Conditions for global stability of two-species population models with discrete time delay

Conditions for global stability of two-species population models with discrete time delay

Bulletin of Mathernatieal Biology Vol.45, No. 5, pp. 793-805, 1983. Printedin GreatBritain 0092-8240/8353.00 + 0.00 PergamonPressLtd. © 1983 Society...

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Bulletin of Mathernatieal Biology Vol.45, No. 5, pp. 793-805, 1983.

Printedin GreatBritain

0092-8240/8353.00 + 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, Kanpur-208016, 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, Lotka-Volterra 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 Volterra-Gause-Witt (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. 106-109), based on a theorem on eigenvalues given in Bellman and Cooke (1963, pp. 449-450). 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 Krylov-Bogoliubov-Mitropolski perturbation method based on the small Present address: Research Officer, Computer Division, Central Water & Power Research Station, Pune-411024, India. 793

794

V.P. SHUKLA

parameter expansion. It has been shown by the authors that the prey-predator 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 prey-predator systems with discrete time lags, and efforts have not been made to study the global stability of the prey-predator and other population systems. In this paper we consider several two-species prey-predator, 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 Lotka-Volterra prey-predator 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 non-negative constants with their usual meanings and T is the constant time lag in the interactions. System (2) describes many prey-predator 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)~- (°/12Jl-fli2)(o~2i-4-~21) e1(o~21 -I-f121)- 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 TWO-SPECIESPOPULATIONMODELS

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)rt2-t-810q12 n2 (t-- T)}

--|Cl~12nl?12(t--T) + 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 3t-c2a21_[_c1/~12 ~22 __C2 ~11~21 - - .1 nln2. a22 al 1 ..]

(10)

The expression on the right-hand 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. 69-70, Theorem 13.1) for various prey-predator 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 el-a22

/~11~21 c2~=0 ~II

(12)

CONDITIONS FOR GLOBAL STABILITY OF TWO-SPECIES 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 interspecies-delayed 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 prey-predator 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 prey-predator 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 prey-predator 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 prey-predator 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 non-negative 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 TWO-SPECIES 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 I--T, 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]~(t-Z

(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 TWO-SPECIES 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 self-competition 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 self-competition 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 TWO-SPECIES 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 right-hand 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 above-mentioned authors analyse the stability of approximated prey-predator 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 prey-predator 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 prey-predator 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, 109-117. 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 41,325-342. 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. 63,385-395. CasweU, H. 1972. "A Simulation of a Time Lag Population Model." J. theor. Biol. 34, 419-439. Goel, N. S., R. S. Maitra and R. S. Montrol. 1971. Nonlinear Models o f interacting Populations. New York: Academic Press.

CONDITIONS FOR GLOBAL STABILITY OF TWO-SPECIESPOPULATION MODELS

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Hale, J. 1971. Functional Differential Equations. New York: Springer-Verlag. Hutchinson, G. E. 1948. "Circular Causal Systems in Ecology." Ann. N.Y. Acad. Sci. 50, 221-246. MacDonald, N. 1976. 'q?ime-delay in Prey-Predator Models." Math. Biosei. 28, 321-330. MacDonald, N. 1978. Time Lags in Biological Models. New York: Springer-Verlag. 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, 329-338. Volterra, V. 1931. Lemons sur la Theorie Math6matique de la Lutte pour la Vie, GauthierVillars, Paris. R E C E I V E D 2-16-82