Available online at www.sciencedirect.com
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F_J_SEVIFR
MATHEMATICAL
AND
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COMPUTER
MODELLING
Mathematical and Computer Modelling 40 (2004) 1509-1525
www.elsgvier.com/locate/mcm
Delay Differential Logistic Equation
with Harvesting
L. BEREZANSKY*
Department of Mathematics and Computer Science
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel
E. BRAVERMAN t
Department of Mathematics and Statistics
University of Calgary, 2500 University Drive N.W.
Calgary, A l b e r t a T2N 1N4, Canada
maelena©math, ucalgary, ca
L. IDELS*
Mathematics Department
Malaspina University-College, Building 360/304
900 Fifth Street Nanaimo, BC VgR 5S5, Canada
(Received December 2002; revised and accepted October 2003)
Abstract--The
logistic delay equation with a linear delay harvesting term
l~(t) = r(t)N(t) a -
bkN(hk(t)) -
cz(t)N(gl(t)),
t >_O,
l=l
N(t)=~(t),
t<O,
N(O)=No,
is considered. The existence and the bounds of positive solutions are studied. Sufficient conditions
for the extinction of the solution are presented. © 2005 Elsevier Ltd. All rights reserved.
K e y w o r d s - - D e l a y logistic equations, Linear harvesting, Positive solutions, Extinction of the population, Solution bounds.
1. I N T R O D U C T I O N
The simplest model of the population growth is described by the equation
dN
dt
m
rN(t).
*Partially supported by Israeli Ministry of Absorption.
tAuthor to whom all correspondence should be addressed. Partially supported by the NSERC Research Grant
and the AIF Research Grant.
:~Partially supported by MI%F (Malaspina University-College Grant).
0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved.
doi: 10.1016/j.mcm.2005.01.008
Typeset by ~4J~S-TEX
1510
L. BEREZANSKYst al.
In more sophisticated models of population dynamics, the growth rate is assumed to be a function
of the population N. If the growth rate is denoted by r(N), then the differential equation
describing population growth takes the form
dN
d--T = r ( N ( t ) , t ) N ( t ) .
It is legitimate to regard r ( N ( t ) , t) as formally describing a self-regulatory mechanism. Of course,
most populations do not live a life unmolested by outside influences. We will study the dynamics
of a population affected by harvesting. The following general differential equation
dN
dt = r ( N ( t ) , t ) N ( t ) - E ( N ( t ) , t)
(1)
will be considered, where E ( N , t) is a harvesting strategy for the population. Function E represents the rate at which individuals are harvested.
In 1959, Holling [1] identified three basic types of functional "predator" responses:
• Type I (linear): E ( N , t ) = a N +~,
• Type II (cyrtoid): E ( N , t) = a N / ( N + fl),
• Type III (sigmoid): E ( g , t ) = aN2/(72 + •Y 2) (a, ~, and 7 are positive functions of t).
In this paper, we will focus on the response of Type I.
If r ( N ) = N ( a - bN), where a > 0, b > 0 are constants, then we have a logistic-like equation
with the harvesting strategy
dN
dt = g ( a - bN) - E ( N , t).
(2)
Below, some applied models of population dynamics are described.
EXAMPLE 1.1. METAPOPULATION MODEL. One of the first metapopulation models was developed in 1969 by Levins [2]. It considered immigration of organisms (e.g., birds) from a continent
to islands in the ocean. The proportion of islands colonized by a species, N, is given by a natural
logistic growth term offset by losses linearly proportional to the birds population (Type I)
d--N = c ( t ) N ( 1 -
- e(t)N,
where c is a colonization rate, e is an extinction rate.
EXAMPLE 1.2. LOBSTER FISHERY. For fishery management and many other harvesting situations, it is unreasonable to assume that the harvesting rate is constant (i.e., independent of the
population). Thus, for modeling the marine fishery, we will assume that the harvesting rate H
is proportional to the population N (Type 1).
The differential equation [3-5] is then
dN
dt - r N ( K - N ) - Q C N ,
where Q is a biological characteristic of the population which is not subject to human control,
while C is under the control of the lobster fishery. Note that a loss rate due to harvesting, in
general, depends both on the fishing effort and on the fish population density. The same is true
for whaling industries and commercial forestries.
The last few decades have seen an expanding interest in retarded models of population dynamics
involving logistic differential equations with delay. There have been many attempts [6-10] to find
reasonable mathematical models with time lags to describe certain complex biological systems.
For example, the delay logistic differential equation in population ecology
d N _ r ( t ) Y ( t ) (1
dt
\
Y(h(t)) ~
K
/ '
h(t) < t,
-
Delay Differential Logistic Equations
1511
is known as Hutchinson's equation [11], where r and K are positive constants and h(t) = t - T
with a positive constant T.
The delays are used in immunology to represent the time needed for immune cells to divide, or
become destined to die. In modeling the spread of infections, Cooke et al. [9] used the following
delay differential equation
dN
d---t- = r ( N ( t - T) ) N ( $ -- ~-) -- cN(t),
where c > 0 and T > 0 are suitable constants.
