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Persistence and Global Stability in a Population Model
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
224, 59]80 Ž1998.
AY985984
Persistence and Global Stability in a Population Model K. Gopalsamy U and Pingzhou Liu Department of Mathematics and Statistics, The Flinders Uni¨ ersity of South Australia, G.P.O. Box 2100, Adelaide 5001, Australia Submitted by G. F. Webb Received September 27, 1996
A difference equation modelling the dynamics of a population undergoing a density-dependent harvesting is considered. A sufficient condition is established for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium. Elementary methods of differential calculus are used. The result of this article provides a generalization of a result known for a simpler special model with no harvesting. Q 1998 Academic Press
1. INTRODUCTION This article is concerned with the dynamics of a logistically growing population subjected to a density-dependent harvesting modelled by the following autonomous differential equation with a piecewise constant argument: 1
dN Ž t .
NŽ t.
dt
s r 1 y aN Ž t . y bN Ž w t x . 4 ,
t / 0, 1, 2, . . . , t g Ž 0, ` . .
Ž 1.1. It is assumed in Ž1.1. that N Ž t . denotes the biomass Žor population density. of a single species where r, a, b denote positive numbers and w t x denotes the integer part of t g Ž0, `.. Equation Ž1.1. in its present form is not convenient for further study and we shall obtain an equivalent difference * E-mail: [email protected]. 59 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
60
GOPALSAMY AND LIU
equation. For t g w n, n q 1., n s 0, 1, 2, . . . , one can rewrite Eq. Ž1.1. as follows: dN Ž t . dt
s rN Ž t . Ž 1 y bN Ž n . . y aN Ž t . 4 , t g w n, n q 1 . , n s 0, 1, 2, . . . .
Ž 1.2.
It follows from Ž1.2. that N Ž t . s N Ž n . exp r
t
½ HŽ n
1 y aN Ž s . y bN Ž n . . ds ,
n F t - n q 1.
5
Ž 1.3. It is found from Ž1.3. that N Ž n. ) 0 « N Ž t . ) 0
for t g w n, n q 1 . , n s 0, 1, 2, . . . .
Allowing t ª n q 1, in Ž1.3., we note that N Ž n q 1. ) 0 and by an inductive procedure it can be shown that N Ž t . ) 0 for t g w0, `., if N Ž0. ) 0. Eq. Ž1.1. can be integrated on intervals of the form w n, n q 1.; first we rewrite Ž1.1. as follows: d dt
½
1 NŽ t.
5
exp r 1 y bN Ž n . t 4 s ar exp r 1 y bN Ž n . t 4 , t g w n, n q 1 . .
Ž 1.4.
Integrating both sides of Ž1.4. with respect to t on w n, t ., one obtains NŽ t. s
N Ž n . exp r 1 y bN Ž n . Ž t y n . 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . Ž t y n . 4 y 1 . r Ž 1 y bN Ž n . . . n F t - n q 1.
,
Ž 1.5.
We let t ª n q 1 in Ž1.5. and obtain N Ž n q 1. s
N Ž n . exp r 1 y bN Ž n . 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n .
4
4 y 1 . r Ž 1 y bN Ž n . . . n s 0, 1, 2, . . . .
,
Ž 1.6.
It is now possible to obtain N Ž1., N Ž2., N Ž3., . . . from Ž1.6. which together with Ž1.5. provides a solution of Ž1.1.. Such a solution process does not, however, reveal the rich dynamical characteristics and the asymptotic behaviour of the dynamical system Ž1.1. for a variety of parameter values
61
PERSISTENCE AND GLOBAL STABILITY
of r, a, b. It is the purpose of this article to study in detail the discrete dynamical system Ž1.6.. First we rescale the dependent variable N by setting bN Ž n . ' x Ž n . so that Ž1.6. simplifies to x Ž n q 1. s
x Ž n . exp r 1 y x Ž n .
4 , 1 q a x Ž n . Ž Ž exp r 1 y x Ž n . 4 y 1 . r Ž 1 y x Ž n . . . n s 0, 1, 2, . . . ,
Ž 1.7.
where a s arb; note that b / 0 since b s 0 results in the trivial dynamics of Ž1.1. leading to those of the elementary logistic equation. The equilibria of Ž1.7. are the roots of xs
x exp r Ž 1 y x . 1 q a x Ž Ž exp r Ž 1 y x . y 1 . r Ž 1 y x . .
Ž 1.8.
and Ž1.8. simplifies to x xŽ1 q a . y 1
ž
exp r Ž 1 y x . y 1 1yx
/
s 0,
r g Ž 0, ` . . Ž 1.9.
Thus the nonnegative equilibria are given by xUo and xU where xUo s 0
and
xU s
1 1qa
.
Ž 1.10.
In the following we establish the persistence of the species under the type of harvesting considered and also obtain conditions for the global asymptotic stability of the positive equilibrium of the discrete dynamic system Ž1.7.. Equation Ž1.1. is a differential equation with a piecewise constant argument. We refer to the monographs of Wiener w19x and Carvalho and Cooke w2x for a discussion of differential equations with piecewise constant arguments. We remark that the initial condition needed for the solution of Ž1.1. is a number and hence Ž1.1. is not in the category of delay differential equations. For an extensive discussing of the stability of delay differential equations Žincluding piecewise constant arguments. modelling the dynamics of interacting populations, we refer to the monograph of Gopalsamy w7x.
62
GOPALSAMY AND LIU
2. PERSISTENCE AND LINEAR STABILITY One of the important problems in the study of the dynamics of populations subjected to harvesting pressures is related to the persistence of the population. By persistence of a population governed by Ž1.7., we mean the following: x Ž 0 . ) 0 « lim inf x Ž n . ) 0. nª`
Thus if persistence holds then the population is not driven to eventual extinction by harvesting or intraspecific competition. In this section we show first that the population governed by Ž1.7. persists and then discuss the linear stability of the positive equilibrium. For convenience of notation, we reformulate Eq. Ž1.7. as follows: x Ž n q 1 . s fa Ž r , x Ž n . . ,
n s 0, 1, 2, . . . ,
Ž 2.1.
x g w 0, ` . ,
Ž 2.2.
where fa Ž r , x . s
x exp r Ž 1 y x .
Fa Ž r , x . s 1 q a x
Fa Ž r , x .
ž
,
exp r Ž 1 y x . y 1 1yx
/
,
x g w 0, ` . .
Ž 2.3.
We have from Ž2.2. and Ž2.3. that Fa Ž r , x . fa Ž r , x . s x exp r Ž 1 y x . .
Ž 2.4.
Differentiating both sides of Ž2.4. with respect to x where Ž drdx . fa Ž r, x . s faX Ž r, x ., Fa Ž r , x . faX Ž r , x . q FaX Ž r , x . fa Ž r , x . s Ž 1 y rx . exp r Ž 1 y x . . Ž 2.5. If we let x s 0 in Ž2.5., then we have Fa Ž r , 0 . faX Ž r , 0 . q FaX Ž r , 0 . fa Ž r , 0 . s e r , which with Ž2.2. and Ž2.3. leads to faX Ž r , 0 . s e r ,
a g w 0, ` . .
Ž 2.6.
We let x s xU s 1rŽ1 q a . in Ž2.5. and obtain Fa Ž r , xU . faX Ž r , xU . q FaX Ž r , xU . fa Ž r , xU . s Ž 1 y rxU . exp r Ž 1 y x . ,
63
PERSISTENCE AND GLOBAL STABILITY
which again with Ž2.2. and Ž2.3. leads to faX Ž r , xU . s
exp yr ar Ž 1 q a . Ž 1 q a . y 1
a
.
Ž 2.7.
The value of faX Ž r, xU . obtained in Ž2.7. will be used later in this section to discuss the linear stability of the equilibrium xU . We now establish the persistence of the population governed by Ž2.1. ] Ž2.3.. THEOREM 2.1. Let r g Ž0, `. and a g w0, `.. Let x Ž n. denote any solution of Ž2.1. ] Ž2.3.. If x Ž0. ) 0 then lim inf x Ž n . ) 0. nª`
Ž 2.8.
Proof. It is sufficient to show that there exists e such that if x Ž j . g Ž0, e . for some positive integer j, then x Ž j q k . ) e for a positive integer k. Figure 1 will enable us to clarify the idea behind the proof. We know that faX Ž r, 0. ) 1 and, by the continuity of faX Ž r, ? ., it follows that there exists e 0 ) 0 such that faX Ž r, x . ) 1 for x g Ž0, e 0 .. Let mŽ a , r . s sup x ) 0 fa Ž r, x . and let e 1 be the smallest positive number such that fa Ž r, e 1 . s fa Ž r, m.. As fa Ž r, x . is continuous with respect to x on w0, `. and furthermore f Ž x . ª 0 as x ª `, mŽ a , r . is finite for any positive number of a and r. It follows that fa Ž r, x . ) e 1 for x g w e 1 , m x Žit is possible for e 1 s m.. Now choose e s min e 0 , e 14 . Suppose now that x Ž j . - e . Then we have by using the mean value theorem of differential calculus, x Ž j q 1 . s fa Ž r , x Ž j . . s x Ž j . faX Ž r , j j . ,
FIGURE 1.
j j g Ž 0, x Ž j . . ; Ž 2.9.
64
GOPALSAMY AND LIU
if x Ž j q 1. ) e we are finished; otherwise, we can repeat the above procedure and obtain x Ž j q 2 . s x Ž j q 1 . faX Ž x, j jq1 . ,
j jq1 g Ž 0, x Ž j q 1 . . . Ž 2.10.
Continuing this procedure, jqky1
xŽ j q k. s xŽ j.
½Ł isj
faX Ž r , j i . ,
5
j i g Ž 0, x Ž j q i . . . Ž 2.11.
