𝑀12,𝑢=𝑈/𝐶,
[s2tr6u,c2t8u]r.eMdatghnreu´es-ssopnepcireospporseeddaatnodr-pinrveeystmigoadteedl,thwehfeorlelo𝑋wi(n𝑡)g, 𝛾=𝛾󸀠𝑀1/𝐴,𝑠=𝑆/𝑉.
This yields the following nondimensionalised system,
𝑌(𝑡),and𝑍(𝑡)arethebiomassofanadultpredator,ajuvenile
withfewerparameters
predator,andprey,atagiventime𝑡.Theinteractionbetween
thesethreeclassesisdescribedbelow: 𝑑𝑢1 =−𝑢1+V1+𝑢1𝑟1+𝛾𝑠𝑢1V1
𝑑𝜏
𝑑𝑋
=−𝑀 𝑋+𝐴𝑌+𝐶𝑋𝑍+𝛾󸀠𝑆𝑋𝑌 𝑑V1
𝑑𝑡 1 =𝑟𝑢1−𝑚V1−𝑠𝑢1V1 (3)
𝑑𝜏
𝑑𝑌
𝑑𝑡 =𝑅𝑋−𝐴𝑌−𝑀2𝑌−𝑆𝑋𝑌 (1) 𝑑𝑟1 =𝑡 𝑟1−𝑢(𝑟1)2−𝑢1𝑟1.
𝑑𝜏 1
𝑑𝑍
=𝑇𝑍−𝑈𝑍2−𝑉𝑋𝑍. Here typically 𝛾 < 1, but not necessarily. Now the can-
𝑑𝑡
nibalismparameter𝑠canbefollowedclosely.Infactonecan
tractablytakethelimitas𝑠→0,toobtainthemodelwithout
Essentiallythepredatorspeciesisstructuredintoadults cannibalism,
𝑋(𝑡)andjuveniles𝑌(𝑡),withtheadultpredatorsfeedingona 𝑑𝑢1
preyspecies𝑍(𝑡).Noteallparametersarepositiveconstants, =−𝑢1+V1+𝑢1𝑟1
𝑑𝜏
and the system goes along with suitable positive initial
cthoenddietaiothnsr.a𝑀te1ofisjutvheenidleeapthredraatteorosf, 𝐴adiusltthperegdroatwotrhs,r𝑀ate2oisf 𝑑𝑑V𝜏1 =𝑟𝑢1−𝑚V1 (4)
adultspredators(juvenilesgrowintoadults),𝑅isthegrowth
rate of juveniles due to adults, 𝐶 is the predators growth 𝑑𝑟1 =𝑡 𝑟1−𝑢(𝑟1)2−𝑢1𝑟1.
rateduetopreying,𝑇isthebirthrateofprey,and𝑉isthe 𝑑𝜏 1
predatorsattackrateonprey.𝑆isthecannibalismrate,and
𝛾󸀠 istheconversionefficiencyofeatenjuvenilesintoadults. TheresultsthatKaewmaneeandTangfindareinaccor-
dance with Magnu´sson; that is increasing the cannibalism
For further details see [18]. Magnu´sson finds that, in this
parameter can lead to the stable interiorequilibriumlosing
particular model, the effect of including cannibalism is to
stability, leading to limit cycle dynamics. Kaewmanee and
destabilise the interior equilibrium, leading to limit cycle
Tang’s results, unlike Magnu´sson, are not limited to the
dynamics,throughaHopfbifurcation.Thisresultiscontrary
casewherethereishighjuvenilemortalityand/orlowadult
tomanyothersintheliterature,wherecannibalismisfound
recruitment. However, the Hopf bifurcation analysis as
to be stabilising. Note one of the problematic issues in [18]
isthescalingofthepredatorattackrateparameter,𝑉,which workedoutin[26]isincorrect.
scaleslike𝑉≈1/𝑆,where𝑆isthecannibalismrateparameter. Ourprimarycontributionsinthecurrentmanuscriptare
Apossiblereasoninghereisthatif𝑉 ≈ 1/𝑆,then𝑆 ≈ 1/𝑉. thefollowing:
Soifpredation/attackrateonregularpreyishigh,or𝑉 ≫ 1, (i)Weexplorethespatiallyexplicitformsofthemodels,
thereisalmostnoneedforcannibalism,and𝑆≪1.Acloser (3)and(4),andshowglobalexistenceofsolutionsvia
examinationofthemodelrevealstwocausesforconcern: Theorem2andCorollary3.
Complexity 3
(ii)We show that the spatially explicit model without Here 𝑢1(𝑥,⋅), V1(𝑥,⋅), 𝑟1(𝑥,⋅) are the concentrations/
cannibalism does not possess Turing instability in populationdensitiesoftheprey,thejuvenilepredator,andthe
anyparameterregimeviaTheorem4.However,intro- adultpredator,atanygiventime𝑡,respectively.Wenextstate
ducing cannibalismcancertainlybringaboutTuring theformoftheone-dimensionalspatiallyexplicitmodelwith
instability,viaTheorem14.Tothebestofourknowl- cannibalism
edge, this is the first cannibalism induced Turing
instabilityresult,inagestructuredthree-speciesPDE 𝑢1 =𝑑 𝑢1 −𝑢1+V1+𝑢1𝑟1+𝛾𝑠𝑢1V1 (8)
𝑡 1 𝑥𝑥
models.
V1 =𝑑 V1 +𝑟𝑢1−𝑚V1−𝑠𝑢1V1 (9)
𝑡 2 𝑥𝑥
(iii)WederivethecorrectformofHopfbifurcationinthe
classicODEmodelproposedin[18]viaTheorem15. 𝑟1 =𝑑 𝑟1 +𝑡 𝑟1−𝑢(𝑟1)2−𝑢1𝑟1, (10)
𝑡 3 𝑥𝑥 1
NoteagainthatthepreviousHopfbifurcationanalysis
asworkedoutin[26]isincorrect.
𝛾<1,andourdomainΩ=[0,𝜋].
WeprescribeNeumannboundaryconditions
(iv)Wenumericallyshowthatcannibalismcanactasalife
boatmechanisminthepreyandconjecturethatitcan
actasalifeboatmechanisminbothpredatorandprey 𝑢1 =V1 =𝑟1 =0 (11)
𝑥 𝑥 𝑥
viaConjectures16and18.
andsuitablepositiveinitialconditions
(v)We discuss ecological interpretations, based on our
work. 𝑢1(𝑥,0)>0,
We relegate the details of the ODE stability analysis, as V1(𝑥,0)>0, (12)
wellasgenericTuringinstabilityconditions,totheAppendix.