Consider a logistic-like equation with a harvesting (hunting) strategy where we do not know
the population N(t) at the exact time. However, we need (Example 1.2) it to determine the
hunting quota. There is always a delay in processing and distributing field information. The
quota must be set long before the hunting season begins.
The time lags in ecological systems can be justified as a discovery time: predator requires time
to discover the prey is very abundant. In the case of metapopulation (Example 1.1), time lags
can be a result of the situation where prey population becomes sufficiently rare, such that the
predator switches to alternative prey or an alternative source of prey (different island).
It might be reasonable to consider a predation function as a function of the delayed estimate
of the true population. It is important to be able to model this delay because it has serious
implications for the long-term behavior of populations. Time lags yield specific insights into the
management of complex ecological systems. It is considered that the major effect of delays is to
make them less stable than the analogous models without delays.
The introduction of delays into existing mathematical ecology equations is supported by general
arguments that the interacting species somehow rely on resources and harvesting that have been
accumulated in the past. It is now a well-known fact that small delays may cause some otherwise
inexplicable periodic oscillations.
Equation (1) with delays can be rewritten in the following form
dN
dt-
r(g(h(t)),t)Y(t)
- c(t)g(g(t),t),
h(t) < t,
g(t) < t.
(3)
In this paper, we consider equations of type (3) with several delays in the logistic and the
harvesting parts. Here we restrict ourselves to the linear harvesting function E = c ( t ) N ( t )
(Type I of response). Such functions were used in Examples 1.1 and 1.2. We will obtain here
sufficient conditions for positiveness, boundedness, and extinction of solutions of equation (3).
Similar problems for the equation with constant harvesting were studied in [12,13]. In [14,15],
a general logistic delay equation without harvesting was investigated, a priori upper bounds
of solutions were obtained and applied to derive explicit conditions of global[ stability for these
equations. In the present paper, we also find a priori upper bound of solutions for equations with
harvesting and use this estimate to study the positiveness of solutions. Our theoretical results
are illustrated by numerical simulations.
The paper is organized as follows. Section 2 includes relevant results for linear and nonlinear
delay differential equations. Section 3 contains main results on the existence of positive solutions
which do not tend to zero. This corresponds to nonextinction of the population. Some estimates
for positive solutions are also presented. In the end of Section 3, we apply these results to the
metapopulation model and to the lobster fishery equation with delay (see Examples 1.1 and 1.2).
In Section 4, the results are discussed and illustrated by numerical examples.
2. P R E L I M I N A R I E S
Consider a scalar delay differential equation
N(t) = r(t)N(t)
a - ~_~ b k N ( h k ( t ) )
k=l
-
cz(t)N(gl(t)),
/=1
t _> o,
(4)
1512
L. BEREZANSKY et al.
with the initial function and the initial value
N(t) = !o(t),
t < 0,
N(0) = No,
(5)
under the following conditions:
(al) a > 0 , bk>O;
(a2) r(t) >_ O, cl(t) >_ 0 are Lebesgue measurable and locally essentially bounded functions;
(a3) hk(t), gl(t) are Lebesgue measurable functions, hk(t) <_ t, gl(t) <_ t, limt--.oo ha(t) =
0% limt__,~ gl(t) = co;
(a4) to : (-0% 0) ---* R is a Borel measurable bounded function, ¢p(t) __ 0, No > 0.
DEFINITION. An absolutely continuous on each interval [0, b] function N : R --~ R is called a
solution of problem (4),(5), if it satisfies equation (4), for almost all t C [0, oo), and equalities (5)
for t <_ O.
We will present here lemmas which will be used in the proof of the main results.
Consider the linear delay differential equation
+
o,
t >_ o,
(6)
_< o,
t >__o.
(7)
=
and a corresponding differential inequality
n
9(t) +
1=1
DEFINITION. We say that a function is nonoscillatory if it is eventually positive or eventually
negative.
LEMMA 1. (See [16].) Suppose that for the functions cl, g~ Hypotheses (a2) and (a3) hold. Then,
(1) if y(t) is a positive solution of (7) for t >_ to >_ O, then y(t) <_ x(t), t >_ to, where x(t) is a
solution of (6) and x(t) = y(t), t <_ to,
(2) for every nonoscillatory solution x(t) of (6) we have hmt-,oo x(t) ~- O,
(3) if
sup
±I;
t>O /=1
cz(s) ds ~ - ,
ink gk(t)
(8)
e
then equation (6) has a nonoscillatory solution.
If in addition, 0 ~_ ~(t) < No, then the solution of initial value problem (6),(5), where N(t)
in (5) is replaced by x(t), is positive.
Consider also the following linear delay equation with positive and negative coefficients
5~(t) + ~ e l ( t ) x ( g l ( t ) ) -- a(t)x(t) ----O,
t >__O.
