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Stochastic Stability Analysis of Model Ecosystems with Time-Delay*
of Lyapu nov. In this work. by applyi ng the direct method Abstr act. under stoch astic unities comm pecies multis of sis analy lity the stabi The stabi lity condi tions and hered itary effec ts is inves tigate d. al prope rty of are expres sed in terms of the domin ant diagon mecha nism to formule suitab a es provid which ces. commu nity matri that exist in issues tant impor al late and partia lly resolv e sever "the compl exity ce. instan for s. system itary hered astic the stoch itary" . the "here ditary vs. stabi lity". the "nonh eredit ary vs. hered astic" . stoch vs. stic vs. stabi lity". and the "dete rmini s; random Delay s; ecosy stems ; Lyapun ov metho ds; model Keywo rds. distur bance s; stabi lity; time lag system s. LINEAR MODELS
INTRODUCTION The objec tive of the presen t work is to initia te the stabi lity study of multis pecies comm unities with pastmemory in stoch astic enviro nment by equati on using the Ito's diffe rentia l (Arno ld. 1972) as commu nity model s and applyi ng the Lyapu nov's direc t method Linea r (Arno ld. 1972; Kushn er. 1967) . y densit with unities comm of models It depen dent speci es, are studie d . is shown that a diago nal domin ance prope rty of commu nity matric es guaran tees stabi lity of the commu nity under non-h eredit ary. determ inisti c hered itary and stoch astic hered itary struc tural The algeb raic stabi lpertu rbatio ns. su1tab le tor reare tions condi ity solvin g sever al proble ms in stoch astic The stahered itary model ecosy stems . bility condi tions provid e an estim ate for time-d elay that preser ves the stability of the corres pondi ng non-h eredIt has been itary model ecosy stems . confir med that time-d elay as well as random distur bance s are desta bilizi ng The presen ted staagents (May. 1973) . s the determ inextend sis analy bility istic non-h eredit ary(S iljak.1 975) determin istic hered itary( Ladde .1976 ). and stoch astic non-h eredit ary (Ladde . Siljak . 1975; Ladde . Siljak , 1975) stabi lity analy sis in a unifie d way.
Let us consid er a commu nity model of of n intera cting specie s descri bed by a simple system of stoch astic linear functi onal diffe rentia l equati ons of Ito type dx = [Ax + B(xt) ]dt + C(xt)d z (1)
where x= x(t) is an n-vec tor. x = (x'. x 2 •...• xn)T and T stands for transp ose of a vecto r; Band Care bounde d linear functi onals define d on e n .= C[[-T. oJJ. Rn] into Rn; C[[-T .O].Rn ] is a space of contin uous functi ons define d on [-T.O] into Rn with a norm 11
O. By Riesz where 1 = {1.2 •...• n}. Theore m (Natan son. 1964) . (1) can be rewri tten as o dx=[A x+[f Tda\s )x(t+s )]]dt+ (2) [fO d 8 (s)x(t +s)]d z
*The resear ch report ed herein was suppo rted by SUNY Resea rch Found ation Facul ty Fellow ship No. 0407- 01-03 076-0. 1149
-T
where the integ rals in (2) are Stielt jes integ rals (natan son. 1964) A=(ai j) is the nxn consta nt nonhered 1tary commu nity matrix that reflec ts the instan taneo us interaction s among the popul ations ; is the nxn deter(s» da(s)= (da minis tic ~jreditary commu nity matrix functi on on [-T.O] that reflec ts the hered itary interdeterm inisti c
G. S. Ladde
1150
actions among the populations in the community on the interval of length T>O; z=z(t) is a scalar function, representing the random environmental fluctuations, which is a normalized Wiener process; d8 (s) = (d8 ij (s)) is the nxn diffusion hereditary community matrix function which specifies how the random variable z(t) influences the community on the length of timeinterval T>O. The elements daij(s) and d8ij (s) of matrices da(s) and d8(s) are Stieltjes measures on [-T,OJ, respectively; aij(s) and 8i;(s) are functions of bounded variation on -T~S~O, for i,jEI. We assume that all species are density dependent (May,1973) so that the diagonal elements of A are negative. This assumption reflects the resource limitation in the community. In the case of off-diagonal elements of A, we do not specify their signs, thus allowing for "fixed" competitive-predator-symbiotic-saprophytic interactions among species. The coefficients of the deterministic hereditary community matrix da(s) and the hereditary diffusion community matrix d8(s) on [-T,OJ can have arbitrary signs which allows a good deal of freedom in t h e ~eter ministic hereditary and stochastic hereditary interactions among the species. The equilibrium ~(s) = 0, -T
= ~
IXil2
(3 )
and V (x) = (a2v/ax ax·) is the Hessian m~~rix relative io ~(x), and the elements a2v/aXiaXj are real-valued functions; tr stanijs for the trace of the matrix. We note that Vex) is a postive definite function.