Someofthesearealsoreadilyavailableintheliterature[19, 𝑟1(𝑥,0)>0.
26],sotheyareincludedintheAppendixonlyforcomplete-
nessandbenefitofthereader. Here 𝑢1(𝑥,⋅), V1(𝑥,⋅), 𝑟1(𝑥,⋅) are the concentrations/
populationdensitiesoftheprey,thejuvenilepredator,andthe
2.SpatiallyExplicitModel adultpredator,atanygiventime𝑡,respectively.
Since species disperse in space in search of food, shelter,
2.1. Functional Preliminaries and Local Solutions. We now
and mates and to avoid predators, spatially dispersing pop-
presentvariousfunctionspacenotationsanddefinitionsthat
ulations are often modeled via partial differential equations willbeusedfrequently.TheusualnormsinthespacesL𝑝(Ω),
(PDE)/spatiallyexplicitmodelsofinteractingspecies[19–22]. L∞(Ω),andC(Ω)are,respectively,denotedby
In this section we shall investigate the effects of diffusion
induced instability in the models (4) and (3). We state the 1
formoftheone-dimensionalspatiallyexplicitmodelwithno |𝑢|𝑝 = ∫ |𝑢(𝑥)|𝑝𝑑𝑥,
𝑝 |Ω|
cannibalism, Ω (13)
|𝑢|∞ =max|𝑢(𝑥)|.
𝑢1 =𝑑 𝑢1 −𝑢1+V1+𝑢1𝑟1 𝑥∈Ω
𝑡 1 𝑥𝑥
V1 =𝑑 V1 +𝑟𝑢1−𝑚V1 It is well known that, under the “regularizing effect
𝑡 2 𝑥𝑥 (5)
principle”toproveglobalexistenceofsolutionsto(8)–(10),
𝑟1 =𝑑 𝑟1 +𝑡 𝑟1−𝑢(𝑟1)2−𝑢1𝑟1, [29],itsufficestoderiveuniformestimateson𝐿𝑝,normsof
𝑡 3 𝑥𝑥 1 thereactionterms,on[0,𝑇max[,forsome𝑝 > 𝑛/2,where𝑛
is the spatial dimension of the domain Ω, and the reaction
𝛾<1,andourdomainΩ=[0,𝜋]. termsinoursettingaregivenin(8)–(10).Here𝑇maxdenotes
WeprescribeNeumannboundaryconditions theeventualblowing-uptimeinL∞(Ω).Thefollowinglocal
existenceresultiswellknown[30].
𝑢𝑥1 =V𝑥1 =𝑟𝑥1 =0 (6) Lemma1. System(8)–(10)admitsaunique,classicalsolution
(𝑢1,V1,𝑟1)on[0,𝑇max[×Ω.If𝑇max <∞then
andsuitablepositiveinitialconditions
𝑢1(𝑥,0)>0, 𝑡↗li𝑇mmax{󵄩󵄩󵄩󵄩󵄩𝑢1(𝑡,⋅)󵄩󵄩󵄩󵄩󵄩∞+󵄩󵄩󵄩󵄩󵄩V1(𝑡,⋅)󵄩󵄩󵄩󵄩󵄩∞+󵄩󵄩󵄩󵄩󵄩𝑟1(𝑡,⋅)󵄩󵄩󵄩󵄩󵄩∞}=∞. (14)
V1(𝑥,0)>0, (7) Proof. Since the reaction terms are continuously differen-
tiableinthepositiveoctant,thenforanyinitialdatainC(Ω)
𝑟1(𝑥,0)>0. or L𝑝(Ω), 𝑝 ∈ (1,+∞), it is easy to check directly their
4 Complexity
Lipschitz continuity on bounded subsets of the domain of the𝑢1V1term.However,integrating(9)inΩ×[0,𝑇]=Ω𝑇we
a fractionalpower of theoperator 𝐼3(𝑑1,𝑑2,𝑑3)𝑡Δ, where 𝐼3 obtain
is the three-dimensional identity matrix, Δ is the Laplacian
operator,and()𝑡denotesthetransposition. ∫ V1(⋅,𝑇)𝑑𝑥+𝑠∫ 𝑢1V1𝑑𝑥𝑑𝑡
Ω Ω
𝑇
(19)
2.2.ControlofMassand𝐿1(Ω)Estimates. Note,inorderto
=𝑟∫ 𝑢1−𝑚∫ 𝑟V1+∫ V1𝑑𝑥.
deriveglobalexistenceviaapplicationof[29],wearerequired 0
toderiveuniformintime𝐿𝑝estimatesonthereactionterms Ω𝑇 Ω𝑇 Ω
in(8)–(10),for𝑝>1/2.Wewillrestrictourselvestothecase
𝑛 = 1.Tothisend(8)posesadifficultyviathe𝛾𝑠𝑢1V1 term. Thus
Weproceedbymultiplying(9)by𝛾andaddingupequations
(8)and(9)toyield 󵄩󵄩󵄩󵄩󵄩𝑢1V1󵄩󵄩󵄩󵄩󵄩Ω ≤ 𝑟𝑠 ∫ 𝑢1+ 1𝑠 ∫ V01𝑑𝑥
𝑇 Ω Ω
𝑇
𝑢t1+𝛾V𝑡1 =𝑑1𝑢𝑥1𝑥+𝑑2𝛾V𝑥1𝑥+(𝑟𝛾−1)𝑢1 ≤ 𝑟𝑠 ∫𝑇󵄩󵄩󵄩󵄩󵄩𝑢01+𝛾V01󵄩󵄩󵄩󵄩󵄩𝐿1(Ω)𝑒(𝐶max((𝑟𝛾−1),(1−𝛾𝑚),‖𝑟1‖∞))𝑡𝑑𝑡
(15) 0
+(1−𝛾𝑚)V1+𝑢1𝑟1.