Since faX Ž r, j i . ) 1, it will follow that there exists a k for which x Ž j q k . ) e and once x Ž j q k . ) e then x Ž n. ) e for n G j q k. The persistence in the sense of Ž2.8. follows and this completes the proof. We now proceed to examine the linear stability of the positive equilibrium of Ž2.1. given by xU s 1rŽ1 q a .. It is well known Žsee, for instance, ¸cak w9x. that the positive equilibrium of Ž2.1. Sandefur w16x or Hale and Ko is linearly stable if < faX Ž r, xU .< - 1 or, equivalently, when a ) 0,
Ž 1 q a . exp yr ar Ž 1 q a . y1 a
- 1;
we note that the above inequality can be simplified to the form 1ya 1qa
- exp Ž yr ar1 q a . - 1
Ž 2.12.
It is easily seen from Ž2.12. that if a G 1, then the inequality Ž2.12. holds for all r g Ž0, `.. Hence let us consider in the following a g w0, 1.. For this case Ž2.12. holds if and only if r-
ž
1qa
a
/ ž ln
1qa 1ya
/
,
a g w 0, 1 . .
Ž 2.13.
It is known from the linear variational system corresponding to xU s 1rŽ1 q a . given by y Ž n q 1 . s faX Ž r , xU . y Ž n .
Ž 2.14.
that the convergence of y Ž n. to 0 as n ª ` is monotonic Žnonoscillatory. if 0 - faX Ž r , xU . - 1
Ž 2.15.
y1 - faX Ž r , xU . - 0.
Ž 2.16.
and oscillatory if
65
PERSISTENCE AND GLOBAL STABILITY
We note from the nature of faX Ž r, xU . that when xU is linearly stable, the eventual convergence of solutions to xU is monotonic if
r-
ž
1qa
a
/
ln Ž 1 q a .
Ž 2.17.
and damped oscillatory if
ž
1qa
a
/
ln Ž 1 q a . - r -
ž
1qa
a
/ ž ln
1qa 1ya
/
.
Ž 2.18.
Some of these observations are illustrated in the plane of the parameters a and r in Fig. 2.
3. GLOBAL STABILITY Derivation of necessary and sufficient conditions for the global asymptotic stability of positive equilibria of discrete dynamical systems modelling single-species dynamics is quite involved in most cases. For a number of special cases of systems having globally asymptotically stable equilibria, we refer to Cull w4x, and Kocic and Ladas w11x, and Agarwal w1x. While a necessary condition for global asymptotically stability is easily obtained
FIG. 2. Regions of linear stability and instability of xU . I: monotonic damping; II: damped oscillation; III: unstable.
66
GOPALSAMY AND LIU
from a linear variational system, establishing that the necessary condition is also sufficient for the global asymptotic stability is usually difficult. In this section we establish sufficient conditions for all positive solutions of
x Ž n q 1. s
x Ž n . exp r 1 y x Ž n .
4 , 1 q a x Ž n . Ž exp r 1 y x Ž n . 4 y 1 . r Ž 1 y x Ž n . . 4 n s 0, 1, 2, . . . , a g Ž 0, ` . , r g Ž 0, ` . ,
Ž 3.1.
to converge to the positive equilibrium xU s 1rŽ1 q a .. The result will be established in two portions, corresponding to a g w1, `. and a g Ž0, 1. respectively. THEOREM 3.1. Suppose that a g w1, `., r g Ž0, `.. Then the positi¨ e equilibrium xU s 1rŽ1 q a . of Ž3.1. is globally asymptotically stable. Proof. We consider a Lyapunov function V Ž n. defined by V Ž n . s x Ž n . y xU , 2
n s 0, 1, 2, . . . .
Ž 3.2.
We calculate the change DV Ž n. in V Ž n. along the solutions of Ž3.1.: DV Ž n . s V Ž n q 1 . y V Ž n . s V Ž x . Ž n q 1 . y V Ž x . Ž n . s x Ž n q 1 . y xU
2
y x Ž n . y xU
s x Ž n q 1 . q x Ž n . y 2 xU
2
x Ž n q 1. y x Ž n . .
Ž 3.3.
It is found from Ž3.1. that
x Ž n q 1. y x Ž n . s s
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . . x Ž n. Fa Ž r , x Ž n . .
4
y x Ž n.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
= Ž 1 q a . xU y x Ž n . ,
Ž 3.4.
67
PERSISTENCE AND GLOBAL STABILITY
where Fa Ž r, x Ž n.. is defined by Ž2.3.. Again we derive from Ž3.1. x Ž n q 1 . q x Ž n . y 2 xU s s
x Ž n . exp r 1 y x Ž n .
4
q x Ž n . y 2 xU
Fa Ž r , x Ž n . . 1 Fa Ž r , x Ž n . .
x Ž n . exp r 1 y x Ž n .
½
= 1 q a x Ž n. s
1 Fa Ž r , x Ž n . .
ž
4q
exp r 1 y x Ž n .
x Ž n . Ž exp r 1 y x Ž n .
exp r 1 y x Ž n .
x Ž n. 1ya
=
1qa
y
4 y1
1 y x Ž n.
Fa Ž r , x Ž n . .
2
4 y1
1 y x Ž n.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
Ž 1 q a . xU y x Ž n . 1 y x Ž n.
ž
/5
4 y 1 . q 2 x Ž n . y xU
q x Ž n . y 2 xU a x Ž n . s
x Ž n . y 2 xU
Ž 1 q a . x Ž n . exp r 1 y x Ž n . 4 y 1
/
.
Ž 3.5.
Now combining Ž3.4. and Ž3.5., we obtain DV Ž n . s x Ž n q 1 . y x Ž n . s
ž
F0
exp r 1 y x Ž n .
x Ž n.
1 y x Ž n.
Fa Ž r , x Ž n . .
=
1ya 1qa
x Ž n q 1 . q x Ž n . y 2 xU
y
2
ž
4 y1
2 U
Ž 1 q a . x y x Ž n.
1 y x Ž n.
Ž 1 q a . x Ž n . exp r 1 y x Ž n . 4 y 1
for a G 1.
/
/ Ž 3.6.
We have from Ž3.6. that for a G 1, DV Ž n. F 0; also we know from the definition of V that V Ž n. is nonnegative; it follows from Ž3.6. that V Ž n. is nonincreasing for n s 0, 1, 2, 3 . . . . It is known from LaSalle’s invariance principle w12x that positive solutions of Ž3.1. approach, as n ª `, the largest invariant set M contained in the set x g Ž0, `.< DV Ž n. s 04 and such a set consists of the points 0 and
68
GOPALSAMY AND LIU
xU . By the persistence result of Theorem 2.1, we know that 0 cannot be approached. It follows that x Ž0. ) 0 « x Ž n. ª xU as n ª `. Combining this with the local stability results in the previous section, we get the conclusion of the theorem. The next result is concerned with the case a g Ž0, 1.; note that a s arb and hence this case corresponds to that of the negative feedback due to harvesting dominating the intraspecific self-regulating negative feedback of the logistic growth. THEOREM 3.2.
Suppose a g Ž0, 1. and r g Ž0, `.. If
rF
1
a
ln Ž 1 q 2 a . q ln
ž
1qa 1ya
/
,
Ž 3.7.
then the positi¨ e equilibrium xU s 1rŽ1 q a . of Ž3.1. is globally asymptotically stable. Proof. We note that the local stability of xU under condition Ž3.7. is valid from the previous section and the relation 1
a
ln Ž 1 q 2 a . q ln
ž
1qa 1ya
F
/ ž
1qa
a
/ ž ln
1qa 1ya
/
,
0 - a - 1.
Ž 3.8. The validity of Ž3.8. can be seen from the following. Let r and w be as follows:
rŽ a. s wŽ a. s
1
a
ž
ln Ž 1 q 2 a . q ln
1qa
a
/ ž ln
1qa 1ya
ž /
1qa 1ya .
It can be found that lim r Ž a . s limq w Ž a . s 2,
aª0 q
a ª0
lim r Ž a . s limy w Ž a . s q`.
aª1 y
a ª1
/
,
69
PERSISTENCE AND GLOBAL STABILITY
We have also
wŽ a. y rŽ a. s s
1qa
ž
a
1
a
G0
ln
ž ž
1qa
/ ž ln
1ya
1qa 1ya
/
1
a
ln Ž 1 q 2 a . y ln
ž
1qa 1ya
/
y ln Ž 1 q 2 a .
1qa
since
/
y
1ya
) 1 q 2 a and a g Ž 0, 1 . ,
/
from which the relation Ž3.8. follows. In order to prove the global asymptotic stability of xU , we only need to prove the global attractivity of xU . We consider again the Lyapunov function V Ž n. s V Ž x .Ž n. defined by V Ž x . Ž n . s x Ž n . y xU
2
Ž 3.9.
and obtain from Ž3.1. and Ž3.9. the following: DV Ž n . s V Ž x . Ž n q 1 . y V Ž x . Ž n . s x Ž n q 1 . y xU
2
y x Ž n . y xU
s x Ž n q 1 . q x Ž n . y 2 xU x Ž n q 1. y x Ž n . s
x Ž n.
2
x Ž n q 1 . y x Ž n . , Ž 3.10.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
Fa Ž r , x Ž n . .
= Ž 1 q a . xU y x Ž n . , x Ž n q 1 . q x Ž n . y 2 xU s
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . .
4
Ž 3.11. q x Ž n . y 2 xU .
Ž 3.12. It follows from Ž3.10. ] Ž3.12. that DV Ž x . Ž n . s
½
x Ž n. Fa Ž r , x Ž n . .
ž
exp r 1 y x Ž n .
4 y1
1 y x Ž n. = Ž 1 q a . xU y x Ž n .
=
½
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . .
4
/
5
q x Ž n . y 2 xU
5
if x Ž n . G 2 xU .
F0
Ž 3.13.