~et u~ define the_matri~es A=(~ij)' B(T)=bij(T)) and C(T)=(cij(T)) as
I-Ia
LV(x)=V (x).[Ax+B(x )J+ x t
~tr(C(xt)CT(xi)Vxx(x))
where vx:avi/ax j
(4) is the gradient vector
l
ajj=Llaijl
i
j
i#j
bij (T)=T(a ij ) and c ij (T)= n
T(8 ij )(k f l T(8 ik ))
(5)
where T(aij) and T(8 ij ) are total variations of a' j and 8. on the interval [-T,OJ; One of~~he objectives is to estimate how much of random perturbations as well as past -history can be absorbed by the nonhereditary deterministic version of the model (1). Therefore, we assume that the non-hereditary deterministic community matrix A is stable. Since A is negative diagonal. we assume that the matrix A satisfied the diagonal dominance conditions
(6)
for all JEI, where kc and kr are positive real numbers. In order to guarantee the stochastic stability property of (1) in the sense of the mean, we assume that the matrices A B(T) and C(T) satisfy the following' conditions 2~jj + bjj(T) < 0, JEI, (7) and n
[12~jj+~jj(:)li~} (~ij+~ji+bji (T))J>
L
L
(bij(T)+Cij(T)).
j=l i=l
(8) We observe that the stability conditions (7) and (8) are expressed ~xpli~itly in terms of the elements aii' b ij (T), C i · (T) of the model matrices A, da(a), d8(s) and a time delay T>O. For any ~ E [0, 00 ), the function F(~)=~+sup[2~jj+bjj (T)+
i=l
as a candidate for Lyapunov's function for the system (1). Using the Ito's calculus we wxamine the following expression
ii
i!l i#j j
(~ij+~ij+bji(T))J
~lit (b ij
(T)+Cij (T)
+
)exp[~TJ
(9)
is defined and continuous on [0,00). From (7), (8), and the definition of F, we abserve that F(O)
Sto c hastic Stability Anal ysis
1151
we can find a postive number A such that F (A) .s. 0, whenever O
We take A(t) = exp [At], where A is defined in (10), and define n A={~ £ en = sup
l:
_
_
(b ij (1)+Cij (1»
i=l exp [A 1]) V (x)
A(t+s)V(~(s»
-T
]
.s.r~£[2~jj+bjj(1)+
=A(t)V(~(s»)}-:-
n
Now by using the relations (sup Ixi(t+s)I)2= sup Ixi(t+s) I~ -T < S
if 1 (~ij+~ji+bji(1»] ilj n n +A+jh (bij(1)+cij(1»]
2ab .s. a 2 + b 2 , we compute L[A(t)V(x)]
A(t)V(x) .s. 0
dh
(14 )
as L[A(t)V(x)] = A(t)V(x) + A(t) L V(x) whenever Xt £ n A and t > to, where to is an initial time. By the appli=A (t)[ ~ Alxil2 + cation of Theorem 8.2.2. (Lakshmikani=l tham,Leela, 1969) and Theorem 3.2 + ~ [2aulxil2 + (Ladde,1974), one can obtain i=l I
+
~
E[lIxto,~o)(t)II].s. 1I~0110
j=l 2 1a i j 1 1xiii xj 1 ]
Hi
exp [-~ (t-to)], t~to
0 + l: [2Ix i l( ~ f 11dOij i=l j=l (s)llxj(t+s)I)] +
(15)
n
+
¥ (.¥ L~ldBij(s)1 1=1 J=l Ix j (t+s)I) 2 ].