+ 1𝑠 󵄩󵄩󵄩󵄩󵄩V01󵄩󵄩󵄩󵄩󵄩𝐿1(Ω) (20)
inteWgraetenothwedaebfionveeethqeuagtrioounpiendspvaacrieatboleo𝑊btai=n 𝑢1 +𝛾V1 and = 𝑟𝑠󵄩󵄩󵄩󵄩󵄩𝑢(01𝐶+m𝛾aVx01󵄩󵄩󵄩󵄩󵄩(𝐿(1𝑟(𝛾Ω)−𝑒(1𝐶)m,a(x1((𝑟−𝛾−𝛾1𝑚),(1)−,𝛾󵄩󵄩󵄩󵄩𝑚𝑟)1,‖󵄩󵄩󵄩󵄩𝑟1‖∞))))𝑇
∞
1
𝑑
𝑑𝑡∫Ω𝑊(𝑥,𝑡)𝑑𝑥 − 󵄩󵄩󵄩󵄩𝑢01+𝛾V01󵄩󵄩󵄩󵄩𝐿1(Ω).
=∫ ((𝑟𝛾−1)𝑢1+(1−𝛾𝑚)V1+𝑢1𝑟1)𝑑𝑥 This follows from the 𝐿1(Ω) estimate on 𝑢1, via (17).
≤mΩax((𝑟𝛾−1),(1−𝛾𝑚),󵄩󵄩󵄩󵄩󵄩𝑟1󵄩󵄩󵄩󵄩󵄩 ) ThThuusswweehsaevee𝐿w1e(Ωc𝑇an)counntirfoorlmoflythebopuronbdleamllatoifcnthoenlrineaecatriiotyn.
∞
termsin𝐿1(Ω𝑇)andbybootstrapargumentthenboundthem
⋅∫ (𝑢1+V1)𝑑𝑥≤𝐶 (16) uniformly in 𝐿𝑝 to deduce global existence [29]. So we can
Ω statethefollowingtheorem.
⋅max((𝑟𝛾−1),(1−𝛾𝑚),󵄩󵄩󵄩󵄩󵄩𝑟1󵄩󵄩󵄩󵄩󵄩∞) Theorem2. Considerthespatiallyexplicitthree-speciesmodel
withcannibalism(8)–(10),wherethespatialdimensionofthe
⋅∫ (𝑢1+𝛾V1)𝑑𝑥=𝐶 physicaldomainis𝑛 = 1.Solutionstothismodelareclassical,
Ω thatis,(𝑢1,V1,𝑟1)∈𝐶1(0,𝑇;𝐶2(Ω)),andexistgloballyintime.
⋅max((𝑟𝛾−1),(1−𝛾𝑚),󵄩󵄩󵄩󵄩󵄩𝑟1󵄩󵄩󵄩󵄩󵄩∞)∫ 𝑊(𝑥,𝑡)𝑑𝑥. AsimplecorollarytoTheorem2canbestatedconcerning
Ω
themodelwithoutcannibalism.
Now the uniform in time 𝐿1 bound on 𝑊 and thus on Corollary3. Considerthespatiallyexplicitthree-speciesmod-
𝑢1,V1 easily follows via applying Gronwall’s lemma on (16), elwithoutcannibalism(5).Solutionstothismodelareclassical,
onanytimeinterval[0,𝑇],toyield thatis,(𝑢1,V1,𝑟1)∈𝐶1(0,𝑇;𝐶2(Ω)),andexistgloballyintime.
󵄩󵄩󵄩󵄩󵄩𝑢1󵄩󵄩󵄩󵄩󵄩𝐿1(Ω) ≤󵄩󵄩󵄩󵄩󵄩𝑢01+𝛾V01󵄩󵄩󵄩󵄩󵄩𝐿1(Ω)𝑒(𝐶max((𝑟𝛾−1),(1−𝛾𝑚),‖𝑟1‖∞))𝑇, (17) 2C.a3n.nNiboanliesxmis.teWnceesotaftTeuarnindgpIrnosvteabthileityfoilnlowthiengMtohdeeolrwemith.out
󵄩󵄩󵄩󵄩󵄩V1󵄩󵄩󵄩󵄩󵄩𝐿1(Ω) Theorem4. Considerthethree-speciesspatiallyexplicitmodel
withoutcannibalism(5).Thespatiallyhomogenoussteadystate
≤󵄩󵄩󵄩󵄩󵄩𝑢01+𝛾V01󵄩󵄩󵄩󵄩󵄩𝐿1(Ω) 𝛾1𝑒(𝐶max((𝑟𝛾−1),(1−𝛾𝑚),‖𝑟1‖∞))𝑇 (18) f(o𝑢r∗1a,Vn∗1y,p𝑟∗a1r)aomftehtiesrmreogdimeleca.nnotbedrivenunstablebydiffusion,
Proof. Weshowthisdirectlybycheckingthenecessarycon-
withthebounddependingononlythefinaltime𝑡 = 𝑇.This ditionsforTuringinstabilityfrom[19].NoteforTuringinsta-
followsastheequationfor𝑟1isamenabletomakeessentially bility to occur we have certain necessary conditions on the
any estimate on 𝑟1, by comparing it to the diffusive logistic coefficients of the characteristic polynomial, gotten by lin-
equation. We now deal with the problematic term which is earizing (5), around the spatially homogenous equilibrium
Complexity 5
Table1:Coefficientsofcubicfunctions𝐴1(𝑘2)𝐴2(𝑘2)−𝐴3(𝑘2)usedindeterminingconditionsforTuringinstability.
