System Mathematics Front Side

8/12/2019 System Mathematics Front Side

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StabiiityAnal3gsis

2 , 1 I N T R O D U C T I O NOneof the mclst mportant'eaturesf anyecologicalyslem,whetherpopulationrcomrnunity,s its inherentstabilityor lackingof it it is also b,ecausef itsoverwhelmIngmplicationsn both theoreticalnd erppliedtudies f naturalandmanipulatedy'stems,ne of the ecologicalarametersroundwhich herehasbeenthe mostdiscur;sionndcontroversyboutwhichmrost asbeenwritten.Ecologicaltabilitys variously efinedbut may be summarizeds the dynanricequilibriumf population,onnmunityr ecosystemize ncl tructure.hestabilityanbedefinedn he ollowing anner.(1) "Theabilityof both populationsnd communitieso withstand nvironmentalperturbationso accommodatehange" MacArthur,(2) Oriens has dentified number f different lements;hichmaybe recognisr:dwithin heoverall onceptf stability.Thestabilit 'ys not a simple haracterut n facta nrultiplicityf distinct ttributers.Muchof the confusionngenderedn the past n,discuss;ionsbouts;tabil i tyn theliterature as come rom failure o conrpareikewith likr:, rom confusion f thesedifferentormsof stability.The hreebasic ypesof statrilitytselfmay beconrsideredsconstancy, esilienceand ineftia. By constancy, e refer o a lackof change n somepararneterf asystem, uchas number f :;pecies,oxonomicompcrsiticln,ife from structure f acommunity,izeof a populationr feature f thephysicalnvironment.{esilienceaybe considereds the abilityof a systerno recorrerandcontinueunctioning fterdisturbancevernhought mayhave hangedts brm.Therefore,community aybe describeds; esilientf, duringor afterdisturbernce,,ven hough G; onstancyfspecies tructuremayhavechangedmarkeC 1,,t is able : cc;':i;luec operaie s aviablesystem.Finally,nertia s the abilityof a systernrt withstandlr resist uchperturbationsrr he irstplace. ttributesf such ;tabilityunctionsreelasticityrrdamplitude. herefore, lasticitys a measure f the speedwith which he systemreturnso its ormer tate ollowing perturbation.mplitudef a systemlefine:;fre

I Stabilin'Anirlysis l - 1l-J r-J


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incs-lrablv'i t ism o r

wrng

ntraci lu

,,i1 1:lon the otherhand, f (x*,yx) is a critica| ointof the S'y:;tem'hen he'valuedfuncticn

x ( t ) = 1 + nY ( t ) = Y *satisFyingys;terTl1)andarecalledquilibrium olutiorrsfthesystem'

. r\e) The: ra.iectoryf the equilibriumolution n equatiori 3) consists f th e\\ (x*,Y*).

2.2 ALMOST LINEAR SYSTEMCcnsiderhe: orn-linearystemsof the orm

dxi = u * * + b ' Y + f ( x ' ' Y )a = u r * + b 2 y + g ( x , y )d t ' ' '

In matrixorm .1) (2)canbewritten so x) (u ' u ' ) ( * ) ( r ( * ,Y) )- l l = l l l + l IdtIv J [u, bz \.y, [s(x, ) iwhere, 1, 1, t|,bz re onstants'Then, y gnoringhenon-l inearerms(x,y)ancl (x'y)linear ysternlso f x \ f a ro ' ] [ * lotly , i- [a2 ur) vForsystem3), we mayassurnehat

d1 b'1dy l, 'z

( i i )

( i i i ) . . f (x' Y)lriil(x , ) ' (0, ) Jxz+ yz=o and lim. , P9lI =o(x , ) * (0 ,0) . r 'Xz r 2

r:onstant{4 \

stnr]lepoinf:II___J

r ir')

/ l \

in (3),we getat'. ,slated(1 )

i t ly,- i ' r) i l Ithe

,. (4)

querval(2)r r r h

by

( i ) + 0Theref'ore,herelatedinear ystem4) has 0, 0) asa critical rint'f ( x , y ) a n d g ( x , y ) a r e co n t i nu o u sa n d h a veco n t i n u o u sp a r t i aI d e r i val l x,y).

