Bulletin of Mathernatieal Biology Vol.45, No. 5, pp. 793805, 1983.
Printedin GreatBritain
00928240/8353.00 + 0.00 PergamonPressLtd. © 1983 Societyfor MathematicalBiology
CONDITIONS FOR G L O B A L STABILITY OF TWO SPECIES P O P U L A T I O N MODELS WITH DISCRETE TIME D E L A Y V. P. SHUKLAt Department of Mathematics, Indian Institute of Technology, Kanpur208016, India By c o n s t r u c t i n g a p p r o p r i a t e L i a p u n o v f u n c t i o n a l s , a s y m p t o t i c b e h a v i o u r o f t h e solutions of various delay differential s y s t e m s describing p r e y  p r e d a t o r , c o m p e t i t i o n and symbiosis models has b e e n s t u d i e d . It has b e e n s h o w n t h a t e q u i l i b r i u m states o f these m o d e l s are globally stable, p r o v i d e d certain c o n d i t i o n s in t e r m s o f i n s t a n t a n e o u s and delay i n t e r a c t i o n coefficients are satisfied.
1. I n t r o d u c t i o n . In the biological process of evolution of interacting populations there are certain perturbations whose effects appear long after they are applied. These perturbations may account for the past history of populations and can be described by time delay mechanisms in population interaction models. There have been several investigations to study the role of time delay on the stability of, particularly, LotkaVolterra population systems (e.g. Volterra, 1931 ; Hutchinson, 1948; Wangersky and Cunningham, 1954; Goel et al., 1971 ; Caswell, 1972; May, 1973; Smith, 1974; Beddington and May, 1975; Bush and Cook, 1976; MacDonald, 1976, 1978). The VolterraGauseWitt (VGW) model with discrete time lag in only the predator rate equation is given by
]Q1 ~ a l N l ( O   N1)/O  a12N1N2 '
(1) IV2 = a2N2 + a 2 1 N l ( t 
A)N2 ( t  A).
The asymptotic stability of the model (1) with small time lag has been studied by Wangersky and Cunningham (1954) for small perturbations around the equilibrium state. However, for significant time lag the local stability of system (1) has been presented by Goel et al. (1971, pp. 106109), based on a theorem on eigenvalues given in Bellman and Cooke (1963, pp. 449450). Recently, model (1) and several other VGW models with both small and significant time lags have been studied by Bojadziev and Chan (1979)using the KrylovBogoliubovMitropolski perturbation method based on the small Present address: Research Officer, Computer Division, Central Water & Power Research Station, Pune411024, India. 793
794
V.P. SHUKLA
parameter expansion. It has been shown by the authors that the preypredator models remain stable for small time lag, but they may become unstable for significant time lag. Most of the investigations mentioned above are confined to the linear stability analysis of only preypredator systems with discrete time lags, and efforts have not been made to study the global stability of the preypredator and other population systems. In this paper we consider several twospecies preypredator, competition and mutualistic models with both instantaneous and delayed interactions. By constructing appropriate Liapunov functionals for the models, we study the global stability of the equilibrium states and obtain sufficient conditions for asymptotic stability in terms of instantaneous and delayed interaction coefficients. Consider both instantaneous and delayed interactions in the LotkaVolterra preypredator model so that following May (1973), MacDonald (1978) and Bozadziev and Chan (1979), the general delay differential equations governing the evolution of prey and predator populations N 1 (t) and N 2 ( t ) at time t are 2. Global S t a b i l i t y o f P r e y  P r e d a t o r M o d e l s .