70
GOPALSAMY AND LIU
From Ž3.13., we note that we have to investigate the sign of DV Ž x .Ž n. for x Ž n. g Ž0, 2 xU .. Proceeding further on from Ž3.12., one can simplify Ž3.12. to obtain x Ž n q 1 . q x Ž n . y 2 xU s
x Ž n.
1 y Ž 1 q a . x Ž n.
Fa Ž r , x Ž n . .
1 y x Ž n.
1ya
=
1qa
exp r 1 y x Ž n .
4y
2 y Ž 1 q a . x Ž n.
Ž 3.14.
Ž 1 q a . x Ž n.
so that DV Ž x .Ž n. is given by DV Ž x . Ž n . s
ž
1 y x Ž n.
Fa2 Ž r , x Ž n . .
=
ž
4 y1
exp r 1 y x Ž n .
x 2 Ž n.
1
1ya
1 y x Ž n.
1qa y
1 y Ž 1 q a . x Ž n.
exp r 1 y x Ž n .
2
/
4
2 y Ž 1 q a . x Ž n.
Ž 1 q a . x Ž n.
/
.
Ž 3.15.
It is found from Ž3.15. that the sign of DV Ž x .Ž n. for x Ž n. g Ž0, 2 xU . is determined by that of W Ž x .Ž n. where W Ž x . Ž n. s
1
1ya
1 y x Ž n.
1qa
y
exp r 1 y x Ž n .
2 y Ž 1 q a . x Ž n.
Ž 1 q a . x Ž n.
4 x Ž n . g Ž 0, 2 xU . . Ž 3.16.
,
We examine the sign of W Ž x .Ž n. for x Ž n. g Ž0, 2 xU . in several stages. Ži. x Ž n. g Ž0, xU .. On this interval the sign of W Ž x .Ž n. is governed by that of PŽ x. s s
1ya 1qa
ž
exp r Ž 1 y x . y
1ya 1qa
/½
1 y Ž1 q a . x
Ž1 q a . x
exp r Ž 1 y x . y =
ž
ž
1qa
/ /ž
1ya
2 y Ž1 q a . x 2 y Ž1 q a . x q 2 a
2 x
y1
/5
.
Ž 3.17.
71
PERSISTENCE AND GLOBAL STABILITY
This P Ž x . F 0 is equivalent to 1
1qa
2 y Ž1 q a . x
1ya
2 y Ž1 q a . x q 2 a
½ž / ž /ž
rF
ln
1yx
q
ž
1 1yx
/ ž ln
2
y1 ,
x g 0,
ž
/
x
1
/5
Ž 3.18.
.
1qa
As x g Ž0, 1rŽ1 q a .x, the first term on the right-hand side of Ž3.18. is greater than ln
½ž
1qa 1ya
/ž
1 1 q 2a
/5
.
Ž 3.19.
We let hŽ x . s
ž
1 1yx
2
y1 ,
Ž 3.20.
lim h Ž x . s 2.
Ž 3.21.
/ ž ln
x
/
and note that lim h Ž x . s `,
xª0 q
xª1 y
By Taylor expansion, hŽ x . s s
ž ž
s2
1 1yx 1 1yx `
/ /
ln Ž 2 y x . y ln x 4 ln 1 q Ž 1 y x . y ln 1 y Ž 1 y x .
Ž1 y x.
Ý
ky 1
.
k!
ks1
4 Ž 3.22.
Since 0 - 1 y x - 1 for 0 - x - 1rŽ1 q a ., we obtain that X
h Ž x . s y2
`
Ž k y 1. Ž 1 y x .
Ý
k!
ks2
ky 2
- 0.
Ž 3.23.
Using the decreasing nature of h, hŽ x . ) h
ž
1 1qa
s
/ ž
1qa
a
/
ln Ž 1 q 2 a . ,
xg
ž
1 1qa
/
,1 .
72
GOPALSAMY AND LIU
Thus we have verified that ln
½ž
1qa 1ya F
ž
/ž
1 1 q 2a
1 1yx
/½ ž ln
/5 ž q
1qa
1qa
a
/ž
1ya
/
ln Ž 1 q 2 a .
2 y Ž1 q a . x 2 y Ž1 q a . x q 2 a qln
2
ž
x
y1
/5
/ ,
0-x-
1 1qa
.
Hence if r-
1
a
ln Ž 1 q 2 a . q ln
ž
1qa
/
1ya
,
then P Ž x . F 0 for x g Ž0, xU .. Žii. x Ž n. g Ž xU , 1.. Again we note that the sign of W Ž x .Ž n. is governed by that of P Ž x . for x g Ž1rŽ1 q a ., 1.; we rewrite P Ž x . as follows: PŽ x. s
1 y a 2 y Ž1 q a . x 1qa =
Ž1 q a . x
1qa Ž1 q a . x exp r Ž 1 y x . y 2 y Ž1 q a . x 1ya
Ž 3.24.
and examine the sign of q Ž x . where Žnote that 1 - Ž1 q a . x - 2 since a g Ž0, 1.. qŽ x. s
1qa Ž1 q a . x exp r Ž 1 y x . y , 2 y Ž1 q a . x 1ya
xg
1 1qa
,1 ;
Ž 3.25. it follows from Ž3.25. and our hypothesis on r that q
ž
1 1qa
/
s exp s
q Ž 1. s
ž
ra 1q
a/
y
1qa 1ya 1ya 1qa 1qa 1ya
y
1qa 1ya
exp
1qa 1ya
ž
ra 1qa
s 0.
/
y 1 - 0,
Ž 3.26.
73
PERSISTENCE AND GLOBAL STABILITY
We will now show that q Ž x . - 0 for x g Ž1rŽ1 q a ., 1.; we consider two cases corresponding to r g Ž 0, 2 Ž 1 q a .
r g 2Ž 1 q a . ,
ž
and
1qa
a
ln
ž
1qa 1ya
//
.
Suppose that r g Ž0, 2Ž1 q a ..; it is found from the definition of q that qX Ž x . s
Ž 1 q a . exp r Ž 1 y x . 2 y Ž1 q a . x
)0
for x g
qX Ž x . s
ž
2 y 2 rx q r Ž 1 q a . x 2
2
1 1qa
, 1 and r g Ž 0, 2 Ž 1 q a . . , Ž 3.27.
/
Ž 1 q a . exp 2 Ž 1 y x . 2 y Ž1 q a . x
)0
for x g
ž
Ž1 q a . x y 1
2
1 1qa
2
, 1 , r s 2Ž 1 q a . .
/
Ž 3.28.
It follows from Ž3.24. ] Ž3.28. that q Ž x . - 0 and hence P Ž x . - 0 for x g Ž1rŽ1 q a ., 1. and r g Ž0, 2Ž1 q a ... We proceed to consider the sign of P Ž x . for xg
ž
1 1qa
,1
/
r g 2Ž 1 q a . ,
ž
and
1qa
a
ln
1qa
ž
1ya
//
.
Note that P
ž
1 1qa
/
s
1ya 1qa
exp
ž
ra 1qa
/
y 1 - 0,
P Ž 1 . s 0, Ž 3.29.
our strategy is to show that P cannot have local maxima on Ž1rŽ1 q a ., 1. which with Ž3.29. will show that P Ž x . - 0 for x g Ž1rŽ1 q a ., 1.. We derive from the definition of P in Ž3.24. that PX Ž x . s Y
P Ž x. s
1ya 1qa 1ya 1qa
exp r Ž 1 y x . Ž yr . q exp r Ž 1 y x . r y 2
2
Ž1 q a . x2 4
Ž1 q a . x3
.
Ž 3.30.
74
GOPALSAMY AND LIU
If x provides a local extremum of P on Ž1rŽ1 q a ., 1., then we deduce from Ž3.30. that PY Ž x . s s s ) )
1ya 1qa 1ya 1qa 1ya 1qa 1ya 1qa 1ya 1qa
exp r Ž 1 y x . r 2 y exp r Ž 1 y x . r 2 y
4
Ž1 q a . x3 2 x
exp r Ž 1 y x . r r y
1ya
½
1qa
exp r Ž 1 y x . r
5
2 x
exp r Ž 1 y x . r r y 2 Ž 1 q a . 1
exp r Ž 1 y x . r Ž 1 q a .
a
ln
since x )
1qa 1ya
1 1qa
y 2 ) 0.
Ž 3.31.
Hence any local extremum of P Žif one exists. corresponds to a local minimum only. If P Ž x . ) 0 for some x g Ž1rŽ1 q a ., 1., then since P Ž1rŽ1 q a .. - 0, P Ž1. s 0 will imply that P Ž x . will have to have a local maximum; since local maxima are not possible for P, we conclude that P Ž x . F 0 for x g Ž1rŽ1 q a ., 1.. Our investigation of the sign of W Ž x .Ž n. for x Ž n. g Ž1rŽ1 q a ., 1. is now complete and we have W Ž x . Ž n. F 0
for x g
ž
1 1qa
, 1 and 0 - r -
/
ž
1qa
a
/ž ln
1qa 1ya
/
.
Žiii. x Ž n. s 1. It is found from the definition of W Ž x .Ž n. that W is not defined when x Ž n. s 1; however, W has removable singularity at x Ž n. s 1 and hence we examine the limiting value of W as x Ž n. ª 1. For convenience we write WŽ x. s
1ya
1
Ž1 y x. 1 q a
exp r Ž 1 y x . y
2 y Ž1 q a . x
Ž1 q a . x
By L’Hopital’s rule one can determine that lim W Ž x . s
xª1
F
1ya 1qa
ry
2 1ya
1ya 1qa 1qa
F 0.
a
ln
ž
1qa 1ya
/
y
2 1ya
.
75
PERSISTENCE AND GLOBAL STABILITY
Živ.
x Ž n. g Ž1, 2 xU .. We note from Ž3.16. that for x Ž n . g Ž 1, 2 xU .