(11)
The third and fourth terms in the right hand side of the inequality (11) can be written as
i~l
[2lxil
(sup -1 <5 <0 and
n .l:
(jfl T( Oij)
1x j (t+s) I )]
~£ en.
To appreciate the stability conditions (7) and (8) imposed on the community matrices A, do(s), dB(s) and a time-delay T>O, we assume that Stieltjes Measures do (s) and dB(s) have the following forms Bds and Dds, where B = (bii) and D (dij) are constant matrIces. Under these specific forms, the stability conditions (7) and (8) are replaced by 2
2~jj + b jj < 0, j£l
n <'~l T(Bij)(sup
I
1=1 J-T
x j (t+s)
1
»
2
_
and
_
n
_
(16) _
_
[12ajj+b jj 1 i~l (aij+aji+bji)] n
n
n
_
ilj _
> 1[j~1 i~l(bij+Cij)]' j£l,
l:
i=l and
which establishes global exponential stability in the mean of the equilibrium ~0=0 of the system (I), where
n n n iL j h T( Bij)kh T(Bik)lxjl ~ J, (13)
respectively. From (5), (7) -, (8), (10), (12), (13) and x t £ n , the A relation (11) reduces to L[A(t)V(x)] .s. A (t)[ n
l:
j=l
¥
AIx
j
I2
j=l
(2a j j +b ' j (1) J
respe~tively,_where ~ii as defined in (5); b ij and c ij are d~fined by
b ij
(9),
Ib ij 1 and [
n
Cij = 1d ij 1 ]
kh
+
(17)
Idik l
'
(18)
By following the previous argument and using the Cauchy-Bunyakovski -Schwarz inequality [J
o -1
< T
-
Ixi(t+s)ldsF
fO-1 Ix i (t+s)1 2 ds], i£l,
the third and fourth terms in the right-hand side of the inequality (11) can be written as Jl
[j~l
1b ij 1 (I xi 12
1152
G. S. Ladde
and
LE i d l.J .. I( Idikl)[J~"[lxj(t+s)ldsF] l.=l J=l k=l which can be replaced by
.¥
£
n
n
I
-
i~l [j~l b ij ( x and
il
2
j 0 2 + "[I_"[lx (t+s)1 ds])] (19)
I~"[lxj(t+s)12dS]],
J l j2l Cij["[
(20) respectively. We note that the function F 41) defined in (9) is replaced by F()J) = ~
+ sup[2ajj+bjj j£I _ _ _
n
+ih i,;j
(aij+aji+b ji )] n
n
+"[[j~l i~l (b ij + (exp
cij )
[~"[]-1)/~]
(2l)
which is defined and continuous on (0, 00 ). From (16) and (17), i t is obvious that lim F(~) exists and the limit is nega~i~~. Furthermore, F' ()J) exists on (0, 00 ) and F'41 »0 for ~£(O,l). This together with the continuity of F implies that there exists positive number A >0 such that feA)
A > O.