]V1 = elN1  allN~l  a i 2 N 1 N 2   [311NiNi(t   T)  [312N1N2(t  7')
(2) N2 =   e 2 N 2 + a 2 1 N a N 2 
a22N~2 + l~21Nl(t  T)N2   ~22N2N2(t  T),
where ei, ~ij, /3ij, g ] = 1, 2 are nonnegative constants with their usual meanings and T is the constant time lag in the interactions. System (2) describes many preypredator models depending on various combinations of aifs, 13q's which m a y . b e positive, or zero. The positive equilibrium populations N* and N* (for N1 = 0 and N2 = 0) are given by =
e1(o~22 + ~22) + e~(~12 + ~i2) (°/11 nt~11)(°/22 ~ fl22)~ (°/12Jlfli2)(o~2i4~21) e1(o~21 If121) e2(oql + fill) (~ll + ~i)(~22 +/322) + (~12 + ~12)(~21 + ~2~)
(3)
provided that
ei(a2i + ~21) > ez(~ll + ~il).
(4)
For computational convenience we use the transformation N~(t) = N * + n~(t), i = 1, 2
(5)
CONDITIONSFOR GLOBALSTABILITYOF TWOSPECIESPOPULATIONMODELS
795
SO that system (2) becomes r[i = (N* + nl)[cqlns

ozl2n
2 
[311nl(t  T)  [312n2(t  T)]
' .
(6)
n2 = (N* + n 2 ) [ ~ 2 1 n l  oe22n2 + ~ 2 1 n l ( t   T)  ~22n2(t  r ) l .
To study the asymptotic stability of the solution of system (2), we consider the Liapunov functional V ( n l t, n2t) defined by
V,(n,t(s),n2t(s))=,:,~c,
,t(O)NTln
i ÷ N~.* ]J
2 0 2 [ ½~ CiOdiif T nit (s)ds, i=1
(7)
where nit(s) = ni(t + s), nit(s) E C [   T , 0], i = 1, 2 and c{s are positive constants to be chosen. The Liapunov functional V1 given by (7) is zero for n i t ( s ) = 0 = n2t(s) and positive definite for bounded nit(s) >   N * , s E [T, 0]. The time derivative of V along the solution of (6) is given by
2
~1 ci d n i + ½ ~ c t a i i [ n ~ ( t )   n~(t  T)]. V l ( n l t ( s ) , n2t(s)) = _ N * + ni dt i=1
(8)
dn i
Substituting ~t' i = 1, 2 from (6) in (8), we get
G11
2
G22
2
1 cl~11nln1(t r )   c2~2'nl(t  r)rt2t810q12 n2 (t T)}
Cl~12nl?12(tT) + c2~22/'/2F/2(t T) +
C20L22n 2 rt  T)}. 2 2~
(9)
796
V.P. SHUKLA
By making perfect squares with the expressions in both the braces { }, we see that
I71(nlt(S), n2t(S)) <~
all
2
22
all
C2 a22_1
a22
el a l l
.1 [cla12 3tc2a21_[_c1/~12 ~22 __C2 ~11~21   .1 nln2. a22 al 1 ..]