/=1
DEFINITION. A solution X (t, s) of the problem
it(t) + E cz(t)x(gl(t)) -- a(t)x(t) = 0,
/=1
x(t)=O,
is called a fundamental function of (9).
Denote G(t) = maxl gt(t).
t < s,
x(s)= l
t>s,
(9)
Delay Differential Logistic E q u a t i o n s
1513
(See [17,18].) Suppose
that for the functions ct, gl Conditions (a2) and (a3) hold, a
is a locally bounded function, a(t) >_ O,
LEMMA 2.
el(t) > a(t),
cl(t) - a(t)
1=1
dt = co,
(10)
I=1
and
lim sup
a(t)(t
a(t)) + ~__,c~(t)(a(t) -gdt))
-
t--*~
< 1.
(11)
1=1
Then,
(1) if there exists a nonoscillatory solution of (9), then for some to and t >_ to we have
X ( t , s) > O, t > s > to, where X ( t , s) is a fundamental function of (9);
(2) for every nonoscillatory solution x(t) of (9) we have limt-.oo x(t) = O.
3. M A I N
RESULTS
Denote
h(t) = ~n{hk(t)},
g(t) = ~in{gl(t)}.
In addition to (al)-(a4), consider the following hypothesis:
(aS) h(t) is a nondecreasing continuous function.
If in equation (4), we neglect harvesting terms, i.e., assume ct -= 0, then the positive equilibrium
becomes a/~'~k~=l bk. Suppose the initial function and the initial value are less than this equilibrium. Then, under certain conditions, the following result presents lower and upper bounds of
the solution.
THEOREM 1. Suppose (al)-(ah) hold,
a
~(t) < No < 7 - - ,
t < 0,
(12)
bk
k=l
and
sup
t>0 I=1
(t) ct(s)exp
a exp l, asup
t>0
(t) r(~)d~
- 1
z(~)
r(T) d7
ds _<- .e1
(13)
Then, for ant solution of (4),(5), we ha~e
a
0 < N(t) < - -
bk
t
}
expasupL./ds
~,
(14)
t>O
k=l
PROOF. Suppose (14) is not valid. Then, either there exists t > 0 such t h a t
a
0 < N(t)<_ - -
~ bk
exp_asup
( t>o
(t)
r(s) ds
,
o_<t<~,
k=l
a
Y(t-) -
- m-
~2 bk
k=l
ex(asuL
t>O
(15)
r(s)
ds
,
N ( t - ) > 0,
1514
et al.
L. BEREZANSKY
or there exists t > 0 such that
O < N ( t ) <_ ma
exp
bk
{ ;
asup
t>0
r(s) ds
(t)
}
,
O<_t<t,
N(t-)=O.
(16)
k----1
Suppose we have the first possibility for a solution N ( t ) of (4),(5). Denote by
Q <t2 <... <tk <...
a sequence of all points tk, such that
a
JV(h(tk)) > O.
N(h(tk)) = -~ bi'
i=1
Inequalities
a
a
N(0) = No < - -
N(t-) > - - ,
,
k=l
k=l
and (a5) imply t h a t the set {tk} is not empty.
Suppose t* is a point of a local maximum for N(t).
We will prove that if N(t*) > a/~i~=l hi, then t* E Uk[h(tk),tk].
Let tk be the greatest among all points of the sequence {tk} satisfying h(tk) < t*.
m
Suppose first N ( t ) < a / ~ i = 1 bi for some t and h(tk) < t < tk. The definition of tk and t*
imply t* < t, and hence, t* E [h(tk), tk].
Now suppose N ( t ) > a / ~ i m l bi for h(tk) < t ~ tk.
Let there exist the smallest point t' > t* such t h a t N ( t ' ) = a / ~ i ~ 1 bi. Then, (4) implies
N(t) < 0, tk <_ t < t'. Hence, in this interval, N ( t ) has no maximal points. Thus, h(tk) < t* < tk.
If such t' does not exist, then fi/(t) < 0, t > tk, and again h(tk) < t* < tk.
Equation (4) implies now that
a
IV(t) ~ a r ( t ) N ( t ) ,
h(tk) ~_ t ~_ t*,
g(h(tk)) = m
i=l
Then,
N(t*) < -m---exp
bi
--e./or
a
k)
bi
i=l
[, Jh(tk)
i~l
air
<~exp
E bi
asup
[
r(s) ds
t>0 Jh(t)
;
,
J
I-=1
which contradicts our assumption (15).
Suppose now there exists t > 0 such that (16) holds. After substituting
(17)
in (4),(5), we have the following system:
~:(t)=-Ecl(t)ex
l=1
p
r(s) a -
z(t)
x(t) = ~(t),
(we assume r(t) -= O, t < 0).
bkN(hk(s))
ds
x(gt(t)),
t > 0,
(is)
_
t < O,
x(O) = No
(19)
Delay Differential Logistic E q u a t i o n s
1515
Consider now an initial value problem for a linear delay differential equation
~(t) = - E p 1 ( t ) y ( g 1 ( t ) ) ,
t > 0,
(20)
I=1
y(t) = ¢(t),
where
pl(t) = el(t)exp
{r
t < 0,
t
r(s)
-
Jgz(t)
y(0) = Yo,
[
m
a - ~-"~bkN(hk(s))
k=l
(21)
}
ds
.