Coefficient 𝐴 (𝑘2) [𝐴 𝐴 −𝐴 ](𝑘2)
3 1 2 3
𝐽 𝐽 𝐽 −(𝐽 +𝐽 +𝐽 )(𝐽 𝐽 −𝐽 𝐽 +𝐽 𝐽 +𝐽 𝐽 −
ℎ 𝐽 𝐽 𝐽 +𝐽 𝐽 𝐽 −𝐽 𝐽 𝐽 11 22 33 11 22 33 11 22 12 21 11 33 22 33
11 32 23 12 21 33 11 22 33 𝐽 𝐽 ) −𝐽 𝐽 𝐽 −𝐽 𝐽 𝐽
23 32 11 23 32 12 21 33
𝑑 (2𝐽 𝐽 +2𝐽 𝐽 +2𝐽 𝐽 +𝐽 𝐽 +𝐽 𝐽 −𝐽 𝐽 )+
1 11 33 11 22 22 33 33 33 22 22 12 21
𝑑 𝑑 (𝐽 𝐽 −𝐽 𝐽 )+𝑑 𝐽 𝐽 +𝑑 (𝐽 𝐽 −𝐽 𝐽 ) 𝑑 (2𝐽 𝐽 +2𝐽 𝐽 +2𝐽 𝐽 +𝐽 𝐽 +𝐽 𝐽 −𝐽 𝐽 −
1 22 33 32 23 2 11 33 3 11 22 12 21 2 22 11 22 33 33 11 11 11 33 33 21 12
𝐽 𝐽 )+𝑑 (2𝐽 𝐽 +2𝐽 𝐽 +2𝐽 𝐽 +𝐽 𝐽 +𝐽 𝐽 −𝐽 𝐽 )
23 32 3 22 11 22 33 33 11 11 11 22 22 23 32
−𝐽 (𝑑 +𝑑 )(2𝑑 +𝑑 +𝑑 )−𝐽 (𝑑 +𝑑 )(𝑑 +2𝑑 +𝑑 )−
𝑐 −𝑑 𝑑 𝐽 −𝑑 𝑑 𝐽 −𝑑 𝑑 𝐽 11 2 3 1 2 3 22 1 3 1 2 3
1 2 33 1 3 22 2 3 11 𝐽 (𝑑 +𝑑 )(𝑑 +𝑑 +2𝑑 )
33 1 2 1 2 3
𝑏 𝑑 𝑑 𝑑 (𝑑 +𝑑 )(𝑑 𝑑 +𝑑 𝑑 +𝑑 𝑑 +𝑑 𝑑 )
1 2 3 2 3 1 1 2 3 1 2 1 3
(𝑢∗1,V∗1,𝑟∗1).SeetheAppendixandTheoremC.1therein.Note Wenextcheckthecoefficient𝑑of𝐴1𝐴2−𝐴3,
thattheJacobianmatrixofthereactiontermsofthemodel
withoutcannibalismatthespatiallyhomogenousequilibrium 2𝑟
(𝑢∗1,V∗1,𝑟∗1)isgivenby 𝑑=𝑑1(𝑚2𝑢(𝑚−𝑟)+𝑟+2𝑢(𝑚−𝑟)
1 2
𝑟 𝑢𝑟 +( 𝑢(𝑚−𝑟)) +(𝑚)2+𝑢1𝑟1)
− 1 (−𝑢+ +𝑡 ) 𝑚 ∗ ∗
[ 𝑚 𝑚 1 ]
[ ]
𝐽=[[[ 𝑟 𝑟 −𝑚 1 0 ]]]. (21) +𝑑2(𝑚2𝑟2𝑢(𝑚−𝑟)+𝑟+2𝑢(𝑚−𝑟) (25)
−1+ 0 − 𝑢(𝑚−𝑟)
[ 𝑚 𝑚 ]
1 2 𝑟 2 2𝑟
+( 𝑢(𝑚−𝑟)) +( ) )+𝑑 ( 𝑢(𝑚−𝑟)
𝑚 𝑚 3 𝑚2
WenowcheckthenecessaryconditionsforTuringinsta- 𝑟 2
bility from Table1, where we take the 𝐽𝑖𝑖 entries, from the +2𝑟+2𝑢(𝑚−𝑟)+(𝑚)2+(𝑚) +𝑢∗1𝑟∗1)>0.
abovematrix.
Wefirstcheckthecoefficient𝑐of𝐴3,
Sinceallofthesecoefficientsarepositive,theresultfollowsas
anapplicationofTheoremC.1fromtheAppendixandsimple
𝑟 contraposition.Thisprovesthetheorem.
𝑐=−[−𝑑1𝑑2𝑢𝑟∗1 −𝑑1𝑑3(𝑚)−𝑑2𝑑3(𝑚)]>0. (22)
Remark 5. Note that Theorem4 shows that there does not
existTuringinstabilityinthemodelwithoutcannibalism(5),
Wenextcheckthecoefficient𝑐of𝐴1𝐴2−𝐴3, foranyparameterregime.
2.4. ExistenceofTuringInstabilitywithCannibalism. Given
𝑟
𝑐=[ (𝑑 +𝑑 )(2𝑑 +𝑑 +𝑑 ) that there is no Turing instability in the model without
𝑚 2 3 1 2 3
cannibalism (5), we now investigate the effect of including
+𝑚(𝑑1+𝑑3)(𝑑1+2𝑑2+𝑑3) (23) cannibalism,onTuringinstabilityinthemodel.Tothisend
we investigate the spatially explicit model (8)–(10). We first
+𝑢𝑟1(𝑑 +𝑑 )(𝑑 +𝑑 +2𝑑 )]>0. introduce several structure theorems from linear algebra in
∗ 2 1 1 2 3 theliterature[31–33].
Definition 6 (strongly stable). A real matrix 𝐴 is said to be
Wenextcheckthecoefficient𝑑of𝐴3, stronglystableif𝐴−𝐷isstableforalldiagonalmatrices𝐷.
Definition7. Let𝑃denotetheclassofmatriceswhosesigned
𝑑=𝑑 (𝑚(𝑢𝑟1))+𝑑 ((𝑟1 −1)(−𝑢𝑟1)+𝑢1𝑟1) principal minors are all positive and 𝑃0+ the class whose
1 ∗ 2 ∗ ∗ ∗ ∗ signedprincipalminorsareallnonnegative,withoneofeach
+𝑑 (−𝑚(𝑟1 −1)−𝑟) orderpositive.
3 ∗
(24) Theorem8. A3×3realmatrix𝐴isstronglystableifandonly
=𝑑1(𝑚(𝑢𝑟∗1))+𝑑2((𝑟∗1 −1)(−𝑢𝑟∗1)+𝑢∗1𝑟∗1) if𝐴∈𝑃+,and𝐴isstable.
0
>0.
Theorem9. 𝐴isstronglystableifandonlyif𝐴iss-stable.
6 Complexity
Theorem10. Ifthekineticsystem by(28).Consideraparametersetforwhich𝐽isstable,whilst
thecannibalismparameter𝑠satisfies
𝑑𝑢
=𝑓(𝑢,V,𝑟) 𝑟1 −1+𝛾𝑠V1 >0,
𝑑𝑡 ∗ ∗
𝑑V 𝑟∗1 −1+𝛾𝑠V∗1 −𝑚−𝑠𝑢∗1 >0, (29)
=𝑔(𝑢,V,𝑟) (26)
𝑑𝑡
𝑜𝑟 𝑟1 −1+𝛾𝑠V1 +𝑡 −2𝑢𝑟1 −𝑢1 >0.
