PREDATOR-PREY MODELS WITH DELAYS AND PREY
HARVESTING
S UBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DECREE OF
MASTERO F SCIENCE
AT
DALHOUSIE
UNWERSITY
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SEPTEMBER
1999
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Contents
AcknowIedgements
Abst ract
vii
1 Introduction
1.1
1
Preliminaries . . . . . . . . .
1.1.1
- .
. .
. ... ..... .......
+
The Logistic Growt h Mode1 . . . . . . . . . . . . . . . . . . .
- .. . - .. - .-
.5
t -1-3 Delay Modeis . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Predator-Prey Models
2.1
II
Funct ional Responses . . . . . . . . . . . . . . . . . . . . . . . . . - -
tl
...... -
- . .... . ..- .. . ..
14
- ....
17
Wangersky-Cunningham Mode1 . . . . . . . . . . . . . . . . . . . . .
18
2 2 Generdzed Gause Models
.
2.3 Generalized Gause SIodel wit h Prey Harvesting . . . . . . .
2.4
3
.. .
1-12 Hamesting..
. . . - .. . . . .
.. .
3
3 Generabed Gause Mode1 with Prey Harvesting and Delay in the
Prey SpecBc Growth
3.1 The Mode1 . . . . . . . . - . .
20
- .- .- ....- ..... .. ...
20
3.2 Criteria of stability: Brauer . . . . . . . . . . . . . . . . . . . . . . .
22
+
3.3 Criteria of Stability and Bifurcation . . . . . . . . . . . . . . . . . . .
24
3.4 Examples . . . . . . . . . . . . .
29
. . .. ... . .. . .. .. . . .. .
4 Generaüzed Gause Mode1 with Prey Harvesting and Delay in the
Predator Response Function
o d e 1 .................
36
......-
4.1
e
... ...
36
4.3
Criteria of Stability and Bifurcation . . . . . . . . . . . . . . . . . . .
38
.. . .. .... .. .... . . . .. . ..... .
41
4.3 Euamples . . . . . .
5 Wangersky-Cunningham Mode1 wit h Prey Harvest ing
47
The Mode1 . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . .
47
5.2 Condition for Stability and Bifurcation . . . . . . . . . . . . . . . . .
49
5.1
5.3 Ekamples . .
Bibiiography
. . . . . ... . .. .. .. ... . . ...... . .
.. .
53
Acknowledgements
I would like to th&
my supervisor.
Dr. Ruan. who üstened to my enciles niunber of
questions. answering t hem somet imes more than once. He hm been a great inspirat ion
to me. and 1 wish him the best of iuck in dl his Future projects.
I dso want to th&
Dalhousie University for the financial support. which made
this thesis and d that corne nith it possible.
Thanks to an the friends 1 made this yeur: I do not think I wotdd have made it
without the spices e u put in my days.
navis. th&
p u for listening to my compiauits. night iifter uight dter night.
Every minute p u spent üstening was one that made me feel better.
P'pa. m'man. Merci de m'avoir donné tout l'amour dont tm enfiuit peltt river.
Merci aussi de màvoir toujours aidé 8 prendre des décisions éclairés tout en respectant
et supportant mes choix quelques fois douteu. Vous avez fait de moi un être übre.
Je vous aime.
Predator-Prey Models with Delays and Prey Hanresting
h m i k Martin
Master of Science
Department of Mathematics and Statistics
Dalhousie University
1999
Abstract
In this
thesis. we wiiI k s t have an overview of some cornponents of predator-prey
rnodels. t hat is population growt h. harvest hg, del-.
will
&O
and fiuict ional responses. We
have a look iit the Wmgerçky-Cunningham mode1 and variations of the
Gause model. models thst have aire*
been malqzecl. W e will
mode! (generalized Gause model with prey harvestuig and del-
&O
an-ed
ü
in the prey specific
growth) that hos alreiidy been studied by Bruuer. but we wil1 add to his resdts. CVe
will dso study the above model but we wiil put the cLelay in the predator Fuoctionül
response iostead of in the prey specific growth. We will h d y look at the LVangersh-
Cunningham model with prey hurvesting and compare the resdts obtained for the
three rnodels mdyzed.
Chapter 1
Introduction
Predator-prey models play a crucial rote in bioeconomics. that is the management of
renewable resources. FVhen practiced. the management of renemble resowces has
been based on the MSY. abbreviation For maximum sustainable yielcls. The SISY
is a simple way to manage resources taking into consideration thüt over-e-uploiting
resources lead to a Ioss in productivity
Biwd on a biological p w t h rnodel. the 5ISY depends upon the environmental
carrying capacity K. .As the population approaches the d u e K. the surplus production approaches zero. Therefore. the aim is to determine how much ciln we have>%
wit hout altering dangerously the hamested population.