DV Ž x . Ž n . F 0
if and only if P Ž x . defined in Ž3.24. satisfies P Ž x . ) 0 for x g Ž1, 2 xU .. We note that P Ž 2 xU . s P
P Ž 1 . s 0,
ž
1 1qa
/
1ya
s
1qa
exp yr
ž
1ya 1qa
/
) 0.
Also we have PX Ž 1. s
1ya
2
ž
y r ) 0.
/
1qa 1ya
If P Ž x . - 0 for some x g Ž1, 2 xU ., then P Žx. must have at least one local maximum point on Ž1, 2 xU . and this is due to the fact that P X Ž1. ) 0, P Ž1. s 0, P Ž2 xU . ) 0. Let x be a local extremum point of P; then P X Ž x . s 0 and PY Ž x . s r 2 s r2 s
ž ž
1ya 1qa 1ya 1qa
r Ž1 y a .
)r
1qa 1ya 1qa
/ /
exp r Ž 1 y x . y exp r Ž 1 y x . y
exp r Ž 1 y x .
ž
4
Ž1 q a . x3
ž
ry
1 y a 2r 1qa 2 x
/
x
exp r Ž 1 y x .
/
exp r Ž 1 y x . Ž r y 2 . .
If r ) 2, we have P Y Ž x . ) 0, which means that P Ž x . has no local maximum and hence P Ž x . ) 0 when r ) 2 and x g Ž1, 2rŽ1 q a ... Hence P Ž x . ) 0 on Ž1, 2 xU .. Next consider the case r F 2. If a local minimum exists let the minimum be at x. Then P X Ž x . s 0 and hence from Ž3.30. PX Ž x . s 0
«
r
1ya 1qa
exp r Ž 1 y x . s
2
Ž1 q a . x2
.
76
GOPALSAMY AND LIU
Then for x ) 1, we have from the definition of P Ž x . in Ž3.17. PŽ x. s G s ) s
2 r Ž1 q a . x 1
Ž1 q a . x 1 1qa 1 1qa 1 1qa
ž ž ž
2
1 x
2
1 x 1 x
2
2
2
y
Ž1 q a . x 2
y
Ž1 q a . x 2
y
x 2
y
x
q1qa q1
q1
q1
/
/
2
y1
) 0.
/
It follows that P Ž x . ) 0 for x g Ž1, 2 xU .. From Ži. ] Živ., we conclude that for x Ž n . g Ž 0, 2 xU . .
DV Ž n . F 0
We have already verified that DV Ž n. F 0 for x Ž n. g w xU , `.. One can now argue as in the case of the proof of Theorem 2.1 and derive that lim x Ž n . s xU s
nª`
1 1qa
.
This completes the proof of Theorem 3.2. The following corollary is concerned with the asymptotic behaviour of positive solutions of Ž1.2. and is a consequence of Theorems 3.1 and 3.2. COROLLARY 3.3. Let r, a, b be positi¨ e numbers. Assume that any one of the following two conditions hold: Ži. a G b; Žii. 0 - a - b and 0-r-
b a
ln 1 q
ž
2b a
q ln
/ ž
Then all positi¨ e solutions of Ž1.2. satisfy lim N Ž t . s
tª`
1 aqb
.
aqb ayb
/
.
77
PERSISTENCE AND GLOBAL STABILITY
Proof. We have from Ž1.6. that any solution of Ž1.2. is given by NŽ n q t. s
N Ž n . exp r 1 y bN Ž n . t 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . t 4 y 1 . r Ž 1 y bN Ž n . . .
,
0 F t F 1, n s 0, 1, 2, . . . . We note from Theorems 3.1 and 3.2 that when the condition of Corollary 3.3 holds, then lim bN Ž n . s xU s
nª`
1 1qa
,
where a s
a b
.
We have from the above that lim N Ž t . s lim N Ž n q t .
tª`
nª`
s lim
nª`
s s
1
N Ž n . exp r 1 y bN Ž n . t 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . t 4 y 1 . r Ž 1 y bN Ž n . . .
ž
1
b 1qa 1 aqb
s
1
/ ž
1
b 1 q arb
/
.
and the proof is completed. A result similar to that of Corollary 3.3, but concerned with an equation containing a more general class of delays, has been established by Cooke and Huang w3x under the assumption a ) b. A condition of this type means that the stabilizing undelayed negative feedback dominates other negative feedbacks with delays Žfor more details of this type of conditions, we refer to Gopalsamy w7x..
4. A FEW COMMENTS We have established that all positive solutions of the discrete dynamic system Ž1.7. converge to the positive equilibrium if a G 1; if a g Ž0, 1., then a sufficient condition for such a convergence is given by the requirement of Ž3.7.. From the proof of Theorem 3.2 we see that the condition Ž3.7. is not necessary. If rs
ž
1qa
a
/ ž ln
1qa 1ya
/
,
a g Ž 0, 1 . ,
Ž 4.1.
78
GOPALSAMY AND LIU
then the linear stability criterion is not applicable since in such a case we have faX Ž r , xU . s y1.
Ž 4.2.
It is, however, known Žfor instance, see Sandefur w16x. that when Ž4.2. holds, if Da Ž xU . s y2 faZ Ž r , xU . y 3 faY Ž r , xU . rs
ž
1qa
a
/ ž ln
1qa 1ya
/
,
2
- 0,
Ž 4.3.
then xU is locally stable. Calculation of the derivatives involved in Ž4.3. is tedious; we have used the Maple symbolic program to calculate Da Ž xU . of Ž4.3. for the case where r satisfies the equality in Ž4.3.. The resulting function Da Ž xU . of the parameter a g Ž0, 1. is plotted using Maple’s plotting procedure and the graph is displayed in Fig. 3. From the graph it is found that Da Ž xU . - 0 for a g Ž0, 1. and hence we conclude that xU of Ž2.1. ] Ž2.3. is asymptotically stable provided rF
ž
1qa
a
/ ž ln
1qa 1ya
/
.
Ž 4.4.
If Ž4.4. is violated then xU is not linearly stable. One can see from Ž2.7. that faX Ž r, xU . - y1 is sufficient for instability of xU . Thus Ž4.4. provides a necessary condition for the asymptotic stability of xU when a g Ž0, 1.. The authors believe that Ž2.13. is sufficient for the global asymptotic stability of xU ; however, they have not been able to prove this.
FIGURE 3.
PERSISTENCE AND GLOBAL STABILITY
79
When a s 0, the system Ž2.1. ] Ž2.3. reduces to the simpler dynamic system x Ž n q 1 . s x Ž n . exp r 1 y x Ž n .
n s 0, 1, 2, . . . .
4,
Ž 4.5.
Several authors Žsee May w13x, May and Oster w14x, Fisher and Goh w6x, Seifert w17, 18x, Norris and Soewono w15x, Guckenheimer, Oster, and Ipaktchi w8x, and Dohtani w5x. have investigated the stability and oscillatory characteristics of Ž4.5.. It is known that if 0 - r F 2, then all positive solutions of Ž4.5. converge to the positive equilibrium of Ž4.5.. One can obtain this special result from our investigation by evaluating the following limit: lim
aª0
ž
1qa
a
/ ž ln
1qa 1ya
/
s 2.
Ž 4.6.
The global asymptotic stability of xU implies that the dynamics of the system has no persistent periodic solutions or other types of complex solutions of the ‘‘chaotic type.’’ In fact, when the requirement in Ž4.1. is violated, period-doubling bifurcations and complex dynamics of the system Ž1.7. emerge and it has not been the intention of the authors to consider this in this article. We conclude with the remark that there exists vast literature concerned with singular perturbations of delay differential equations and their connection with certain associated difference equations with a continuous argument Žfor a survey see Ivanov and Sharkovsky w10x and the references therein .. We remark that the discrete dynamic system Ž1.7. is not derived from the differential equation Ž1.1. through any regular or singular perturbation.
ACKNOWLEDGMENTS The authors wish to thank a referee for suggesting improvements in the proofs of Theorems 2.1 and 3.2.
REFERENCES 1. R. P. Agarwal, ‘‘Difference Equations and Inequalities, Theory, Methods, and Applications,’’ Dekker, New York, 1992. 2. L. A. V. Carvalho and K. L. Cooke, A nonlinear equation with piecewise continuous argument, Differential Integral Equations 1 Ž1988., 359]367. 3. K. L. Cooke and W. Huang, A theorem of George Seifert and an equation with state-dependent delay, in ‘‘Delay and Differential Equation’’ ŽA. M. Fink, K. Miller, and W. Kliemann, Eds.., World Scientific, Singapore, 1991.
80
GOPALSAMY AND LIU
4. P. Cull, Global stability of population models, Bull. Math. Biol. 43 Ž1981., 47]58. 5. A. Dohtani, Occurrence of chaos in higher dimensional discrete time systems, SIAM J. Appl. Math. 52 Ž1992., 1707]1721. 6. M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol. 19 Ž1984., 147]156. 7. K. Gopalsamy, ‘‘Stability and Oscillations in Delay Differential Equations of Population Dynamics,’’ Kluwer Academic, Dordrecht, 1992. 8. J. Guckenheimer, G. Oster, and A. Ipaktchi, The dynamics of density dependent population models, J. Math. Biol. 4 Ž1977., 101]147. 9. J. K. Hale and H. Koc ¸ak, ‘‘Dynamics and Bifurcations,’’ Springer-Verlag, New York, 1991. 10. A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in ‘‘Dynamics Reported’’ ŽC. K. R. T. Jones, U. Kirchgraber, and H.-O. Walther, Eds.., Vol. 1, pp. 164]224, Springer-Verlag, Berlin, 1991. 11. V. K. Kocic and G. Ladas, ‘‘Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,’’ Kluwer Academic, Dordrecht, 1993. 12. J. P. LaSalle, Stability theory for difference equations, in ‘‘Studies in Ordinary Differential Equations,’’ ŽJ. K. Hale, Ed.., Math. Assoc. of America, Washington, DC, 1977. 13. R. M. May, Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theoret. Biol. 51 Ž1975., 511]524. 14. R. M. May and G. F. Oster, Bifurcations and dynamic complexity in simple ecological models, Amer. Natur. 110 Ž1976., 573]599. 15. D. O. Norris and E. Soewono, Period doubling and chaotic behaviour of solutions to X y Ž t . s m y Ž t .w1 y y Ž d wŽ t q a .rd x.x, Appl. Anal. 40 Ž1991., 181]188. 16. J. T. Sandefur, ‘‘Discrete Dynamical Systems, Theory and Applications,’’ Clarendon, Oxford, 1990. 17. G. Seifert, On an interval map associated with delay logistic equation with discontinuous delays, in ‘‘Delay Differential Equations and Dynamical Systems,’’ ŽS. Busenberg and M. Martelli, Eds.., Lecture Notes in Mathematics, Vol. 1475, Springer-Verlag, BerlinrNew York, 1991. 18. G. Seifert, Certain systems with piecewise constant feedback controls with time delay, Differential Integral Equations 6 Ž1993., 937]947. 19. J. Wiener, ‘‘Generalized Solution of Functional Differential Equations,’’ World Scientific, Singapore, 1993.