< 0
(22)
From (16), (17), (18), (19), (20), (21), (22) and by imitating preceding argument, the relation (11) can be replaced to the relation (14). Rence, one can conclude that the equilibrium ~=O of (1) is exponentially stable in the mean. The presented stability conditions can be utilized in order to study the connective stability properties (Ladde, Siljak,1975; Ladde,Siljak,1975) of (l). To study the stability anal y sis of (1) under structural perturbations, we need to redefine the community matrices A=(aij)' d a (s)=(d a ij(s» and d Bij(S» in (2) as f 0 11 ow s :
aij=
r
ii Yi + e Yii' i=j
el.J Yij
fhii(s)
(23)
ih d OH (s), i=j (24)
d a ij(s)= Lhij(s) d Oij (s), i,;j and i=j e ii (s)d ll H (s), d Bij (s)= .. i,;j Lel. J (s)d ll ij (s),
r"
(25)
where Yi > 0 Y . are real numbers, and i 0 ij (s), llij (s) are fun<;:pons of bounded variation on [-"[,0]; el. J are elements of the nxn non-hereditary interconnection matrix E = (eij) (Ladde,Siljak, 1975; Ladde,Siljak,1975), and can take
on any values between zero and one: hij(s) and eij(s) are elements of the leterministic hereditary and sto: hastic hereditary interconnection matrix functions R(s) = (hij(s» and L(s) = (eij(s» whose elements are defined and continuous on [-"[,0] with values in [0,1]. By e ij , we represent the degree of non-hereditary connection of the j-th species xj with the i-th species xi ranging from a disconnection (eij=O) to a full connection (eij=l). Furthermore, the nonhereditary interconnection matrix E represents non-hereditary structural changes in the non-hereditary interactions among the species. In particular, the shut down of a link from the i-th species to the j-th species in the community is represented by eji=O. This implied that the rate of change of xj is unaffected by the instantaneous size of xi. The disappearance of the non-hereditary links between the i-th species and the j-th spec~~s in the community is shown by el.J=eji = O. The disconnection of the non-hereditary link between the p-th species and the rest of the speries ~n the syste is pep resented by e P=e PJ =0 for all i,j £ I. If E=O, then the species in the community are isolated relative to non-hereditary links. Similarly, the elements hij=hij(s), eij=eij(s) for s £ [-"[,O] reflect the magnitude of the deterministic hereditary and stochastic hereditary influence of the hereditaryand environmental perturbations on the population xi during the time-interval of length "[> 0. R(s) and L(s) represent the deterministic hereditary and stochastic hereditary structural changes in the heredtiary and stochastic interactions among the species in (1). Further detail structural changes relative to hereditary and stochastic interactions can be discussed similar to the deterministic non-hereditary structural changes. In order to investigate the sta -bility properties of (1) under deterministic non-hereditary, deterministic hereditary and stochastic hereditary structural perturbations, we need the concepts of non-hereditary (Silajk,1975; Ladde,Siljak,1975: Ladde,Siljak,1975), deterministic hereditary (Ladde), and stochastic hereditary fundamental interconnection matrices E = (eij)l R(s) = (hij(s» and L(s) = (eiJ(s», in ~~~ch elements el.j, hij(s) and e J (s) take on binary values 0 or l. We note that 0 and 1 are used for both real numbers as well as functions on [- "[,0], and these are understood in the context of the
Stocha stic Stabil ity Analys is
e
The ij , hij(s) and eij(s) . use of binary are L(s) and R(s) E, es matric matric es which reflec t the basic structure of the stoch astic hered itary sysThe matric es E, R(s) and L(s) tem (1). provid e a suitab le mean for formu lating the conne ctive stabi lity conce pt (Silja k, 1975; Ladde ,1976; Ladde ,Siljak ,1975 ; Ladde ,Siljak ,1975 ; Ladde ), i.e., the equili brium ~=o of (1) is conne ctivel y stable if it is stable for all interconne ction matric es E, R(s) and L(s). The conne ctive stabi lity in the mean of the equili brium ~=o of (1) follow s immed iately, if the matric es in (23), (24), (25) and the time-d elay T>O, satisf y either the condi tions (7) and (8) or