Accordîng to Clark [51: the MSY IevelLhas been found t o be situateci between 40%
and 60% of the carrying capacity in most population where the biological g o m h
mode1 appiies. The main problem of the MSY is economical irrelevance. It is so
since it takes into consideration the bene&
of resource exploitation. but completeiy
disregard the cost operation of resource euploitation. For euampk. it ignores the fact
'The MW levei is the level with maximked growth rate. that is mhere the surplus production k
the greatest-
that if a species is hanrested such that its population decreases to a certain level. then
the cost of harvesting can become e~orbitantbecause hduig the desirable resource
becomes more time consumhg. This might l e d to a situation where the cost of
hmesting is higher than the benefit.
ConEronted with the inadequacy of the MSY. people tried to replace it by the
OSY. that is. the optimum sustainable yield. which is biised on the standard costbenefit criterion used to muvimize the revenues. Historicnlly. however. few real cases
have been rnanaged usuig the OSY. Actually. meny population harvesting üctivities
have not been rnanuged at an. If it had been rnanaged. it wos r n d y ushg the MSY.
which often gave rise to criticai situations.
Renewable resources management is compiicated and const mct ing accurat e mut hematical rnodels about the efFect of harvesting on vegetable or animal popiilat ions is
even more complicuted. This is so because to have ii prefect model we woulcl hiive to
take into account many factors having an effect on the cost-benefit critenon and on
the strvival of the harvested population. For each poptdation we wouId neecl to con-
sider its size. growt h rate. c a a y h g capacity. predators. compet it ors combinecl nrith
the cost of hnnresting and the price obtained for the harvested species. More informations can be found about t hese factors in Clark [5]. but it is obviotts that a perfect
mode1 c w o t be achieved becsuse even if we couid put d
l these factors in ü model.
the model could never predict ecologicd catastrophes or Mot her Nature caprice.
Therefore. the best we can do is to look for ÿnalyzable modeIs that describe
iis
weil as possible the reslity or the effect of harvesting on populations. which is what
we will try to do in this thesis.
1.1.1
The Logistic Growth Model
When studying population dynamics. we usudiy begin by modehg the population
growth and then we udd other features to this model so that it better refiects the
reaüty. The population growth c m be described by discrete models or continuo~is
l focus on continuous models but more information on
rnodeis. In this thesis- we d
discrete models can be found in 161. The bimplest continuous model of popidat ion
growth.
hitû been studied by .Chithus in 1798. In this equat ion. r represents the net proportionai growth rate. that is. r = 6 - cl where 6 is the birth rate and d is the desth
rate. The solution of (1.1) is r(t)= r ( 0 ) e r t . LVe can see that the popdation s goes
to infinity for r > O and p e s to O for r < O. The case r > O is biologically impossible 3llce every popdation is restricted by the camying capacity of its environment.
Taking into account the carqhg capacity. we obtain
where k > O is the c i i g capacity of the environment and r
> O is the intrinsic
growth rate. This mode1 has first been studied by Verhulst in 1838 [36]. It is eass to
see that for O c z < k. dx/dt > O and for x > k. dxldt < O. These resdts are shom
in Figure 1.1 a .
Figure 1.1: (a) Behavior of the logistic h c t i o n : (b) Typicitl solution curves.
Using sepmation of &ables.
~ ( ' 1=
we c m soIve equat ion ( 1.2) iuid obtÿin
k
+ce-~t
k - zf,
.where c = -.
JO
The graph of this Function is shown in Figure 1.1 b.
It has been noticed by FeUer [7] that the logistic growth mode1 ndI aiways give a
good representation of populations increasing to an aqymptotic le&. The growth of
several populations has been studied such as D~osophiOamelanogwter (fniit £iy)p o p
dations (Pearl [BI).
Paramecium and Tiibolium (8our beet Ie) popdations (Gause [L 11)
and other populations such as the ones studied by Pielou [29] and SI- [231. AH
demonstrate that the Iogistic growth reasonably fits euperimental data. Howeuer. a
better fit c m be found in most cases using aiternative models. Other e*b
species
gcowth models have been suggested throughout history svch as the %Iee effect"
(&O
c d e d depensation) and the ;Gompertz Ianrl of population growth. More information
can be found about these different models in [51 or [61.