224, 59]80 Ž1998.
AY985984
Persistence and Global Stability in a Population Model K. Gopalsamy U and Pingzhou Liu Department of Mathematics and Statistics, The Flinders Uni¨ ersity of South Australia, G.P.O. Box 2100, Adelaide 5001, Australia Submitted by G. F. Webb Received September 27, 1996
A difference equation modelling the dynamics of a population undergoing a density-dependent harvesting is considered. A sufficient condition is established for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium. Elementary methods of differential calculus are used. The result of this article provides a generalization of a result known for a simpler special model with no harvesting. Q 1998 Academic Press
1. INTRODUCTION This article is concerned with the dynamics of a logistically growing population subjected to a density-dependent harvesting modelled by the following autonomous differential equation with a piecewise constant argument: 1
dN Ž t .
NŽ t.
dt
s r 1 y aN Ž t . y bN Ž w t x . 4 ,
t / 0, 1, 2, . . . , t g Ž 0, ` . .
Ž 1.1. It is assumed in Ž1.1. that N Ž t . denotes the biomass Žor population density. of a single species where r, a, b denote positive numbers and w t x denotes the integer part of t g Ž0, `.. Equation Ž1.1. in its present form is not convenient for further study and we shall obtain an equivalent difference * E-mail: [email protected]. 59 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
60
GOPALSAMY AND LIU
equation. For t g w n, n q 1., n s 0, 1, 2, . . . , one can rewrite Eq. Ž1.1. as follows: dN Ž t . dt
s rN Ž t . Ž 1 y bN Ž n . . y aN Ž t . 4 , t g w n, n q 1 . , n s 0, 1, 2, . . . .
Ž 1.2.
It follows from Ž1.2. that N Ž t . s N Ž n . exp r
t
½ HŽ n
1 y aN Ž s . y bN Ž n . . ds ,
n F t - n q 1.
5
Ž 1.3. It is found from Ž1.3. that N Ž n. ) 0 « N Ž t . ) 0
for t g w n, n q 1 . , n s 0, 1, 2, . . . .
Allowing t ª n q 1, in Ž1.3., we note that N Ž n q 1. ) 0 and by an inductive procedure it can be shown that N Ž t . ) 0 for t g w0, `., if N Ž0. ) 0. Eq. Ž1.1. can be integrated on intervals of the form w n, n q 1.; first we rewrite Ž1.1. as follows: d dt
½
1 NŽ t.
5
exp r 1 y bN Ž n . t 4 s ar exp r 1 y bN Ž n . t 4 , t g w n, n q 1 . .
Ž 1.4.
Integrating both sides of Ž1.4. with respect to t on w n, t ., one obtains NŽ t. s
N Ž n . exp r 1 y bN Ž n . Ž t y n . 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . Ž t y n . 4 y 1 . r Ž 1 y bN Ž n . . . n F t - n q 1.
,
Ž 1.5.
We let t ª n q 1 in Ž1.5. and obtain N Ž n q 1. s
N Ž n . exp r 1 y bN Ž n . 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n .
4
4 y 1 . r Ž 1 y bN Ž n . . . n s 0, 1, 2, . . . .
,
Ž 1.6.
It is now possible to obtain N Ž1., N Ž2., N Ž3., . . . from Ž1.6. which together with Ž1.5. provides a solution of Ž1.1.. Such a solution process does not, however, reveal the rich dynamical characteristics and the asymptotic behaviour of the dynamical system Ž1.1. for a variety of parameter values
61
PERSISTENCE AND GLOBAL STABILITY
of r, a, b. It is the purpose of this article to study in detail the discrete dynamical system Ž1.6.. First we rescale the dependent variable N by setting bN Ž n . ' x Ž n . so that Ž1.6. simplifies to x Ž n q 1. s
x Ž n . exp r 1 y x Ž n .
4 , 1 q a x Ž n . Ž Ž exp r 1 y x Ž n . 4 y 1 . r Ž 1 y x Ž n . . . n s 0, 1, 2, . . . ,
Ž 1.7.
where a s arb; note that b / 0 since b s 0 results in the trivial dynamics of Ž1.1. leading to those of the elementary logistic equation. The equilibria of Ž1.7. are the roots of xs
x exp r Ž 1 y x . 1 q a x Ž Ž exp r Ž 1 y x . y 1 . r Ž 1 y x . .
Ž 1.8.
and Ž1.8. simplifies to x xŽ1 q a . y 1
ž
exp r Ž 1 y x . y 1 1yx
/
s 0,
r g Ž 0, ` . . Ž 1.9.
Thus the nonnegative equilibria are given by xUo and xU where xUo s 0
and
xU s
1 1qa
.
Ž 1.10.
In the following we establish the persistence of the species under the type of harvesting considered and also obtain conditions for the global asymptotic stability of the positive equilibrium of the discrete dynamic system Ž1.7.. Equation Ž1.1. is a differential equation with a piecewise constant argument. We refer to the monographs of Wiener w19x and Carvalho and Cooke w2x for a discussion of differential equations with piecewise constant arguments. We remark that the initial condition needed for the solution of Ž1.1. is a number and hence Ž1.1. is not in the category of delay differential equations. For an extensive discussing of the stability of delay differential equations Žincluding piecewise constant arguments. modelling the dynamics of interacting populations, we refer to the monograph of Gopalsamy w7x.
62
GOPALSAMY AND LIU
2. PERSISTENCE AND LINEAR STABILITY One of the important problems in the study of the dynamics of populations subjected to harvesting pressures is related to the persistence of the population. By persistence of a population governed by Ž1.7., we mean the following: x Ž 0 . ) 0 « lim inf x Ž n . ) 0. nª`
Thus if persistence holds then the population is not driven to eventual extinction by harvesting or intraspecific competition. In this section we show first that the population governed by Ž1.7. persists and then discuss the linear stability of the positive equilibrium. For convenience of notation, we reformulate Eq. Ž1.7. as follows: x Ž n q 1 . s fa Ž r , x Ž n . . ,
n s 0, 1, 2, . . . ,
Ž 2.1.
x g w 0, ` . ,
Ž 2.2.
where fa Ž r , x . s
x exp r Ž 1 y x .
Fa Ž r , x . s 1 q a x
Fa Ž r , x .
ž
,
exp r Ž 1 y x . y 1 1yx
/
,
x g w 0, ` . .
Ž 2.3.
We have from Ž2.2. and Ž2.3. that Fa Ž r , x . fa Ž r , x . s x exp r Ž 1 y x . .
Ž 2.4.
Differentiating both sides of Ž2.4. with respect to x where Ž drdx . fa Ž r, x . s faX Ž r, x ., Fa Ž r , x . faX Ž r , x . q FaX Ž r , x . fa Ž r , x . s Ž 1 y rx . exp r Ž 1 y x . . Ž 2.5. If we let x s 0 in Ž2.5., then we have Fa Ž r , 0 . faX Ž r , 0 . q FaX Ž r , 0 . fa Ž r , 0 . s e r , which with Ž2.2. and Ž2.3. leads to faX Ž r , 0 . s e r ,
a g w 0, ` . .
Ž 2.6.
We let x s xU s 1rŽ1 q a . in Ž2.5. and obtain Fa Ž r , xU . faX Ž r , xU . q FaX Ž r , xU . fa Ž r , xU . s Ž 1 y rxU . exp r Ž 1 y x . ,
63
PERSISTENCE AND GLOBAL STABILITY
which again with Ž2.2. and Ž2.3. leads to faX Ž r , xU . s
exp yr ar Ž 1 q a . Ž 1 q a . y 1
a
.
Ž 2.7.
The value of faX Ž r, xU . obtained in Ž2.7. will be used later in this section to discuss the linear stability of the equilibrium xU . We now establish the persistence of the population governed by Ž2.1. ] Ž2.3.. THEOREM 2.1. Let r g Ž0, `. and a g w0, `.. Let x Ž n. denote any solution of Ž2.1. ] Ž2.3.. If x Ž0. ) 0 then lim inf x Ž n . ) 0. nª`
Ž 2.8.
Proof. It is sufficient to show that there exists e such that if x Ž j . g Ž0, e . for some positive integer j, then x Ž j q k . ) e for a positive integer k. Figure 1 will enable us to clarify the idea behind the proof. We know that faX Ž r, 0. ) 1 and, by the continuity of faX Ž r, ? ., it follows that there exists e 0 ) 0 such that faX Ž r, x . ) 1 for x g Ž0, e 0 .. Let mŽ a , r . s sup x ) 0 fa Ž r, x . and let e 1 be the smallest positive number such that fa Ž r, e 1 . s fa Ž r, m.. As fa Ž r, x . is continuous with respect to x on w0, `. and furthermore f Ž x . ª 0 as x ª `, mŽ a , r . is finite for any positive number of a and r. It follows that fa Ž r, x . ) e 1 for x g w e 1 , m x Žit is possible for e 1 s m.. Now choose e s min e 0 , e 14 . Suppose now that x Ž j . - e . Then we have by using the mean value theorem of differential calculus, x Ž j q 1 . s fa Ž r , x Ž j . . s x Ž j . faX Ž r , j j . ,
FIGURE 1.
j j g Ž 0, x Ž j . . ; Ž 2.9.