It is evident that if ~b(t) = ~(t), Y0 = No, then the solutions of (18),(19) and (20),(21) coincide.
Inequalities (14) and (13) imply that
~ fgt pl(s) d s =
l=1
l=1 (t)
cz(s) exp
< sup E
t>o ~=1
/r [t
r(T)
bkN(hk(v)) - a
[Jg~(~)
(t)
(t)
cz(s)exp
a exp
asup
[
t>o
(t)
]}
r(~)d~
dT
ds
1
z(8)
r(T)dT
ds
< -1- ,
e
Inequality (12) implies
~(t) _< No.
Thus, Lemma i yields t h a t if ¢(t) = ~(t), yo = No, then y(t) > 0, t > 0. Hence, x(t) > 0, t > 0.
Consequently, by (17), we have N ( t ) > 0, t > 0, which contradicts assumption (16). The theorem
is proven.
COROLLARY 1.1. Consider an autonomous equation
ill(t) = N ( t )
a - EbkN(t
- hk)
- EciN(t
k=-i
- gl),
t >_ o,
(22)
l=l
with initial conditions (5), where a > 0, bk > 0, cl > 0, hk >_ O, gl >_ O.
Denote h = maxk hk, g = maxt gz.
Suppose (12) holds and
n
gE
cl exp {ag, (exp {ah} - 1)} < 1
l=l
e
Then, ~or a solution o~ (22),(5), we ha.e
a e ah
0 < N(t) < - k=l
REMARK. If in equation (4) either g(t) -- t or ck(t) -- 0, then this equation is a logistic one. It is
well known that for the logistic equation positiveness of initial values implies positiveness of its
solutions without any additional constraints.
If g(t) ~ t or c =- 0, then Theorem 1 also implies that N ( t ) > 0. However, together with this
inequality we also obtain an upper bound of these solutions.
L. BEREZANSKYet
1516
aI.
COROLLARY 1.2. Consider the delay logistic equation
N(t)=r(t)N(t)[1
N(~___(t))] ,
(23)
~he~e ~: > O, ~o~ ~ and h Conditions (a2), (a3), and (aS) hold, and
~(t) <_ No < K,
t < o.
Then,
rt
0 < N(t) < K e sups>° Jh(t) r(s) ds
THEOREM 2. Suppose (al)-(a5) hold, then for every eventually positive solution of (4), (5), there
exists to >_ 0 such that (14) holds for t > to.
PROOF. Suppose N(t) is an eventually positive solution of (4),(5). If N(t) <_a/}-~= 1 bk for some
to _> 0 and t >__to, then the statement of the Theorem 2 is true.
Suppose now that N(t) > a / ~ = 1 bk for some tl _> 0 and t > tl. Equality (4) implies that
N(t) <_ - ~ = 1 cl(t)N(gl(t)), t >_ t2 for some t2 >_ tl. Lemma 1 implies that 0 < N(t) <__x(t),
t _> t2, where x(t) is a solution of the equation
2(t)+~cz(t)z(gl(t))=O, t>_tl,
x(t)=N(t), t<_t2,
l=l
and limt-.oo x(t) = 0. Then, limt-~oo N(t) = 0. We have a contradiction with our assumption.
Hence, there exists a sequence {tn}, limn t~ = oo, such t h a t N(h(t~)) = a / ~ = 1 bk. The end
of the proof is similar to the corresponding part of the proof of Theorem 1.
REMARK. The same result for a more general logistic equation without harvesting term was
obtained in [15].
Consider now equation (4) with a nondelay term in the logistic part
,~(t) =r(t)N(t) [ a - boN(t) - Er,b k N ( h k ( t ) ) I n- Ecl(t)N(g~(t)),
k=l
THEOREM
t >>_O.
(24)
/=1
3. Suppose bo > O, Hypotheses (al)-(a4) hold,
a
(25)
qo(t) _< No < bo
and
sup
,>o~=1
(~)
c~(s) exp
~
bo
]
~(~)
r(~) d~
1
ds <_ -.
(26)
e
Then, for any" solution of (25),(5), we have
o < N(t) < ~a.
(27)
PROOF. We apply the scheme of the proof of Theorem
either there exists { > 0 such that
i. Suppose (27) is not correct. Then,
a
0 < N(t)< Vo,
0<t<~,
a
N(t-)=bo,
g (t-) > 0,
(28)
Delay Differential Logistic Equations
1517
or there exists t" > 0 such t h a t
0 < N(t) < a
Suppose
0 < t < t,
bo'
(29)
N(t-) = O.
-
the first possibility (28) holds. Then, for 0 < t < t, we have
1V(t) < r ( t ) N ( t ) [ a - boN(t)],
N(O) = No.