(10)
The expression on the righthand side o f the above expression is a negative definite quadratic form, provided that
all
/321
C1 ~22 > 0
all
C2 a22
(11) a22
_ cA
> o
Cl all
a22
and
I
1~12~22 ~11/321] 2 C2  
~cla12 4 c2a21 ~ C1 
0/22
all
C1C 2
OLll
0/11
a22 C2 {222..[
a22
ClaTi1
,
which may be regarded as the condition for stability (Hale, 1971, pp. 6970, Theorem 13.1) for various preypredator models described by (2). Condition (11) contains Cl, c2 as unknowns which can be properly chosen for the models given by (2) based on various combinations o f ai/'s, {3i/'s, positive or zero, to yield the corresponding stability conditions independent of cl, c2 as follows: Case 1. When a12 = 0, a21 = 0, ~1~22 > 0, ~11~21 > 0, the choice o f c 1 > 0, c 2 > 0 is to be given by
1~12~22 ela22
/~11~21 c2~=0 ~II
(12)
CONDITIONS FOR GLOBAL STABILITY OF TWOSPECIES POPULATION MODELS
797
so that the condition for stability in this case, using (12) in (1 1), can be given b y kl>O, k2>O,
(13)
where
ki = otu     
15, i = 1, 2
(14)
Olii
li = flu/3i]/3£i, i 4= ]; oq~ > O, flu > O, i, ] = 1, 2 . aucg] Since condition (13) is easily satisfied for large ~lx, a22 and small/312, 132a, it can be asserted that the equilibrium state (NT, N~') o f system (2) with ~x2 = 0 = ~2a is asymptotically stable for weak interspeciesdelayed interactions dominated b y strong instantaneous intraspecies competition. Case 2. When/3al/321 = 0,/3x~22 = 0, ~ii > 0, i, j = 1,2. This case includes several preypredator models other than those o f case 1, depending upon the combinations of/3ii's positive or zero, as follows: (i) One of/311,/321 and one of/3x2,/322 are zero. (ii) Both/31x,/321 and one of/312,/322 are zero. (iii) One of/311,/322 and both/312,/322 are zero. In this case we choose the el > 0, c2 > 0 such that ClCq2 c2c%1 = 0
(15)
and therefore condition (13) yields the following sufficient condition for stability k*>O, k*>O,
(16)
where
k* = c~u      l*, i = 1, 2 oqi
l *=°qi/3~j, i ~ j ; a i j > O , i , j = oqi o~/
(17)
1, 2.
798
V.P. SHUKLA
The stability analysis in this case includes eight models, viz. four models in combination (i) and two models each in combinations (ii) and (iii). Using the modified k*'s and l*'s for all these models, in (1 6) the corresponding sufficient conditions for stability can be derived. We see that condition (16) is easily satisfied for large cq]'s and small ~q's. This shows that an otherwise stable preypredator system remains stable with discrete time delay, provided the effect of instantaneous interactions overpowers the effect of delayed interactions so as to satisfy (1 6). However, if condition (16) is violated, the preypredator system with delay may behave like an unstable one. Case 3. When c~ii > 0, f l i i > O, i, ] = 1, 2, we choose e 1 > 0, c2 > 0 by (15) so that sufficient condition for stability in this case [using (15) in (11)] is given by
k*>O,k*>O,
* k2* > oq2c~21kl
( ~lfln/32x Odl 1
a2flxf122/2 0L22
(18)
]
where k~' and k~' are given by (1 7). Thus, we see that the equilibrium state (N*, N~') of (2) is asymptotically stable under condition (18). Similar to the earlier cases, it is evident from condition (1 8) that delayed interactions may develop the instability in a preypredator system. Similar to the preypredator system (2) of Section 2, here we shall also take all instantaneous and delayed interactions for two competing populations N l ( t ) and N 2 ( t ) at time t. Then the evolution of both the populations is governed by the following delay differential equations: 3. Global Stability o f C o m p e t i t i o n Models.
IV1 = e'lNx  otil lV~l  ~I2N1N2  fi'nNi N l ( t  T)  IY12NIN2 ( t  73
(19) 1~2 = e'2N2  ~'2,N2  a'22N~2  ~'21N~ ( t  T)N2  [3'22N2N2 ( t  T),
where ei" s, a~] ' ' s, /3~] ' ' s are nonnegative constants with their usual meanings in a competition model. Various competition models based on different combinations of a~] >~ O, ~] >1 O, i, ] = 1,2 can be described by system (19). The positive equilibrium state populations 5) 1 and N2 are to be given by
CONDITIONS FOR GLOBAL STABILITY OF TWOSPECIES POPULATION MODELS
799
e1(0~22 + fl22)   e'2(Oq2 + ill2) ( ~ ' ~ + ~n)(a22'' + &2)' !
(~2' + ~)(~2~'
?