Tl.ren0, 0) issaid o be simple ritical nintof thr3 ystem3) andsysterrn3)is s3llr3dlmostinear ystem'/ ? \


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lflt'll fsi'ABrtrnY-ffiivsrl----1Now,r i-L nd .z are heeiqenaluesndi:t l ' 1: ' i a'e heeir;enectorslBr I ,Bz. i

+() :J5

en ihe

ed that), v(t)) correspondinEo them, hengeneraioiution i (i) isgiverny

x = Cl Al e^tt + C2A2 e;i2ty = C181e^tt + C282'2 '

since,7,1 nd t-2arc realanddistinct,ealor equal/ r conjr'rgateomplex'herefOre'(6) the nature f the, ritical oint 0, 0) of the system1) isdeterminedy the nature:l'thenumbers'1Bnd "2

rleand Here,we have he ollowingasesI ".o roarrrisrinct r (Node).yx) as Case : Theeigen aluesil", nd ;, df eal, ist in(:trrd 1'theame lglri, case rr : The eigenvahle'/.tt

t,z are real,distinct nd c'|f ppositeir]ns saddlt:

' '(7) Points)'case III : 'Ihe eigenvalue r"1,,; are conjugate ornplex ut not pure maginarl

r 1 \

(Spiral),Case V : Theeigen alues t, )'2 are ealandequerlNode)'CaseV : Theeiqen alues r'v,7-zarepure maginanlCrrntre)'Here,we shalt iscussach f theatroveases' eparately'CaseAssumehat }'1and }'2 arebothnegativend?'1 ',. ' t '20. -I.henthegenera|o|utionof (1) in thiscas;esasgiven Y 3) ,

x = Cr \1 ett t + C2A2 e; ' : tY = Cr131 irt + C282 ei2t

If C2= 0, vrte et hesolution s

.. . . . . r (3)

.. ,t l)

. (s)

rat

x =' c1 A1ertt, Y = C:1 1e;r1tIwhichnrPlies IA1r1 lrx 'If C1= 0 , then hEsolutionsgivett y

x '=CzA2elzr Y= CzB2 iztwhichmPlies

A2t= B2xForany c2 > cl, thesolution4) represenb trajectonlonsistingf uprperalfof the

/ ? \


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[@'YsrCase-IIAssume hat t "1 0'o' thert:fore'all trajectoriespproach0, 0) as t -+ cc Now,we show hat these rajrectorieso lotenter he pointas t -r co,but nstead indaroundt in a spiralikemanner' or his'letus ntroducehe polar oordinate andshow hatalong nypath' rle dt is eitherpositiveor all or negativeor all but t isnever;lero'Weknow .hat o = tan-1l ,X

' ) r 1 /

Since, since

Ittt-"flane fllence,)) withof the

r) with

Suchsamegivenint n

Fi1,,2.2

/ri\


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vlrEll l-sr*rl*Iiy e--l- --*{ 3'ein thiscase, ys;temi1) anbe rewritten s

dx dy_ - : : a x , - - = i Ydt 0iwhose eneial olutionsgivenbY

x =.Ci 3'""Y '- C2 e;'t

The traject,crieseiinedbY i10)are half l ines of all Possibleslopes and since t" c.; Tfrerefore, riticalPointis a nodeand t is asYmPtoticallYstable. If l. :' 0, we get thesamesituationexcept hat thetrajectoriesapproach o ast -+ ca Also, he Point 0, 0) isunstable,which is shown inFi1.2.4.CaseVIf the roots )4 and i"2 arepurelymaginilry,henas n caseIiI, for 1'1and )-2 are now ofthe form ?" ip with 7-= 0 andI + 0. The general olution fsystem 1) is givenbY equation(7) witl"r exponential factormissing.Therefore,x(t) andy(0 are Preriodic nd eachtrajectory is a closed curuesurroundinghe origin' Thesecurves are actuallY elliPses,whichcan be shownbYsolvingthedifferentialquation

dY a2x+ b2Y

I) . . . . . ( r D )d0re' dt

re case

rq\D < 0

ry, t isftenasrtiseor)wn aspt that

e, the0. Let

>x

Fig.2.4

F ig .2 .5

. . . ( 1 1 )dx a1x+ b1Y

In thiscase,he critical oint 0,0) scal|ed entre rtd t is stable ut notasymrpt


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;i;i-l1_":_l

rn

I STABILITY ANAJ,YSIS

Case II : F,or Cz= 0, x = C1e-4t, = -4Cre4t. l 'his lraiectcry s; t l te halfY = - 4 x ; x > C ) ,w h e nC 1 0 a n dh a l i l i n ty = ' - 4 x x < 0 w h e nC 1 0 ' B o t htrajectoriespproachndenter heorigin 5 t -+ :aCase II : If C1 0, C2 0 , Thesolution3) repre;entsuryedrajectoriesonr: fwhich pproaches0,O)?s t -r ". . The ollowingigure ives qualitativeicture,

Fig.2.7E X A M P L . E : }

l l l - rCthe

its

/ 1 ' ,

.(2)

1ra

the

toConsiderhesYstemfequations

d x \- - - = : x Id r lrd y . - . . I- l=l== x.+Zy )LU L(1)