∗ ∗ 1 ∗ ∗
𝑑𝑟
𝑑𝑡 =ℎ(𝑢,V,𝑟) Thenthereexistdiffusioncoefficients(𝑑1,𝑑2,𝑑3),suchthat
the spatially homogenous steady state (𝑢1,V1,𝑟1) of (8)–(10)
∗ ∗ ∗
canbedrivenunstableduetoTuringinstability.
is s-stable, then no Turing bifurcation is possible from the
uniformsteadystatesolution.
Theproofreliesonextractinganunstablesubmatrixfrom
(28). This is equivalent to some signed principal minor not
Theorem11. Ifthekineticsystem belongingto𝑃+andhence𝐽givenby(28)notbeingstrongly
0
stable.
𝑑𝑢
=𝑓(𝑢,V,𝑟)
Proof. Noteifwelookattheminorgottenbyremovingrow3
𝑑𝑡
andcolumn3,weobtainthefollowingsubmatrix:
𝑑V
𝑑𝑡 =𝑔(𝑢,V,𝑟) (27) [𝑟∗1 −1+𝛾𝑠V∗1 1+𝛾𝑠𝑢∗1 ]. (30)
𝑑𝑟 𝑟−𝑠V1 −𝑚−𝑠𝑢1
=ℎ(𝑢,V,𝑟) ∗ ∗
𝑑𝑡
Thisisclearlyseentobeunstableif𝑟1−1+𝛾𝑠V1−𝑚−𝑠𝑢1 >
∗ ∗ ∗
0.Also,ifwelookattheminorgottenbyremovingrow2and
contains an unstable subsystem, then Turing bifurcation is
column2,weobtainthefollowingsubmatrix:
possiblefromtheuniformsteadystatesolution.
𝑟1 −1+𝛾𝑠V1 𝑢1
Remark 12. We will now use the above theory to show that [ ∗ ∗ ∗ ]. (31)
−𝑟1 𝑡 −2𝑢𝑟1 −𝑢1
cannibalism can in fact induce Turing instability. We will ∗ 1 ∗ ∗
first show that the Jacobian matrix of the reaction terms
of the model with cannibalism (8)–(10), given by (A.2), is Thisisclearlyseentobeunstableif𝑟∗1 −1+𝛾𝑠V∗1 +𝑡1 −
not strongly stable, for certain values of the cannibalism 2𝑢𝑟∗1 −𝑢∗1 >0.
parameter 𝑠. This will yield the result as a simple corollary Thus if either of these conditions are met, 𝐽 given by
toTheorems9and10.Notesimilarideaswereusedin[32], (28) contains unstable submatrices. That is, certain signed
toshowthatacertainSEIRtypeepidemicsmodeldoesnot principalminorsarenotin𝑃0+.ThusviaTheorem8 𝐽isnot
possessTuringinstability,aslongasthedisease-causeddeath stronglystable,whichimpliesviaTheorem9that𝐽isnot𝑠-
rate parameter is zero, but can have Turing patterns for stable.Thisthenimpliesthatthereexistdiffusioncoefficients
positivedisease-causeddeathrate. (𝑑1,𝑑2,𝑑3)forwhichTuringinstabilityisacertainty,viaan
Recall the Jacobian matrix of the reaction terms of the application of Theorems 10 and 11. This proves the theo-
modelwithcannibalismisgivenby rem.
𝑟1 −1+𝛾𝑠V1 1+𝛾𝑠𝑢1 𝑢1 3.NumericalAnalysisofTuringInstability
∗ ∗ ∗ ∗
[ ]
𝐽=[[ 𝑟−𝑠V∗1 −𝑚−𝑠𝑢∗1 0 ]] 3.1. The Effect of Cannibalism on Turing Instability. We
[ −𝑟∗1 0 𝑡1−2𝑢𝑟∗1 −𝑢∗1] nThoweornemum14ericaarellymdeetm, ownestrcaatne tfihnadt idfiffthuesiocnoncdoiteioffincsienvitas
(28)
free + + 𝑑1,𝑑2,𝑑3 so as to induce Turing instability in (8)–(10). We
=[[ + − 0]]. c1h0o−8o,s𝑑e3th=e0f.o9l,lo𝑟w=in0g.0p1,a𝑚ram=e0te.1r,s𝛾et=,𝑑01.01=,𝑠10=−011.0,1𝑑,2𝑢==10..713×,
[ − 0 −] 𝑡1 = 0.21.Forthisparametersetweseethat𝑢∗1 = 0.058844,
V1 =0.005850,𝑟1 =1.162736.Thus𝑟1−1+𝛾𝑠V1 =0.1627>0
∗ ∗ ∗ ∗
Remark13. Noticewewritethesignof𝐽11asfree,becausedue and𝑟∗1−1+𝛾𝑠V∗1−𝑚−𝑠𝑢∗1 =0.0614>0.Thustheconditions
tothecannibalismparameter𝑠thesignof𝐽11 =𝑟∗1−1+𝛾𝑠V∗1 ofTheorem14aremet,andwecanexpectTuringinstability.
Thisisdemonstratedintheplotsnext.
couldnowbepositiveornegative.
Theorem14. Considerthethree-speciesmodelwithcannibal- 3.2. Sensitivity Analysis: Turing Space and Dispersion Plots.
ism(8)–(10)anditsJacobianmatrix𝐽ofreactionterms,given SincecannibalismisthesolefactorinbringingaboutTuring
Complexity 7
instability in (8)–(10), we decide to further investigate this show numerically that a life boat mechanism for the prey
result. We perform next a sensitivity analysis, also referred exists.Nextweconjecturethatalifeboatmechanismforthe
to as a Turing space analysis, considering the effect on predatoralsoexists.