the condi tions (16) and (17) with respec t to fundam ental interc onnec tion matric es E, R(s) and L(s). From the above stabi lity analy sis, we can draw a few conclu sions about the invar iabili ty of the stable system under determ inisti c non-h eredit ary, determ inistic hered itary, and stoch astic hereditary struc tural pertu rbatio ns, the measu rabili ty of the past-m emory , the measu rabili ty of the compl exity of the system , the tolera nce of time-l ag by the stable system , the tolera nce of compl exity by the stable system , the tolera nce of stoch astic distur bance s Furthe rmore , by the stable sustem . lity includ es stabi ted presen the we note the earlie r determ inisti c non-h eredit ary (Silja k,197 5), determ inisti c hered itary (Ladd e,1976 ), and stoch astic non-h ereditary (Ladd e,Silja k,197 5; Ladde ,Silja k, 1975) , whene ver, R(s) = L(s) = 0, L(s) =0, and R(s)=O and B(s) = consta nt for S£[-T, OJ and B(O) is differ ent from Furth er the const ant, respe ctivel y. note that the condi tion (7) or (17) implie s that the matrix A+A-T+ B is domin ance diagon al matrix . CONCLUSION Multi specie s stoch astic time-d elay model under struc tural pertur bance s are tested analy ticall y in order to study behav ior of the model in the contex t of Lyapun ov functi ons and The propos ed compa rison theore ms. determ insyudy to us s enable method istic hered itary and stoch astic hered itary effec ts on the commu nity staIt is shown that the diago nal bility . domin ance condi tion provid es a suitab le mecha nism for dealin g with centra l proble ms of "comp lexity vs. stabi lity," "here ditary vs. non-h eredit ary", "stoc hastic vs. determ inisti c" "time delay vs, stabi lity" and "stoc hastic vs. stabi lity" in model ecosy stmes . The presen ted result s includ e the earlie r result s on commu nity models with or witho ut past-m emory as well as with or witho ut random fluctu ations . The approa ch propos ed in thes work,
1153
can be direc tly extend ed by using the conce pt of vecto r Lyapu nov's functi on to study nonlin ear timevaryin g as well as hierar chic model s modes of multis pecies hered itary-n on-he redita ry comm unities in a hered itary- non-h eredit ary random Some of these modif ienviro nment . cation s of the models are curren tly under inves tigati on and will be report ed elsew here. REFERENCES 1. Arnol d, L., "Stoc hastic Diffe rentia l Equat ion: Theory and Appli cation s", John Wiley & sons, Inc., New York, 1972. 2. Kushn er, M. j., "Stoc hastic Stability and Contro l",Aca demic Press , New York, 1967. 3. May, R.M., "Stab ility and Complexit y in Model Ecosy stems Prince ton Unive rsity Press , Prince ton, 1973. 4. Silj ak, D. D., "When is a Compl ex Ecosy stem Stable "? Math. Biosc i., 25 (1975 ), pp. 25-50. 5. Ladde ,G.S., "Stab ility of Model Ecosy stems with Time- Delay ", J. Theor . Bio!., 61 (1976 ), pp. 1-13. 6. Ladde , G.S., and Siljak , D.D., "Stab ility of Multi specie s Comm unities in Random ly varyin g Enviro nment ," J. Math. BioI. , 2 (1975 ), pp. 165-17 8. 7. Ladde , G.S., and Siljak , D.D., "Stoc hastic Stabi lity and Instability of Model Ecosy stems" , Proc. of the 6th IFAC World Congr ess, Boston , Mass. , IFAC (1975 ), pp. 55.4: 1=7. 8. Natan son, I.P., "Theo ry of Functions of Real Varia bles" ,- vol. 1 Frede rick Unger Publ, Co., New York, 1964. 9. Ladde , G.S., "Diff erent ial Inequa lities and Stoch astic Funct ional Diffe rentia l Equat ions", J. Math. Phys. , 15 (1974 ), pp. 738-74 3. 10.La dde, G.S., Lakshm ikanth am, V. and Liu, P.T., "Diff erent ial Inequ alitie s and Ito Type Stochast ic Diffe rentia l Equat ions", Proc. of the Int. Conf. on Nonlinear Diff. and Funt. Eqs., Bruxe lles et Louva in, Belgiu m, (Rerm an, Paris , 1973) , pp. 611640. 11.Lak shmik antham , V.
and Leela , S.,
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G. S. Ladde "Differential and Integral Inequalities, Theory and Applications", Vol. 11, Academic Press, New York, 1969.
12.
Ladde, G.S., "Competitive Processes I: Stability of Hereditary Systems' J. of Nonlinear and Theo. Hath. and Appl., (in press).