(3)

l inelso,

Find her:riticaloint f thesystem. iscusshe ypeandstability f thecritical oint,write dorvn the generalsolutionof the syslem :1)and draw the graph of itstrajectories.solution: clearly,0,0) is thecrit ical oint f tlne ystenr1).Now, heeigen aluesof (i) are h: oots f theequation

1 * ) . = 0


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l

: -

trie

ax|s

- . . ' t 4,7iTt*"*ry1gg" --J-whichpasse$hrougr.,he oi.igin ithsiope1. Each f these i'aiectoriesisoapprc'acir; c i l S t - t 3 r .Wecandraw ll heserajectoriess ollows

Fig.2,8Exa,up le 4Finuall realcrit[icaloints f the following ystem f equationsnddiscussheir ypeandstability

d x , d Yi t = ) l + Y ' ; d , = * * Y

Solution:Clearly,hecritical oints re 0, 0) and (-1,1) arrdpoint 0, 0) :s nodeorspiral oint, hichsunstable.Now, o checkhe stability f (-1, ) , let

X ' = u ' - ta n dY = v + 1Then,given Yl;temeduceso

d t l = , ,+ z v + v z i 9 ] = u * udt dtwhichhas 0, 0) as the critical ointcorrespondinr3o the critical oint(-1,1) of l:hegiven ystem. lso,we caneasily erify hatcritical oirrtsunstableaddle oint'Exa.uPt-E 5

+ v

lencan

r? \

< 0

Find he criticarl ointof the systemd:x dY_ _ _ - _ v : _ = {dt dt


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J

n oi

curye

i StAStLtn Ar.{Ar.ysts ri} 49(i) shour hat he ci-igin nd he point 1.,1) an:eqr-ti l itrrrurnoints f the ah:versvsleiri.(ii) shon,thert0, 0) ise sacjdlerointnd 1, i) iEa t:entre f theabo're ;ystern'

solut ion:Byusing = o, *-= o,,wecaneasi lyind hat 0,0) andt, 1.) re he:

dt dt^ - : r - : . * - - ' l i ' - - ' , ' t - t " - i * ' " " ' r : l ' - a - : " ^ * - - r r - , 1 : * ', . i L i - J , r . : i - ' i . ; ' r ; ' t ' i r - t '

: l l : : : : ' n l t : : i ' : : 1 3 : ' . " : l - t - : i l ) - ' ? : : : ' - ' : : : i 1 ' t - f i t , l l , ' i 5 1 ; ' r g i : i ' r "cix dYl - - . ' X a n d - = - Ydt ct

These quatiortsanbewritten sd ( x \ ( L 0 ) f * )at y j [o - lJ v /

Eigen alues f (1)are he oots f theequation

. (1)

L - ) " 00 - 1 - i .

n l 1

I t l - _ jJ , v

- 0 = , ( i - i " ) ( - 1; i = si.., 7"1=' - l' 'Lz = Lclearly otheigelnalues re ealanddistinct ndareoi opp)site ign.-hu:;,0, J) sa sadcjleoint.I n t h e n e i g h b ou r l r o o d o f ( 1 , 1 ) , t he g i ve n sys t e m ca n b er e d u ce d t o

X = X - 1 , = Y - 1dx dX dY i{e d t = c i f ' d t = d tdX: - = ( x - i ) ( - Y ) = - - Y - K Yot9 1 = r , ( : r + 1 )x + x Ydt

Cn inearizing,egetdX'= -Y,9I =*dt ct

Thus,wecanwrited r ' x \ c - l i i x io t [ v , J[ t o J t v lHence,herequi;'edigen alues regiven y

, ':t

/ 1> \. , . . . 1 _ . r /

: that:, the

- n


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a A . j - i I ' t ' ' '

( - i l f \ t F

/i thatcri t ical

l inea

)wrng

i STABILITT'ANAI,YSIS . . ; ? i l

| .''j I

{ u l

Cx- - = - - I rcrdx;

I = *drrlv. \ . ;- - x - o y ,- d tU L

e. g Deterrninethenaturganc1tabili$Drooertiesf thecritical oint 0. 0l for eilcftof the ollolvinornearutoncmousystem(.d j

rh\

/.1

(e)

- x = - x - z y , , 9 = o r - t ,dt dt9 = r * - 2 u , d Y = - f l x - 5 ydt dtd* = - 3 x+ 4 y ; dY= - zx - 3 yd t

- t 'd t

dx dyf r = + x - z y ;t = 5 x + 2 v$ =,* -zy ro' ,S = t*-ev+s

o@

I@6@

constanr:yefer o a lack f changensome arameterf a system.Resilience:raybe considereds the abilityof il Systemo recover nclcontinuefunctioningtfter isturbancesven hought mayhave hangedts orm'Thesurvi'valimeof a system r some omponentf it is; nown sperrsis;tence'The properlry f a Systemo osciilate roundSorrle :entral oint s knownetscyclical;tability.Fr:r l irrearystem, ritical ointP isstablef for irlrtial opulationxs,y6) c'los;eto P, hepopulationx(t), (t)) remain ear t forall > 0'Theareaov,er hich systemsstabiesknotrynsarnplitucje.