I
/92 = ( a n' + ~11)(a22 ' ' + t3~2) 
t
' + ~'2~)
(20)
F
(a12 + t~) ' + ~12)(a~1 '
provided oq1,+/311 > >oq2 'f ' ~ 'P + t3h W ! ~21 + f121
e2
(21)
0~22 + ~322
or P
oql ' +/3n' < e'1 < cq___~2+ / 3 h ?
r
~21+/321
e2
I
I
a2~+/322
(22)
"
Substituting (23)
N ( t ) =/qi + ni(t), i = 1, 2 in (19), we get f
t
t
ha = (N1 + n l ) [   ~ l l n l + oq2n2 + ~'nnl(t  7) + (312n2(t  7)1
(24) n2 =   ( ~ 2 + n2)[~'21nl + a'22n2 + {3'21n1(t  T) + [3'~2n2(t  T)]. The asymptotic stability of the solution of system (19) can be studied by defining the Liapunov functional V2(nxt(s), n2t(s)) for system (24) as
V2(n,,(s),
= i=, di
.(o)
1+
2
+½y i=1
di u
nit(s) ds, T
(25)
where ali's are positive constants and nit(s) = nt(s + t), se [T, 0] are continu ously differentiable functions which are bounded in IT, 0] for all t > 0 such that nit(s) >   N*, se[T, 0]. Computing directly the derivative of V2 with respect to t along the solution (nl(t), n2(t)) of (24), we get
800
V.P. SHUKLA t 2 . otlln 1
[~'2(nlt(S), n2t(s)) =   a l

~

(d~oth +
 dl~'llnlnl(t
d2ot21)n lnz '

d~ "
2
n~
T) + d2~2anl(t ' l T) n2 + dl

n ](t T)
T) . {dl[fl2nln2(t T) + d~22n2n~(t T) + d2
[email protected]~(tZ
(26) Separating out the perfect squares from the expressions in b o t h { } expression (26) reduces to
n2t(s))
_ 11 <,d.d,_;[~,
, 0/11
Lot22
2
dl

~i~
'12
dl~i~ln ~ d2 otll J
ot22 ,
dl ot.J 7
n22
~!22' +d2
'21
0~22 /
7
nan2,
otll
which is negative definite for ni(t) >   N * , i = 1, 2, provided that ,
fll 2
otll
d,/31~ > 0
!
,
0/22
f
ot11
d2 ot22
~;~
d2~;~>0
ot;2
d I otl I
(27)
and '
dlotl2+
dot'
( dlfll~f122
2 21~,
< did2
°/11
~
t 0/11
+
dfl!  1132'
ot11 /J
t
d2 ot2~
22
p ~22
dl otllJ
•
With help of condition (27), the asymptotic stability of ten distinct competition models, grouped in three cases as in Section 1, can be studied and stability conditions independent of Cl, c2 in each case can be derived as follows:
CONDITIONS FOR GLOBAL STABILITY OF TWOSPECIES POPULATION MODELS r
I
I
f
I
801
F
Case 1. When 0/12 = 0, 0/21 = 0, flxd32:> 0, 13n1321 > O, the constants cl, c2 are chosen b y I
I
dl
!
,
I
d2 ~
0/22
0
0/11
for which condition (27) becomes
4thlthz l 22 t
K1 > O, Ks > O, KIK2 >
!
?
I
t
(28)
I
0/110/22
where ,
Ki = 0/ii
Li, i = 1, 2
!
Olii
Li    ,
0/;iYjj
(29)
i :/=j, i, j = 1, 2.
(30) 1
!
It can be noted that condition (28) is satisfied for large 0/11, 0/22 and small ~2, /3;1. This ensures the asymptotic stability o f the equilibrium state (2V1, f t N2) given b y (20) with 0/12 = 0/21 = 0, provided the effect of instantaneous selfcompetition overpowers the destabilizing effect o f retarded interspecies competitions so that condition (28) holds good. Case 2. When [3~113':1= 0, ~12fl22 = 0, 0/i1 > O, i, ] = 1, 2, the choice Of Cl, c2 can be given b y d l 0 / 1 2  d20/'21 = 0
(31)
so that condition (27) yields K*>0,
K~>0,
' ' KI*K2* > 40/120/21,
(32)
where
'     f  L * , i = K* = 0/u 0/ii
1,2 (33)
= 0/ i b i s ~ ,
ai/0///
i , / = 1 2.