Turing instability with two parameters at a time. Since
cannibalism is the central theme under investigation, we
5.1.CannibalismasaLifeBoatforthePrey. Noticethat,inthe
check the sensitivity of the cannibalism parameter against specialcasethat𝑟 = 𝑚,apreyfreeequilibrium(𝑢1,V1,0)is
∗ ∗
every other parameter. In our simulations we refer to the
possibleinboththenocannibalismmodel(4)andthemodel
current literature [8, 34] to keep the parameters close to
withcannibalism(3).WeshowinFigure8numericallythata
reality.InalloftheplotsinFigures2and3theblueareais
lifeboatmechanismforthepreyalsoexists.Thisisnotproved
theregionwhereTuringinstabilityexists,andthewhitearea
rigorously.Arigorousinvestigationwillhavetoemploythe
istheregionwhereitdoesnot.Theresultsarepresentednext.
normalformtheory,fornonlinearstabilityanalysis,asthere
cannWibeanliesmxtpdaisrpamlayettehre𝑠.dispersion relation, as we vary the aretwozeroeigenvaluesatequilibrium(𝑢∗1,V∗1,𝑟∗1),andthus
this equilibrium point is nonhyperbolic. However we can
WeinterpretanddiscusstheaboveresultsintheDiscus-
makethefollowingconjecture.
sionandConclusion(Section6).
Conjecture 16. Consider model system (3). Under certain
4.HopfBifurcation parametricrestrictions,ifthereisnocannibalism,thatis,𝑠=0,
asmallperturbationtopreyfreestate(𝑢1,V1,0)willstillyield
∗ ∗
4.1.HopfBifurcationinTermsoftheCannibalismParameter. extinctionoftheprey.However,if𝑠 > 0,thepreypopulation
We will now investigate the correct criterion for Hopf willrecover,toreachanontrivial(𝑢1,V1,𝑟1)state.
bifurcation in (3), in terms of the cannibalism parameter𝑠. ∗ ∗ ∗
Thecriterionprovidedin[26]isincorrect.SeeFigures6and Remark17. Whatwenoticeisthat,undercertainparametric
7.InordertoinvestigatetheHopfbifurcationofmodelsystem restrictions,thepreyfreestateisgloballystable,butintroduc-
(3), we follow the method developed by Liu [35]. It can be ingcannibalismresultsincoexistenceofallspecies,andorbits
seenthattheHopfbifurcationat𝑠 = 𝑠∗ canoccurprovided convergetoanontrivialinteriorequilibrium(𝑢1,V1,𝑟1).This
that𝐴1(𝑠∗),𝐴3(𝑠∗),andΦ(𝑠∗) = 𝐴1(𝑠∗)𝐴2(𝑠∗)−𝐴3(𝑠∗)are showsthatalifeboatmechanismexistsforthe∗pr∗ey∗aswell.
smoothfunctionsof𝑠inanopenintervalof𝑠∈Rsuchthat AlsoseeFigure8.
(1)𝐴1(𝑠∗)>0,𝐴3(𝑠∗)>0,andΦ(𝑠∗)=𝐴1(𝑠∗)𝐴2(𝑠∗)−
𝐴3(𝑠∗)=0; 5.2. CannibalismasaLifeBoatforthePredator
(2)(𝑑Φ(𝑠)/𝑑𝑠)|𝑠=𝑠∗ ≠0. Conjecture 18. Consider model system (3). Under certain
parametric restrictions there exist cannibalism rates 𝛾,𝑠 such
Thuswecanstatethefollowingtheorem. thatifthereisnocannibalism,thatis,𝑠 = 0,asmallpertur-
bation to the predators in the (0,0,𝑡1/𝑢) state will still yield
Theorem15. Under conditions (B.2), (B.3), (B.5), (B.7), and extinction of the predator. However, if 𝑠 > 0, the predator
(B.9) there is a Hopf bifurcation of the positive equilibrium populationwillrecover,toreachanontrivial(𝑢1,V1,𝑟1)state.
point(𝑢∗1,V∗1,𝑟∗1)at𝑠=𝑠∗ofmodelsystem(3),where𝐴1(𝑠∗)> ∗ ∗ ∗
0,𝐴3(𝑠∗)>0,and𝐴1(𝑠∗)𝐴2(𝑠∗)−𝐴3(𝑠∗)=0. Remark 19. The Jacobian matrix for the no cannibalism
model(4)at(0,0,𝑡1/𝑢)[26]isgivenby
Thecoefficients𝐴1,𝐴2,and𝐴3anddetailsofconditions
(B.2)–(B.9)areprovidedintheAppendix. 𝑡
1 −1 1 0
Wenownumericallyillustrateacorrectbifurcationpoint [𝑢 ]
ftohratHΦo(p𝑠f∗i)ns=ta𝐴bi1li(t𝑠y∗.)W𝐴e2(b𝑠e∗g)in−b𝐴y3t(r𝑠y∗i)ng=to𝑎(fi𝑠n∗d)2a+n𝑠𝑏𝑠=∗𝑠+∗,𝑐su=ch0 𝐽=[[[[ 𝑟𝑡 −𝑚 0 ]]]]. (32)
whilst(𝑑Φ(𝑠)/𝑑𝑠)|𝑠=𝑠∗ =2𝑎𝑠∗+𝑏≠0. [ −𝑢1 0 −𝑡1]
The form of Φ is complicated and depends on the
correctlycalculated𝐴1,𝐴2,𝐴3;seetheAppendix.Wechoose Satisfactionofthefollowingconditions,
the same parameters from [26], 𝑟 = 0.5, 𝑚 = 1, 𝑢 = 0.1,
𝛾 = 1.5,𝑡1 = 1.3,butthecorrectformofthecoefficientsas 𝑢>𝑡 ,
wehaveworkedoutintheAppendixandplotΦ(𝑠). 1
𝑡 (33)
𝑟<𝑚(1− 1),
5.CannibalismasaLifeBoatMechanism 𝑢
Cannibalismoftengivesrisetocounterintuitivephenomenon are sufficient to satisfy the Ruth-Hurwitz criteria for local
inpopulationdynamics.Alifeboatmechanismisafeatureof asymptoticstabilityof(0,0,𝑡1/𝑢).Thusifthereexistcannibal-
cannibalism,whereanoncannibalisticpopulationisdoomed ismrates𝛾,𝑠suchthat(𝑢1,V1,𝑟1),thenontrivialequilibrium
∗ ∗ ∗
to extinction but can persist under the act of cannibalism. of the model with cannibalism (3), is globally stable and
Wenowexploresuchfeaturesinourmodelsystems.Wefirst (33) also holds, then the populations will rebound from
8 Complexity
u population v population ×10−3
1000 0.16 1000
10
800 0.14 800
9
600 0.12 600
t t
8
400 0.1 400
7
200 0.08 200
6
0 0.06 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
x x
Turing system at t = 990
0.096
u 0.094
0.092
0.09
0 0.5 1 1.5 2 2.5 3
×10−3 x
9.6
9.4
v 9.2
9
8.8
0 0.5 1 1.5 2 2.5 3
x
0.90093
0.900925
r 0.90092
0.900915
0.90091
0 0.5 1 1.5 2 2.5 3
x
Figure1:HerewedemonstratethatcannibalismcanindeedinduceTuringinstabilityin(8)–(10).WeseeaspatialTuringpatterninthejuvenile
speciesintheleftpanel.Therightpanelshowsthespatialprofileofallthreespecies,afterthesimulationisrunmuchlonger(𝑡=1000).Wesee
Turingpersistsinboththeadultandjuvenilepredators,butthepreyhasveryweakTuring.Thevaluesoftheparametershereare𝑑1=10−11,
𝑑2 = 1.7×10−8,𝑑3 = 0.9,𝑟 = 0.01,𝑚 = 0.1,𝛾 = 0.01,𝑠 = 0.01,𝑢 = 0.13,𝑡1 = 0.21.Forthisparametersetweseethat𝑢∗1 = 0.058844,
V1 =0.005850,𝑟1 =1.162736.