64
GOPALSAMY AND LIU
if x Ž j q 1. ) e we are finished; otherwise, we can repeat the above procedure and obtain x Ž j q 2 . s x Ž j q 1 . faX Ž x, j jq1 . ,
j jq1 g Ž 0, x Ž j q 1 . . . Ž 2.10.
Continuing this procedure, jqky1
xŽ j q k. s xŽ j.
½Ł isj
faX Ž r , j i . ,
5
j i g Ž 0, x Ž j q i . . . Ž 2.11.
Since faX Ž r, j i . ) 1, it will follow that there exists a k for which x Ž j q k . ) e and once x Ž j q k . ) e then x Ž n. ) e for n G j q k. The persistence in the sense of Ž2.8. follows and this completes the proof. We now proceed to examine the linear stability of the positive equilibrium of Ž2.1. given by xU s 1rŽ1 q a .. It is well known Žsee, for instance, ¸cak w9x. that the positive equilibrium of Ž2.1. Sandefur w16x or Hale and Ko is linearly stable if < faX Ž r, xU .< - 1 or, equivalently, when a ) 0,
Ž 1 q a . exp yr ar Ž 1 q a . y1 a
- 1;
we note that the above inequality can be simplified to the form 1ya 1qa
- exp Ž yr ar1 q a . - 1
Ž 2.12.
It is easily seen from Ž2.12. that if a G 1, then the inequality Ž2.12. holds for all r g Ž0, `.. Hence let us consider in the following a g w0, 1.. For this case Ž2.12. holds if and only if r-
ž
1qa
a
/ ž ln
1qa 1ya
/
,
a g w 0, 1 . .
Ž 2.13.
It is known from the linear variational system corresponding to xU s 1rŽ1 q a . given by y Ž n q 1 . s faX Ž r , xU . y Ž n .
Ž 2.14.
that the convergence of y Ž n. to 0 as n ª ` is monotonic Žnonoscillatory. if 0 - faX Ž r , xU . - 1
Ž 2.15.
y1 - faX Ž r , xU . - 0.
Ž 2.16.
and oscillatory if
65
PERSISTENCE AND GLOBAL STABILITY
We note from the nature of faX Ž r, xU . that when xU is linearly stable, the eventual convergence of solutions to xU is monotonic if
r-
ž
1qa
a
/
ln Ž 1 q a .
Ž 2.17.
and damped oscillatory if
ž
1qa
a
/
ln Ž 1 q a . - r -
ž
1qa
a
/ ž ln
1qa 1ya
/
.
Ž 2.18.
Some of these observations are illustrated in the plane of the parameters a and r in Fig. 2.
3. GLOBAL STABILITY Derivation of necessary and sufficient conditions for the global asymptotic stability of positive equilibria of discrete dynamical systems modelling single-species dynamics is quite involved in most cases. For a number of special cases of systems having globally asymptotically stable equilibria, we refer to Cull w4x, and Kocic and Ladas w11x, and Agarwal w1x. While a necessary condition for global asymptotically stability is easily obtained
FIG. 2. Regions of linear stability and instability of xU . I: monotonic damping; II: damped oscillation; III: unstable.
66
GOPALSAMY AND LIU
from a linear variational system, establishing that the necessary condition is also sufficient for the global asymptotic stability is usually difficult. In this section we establish sufficient conditions for all positive solutions of
x Ž n q 1. s
x Ž n . exp r 1 y x Ž n .
4 , 1 q a x Ž n . Ž exp r 1 y x Ž n . 4 y 1 . r Ž 1 y x Ž n . . 4 n s 0, 1, 2, . . . , a g Ž 0, ` . , r g Ž 0, ` . ,
Ž 3.1.
to converge to the positive equilibrium xU s 1rŽ1 q a .. The result will be established in two portions, corresponding to a g w1, `. and a g Ž0, 1. respectively. THEOREM 3.1. Suppose that a g w1, `., r g Ž0, `.. Then the positi¨ e equilibrium xU s 1rŽ1 q a . of Ž3.1. is globally asymptotically stable. Proof. We consider a Lyapunov function V Ž n. defined by V Ž n . s x Ž n . y xU , 2
n s 0, 1, 2, . . . .
Ž 3.2.
We calculate the change DV Ž n. in V Ž n. along the solutions of Ž3.1.: DV Ž n . s V Ž n q 1 . y V Ž n . s V Ž x . Ž n q 1 . y V Ž x . Ž n . s x Ž n q 1 . y xU
2
y x Ž n . y xU
s x Ž n q 1 . q x Ž n . y 2 xU
2
x Ž n q 1. y x Ž n . .
Ž 3.3.
It is found from Ž3.1. that
x Ž n q 1. y x Ž n . s s
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . . x Ž n. Fa Ž r , x Ž n . .
4
y x Ž n.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
= Ž 1 q a . xU y x Ž n . ,
Ž 3.4.
67
PERSISTENCE AND GLOBAL STABILITY
where Fa Ž r, x Ž n.. is defined by Ž2.3.. Again we derive from Ž3.1. x Ž n q 1 . q x Ž n . y 2 xU s s
x Ž n . exp r 1 y x Ž n .
4
q x Ž n . y 2 xU
Fa Ž r , x Ž n . . 1 Fa Ž r , x Ž n . .
x Ž n . exp r 1 y x Ž n .
½
= 1 q a x Ž n. s
1 Fa Ž r , x Ž n . .
ž
4q
exp r 1 y x Ž n .
x Ž n . Ž exp r 1 y x Ž n .
exp r 1 y x Ž n .
x Ž n. 1ya
=
1qa
y
4 y1
1 y x Ž n.
Fa Ž r , x Ž n . .
2
4 y1
1 y x Ž n.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
Ž 1 q a . xU y x Ž n . 1 y x Ž n.
ž
/5
4 y 1 . q 2 x Ž n . y xU
q x Ž n . y 2 xU a x Ž n . s
x Ž n . y 2 xU
Ž 1 q a . x Ž n . exp r 1 y x Ž n . 4 y 1
/
.
Ž 3.5.
Now combining Ž3.4. and Ž3.5., we obtain DV Ž n . s x Ž n q 1 . y x Ž n . s
ž
F0
exp r 1 y x Ž n .
x Ž n.
1 y x Ž n.
Fa Ž r , x Ž n . .
=
1ya 1qa
x Ž n q 1 . q x Ž n . y 2 xU
y
2
ž
4 y1
2 U
Ž 1 q a . x y x Ž n.
1 y x Ž n.
Ž 1 q a . x Ž n . exp r 1 y x Ž n . 4 y 1
for a G 1.
/
/ Ž 3.6.
We have from Ž3.6. that for a G 1, DV Ž n. F 0; also we know from the definition of V that V Ž n. is nonnegative; it follows from Ž3.6. that V Ž n. is nonincreasing for n s 0, 1, 2, 3 . . . . It is known from LaSalle’s invariance principle w12x that positive solutions of Ž3.1. approach, as n ª `, the largest invariant set M contained in the set x g Ž0, `.< DV Ž n. s 04 and such a set consists of the points 0 and
68
GOPALSAMY AND LIU
xU . By the persistence result of Theorem 2.1, we know that 0 cannot be approached. It follows that x Ž0. ) 0 « x Ž n. ª xU as n ª `. Combining this with the local stability results in the previous section, we get the conclusion of the theorem. The next result is concerned with the case a g Ž0, 1.; note that a s arb and hence this case corresponds to that of the negative feedback due to harvesting dominating the intraspecific self-regulating negative feedback of the logistic growth. THEOREM 3.2.
Suppose a g Ž0, 1. and r g Ž0, `.. If
rF
1
a
ln Ž 1 q 2 a . q ln
ž
1qa 1ya
/
,
Ž 3.7.
then the positi¨ e equilibrium xU s 1rŽ1 q a . of Ž3.1. is globally asymptotically stable. Proof. We note that the local stability of xU under condition Ž3.7. is valid from the previous section and the relation 1
a
ln Ž 1 q 2 a . q ln
ž
1qa 1ya
F
/ ž
1qa
a
/ ž ln
1qa 1ya
/
,
0 - a - 1.
Ž 3.8. The validity of Ž3.8. can be seen from the following. Let r and w be as follows:
rŽ a. s wŽ a. s
1
a
ž
ln Ž 1 q 2 a . q ln
1qa
a
/ ž ln
1qa 1ya
ž /
1qa 1ya .
It can be found that lim r Ž a . s limq w Ž a . s 2,
aª0 q
a ª0
lim r Ž a . s limy w Ž a . s q`.
aª1 y
a ª1
/
,
69
PERSISTENCE AND GLOBAL STABILITY
We have also
wŽ a. y rŽ a. s s
1qa
ž
a
1
a
G0
ln
ž ž
1qa
/ ž ln
1ya
1qa 1ya
/
1
a
ln Ž 1 q 2 a . y ln
ž
1qa 1ya
/
y ln Ž 1 q 2 a .
1qa
since
/
y
1ya
) 1 q 2 a and a g Ž 0, 1 . ,
/
from which the relation Ž3.8. follows. In order to prove the global asymptotic stability of xU , we only need to prove the global attractivity of xU . We consider again the Lyapunov function V Ž n. s V Ž x .Ž n. defined by V Ž x . Ž n . s x Ž n . y xU
2
Ž 3.9.
and obtain from Ž3.1. and Ž3.9. the following: DV Ž n . s V Ž x . Ž n q 1 . y V Ž x . Ž n . s x Ž n q 1 . y xU
2
y x Ž n . y xU
s x Ž n q 1 . q x Ž n . y 2 xU x Ž n q 1. y x Ž n . s
x Ž n.