Similar to Case 2 of Section 2, here also eight distinct competition models are included. The stability conditions for them can be derived from (32) by
802
V.P. SHUKLA
using the modified K*'s and Li*' s, and similar biological interpretation of the results as in Section 2 can be obtained. Case 3. When aq > 0, /3e > 0, i, / = 1, 2, we can choose c~, c~ by (32) for which condition (29) gives [] ? t K* > 0, K~ > 0, a120/211r~1A2 ~* [20/120/Zl ¢
?
rT,
I
t
t
t
~t
~t
I 2
0/I~I~22 a2iPi~22[
r:,
0/;2
ala
.1 , ( 3 4 )
where K~', K~ are given by (33). Condition (34) is the condition for asymptotic stability in this case. The stabilizing role of selfcompetition and destabilizing role of retarded interspecies competitions are evident from condition (34). 4. Global Stability o f Symbiosis Models. The population model with instantaneous and delayed mutualistic interactions in populationsNa andN2 is given by IV1 = e**N1  0/~{1N~1+ 0/~{2N1N2 /3~IN1NI(t  T) + ~**2NIN2(t  T) N2 = e~N2 + 0/~IN, N2  0/'2N~2
(35)
/3~1Nl(t  T)N2 /3*2N2N2 (t  T),
where e* > 0, 0/,/> 0, /3*/> 0 are constants and have their usual meanings in the symbiosis model. The positive equilibrium position (N1, N2) of system (35) is given by el (a22 + ~2) + e2 (a12 + ~*.2) N1  (~'1 + th0(0/.* * + ~ 2 )   (0/*2 + /312)(~21 * * + ¢~i) *
*
* ,
el (~21 + ~1) + e2 (0/. + ~*.1) N2=(~*.1+ * * /311)(0t22 + ~ ' 2 )   (0/**2 + /312)(0/21 * * +/3~1)
(36)
provided (0/**1 t /311)(0/22" * 1 /3~2)   (0/**2 ] /312)(0/21" * + /3~1) > 0.
Using the transformation Ni(t) = Ni + ni(t) in (35), we get
(37)
CONDITIONS FOR GLOBAL STABILITY OF TWOSPECIES POPULATION MODELS
803
h~ = (N1 + n l ) [ ~ T l n l  ~'2n2 + [ 3 T l n l ( t   T) /3~2n2(t  T)]
(38)
n2 = (N2 + n2)[a~lnl + 0L~21'12/3~xnl(t  T) + ~*2n2(t  T)]. Consider the Liapunov functional Va(nlt(S), n2t(s)) defined by
Vs(nlt(s),n2t(s))=g=l ~ e~ , t ( 0 )   N t l n
1 ,~]j
2
+ ½ ~" eio~* f_~ nit (s)ds, el, e2 > 0 i=l
(39)
T
so that on proceeding as in earlier sections, Va along the solution (ha(t), n2(t)) of (38) is given by el [a g3(n,t(s), n2t(s)) ~~ ~1
e2 2
a
"2
/3T~ o~*,
ea /3T] ] 2 ez a~*22Jnl
/3~] oe~2
e2 ~21 / 2 emO~TaJnz
~,2n
+ [ e l ( a ~ 2 + /31~fl22) * • + e2(o~*1 + flllfi21)]nln2. , • (40) Choosing el > 0, e= > 0 such that
el(a~2 + fl*1~2fl22) * * * > 0. e2(o~'1+/321/322) 

(41)
We see that the quadratic form on the righthand side of (40) is negative definite for ni(t) > N*, i = 1, 2, provided
K*>O,K*>O,
* * >4(ot~2 +/31~322)(o~zl K1K2 * * * +/3n/32a), * *
(42)
where ~*2(rv*
K * = ~ *   ~"* "~J *
+ (3"/3") * *
i 4= ], i, ]  1, 2.