Denote by x a solution of the following problem
5~(t) = r(t)x(t)[a - box(t)],
Then,
(30)
x(O) = No.
a
N ( t ) <_ x(t) < bo'
0 < t < t,
since the solution of equation (30) tends to a/bo and is always less than a/bo.
We have a contradiction with assumption (28).
Suppose now that for t > to (29) holds. Substituting in (24),(5)
N ( t ) = exp
r(s)
(/:I
a - boN(s) -
bkN(hk(s))
k=l
1}
ds
(31)
x(t),
we have the following system:
Tt,
t > 0,
5:(t) = - E p t ( t ) x ( g t ( t ) ) ,
(32)
l=1
x(t) = vp(t),
where
Pt(t) = cl(t) exp
t < O,
x(O) = No,
{/: [
-
r(s)
e(t)
m
a - boN(s) - E b k N ( h k ( s ) )
k=a
]}
ds
.
Inequalities (27) and (26) imply that
cl(s) exp
pt(s) ds <
I----1
(t)
/=i
< sup
r('r)
(t)
b k N ( h k ( T ) ) + boN(v) - a dT
I(s)
ds
k=l
cl(s)exp
r(~-)dT
ds < - .
Similarly to the proof of Theorem 1, Lemma 1 implies N ( t ) > 0, 0 < t < L This contradiction
proves the theorem.
COROLLARY 3.1. Consider an autonomous equation
IV(t) = N ( t )
a - boN(t) - E
b k N ( t - hk)
- E
k=l
e~N(t - gz),
t >_ O,
(33)
l=l
with initial conditions (5), where a > 0, bk > 0, cz > 0, hk _> 0, gl >_ O. Suppose (25) hoIds and
n
mkaxgk E cl exp
l=l
agt ~ bk
k=l
bo
Then, [or every soIution of (33), (5), we have
a
0 < N i t ) < ~o"
1
< _
-- e"
1518
L. BEREZANSKY et al.
COROLLARY 3.2. Consider an autonomous equation
IV(t) = N(t)[a - bN(t)] - £
ctN(t - gl),
t >_ 0,
(34)
/-----1
with initiM conditions (5), where a > O, b > O, cl > O, g~ >_ O. Suppose (25) holds and
75
maxgk
e, < !.
e
l--1
Then, for every solution of (33),(5), we h ve
0
a
< N(t)
-~.
<__
THEOREM 4. Suppose bo > O, (al)-(a4) hold. Then, for every eventually positive solution of
(25),(5), there exists to k 0 such that (27) holds for t >_to.
The proof is similar to the proof of Theorem 2.
Now let us obtain sufficient extinction conditions for solutions of logistic equations with harvesting. To this end, consider the following equation which is more general than (4) and (25):
N(t) = N(t)
a(t) -
bk(t)N(hk(t))
-
el(t)N(gz(t)),
t > O.
(35)
l=l
THEOREM 5. Suppose a(t) _> 0, bk(t) >_ 0 are locally essentially bounded functions and t'or
el, hk, 9z, Conditions (32),(33) hold.
Suppose in addition (10),(11) hold. Then, for any solution of (35),(5) either l i m t _ ~ N(t) = 0
or there exists ~ > 0 such that N(t-) < 0.
PROOF. It is sufficient to prove that for every positive solution N(t) of (35),(5), we have l i m t ~
N(t) =
Suppose N(t) > 0 is a solution of (35),(5). Equation (35) implies
O.
n
N(t) + E c l ( t ) N ( g ~ ( t ) ) - a(t)N(t) <_ O.
/=1
Lemma 2 yields that there exists to > 0, such that the fundamental function X ( t , s) of the
equation
2(t) + E
cl(t)x(gl(t)) - a(t)x(t) = 0
(36)
/=1
is positive for t > s > to. Then, variation of constant formula [16] implies
N(t) = x(t) +
X ( t , s)f(s) ds,
where x(t) is a solution of (36) with the initial condition x(t) = N(t), t <_ to and f(t) is a
nonpositive function. Hence, 0 < N(t) <_ x(t).
Lemma 2 implies limt-~oo x(t) = 0. Then, also limt-~oo N(t) = O.
COROLLARY 5.1.
Suppose for equation (4), (al)-(a4) hold,
cl(t) > ar(t),
/-----1
cl(t) -- at(t)
]
dt = oo
and
lim sup
t--+oo
[ar(t)(t-G(t)) + £l = l c,(t)(G(t)-gl(t)) !
<1.
Then, for any solution of (4),(5), either limt~oo N(t) = 0 or there exists { > 0 such that N(t-) < O.
Delay Differential Logistic Equations
1519
COROLLARY 5.2. Suppose conditions of Theorem 1 and Corollary 5.1 hold. Then, any solution
N(t) of (4),(5) is positive and satisfies limt-.oo N(t) = O.
Denote g = roan gz.