1.1.2
Harvesting
Consider the mode1
where F ( r ) = r r ( l - rlk) represents the logistic growth iuid h ( t ) the hanesthg
rate. The harvesting rate h ( t ) c m be defined differently depending on the species
studied. In tishery for example. the hiwesting rate is norrndy definecl ÿs h = qEr
nrhere q is the catchubility of the species. E is the hürvesting effort and r is the fish
stock (see Clark [51 For more detail). In this thesis. we wiil consider the case where
h = constant:
In this case. rkl4
is the maximum sustainable yield (ZIISY). For h > k/4. rl = O
is the only equilibrium. that is the population is driven to extinction for m y initial
- rl For
popidation ~ ( 0 ) .For h = rk1.l. ri = k / 2 is the o d y equilibrium and r(t)
x(0) > rit r(t)-r O for x(0) < XI. For h < rk/L there are two positive equilibria:
Equation (2.3) c a n be rewritten as
Figure 1.2: Logistic equation wit h constant harvest hg.
where for O < r < El. drr/dt < 0. for
EL < r < &. d.r/dt > O and for r > E2.
dxldt > O. Therefore. ELis unstüble iud & is stable (see Figure I.2).
This mode1 has three main properties. First. it has a rn~ximumsustainable yield
(blSY). hbrsY = rnarcF(r). for which any larger harvesting rate wilI leiad the p o p
dation to extinction. Second. the population level. r = rsrsy. is not the nat tiral
equilibrium level k even t hough it is the level at which the product ivity of the species
is rn~uimized:in this model. x~~~ = k / 2 . Third. if the harvested species r decreases
so t hat it is below r
, k ~ =
~ ylc/2.
the only woy tu recovery is to have a hmesting rate
smder thaa the bISY. the fastest way being h = 0.
Some conhision occurs when tryïng to determine if a species has been overexploited. Indeed. since an harvested population has Iess members t han before it was
harvested. it is easy for ine'cperienced eyes to believe that an hanrested popdation
is overexptoited. However. hanresting ohen decreases the equilibnum of the population levd since it increases the mortaüty rate. In the bioIogÏcaI c o a ~ ~ ~ u n ïitt yis
.
generdy accepted that overexploitation occurs when a popdation declines beIow the
maximum sustainable yield (MÇY). Aowever. for people living kom harvesting, a
population level close to the MSY means higher harvesting effort which increases the
cost of harvesting and can be critical for individuai and101 community Living boom
harvesting.
1.1.3
Delay Models
h o t h e r factor to be taken int O account when modeling dpomical systems is the pres-
ence of delays. Delay is a general concept that c m represent ciiffereut phenornena sitch
as the time it takes for the progeniture to reach maturity or the finite gestation perioti
of a species. 'ufathematicd delays are input in modeh to correct the classicd logistic
model. which assumes that the growt h rate of a poptdat ion at time t is deterrnined by
the number of individu& at that tirne. Of course. bioIogicd del-s
are cornplex
and
the mathematical representation iç often a simplification of r e d i t . Biologicd delays
c m be represented by discrete (shzwp) delays or continuous (dis~ributeci)delays. In
this t hesis. we will focus on discrete delays. but more information can be found about
continuous delays in [37] and [;II.
Discrete Delay in Logistic Equation
In 1948. Hutchinson [I8] proposed û rather redistic logistic equation Nith delay:
where r and Ic are as described before and the delay r > O is a constant. Equation (1.4)
Ïs
known as the dehyed logistic epation or Hutchanson's equation. Hutchinson came
up with this equation when trying to mode1 a DaphniG population. hdeed. the size
of a Daphnia clutch is believed to be detennined by the amount of food a d a b l e
during the formation of the eggs. before their passage into the broad pouch. This
characteristic is not present in the classical logistic model. which assumes that the
size of a citttch depends on the amount of food available when the eggs hatch. It is
important to ûdd this particulmity to the classical model because not doing so could
aUow. in extreme cases. overproduction. which in turn could leiid to a population
above the carrying capltcity k. This couid have drastic consequences on the habitat
and the population itself. So in order to obtain equütion (1.4). Hutchinson asstuned
that egg formation occurs r units of tirne before the eggs hntch.
T h e initiai val~teof equation (1.4) is r ( 0 ) = o(0)
and a> is continuous on
[-T.
> O. where 6, belongs to
[-T.
O]
O]. It is eusy tu see that the eqtdibria of (I.4) are r = O
and r = k . We can determine the stability of the eqtulibria of (L.4). .An equilibrium
x = r* of (1.4) is stable if for dl i1l > 0. there exists a d > O buch that lo(B) - r'l 5 d
-
for û E [-TA] implies that lx@)
t
- x.1 < c
-
for t > O. Moreover. if r ( t )
r' wtien
x. then the equilibrium '
r is said to be asymptotically stable.
The equilibrium r = O is unstable since a n d perturbations From r = O satiçti.
the h e a r equation d+t
= rx. Therefore. the case of interest is r = k for which we
have the foollowing theorem [14]:
Theorem 1
(2)
if 0 5 r r < r/2. then the positive equilibrium r = k is
asymptot-
ically stable.