(43)
Similar to earlier sections, here also stability of many symbiosis models can be studied by means of the modified stability condition (42) with modified K~, K* given by (43) for each model. The solution (N1, N2) of each model falling in the three cases (Sections 1 and 2) asymptotically tends to the
804
V.P. SHUKLA
corresponding equilibrium positions under the modified conditions (42) in the whole positive quadrant o f (Na, N2) state plane. The approach of stability analysis in this article differs from Wangersky and Cunningham (1954), Goel et al. (1971) and Bojadziev and Chan (1979) in the sense that it studies the nonlinear models to give the global stability results in terms of instantaneous and delay interaction coefficients, while the investigations b y the abovementioned authors analyse the stability of approximated preypredator systems around the equilibrium state and give the stability conditions for small and significant time lags. It can be noted from the stability conditions (13), (16) and (18) for p r e y predator systems, conditions (28), (32) and (34) for competition systems, and condition (42) for mutualistic systems that delayed interactions may destabilize the otherwise stable population systems. However, when instantaneous interactions are strong enough to overpower the destabilizing effect of time delay, the system can remain stable. Also, we see that for preypredator systems, condition (18) in case 3 is more restrictive than conditions (13) and (16) in cases 1 and 2. The reason for this is attributed to the increased complexity of both instantaneous and delayed preypredator interactions in case 3. However, the stability of similar models with competitive and mutualistic interactions demand all the three inequalities in either of conditions (28), (32), (34) and (42) to be satisfied; no such straightforward effect o f complexity due to delayed interactions can be reported. Nevertheless the quantitative effect of complexity o f interactions in competition and symbiosis models can be observed by the corresponding sufficient conditions for stability [(28), (32), (34) and (42)].
5. R e m a r k s and R e s u l t s .
This work has been supported b y R. A. scholarship o f the Indian Institute of Technology, Kanpur, India. LITERATURE Beddington, J. R. and R. M. May. 1975. "Time Delays Are Not Necessarily Destabilizing." Math. Bios ci. 27, 109117. Bellman, R. and K. L. Cooke. 1963. Differential Difference Equations. New York: Academic Press. Bojadziev, G. and S. Chart. 1979. "Asymptotic Solutions of Differential Equations with Time Delay in Population Dynamics." Bull. math. BioL 41,325342. Bush, A. W. and A. E. Cook. 1976. "The Effect of Time Delay and Growth Rate Inhibition in the Bacteria Treatment of Wastewater." J. theor. Biol. 63,385395. CasweU, H. 1972. "A Simulation of a Time Lag Population Model." J. theor. Biol. 34, 419439. Goel, N. S., R. S. Maitra and R. S. Montrol. 1971. Nonlinear Models o f interacting Populations. New York: Academic Press.
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Hale, J. 1971. Functional Differential Equations. New York: SpringerVerlag. Hutchinson, G. E. 1948. "Circular Causal Systems in Ecology." Ann. N.Y. Acad. Sci. 50, 221246. MacDonald, N. 1976. 'q?imedelay in PreyPredator Models." Math. Biosei. 28, 321330. MacDonald, N. 1978. Time Lags in Biological Models. New York: SpringerVerlag. May, R. M. 1973. Stability and Complexity in Model Ecosystems. Princeton: Princeton U.P. Smith, M. J. 1974. Models in Ecology. Cambridge: Cambridge U.P. Wangersky, P. J. and W. J. Cunningham. 1954. "Time Lags in Population Models." Cold Spring Harbor Syrup. Quant. Biol. 22, 329338. Volterra, V. 1931. Lemons sur la Theorie Math6matique de la Lutte pour la Vie, GauthierVillars, Paris. R E C E I V E D 21682