∗ ∗
(𝜖,𝜖,𝑡1/𝑢 + 𝜖) and be driven up to (𝑢∗1,V∗1,𝑟∗1). This tells in spatially explicit three-species age structured predator-
us that a small amount of adult and predator populations prey systems. That is, if one considers the spatially explicit
withoutcannibalismcannotsustainthemselvesundercertain version of the three-species age structured ODE model
parametric restrictions, as (0,0,𝑡/𝑢) is locally stable. Thus originallyproposedin[18],thatis,(5),thenwithoutcanni-
(𝜖,𝜖,𝑡/𝑢 + 𝜖) will be pulled back to (0,0,𝑡/𝑢). However, balism there can be no Turing instability in any parameter
the act of cannibalism, under these parametric restrictions regime. This is shown via Theorem4. However, introducing
(assuming one can formally prove global stability of the cannibalismbringsaboutTuringinstability,in(8)–(10),seen
equilibrium), can cause these small amounts of adult and viaTheorem14.AlsonotethatalthoughtheHopfbifurcation
juvenilepredatorpopulationstorise,andso(𝜖,𝜖,𝑡/𝑢+𝜖)can analysis on the model originally proposed by Magnu´sson
bedrivenupto(𝑢1,V1,𝑟1). has been done in [26], the results therein are incorrect. We
∗ ∗ ∗
derive the correct conditions for the Hopf bifurcation via
6.DiscussionandConclusion Theorem15.Inthecaseofthepreyspecies,weseenumerical
evidenceofalifeboatviaFigure1.
Inthecurrentmanuscript,tothebestofourknowledge,we Fromanecologicalpointofview,Theorem14tellsusthat,
reportthefirstcannibalisminducedTuringinstabilityresult, forthisspecificmodel,cannibalisminducesTuringinstability.
Complexity 9
Turing space Turing space
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
s 0.5 s 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.2 0.4 0.6 0.8 1
r u
Turing Turing
Figure2:Thevaluesoftheparametershereare𝑑1 = 10−11,𝑑2 = (a)
1.7×10−8,𝑑3 =0.9,𝛾=0.01,𝑚=0.1,𝑢=0.13,𝑡1 =0.21.Welook Turing space
attheTuringspace(sensitivity)generatedbyvarying𝑠and𝑟,each
intherange[0,1].
0.9
0.8
0.7
Thus in order to investigate the Turing instability result 0.6
further,weperformasensitivityanalysis,wherewelookatthe
s 0.5
Turing space generated by the cannibalism parameter, with
everyotherparameterinthesystem. 0.4
We look at how the Turing space changes as we vary 𝑠
0.3
the cannibalism parameter and 𝛾 the rate of intake by the
0.2
cannibalisticadult.WeseethatTuringoccurseverywherein
𝑠 ∈ [0,1],𝛾 ∈ [0,1], so we do not show a plot of this 0.1
case. This tells us that the interplay between the amount of
0
cannibalismandintakerateofthecannibalalwaysproduces 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Turing instability, in this range (to keep things realistic we t1
varytheparametersonlyin[0,1]).Thatis,ifwekeep𝑠fixed
andincreaseordecrease𝛾,theTuringinstabilitypersists. Turing
In Figure2 we look at how the Turing space changes (b)
with respect to the interactions between the cannibalistic
adult predator and the juvenile predator. Here we vary 𝑠 Figure3:Thevaluesoftheparametersin(a)are𝑑1 = 10−11,𝑑2 =
the cannibalism parameter and 𝑟 the growth rate of the 1.7×10−8,𝑑3=0.9,𝛾=0.01,𝑚=0.1,𝑟=0.01,𝑢=0.13.Welookat
theTuringspace(sensitivity)generatedbyvarying𝑠and𝑢,eachin
𝑟ocseacenc∈unthri(bsa0ati,lf0Tisw.ut0ier3ci]inan.gdcFurooelcrtacsltueohrwa𝑟stufvaopadrldtuaosellsat𝑠obooft∈uh𝑠te,[00sj,.ua21yv,]eb,𝑠nuiiftl∈e𝑟npio(s0ot,pvi0feu.rwl2ya]et,iloofTuwnur.,rthWisnaegyer t𝑑eloha2oec=khraai1nnt.g7ttheh×ee[01Tr,0au1n−r]8gi.n,eTh𝑑g[30sep,=1va]ac0.le.u9(e,ss𝛾eon=fsitt0hiv.e0it1py,a)𝑟rga=emne0et.r1ear,tse𝑢din=b(y0b.v)1a3arr,ye𝑚in𝑑g1=𝑠=0a.1n10.d−W𝑡111e,,
increase𝑟.Furthermore,weobservethatweloseTuring,for
moderatevaluesof𝑟,sayabout0.2ormore,ifweincrease𝑠
past0.5.Thistellsusthattheinterplaybetweentheamount
of cannibalism and growth rate of the cannibalistic adult ofthejuvenileessentiallyalwaysactstogiveTuringformost
essentiallyalwaysactstoremoveTuringforvaluesof𝑠>0.5. valuesintheconsideredrange,soagainwedonotplotthis
HoweverTuringessentiallypersistsforall𝑠,ifsay𝑟∈(0,0.03] case.