COROLLARY 5.3. Suppose for equation (22) a > O, bk > O, el > O, hk > O, gl > O. If
71
n
> a
and
ag +
l=l
c,(g
- g) < 1,
l=l
then for any solution of (22),(5), either limt-.~o N(t) = 0 or there exists t > 0 such that N(t-) < 0.
COROLLARY 5.4. Suppose conditions of Corollaries 1.1 and 5.3 hold. Then, any solution N(t)
of (22),(5) is positive and limt-~oo N(t) = O.
Finally, let us apply the above results to the delayed models considered in the Introduction (a
nondelay equation is a special case with the zero delay in the harvesting term).
EXAMPLE 3.1. Consider a delay metapopulation model
dN(t)-c(t)Ndt
(1
e(t)N(g(t)),
N~))_
(aT)
where g(t) satisfies (a3). Then, Theorems 1 and 2 and Corollary 5.1, respectively, imply the
following results.
(1) If the initial function and the initial value satisfy ~0(t) < No < K and
e(s) ds
sup
t>0
< -,
(t)
e
then for any t a solution of (37) satisfies 0 < N(t) <_K.
(2) For any positive solution of (37), there exists such to that for t _> to we have N(t) < K.
(3) If e(t) >_ c(t) for any t > 0, fo[e(t) - c(t)] dt = oo and limsupt_~oo[c(t)(t - g(t))] < 1,
then all solutions of (37) either become negative or tend to zero.
EXAMPLE 3.2. Consider a delay model of lobster fishery
dN(t) = r N ( t ) ( g - N(t) ) - QCN(t - g).
dt
(38)
Then, Corollary 1.1, Theorem 2, and Corollary 5.3, respectively, imply the following results.
(1) If the initial function and the initial value satisfy ~o(t) _< No < 1 / K and gQC <_ 1/e, then
for a solution of (38) the relation 0 < N(t) <_1 / K holds for any t > 0.
(2) For any positive solution of (38), there exists such to that for t > to we have N(t) < K.
(3) If QC > r K and Krg < 1, then all solutions of (38) either become negative or tend to
zero.
4. N U M E R I C A L
SIMULATIONS
AND
RESULTS
INTERPRETATION
In this section, we will discuss the main results and present numerical illustrations.
Concerning the condition on delays, harvesting, and growth rates like (13) and (26), they can
be described as follows. The greater the growth and the harvesting rates are, the smaller should
be delays providing that the solution is positive which means that there is no extinction of the
population. Or, for prescribed delays and a given natural growth rate r(t), the harvesting rate
should not exceed a certain number to avoid possible extinction.
1520
L. BEREZANSKYet al.
Now, let us consider the case when all the delays are in the harvesting part only (Corollary 3.2).
Then, the nonextinction condition includes only harvesting rates and delays (it is certainly assumed that the harvesting rate does not exceed the growth rate). The higher the harvesting rate
is, the smaller should be a delay between the data collection and the proportional harvesting.
Theorem 4 establishes a rather natural fact that if there is no extinction of the population in a
finite time, then for any initial data, there exists a certain moment of time, such that a solution
does not exceed an equilibrium (we assume that all the delay terms in the logistic part vanish).
This means t h a t asymptotically the population size is decreasing only due to harvesting and not
because of insufficient natural resources.
Finally, Theorem 5 claims that (under certain conditions) if the harvesting rate is higher than
the natural growth rate, then the extinction of the population is inevitable.
The main results and the sharpness of the hypotheses are illustrated below in Examples 4.1-4.5.
Now, let us interpret constraint (12). This inequality means that the initial function (the
prehistory available) is less than the equilibrium state without harvesting. Certainly this is not a
necessary condition, however this hypothesis matches asymptotics of positive solutions and allows
us to avoid extinction of the population in the very beginning of the history.
First of all, under rather natural assumptions a positive solution eventually satisfies this condition (see Theorems 2 and 4), so we can start the prehistory at this point and completely
characterize the asymptotics of solutions. Thus, it remains to discuss the case when the initial
d a t a exceed the equilibrium without harvesting which was not analyzed in our main results.
Numerical simulations in Example 4.3 partially fill up this gap for the case of constant delays
and coefficients. The computations demonstrate t h a t if the initial function is smaller than a
certain number (which depends on the delay in the harvesting term: the smaller the delay is, the
greater can be the initial function), then (if the other conditions for the positiveness of solutions
are satisfied), the solution is positive and tends to a new equilibrium with harvesting. If the
initial function exceeds a certain value, then the solution becomes negative in the beginning of
the process. This can be explained as follows.
If the harvesting rate is based on the size of the population some time ago, then for the survival
of the population it is important that the field data on the population size is collected at the
time when the population is not abundant. In other words, if a population is subject to double
reducing factors:
(a) decrease due to insufficient carrying capacity of the environment,
(b) harvesting based on the overstated estimate on the size of the population, then this can
result in the extinction of the population.