2
x Ž n q 1 . y x Ž n . , Ž 3.10.
exp r 1 y x Ž n .
4 y1
1 y x Ž n.
Fa Ž r , x Ž n . .
= Ž 1 q a . xU y x Ž n . , x Ž n q 1 . q x Ž n . y 2 xU s
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . .
4
Ž 3.11. q x Ž n . y 2 xU .
Ž 3.12. It follows from Ž3.10. ] Ž3.12. that DV Ž x . Ž n . s
½
x Ž n. Fa Ž r , x Ž n . .
ž
exp r 1 y x Ž n .
4 y1
1 y x Ž n. = Ž 1 q a . xU y x Ž n .
=
½
x Ž n . exp r 1 y x Ž n . Fa Ž r , x Ž n . .
4
/
5
q x Ž n . y 2 xU
5
if x Ž n . G 2 xU .
F0
Ž 3.13.
70
GOPALSAMY AND LIU
From Ž3.13., we note that we have to investigate the sign of DV Ž x .Ž n. for x Ž n. g Ž0, 2 xU .. Proceeding further on from Ž3.12., one can simplify Ž3.12. to obtain x Ž n q 1 . q x Ž n . y 2 xU s
x Ž n.
1 y Ž 1 q a . x Ž n.
Fa Ž r , x Ž n . .
1 y x Ž n.
1ya
=
1qa
exp r 1 y x Ž n .
4y
2 y Ž 1 q a . x Ž n.
Ž 3.14.
Ž 1 q a . x Ž n.
so that DV Ž x .Ž n. is given by DV Ž x . Ž n . s
ž
1 y x Ž n.
Fa2 Ž r , x Ž n . .
=
ž
4 y1
exp r 1 y x Ž n .
x 2 Ž n.
1
1ya
1 y x Ž n.
1qa y
1 y Ž 1 q a . x Ž n.
exp r 1 y x Ž n .
2
/
4
2 y Ž 1 q a . x Ž n.
Ž 1 q a . x Ž n.
/
.
Ž 3.15.
It is found from Ž3.15. that the sign of DV Ž x .Ž n. for x Ž n. g Ž0, 2 xU . is determined by that of W Ž x .Ž n. where W Ž x . Ž n. s
1
1ya
1 y x Ž n.
1qa
y
exp r 1 y x Ž n .
2 y Ž 1 q a . x Ž n.
Ž 1 q a . x Ž n.
4 x Ž n . g Ž 0, 2 xU . . Ž 3.16.
,
We examine the sign of W Ž x .Ž n. for x Ž n. g Ž0, 2 xU . in several stages. Ži. x Ž n. g Ž0, xU .. On this interval the sign of W Ž x .Ž n. is governed by that of PŽ x. s s
1ya 1qa
ž
exp r Ž 1 y x . y
1ya 1qa
/½
1 y Ž1 q a . x
Ž1 q a . x
exp r Ž 1 y x . y =
ž
ž
1qa
/ /ž
1ya
2 y Ž1 q a . x 2 y Ž1 q a . x q 2 a
2 x
y1
/5
.
Ž 3.17.
71
PERSISTENCE AND GLOBAL STABILITY
This P Ž x . F 0 is equivalent to 1
1qa
2 y Ž1 q a . x
1ya
2 y Ž1 q a . x q 2 a
½ž / ž /ž
rF
ln
1yx
q
ž
1 1yx
/ ž ln
2
y1 ,
x g 0,
ž
/
x
1
/5
Ž 3.18.
.
1qa
As x g Ž0, 1rŽ1 q a .x, the first term on the right-hand side of Ž3.18. is greater than ln
½ž
1qa 1ya
/ž
1 1 q 2a
/5
.
Ž 3.19.
We let hŽ x . s
ž
1 1yx
2
y1 ,
Ž 3.20.
lim h Ž x . s 2.
Ž 3.21.
/ ž ln
x
/
and note that lim h Ž x . s `,
xª0 q
xª1 y
By Taylor expansion, hŽ x . s s
ž ž
s2
1 1yx 1 1yx `
/ /
ln Ž 2 y x . y ln x 4 ln 1 q Ž 1 y x . y ln 1 y Ž 1 y x .
Ž1 y x.
Ý
ky 1
.
k!
ks1
4 Ž 3.22.
Since 0 - 1 y x - 1 for 0 - x - 1rŽ1 q a ., we obtain that X
h Ž x . s y2
`
Ž k y 1. Ž 1 y x .
Ý
k!
ks2
ky 2
- 0.
Ž 3.23.
Using the decreasing nature of h, hŽ x . ) h
ž
1 1qa
s
/ ž
1qa
a
/
ln Ž 1 q 2 a . ,
xg
ž
1 1qa
/
,1 .
72
GOPALSAMY AND LIU
Thus we have verified that ln
½ž
1qa 1ya F
ž
/ž
1 1 q 2a
1 1yx
/½ ž ln
/5 ž q
1qa
1qa
a
/ž
1ya
/
ln Ž 1 q 2 a .
2 y Ž1 q a . x 2 y Ž1 q a . x q 2 a qln
2
ž
x
y1
/5
/ ,
0-x-
1 1qa
.
Hence if r-
1
a
ln Ž 1 q 2 a . q ln
ž
1qa
/
1ya
,
then P Ž x . F 0 for x g Ž0, xU .. Žii. x Ž n. g Ž xU , 1.. Again we note that the sign of W Ž x .Ž n. is governed by that of P Ž x . for x g Ž1rŽ1 q a ., 1.; we rewrite P Ž x . as follows: PŽ x. s
1 y a 2 y Ž1 q a . x 1qa =
Ž1 q a . x
1qa Ž1 q a . x exp r Ž 1 y x . y 2 y Ž1 q a . x 1ya
Ž 3.24.
and examine the sign of q Ž x . where Žnote that 1 - Ž1 q a . x - 2 since a g Ž0, 1.. qŽ x. s
1qa Ž1 q a . x exp r Ž 1 y x . y , 2 y Ž1 q a . x 1ya
xg
1 1qa
,1 ;
Ž 3.25. it follows from Ž3.25. and our hypothesis on r that q
ž
1 1qa
/
s exp s
q Ž 1. s
ž
ra 1q
a/
y
1qa 1ya 1ya 1qa 1qa 1ya
y
1qa 1ya
exp
1qa 1ya
ž
ra 1qa
s 0.
/
y 1 - 0,
Ž 3.26.
73
PERSISTENCE AND GLOBAL STABILITY
We will now show that q Ž x . - 0 for x g Ž1rŽ1 q a ., 1.; we consider two cases corresponding to r g Ž 0, 2 Ž 1 q a .
r g 2Ž 1 q a . ,
ž
and
1qa
a
ln
ž
1qa 1ya
//
.
Suppose that r g Ž0, 2Ž1 q a ..; it is found from the definition of q that qX Ž x . s
Ž 1 q a . exp r Ž 1 y x . 2 y Ž1 q a . x
)0
for x g
qX Ž x . s
ž
2 y 2 rx q r Ž 1 q a . x 2
2
1 1qa
, 1 and r g Ž 0, 2 Ž 1 q a . . , Ž 3.27.
/
Ž 1 q a . exp 2 Ž 1 y x . 2 y Ž1 q a . x
)0
for x g
ž
Ž1 q a . x y 1
2
1 1qa
2
, 1 , r s 2Ž 1 q a . .
/
Ž 3.28.
It follows from Ž3.24. ] Ž3.28. that q Ž x . - 0 and hence P Ž x . - 0 for x g Ž1rŽ1 q a ., 1. and r g Ž0, 2Ž1 q a ... We proceed to consider the sign of P Ž x . for xg
ž
1 1qa
,1
/
r g 2Ž 1 q a . ,
ž
and
1qa
a
ln
1qa
ž
1ya
//
.
Note that P
ž
1 1qa
/
s
1ya 1qa
exp
ž
ra 1qa
/
y 1 - 0,
P Ž 1 . s 0, Ž 3.29.
our strategy is to show that P cannot have local maxima on Ž1rŽ1 q a ., 1. which with Ž3.29. will show that P Ž x . - 0 for x g Ž1rŽ1 q a ., 1.. We derive from the definition of P in Ž3.24. that PX Ž x . s Y
P Ž x. s
1ya 1qa 1ya 1qa
exp r Ž 1 y x . Ž yr . q exp r Ž 1 y x . r y 2
2
Ž1 q a . x2 4
Ž1 q a . x3
.
Ž 3.30.
74
GOPALSAMY AND LIU
If x provides a local extremum of P on Ž1rŽ1 q a ., 1., then we deduce from Ž3.30. that PY Ž x . s s s ) )
1ya 1qa 1ya 1qa 1ya 1qa 1ya 1qa 1ya 1qa
exp r Ž 1 y x . r 2 y exp r Ž 1 y x . r 2 y
4
Ž1 q a . x3 2 x
exp r Ž 1 y x . r r y
1ya
½
1qa
exp r Ž 1 y x . r
5
2 x
exp r Ž 1 y x . r r y 2 Ž 1 q a . 1
exp r Ž 1 y x . r Ž 1 q a .
a
ln
since x )
1qa 1ya
1 1qa
y 2 ) 0.
Ž 3.31.