At q Glonc,e.,".


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1 I Sr.qstlln' nFr.Ai.YSIS-l

: 5

AN IM PORTAh'ITONCLUSIONLet i.1 an{l )-:zbe the eigen alues f the linear }'stem1) correspDndingc thealmost inear ystem. hen, he bype nd stability f the criticalpoint Ct, ) of thesystemsfsh,owfin the ollclwingable

r system A,lmostinear ystem

N = node; SP= saddle oint;SoF' spiral oint; C = centre2.5 STAtsILITY OF SINGLE SPECIE: ' POPULATION:;Modelsof' population rowth of singlespeciesSlpulatiotts an be created orpopulationsithcontinuousrowthand or thosewith diis:rete rowtlh'iuchrnodelsmaybe refinedo mode uite losely. tudy f natural opulationsehave ugges;tedthat there sat ange f possibleutcomeso populatronrowth, ependingprcnherelativeeprorluctiveateandsizeof the fbunderpcpul;rtlorr.opulations ay

stlowstable quilibriumoints,hathascome o equilibniumt ,a ixedandstableevel,heymayshow table ycles i' imitcycles etween onstant nddefinedimitsor they naybehave npredictablynd rregularlyn a chaotic ray'Bycalculatinghe appropriatearameters,ecanprerJict hichbehavircurill resultngiven ircumstances,heresa tendencyor laboratorl opulationso exhibit tlclic rchaotic ehav,iours,hilenatural opulationsends c,display tatlle oint' quilibrium'These esurtsndicatinghe tendencyor naturar opurertionso exhtibit tablepointequilibrium,espitehe range f possibleehavioursheycould xhibit'This act hat raboratoryopurations,aintainedn artificiar,redator nd competitor-

I Linea. ipan. I_ l

'eiorei f p =;igns,:n bysign,

- 0t,

\- > p

t " 1 > ) . 2 > 0t " 1 < i " 2 < 0

Type S t a b i l i t y l r y p e l s t a o i t i t YN Unstable N I UnstallleN Asymptoticallystable

UnstableN:^ --5 J* ; t f

Asymptotically ,:t.b9-----l7 . 2 < 0 < i r 1i : , - = l " Z > 0

SP UnstableUnstalrle jUnstablei ' r = 7 " 2 < 0 N Asymptoticallystatlle Nor SrP i AsYmPtoticallYI stable

" t .1,7"p 7' iPtr " ) U soP Unstable (: p Unstablei " < 0 SoP Asymptoticallystable

C , DY Asymptoticallystabler

^)'1= i11, )'Z =' -ift C Stable Clor SoP Indetr:rminate


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i

4S

fy ofunityb e afoodarentity ofVilsorKeystotalh there of30 toue oiome;table

j ST/iBiLIfi ANAt.Ysls . : . t 13

Veriiy that {0, 0) is a criticaip.eint,Show hat the systent s etlrnostiilr:arand citsct.issrhe typearrd iabi i i ty i the cri t ica, oint 0, 0).Solution: It can be eesilyshown hat (0, 0) is a criticalpointof the sy'stemi) Ncur,system1)canbewritten s

- l ' = x - y + f ( x , y ) Ir f i9; = - - zy+s(x,) Id t /

where,frix,Y) = xYd0dg(:x, ) -- xY

FCr singhepolarcOordinate,utting = rcosg andY= 1. in0,wegetf(x, '.)_ rzcos sir9= r cos sinf r

which ends r: 0 asr tends o 0.Similarly,

gl(x, ' /)_ _r2coso ino=, r sino os0r rwhichagainendso 0 asr tendso 0.Therefore,ystem1) isalrnostinear.Also, he relatedinear ystemn theneighbourhoodf (0, 0) is

o ( : x ) ( L - t ) ( t ) , ' . ,I t - | | I . . . . . ' \ J . /- ; l t - l | | |o r l y J 3 _ z ) \ v )Nowo igelnalues f (3)are he roots f theequation

1 - i - 13 - 2 - ) "

which mPlies; - 1r J '= > ; . = - f'Therefore,

_ r + i . 6 _ l _ i \ 6)"t='--T- ang Lz = -T-We obseruehat the eigen alues re complex on:rugatef the form )" iF Where)',V are real.Critica|p

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