If both the initial function and the initial value in the interval are less than the equilibrium
state, then the harvesting effect which reduces the size of the population can be balanced by the
natural growth. In addition, the first inequality in (12) means that the initial function does not
exceed the initial value. Otherwise, the population can be brought to extinction by the rapacious
harvesting based on the overstated size (more than the present size) of the population. Since the
initial function is not assumed to be continuous and does not necessarily continuously match the
initial value, then one can imagine a situation when there is a jump N0 - ~ ( 0 - ) which is less
than zero which means that the population is reduced during a short period of time. This can be
caused by some disaster (for example, an epidemic). Obviously, if in the following the harvesting
rate is based on the d a t a before the disaster, this can also lead to the complete extinction of the
population.
As a basic model to illustrate our theoretical results, let us consider the following equation
with constant delays and coefficients
N ( t ) = N ( t ) [a - b N ( t ) ] - c N ( t - g),
which is a special case of equation (22).
t > O,
(39)
Delay Differential Logistic Equations
1521
1.5
' g=0.5
g=2
equilibrium N = O . 4
bound N=I
•7%•%'
0.5
HOHOHHOHHO00
HOH'O=OU''==
........
~"
.......
........
................
................
%%%%
\
\
-0.5
0
,
i
,
2
4
6
I
i
8
10
Figure 1. T h e behavior of solutions of (39), with a -- 1, b -- 1, c ---- 0.6, N(0) -0.5, ~ -- 0, a n d g = 0.5, 2, respectively. T h e delay g = 0.5 satisfies t h e inequality
(40), while g = 2 does not.
The following Example 4.1 illustrates Corollary 1.1 of Theorem 1. Let the initial function
be identically equal to zero (in this section, we will consider only examples with the zero initial
function). According to Corollary 1.1, if the initial function is identically equal to zero, the initial
value satisfies 0 < N(0) _< a/b and
1
cg < - ,
(40)
e
then the solution is positive for t > 0.
EXAMPLE 4.1. Let
a=l,
b=l,
c=0.6,
N(0)=0.5,
the initial f u n c t i o n ~ = 0 .
Then, (40) is satisfied if g < 1/ec ,.~ 0.61. In Figure 1, the solution which becomes negative
(extinction of the population) corresponds to g -- 2 (the hypotheses of Corollary 1.1 are not
satisfied), the solution which tends to the equilibrium (a - c)/b = 0.4 corresponds to g = 0.5 (the
hypotheses of Corollary 1.1 are satisfied). The equilibrium solution and the prescribed bound
a/b are also presented in Figure 1. In the case when the hypotheses of Corollary 1.1 are satisfied,
the solution does not exceed the a priori bound a/b.
In addition to Example 4.1, let us illustrate the sharpness of inequality (40) for equation (39).
For any a and c < a, we axe looking for such ~ that all the solutions with the zero initial function
and a positive initial value are positive in the hairline t > 0 for g _< ~ while there exists a solution
which becomes negative for g > ~.
EXAMPLE 4.2. Figure 2 illustrates the sharpness of inequality (40) for the positiveness of all
solutions. In Figure 2, the results of the following simulations are presented: the bounds for g for
any c, such t h a t for a fixed c, all the solutions considered remain positive in the hairline (below
the curve); there exists a solution which becomes negative for t > 0 (above the curve). The
upper curve in Figure 2 corresponds to the family of solutions satisfying conditions of Theorem 1
(No < a/b), with ~ -- 0, a = b -- 1. The middle curve corresponds to the case when b and No are
arbitrary (condition No < a/b is not necessarily satisfied), while the lower curve gives the delay
bound of Corollary 1.1 (see (40)) g = 1/ce.
1522
L. BEREZANSKYet al.
9
'any initial value and
theoretical curve g=l/(ce) . . . . . . .
initial value < a/b
8
7
6
%
5
03
4
%
%.
%%%
3
%%%
2
....................................................
1
i i
0
0.2
I
I
I
I
I
I
I
0.3
0.4
0.5
0.6
0.7
0.8
0.9
C
Figure 2. The bounds ~ of delay g for any c are presented, such that for a fixed c,
all the solutions remain positive for g < ~ and there exists a solution which becomes
negative for g > ~. The upper curve corresponds to the family of solutions satisfying
conditions of Theorem 1 (No < a/b), with ~ =_ 0, a = b = 1. The middle curve
corresponds to the case when b and No are arbitrary (condition No < a/b ks not
necessarily satisfied), while the lower curve gives the theoretical dependence (see (40))
g : 1lee.
E x a m p l e 4.2 illustrates (see Figure 2) t h a t the a b u n d a n t p o p u l a t i o n in t h e beginning incorpor a t e d with delayed p r o p o r t i o n a l harvesting can lead to its extinction.
EXAMPLE 4.3. Let
a----l,
b:l,
c=0.6,
~o(t)-~o=N(0).