thatisverylow.Nowwevary𝑠thecannibalismparameterand InFigure3welookathowtheTuringspacechangeswith
𝑚thelossinthejuvenilepopulation,duetonaturaldeathas respect to the interactions between the cannibalistic adult
wellastransitioningintotheadultclass.WeseethatTuring predator and the prey species. In Figure3(a) we vary 𝑠 the
occursalmosteverywhereexceptforverylowvaluesof𝑠,𝑚, cannibalism parameter and 𝑢 the growth rate of the prey.
that is, 𝑠 ∈ (0,0.3] and 𝑚 ∈ (0,0.1]. This tells us that the This particular interaction is probably the most interesting
interplay between the amount of cannibalism and loss rate in our opinion. As a first analysis we noticed that Turing
10 Complexity
occurs for all 𝑠 ∈ [0,1], for a narrow band of 𝑢 values, say ×106
Dispersion plot
𝑢 ∈ [0.15,0.22].However,uponfurtherdiscretizationofthe 2) 1
k
parameter space, we observed that there are disconnected A(3 0.8
patches where Turing occurs, for roughly 𝑠 ∈ [0.65,0.75] −
and 𝑢 ∈ [0.55,0.6], and then again for certain values of 2k) 0.6
𝑢 ∈ [0.7,1]. This property is not seen in any of the other A(2 0.4
)
simulations, despite various further discretization in all of 2
the parameters. In Figure3(b) we vary 𝑠 the cannibalism A(k1 0.2
parameter and 𝑡1 the competition coefficient for the prey. on 0
Note the competition mediated effects of cannibalism have elati −0.2
beenthefocusofanumberofrecentstudies[8,36].Wesee n r
vtahlalalu𝑡t1eTsvu,arsliaunyegs𝑡,1osc∈acyu[0𝑡r1s.1f1≥o,r00a..21ll],.𝑠fAo∈rlsao[0nT,au1r]rr,ionfwogrobacacnnudarrsoroffow𝑠rvebasalsnuedensto,iaf𝑠l𝑡l∈y1 of dispersio −−−000...864
[0.95,1].Thistellsusthattheinterplaybetweentheamount ff.
ofcannibalismandcompetitionamongstpreyessentiallyacts Coe −10 1000 2000 3000 4000 5000 6000 7000
togiveTuringforrestrictedvaluesofcompetitionparameter Wave number (k)
𝑡1.
s=0.01 s=0.31
Inthesameveinastheabove,wealsoconductadisper-
s=0.21 s=0.41
sionrelationanalysis.Hereweplotthecoefficient𝐴1𝐴2−𝐴3
forvariousvaluesofthewavenumber𝑘.RecallthatTuring Figure 4: Here we plot the dispersion relation of the coefficient
instabilityoccurs,ifthiscoefficientdipsbelowthe𝑘axis.We 𝐴1𝐴2 −𝐴3 versus the wave numbers 𝑘, for various values of the
constructasequenceof4graphs,byvaryingthecannibalism parameter𝑠.RecallthereisTuringinstabilitywhenatleastsomepart
parameter𝑠.Whatwenoticeisthatas𝑠isincreased(forother ofthecoefficient𝐴1𝐴2−𝐴3liesbelowthe𝑘axis.
parametricvaluesfixedasinFigure1),weseethecoefficient
moves upwards and then becomes positive past 𝑠 = 0.31.
Thistellsusthattheeffectofincreasingcannibalismhereis
70
toremovetheTuringinstability.
Inlightofourresultswecansummariseafewimportant 60
findings.
50
(1)In the spatially explicit form of the classical model
consideredbyMagnu´ssonandthenrescaledbyTang, 40
cannibalismcancauseTuringinstability.
30
(2)TheTuringinstabilityinducedbycannibalismseems
tobemostsensitivetothepreygrowthrateparameter 20
𝑢and𝑡1thecompetitioncoefficientfortheprey.Italso
seemstobeleastsensitiveto𝛾therateofintakebythe 10
cannibalisticadultand𝑚thelossrateinthejuvenile X: 0.748
0 Y: −3.951
population.Thistellsusthattheadultpredator-prey
relation is critical, despite the adult being able to −10
cannibalisethejuvenilepredator. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
(3)Magnu´sson and Tang’s findings are that, in the Figure5:HereweseeaplotofΦ(𝑠)versus𝑠fortheotherparameters
ODEmodel,increasingthecannibalismparameter𝑠 asmentionedearlierfrom[26],𝑟=0.5,𝑚=1,𝑢=0.1,𝛾=1.5,𝑡1=
destabilisesthesystem.Giventhatwechoosesimilar 1.3.Weclearlyseethat(𝑑Φ(𝑠)/𝑑𝑠)|𝑠=1.001 >0,and(𝑑Φ(𝑠)/𝑑𝑠)|𝑠=0.75 =
parameters, our result suggests that, in the spatially 0.Therootoccursatapproximately𝑠=1.001.
explicitcase,increasingthecannibalismparameter𝑠
removestheTuringinstabilityandhencestabilisesthe
system; see Figure4. This makes the ODE and PDE
modelsquitedifferentdynamically.
investigation is considering cannibalism among juveniles.
NotetheTuringpatternswecurrentlyobtainarepurely We know for a fact that this does take place in certain
spatial.Itwouldbeinterestingtoinvestigateifspatiotemporal amphibian species [36]. One might also consider the effect
patternsdrivenbyTuring-Hopftypeinstabilitiesexist.Ifone of cannibalism occurring simultaneously amongst adults
can prove otherwise, it raises further interesting questions and juveniles. Lastly, akin to [17] one can investigate the
into the nature of cannibalism and its ecological conse- occurrenceofcannibalismintheprey.Furtherinvestigation
quences,suchaswhywouldconspecificconsumptionhinder ofthedisconnectedTuringspaceseeninFigure3shouldalso
temporal changes in the population, at fixed locations in beafuturedirection.Theissueofwhethercannibalismacts
space? The other interesting question for possible future as a life boat mechanism for the predator, in (3), remains