Hence any local extremum of P Žif one exists. corresponds to a local minimum only. If P Ž x . ) 0 for some x g Ž1rŽ1 q a ., 1., then since P Ž1rŽ1 q a .. - 0, P Ž1. s 0 will imply that P Ž x . will have to have a local maximum; since local maxima are not possible for P, we conclude that P Ž x . F 0 for x g Ž1rŽ1 q a ., 1.. Our investigation of the sign of W Ž x .Ž n. for x Ž n. g Ž1rŽ1 q a ., 1. is now complete and we have W Ž x . Ž n. F 0
for x g
ž
1 1qa
, 1 and 0 - r -
/
ž
1qa
a
/ž ln
1qa 1ya
/
.
Žiii. x Ž n. s 1. It is found from the definition of W Ž x .Ž n. that W is not defined when x Ž n. s 1; however, W has removable singularity at x Ž n. s 1 and hence we examine the limiting value of W as x Ž n. ª 1. For convenience we write WŽ x. s
1ya
1
Ž1 y x. 1 q a
exp r Ž 1 y x . y
2 y Ž1 q a . x
Ž1 q a . x
By L’Hopital’s rule one can determine that lim W Ž x . s
xª1
F
1ya 1qa
ry
2 1ya
1ya 1qa 1qa
F 0.
a
ln
ž
1qa 1ya
/
y
2 1ya
.
75
PERSISTENCE AND GLOBAL STABILITY
Živ.
x Ž n. g Ž1, 2 xU .. We note from Ž3.16. that for x Ž n . g Ž 1, 2 xU .
DV Ž x . Ž n . F 0
if and only if P Ž x . defined in Ž3.24. satisfies P Ž x . ) 0 for x g Ž1, 2 xU .. We note that P Ž 2 xU . s P
P Ž 1 . s 0,
ž
1 1qa
/
1ya
s
1qa
exp yr
ž
1ya 1qa
/
) 0.
Also we have PX Ž 1. s
1ya
2
ž
y r ) 0.
/
1qa 1ya
If P Ž x . - 0 for some x g Ž1, 2 xU ., then P Žx. must have at least one local maximum point on Ž1, 2 xU . and this is due to the fact that P X Ž1. ) 0, P Ž1. s 0, P Ž2 xU . ) 0. Let x be a local extremum point of P; then P X Ž x . s 0 and PY Ž x . s r 2 s r2 s
ž ž
1ya 1qa 1ya 1qa
r Ž1 y a .
)r
1qa 1ya 1qa
/ /
exp r Ž 1 y x . y exp r Ž 1 y x . y
exp r Ž 1 y x .
ž
4
Ž1 q a . x3
ž
ry
1 y a 2r 1qa 2 x
/
x
exp r Ž 1 y x .
/
exp r Ž 1 y x . Ž r y 2 . .
If r ) 2, we have P Y Ž x . ) 0, which means that P Ž x . has no local maximum and hence P Ž x . ) 0 when r ) 2 and x g Ž1, 2rŽ1 q a ... Hence P Ž x . ) 0 on Ž1, 2 xU .. Next consider the case r F 2. If a local minimum exists let the minimum be at x. Then P X Ž x . s 0 and hence from Ž3.30. PX Ž x . s 0
«
r
1ya 1qa
exp r Ž 1 y x . s
2
Ž1 q a . x2
.
76
GOPALSAMY AND LIU
Then for x ) 1, we have from the definition of P Ž x . in Ž3.17. PŽ x. s G s ) s
2 r Ž1 q a . x 1
Ž1 q a . x 1 1qa 1 1qa 1 1qa
ž ž ž
2
1 x
2
1 x 1 x
2
2
2
y
Ž1 q a . x 2
y
Ž1 q a . x 2
y
x 2
y
x
q1qa q1
q1
q1
/
/
2
y1
) 0.
/
It follows that P Ž x . ) 0 for x g Ž1, 2 xU .. From Ži. ] Živ., we conclude that for x Ž n . g Ž 0, 2 xU . .
DV Ž n . F 0
We have already verified that DV Ž n. F 0 for x Ž n. g w xU , `.. One can now argue as in the case of the proof of Theorem 2.1 and derive that lim x Ž n . s xU s
nª`
1 1qa
.
This completes the proof of Theorem 3.2. The following corollary is concerned with the asymptotic behaviour of positive solutions of Ž1.2. and is a consequence of Theorems 3.1 and 3.2. COROLLARY 3.3. Let r, a, b be positi¨ e numbers. Assume that any one of the following two conditions hold: Ži. a G b; Žii. 0 - a - b and 0-r-
b a
ln 1 q
ž
2b a
q ln
/ ž
Then all positi¨ e solutions of Ž1.2. satisfy lim N Ž t . s
tª`
1 aqb
.
aqb ayb
/
.
77
PERSISTENCE AND GLOBAL STABILITY
Proof. We have from Ž1.6. that any solution of Ž1.2. is given by NŽ n q t. s
N Ž n . exp r 1 y bN Ž n . t 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . t 4 y 1 . r Ž 1 y bN Ž n . . .
,
0 F t F 1, n s 0, 1, 2, . . . . We note from Theorems 3.1 and 3.2 that when the condition of Corollary 3.3 holds, then lim bN Ž n . s xU s
nª`
1 1qa
,
where a s
a b
.
We have from the above that lim N Ž t . s lim N Ž n q t .
tª`
nª`
s lim
nª`
s s
1
N Ž n . exp r 1 y bN Ž n . t 4 1 q aN Ž n . Ž Ž exp r 1 y bN Ž n . t 4 y 1 . r Ž 1 y bN Ž n . . .
ž
1
b 1qa 1 aqb
s
1
/ ž
1
b 1 q arb
/
.
and the proof is completed. A result similar to that of Corollary 3.3, but concerned with an equation containing a more general class of delays, has been established by Cooke and Huang w3x under the assumption a ) b. A condition of this type means that the stabilizing undelayed negative feedback dominates other negative feedbacks with delays Žfor more details of this type of conditions, we refer to Gopalsamy w7x..
4. A FEW COMMENTS We have established that all positive solutions of the discrete dynamic system Ž1.7. converge to the positive equilibrium if a G 1; if a g Ž0, 1., then a sufficient condition for such a convergence is given by the requirement of Ž3.7.. From the proof of Theorem 3.2 we see that the condition Ž3.7. is not necessary. If rs
ž
1qa
a
/ ž ln
1qa 1ya
/
,
a g Ž 0, 1 . ,
Ž 4.1.
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GOPALSAMY AND LIU
then the linear stability criterion is not applicable since in such a case we have faX Ž r , xU . s y1.
Ž 4.2.
It is, however, known Žfor instance, see Sandefur w16x. that when Ž4.2. holds, if Da Ž xU . s y2 faZ Ž r , xU . y 3 faY Ž r , xU . rs
ž
1qa
a
/ ž ln
1qa 1ya
/
,
2
- 0,
Ž 4.3.
then xU is locally stable. Calculation of the derivatives involved in Ž4.3. is tedious; we have used the Maple symbolic program to calculate Da Ž xU . of Ž4.3. for the case where r satisfies the equality in Ž4.3.. The resulting function Da Ž xU . of the parameter a g Ž0, 1. is plotted using Maple’s plotting procedure and the graph is displayed in Fig. 3. From the graph it is found that Da Ž xU . - 0 for a g Ž0, 1. and hence we conclude that xU of Ž2.1. ] Ž2.3. is asymptotically stable provided rF
ž
1qa
a
/ ž ln
1qa 1ya
/
.
Ž 4.4.
If Ž4.4. is violated then xU is not linearly stable. One can see from Ž2.7. that faX Ž r, xU . - y1 is sufficient for instability of xU . Thus Ž4.4. provides a necessary condition for the asymptotic stability of xU when a g Ž0, 1.. The authors believe that Ž2.13. is sufficient for the global asymptotic stability of xU ; however, they have not been able to prove this.
FIGURE 3.
PERSISTENCE AND GLOBAL STABILITY
79
When a s 0, the system Ž2.1. ] Ž2.3. reduces to the simpler dynamic system x Ž n q 1 . s x Ž n . exp r 1 y x Ž n .
n s 0, 1, 2, . . . .
4,
Ž 4.5.
Several authors Žsee May w13x, May and Oster w14x, Fisher and Goh w6x, Seifert w17, 18x, Norris and Soewono w15x, Guckenheimer, Oster, and Ipaktchi w8x, and Dohtani w5x. have investigated the stability and oscillatory characteristics of Ž4.5.. It is known that if 0 - r F 2, then all positive solutions of Ž4.5. converge to the positive equilibrium of Ž4.5.. One can obtain this special result from our investigation by evaluating the following limit: lim
aª0
ž
1qa
a
/ ž ln
1qa 1ya
/
s 2.
Ž 4.6.
The global asymptotic stability of xU implies that the dynamics of the system has no persistent periodic solutions or other types of complex solutions of the ‘‘chaotic type.’’ In fact, when the requirement in Ž4.1. is violated, period-doubling bifurcations and complex dynamics of the system Ž1.7. emerge and it has not been the intention of the authors to consider this in this article. We conclude with the remark that there exists vast literature concerned with singular perturbations of delay differential equations and their connection with certain associated difference equations with a continuous argument Žfor a survey see Ivanov and Sharkovsky w10x and the references therein .. We remark that the discrete dynamic system Ž1.7. is not derived from the differential equation Ž1.1. through any regular or singular perturbation.
ACKNOWLEDGMENTS The authors wish to thank a referee for suggesting improvements in the proofs of Theorems 2.1 and 3.2.
REFERENCES 1. R. P. Agarwal, ‘‘Difference Equations and Inequalities, Theory, Methods, and Applications,’’ Dekker, New York, 1992. 2. L. A. V. Carvalho and K. L. Cooke, A nonlinear equation with piecewise continuous argument, Differential Integral Equations 1 Ž1988., 359]367. 3. K. L. Cooke and W. Huang, A theorem of George Seifert and an equation with state-dependent delay, in ‘‘Delay and Differential Equation’’ ŽA. M. Fink, K. Miller, and W. Kliemann, Eds.., World Scientific, Singapore, 1991.
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