T h e n , (40) is satisfied if g ~ 1 / e c ~ 0.61. We assume g = 0.5 a n d ~o > a / b = 1. Numerical
simulations d e m o n s t r a t e t h a t there exist c~ ~ 7.6 such t h a t for ~o < c~ t h e solution is positive while
for ~o > ~ t h e solution becomes negative. In Figure 3, the solution w i t h ~o = N ( 0 ) = 9 becomes
negative at t ~ 0.83, while the solution with ~o = N ( 0 ) -- 7.7 b e c o m e s negative at t ~ 1.5; the
solution with the initial d a t a ~ - N ( 0 ) = 6 is positive and t e n d s t o t h e equilibrium N -- 0.4. T h e
prescribed b o u n d y = a / b = 1 (the equilibrium w i t h o u t harvesting) is also presented in Figure 3.
A n y solution w i t h ~o = N ( 0 ) >__7.7 is negative for t > 1.5.
Now let us proceed t o the e q u a t i o n
IV(t) = N ( t ) [a - b o N ( t ) - b N ( t - h)] - c N ( t - g),
t > 0,
(41)
which is a special case of equations (22) and (33).
B y Corollary 3.1, every solution with the initial value No, 0 < No < a/bo satisfies 0 < N ( t ) <
a/bo for a n y t > 0 if
gcexp
g
} < - .:
e
(42)
EXAMPLE 4.4. Suppose a : b0 = b : 1, c ----0.4, ~ -- 0. Figure 4 illustrates three solutions: the
solution w i t h h ---- 2, g = 0.5 ((42) is satisfied) is positive and oscillating a b o u t the equilibrium
N* = 0.3, t h e solution corresponding to h -- 2, g = 1.2 ((42) is n o t satisfied) becomes negative
and t e n d s t o - c o at infinity, if h = 0.2, g = 0.5 ((42) is satisfied), t h e n the solution is positive and
Delay Differential Logistic E q u a t i o n s
9 l
.
.
.
.
1523
N(0)=7.7N(0)='6 ..... I
N(0)=9 ........
N=0,N=I ................ 1
8
5
4
3
0
.- . . . . . . . . . . . . . . . . . . . . . . . . .
0
0.5
1
1.5
1
2
2.5
3
Figure 3. The behavior of solutions of (39), with a = 1, b = 1, c = 0.6, g = 0.5,
a n d ~o(t) -- ~ = N ( 0 ) = 6, 7.7, 9, respectively. T h e first s o l u t i o n is positive a n d
t e n d s to t h e equilibrium, while t h e second a n d t h e t h i r d s o l u t i o n s b e c o m e n e g a t i v e
for t > 0.83, t > 1.5, respectively.
1
f o",\
|
\
|
m
|
u
h=2',g=0.5
|
h=2,g=1.2 .......
h=0.2,g---0.5 ........
~%
equilibrium
0.5
\,
0
%
\
-0.5
-1
0
I
I
I
I
I
I
I
2
4
6
8
10
12
14
F i g u r e 4. T h e solutions of (41) w i t h a = b0 = b = 1, c = 0.4, No = 0 . 8 , ~ -- 0.
T h e s o l u t i o n w i t h h ---- 2, g ---- 0.5 is positive a n d oscillating a b o u t t h e e q u i l i b r i u m
N = N * = 0.3, t h e solution c o r r e s p o n d i n g to h = 2, g ---- 1.2 b e c o m e s n e g a t i v e a n d
t e n d s to - c o at infinity, if h -- 0.2, g = 0.5, t h e n t h e s o l u t i o n is positive a n d g r e a t e r
t h a n t h e equilibrium.
greater than the equilibrium. In addition to (42), this example presents the oscillations caused
by the delay in the logistic part.
Consider again equation (39). Corollary 5.3 implies that if c > a, a g < 1, then any solution of
this equation either tends to zero at infinity or becomes negative. If in addition to the previous
L. BEREZANSKY et aI.
1524
' g=0.3 '
g=0.5 . . . . . . .
g=0.7
........
0.5
~~wMwmw
°'*°°'*'.,°°.%°°%
"%°°°%o
-0.5 i
ol
0
|
2
!
4
i
6
i
8
Figure 5. The behavior of solutions of (39), with a = i, b = 2, c =
0.2, ~o ~- 0, and values on one of the faces after subtraction and g =
respectively. For g = 0.3 the solution is positive and tends to zero at
g = 0.5 the solution becomes negative and tends to zero at infinity, for
solution becomes negative and tends to -(x~ at infinity.
10
1.2, x(0) =
0.3, 0.5, 0.7,
infinity, for
g = 0.7 the
inequalities c9 <_ 1/e, t h e n by Corollary 5.4 any solution is positive and tends to zero at infinity.
T h e following example illustrates these results.
EXAMPLE 4.5. Let a = 1, b = 2, c = 1.2 > a, N ( 0 ) = 0.2, the initial function p(t) -= 0. Figure 5
illustrates three cases (g < 1): for g = 0.3 the solution is positive a n d tends to zero at infinity,
for g = 0.5 the solution b e c o m e s negative and tends to zero at infinity, for g = 0.7 the solution
b e c o m e s negative and tends to - c o at infinity.
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