(iii) If r~ = lr/2? then a flopf bz/ufcation occvrs at x = k: the pen'odic sohtions
-
-
--
24Daphoia is a tiny. insect-üke. fksh =ter crustacean with a transparent sheii.
Figure 1.3: Bifurcation diagram for equat ion (1.4).
X
Fi,we L -4: Periodic solut ion of ( 1.4).
exist for r r > r / 2 and are stable.
The Iast part of Theorem 1 is represented in Figure 1.3. Aho. the Hutchinson's
equation (1.4) can exhibit stable Limit cycle periodic solutions for a wide r a q e of
values of rT with period 4~ (see [141 for more details). This behavior is shown in
Figure 1.4.
Nthough Hutchinson elaborated the delayed logistic equation ( Z -4) when st udying Daphnia populations. it can be used. and has been used to mode1 other species.
One example is Nicholson's [251 experimentd results of the Aus-traIian sheepblowf-
(Lucila cuprina). Nicholson studied for 2 years the sheepblowfiy population. obsenring a periodic oscülation of about 35-40 days.
To compare with the Hutchinson q u a -
tion. we need to find values for the parmeters. The parameters k and
T
are known
where k depends on the bod a d a b l e and r is approximately the t h e required for
al
m to becorne an addt: the intrinsic growth rate r is unknown. for a period of
40 days. Nicholson's data give ii del-
tion for r r = 2.1 and T
of about II days while the Hutchuison's eqiia-
= 4 . 5 4 ~produces a del-
of about 11 d g . Mthough the
mathematicai model is rather accttrate. it codd be better. Indeed, for the observeci
period of 40 days. Hutchinson*~equation gives us a del- of about 9 days while the
e.xperemental del-
is about 11 d-S. X more realistic model hm later been proposed
and more details cm be found about this improved mode1 in [131.[26].[20].[191.
H u t ~ ~ o nequütion
's
h a dso been used to explain the lemming popuiation in
the region of Churchill in Canada. This lemming popdation hirs been studied by
SheIford [321. iuid is well-known for the regtdarity of its population cycles. SheKord%
data exhibit cycles of either 3 or 4 years. SI- [231 compared the experimental resitits
using the HutchinsonS equat ion with r r = 2.4 and r = 0.72 y
ü
r (9 rnonths). In t his
case. r represents the period eIapsed between the end of a summer and the beginning
of the next one. The cycles obtained with the Hutchinson's equation have a ~tniform
period intermediate between 3 and 4 years.
Chapter 2
Predator-Prey Models
As seen in the examples of the introduction.
ÿnimd
popdations have ÿ. tenciency to
oscillate periodicdy in t ime. Predator-prey systems have the same tencleticy. which
has been observed for more than a centtw. In fact. ils e u i y as 1840. the Hiidson Bay
Company trading anUnal hus in Canada, was keeping records of the number of peltr
traded. given LIS a good indication of animai popdations at that t ime. For exmple.
we cm see
in Figure 2.1 thut the oscillations of the 1 - n and
~ its prey. the snowshoe
hare. are rernukabiy regtdar. both having ü cycle of about 10 y e i s . In this chapter.
we wilI
look at two wd-knom predator-prey modeIs that will be modified in the
next chapters. but b s t we wiU describe an important feature of severai predator-prey
models. the Eunctiond response.
2.1
FunctionaI Responses
When bdding predator-prey models. we want modeis to include the interactions
between the predator and prey populations. More precisely: we want to describe how
the density of prey attacked per unit of t h e per predator changes as the prey density
Figue 2.1: Number of pelts of Lynx and snowshoe hures sold by the Hudson Bay
Company [Zr].
M e s . This relation is c d e d the jbnctional response of the predator to the prey
density.
The sMplest mode1 of functionilr responses is
p ( r ) = ur.
(2.1)
which is obtained by assuming that in the t h e a d a b l e for searching, the totd change
in the prey density is the attack coeEcient x the total search t h e x the prey densit-
In Figirre 3.2.
we c m see that the predator curve obtained from (2.1) is a straight
h e going t hrough the origin and is not bounded. The hinctional response (2.1) bas
been used in the Lotka-Volterra mode1
Figure 2.2: Loth-Volterra response lunction.
where x ( t ) is the prey population. y ( t ) is the predator popdation. and a. b. c and d
me positive constants. CVe cun solve -tem
(2.3) and obtuin the phase trajectories
for some constant kt Therefore. all solutions of y s t e m (2.3) are periodic solutions.
This mode1 has been critizised by several reseichers. most notably by Smith [33].
Later in ths section. we nriU see a variation of the Lotka-Volterra model (3.1).
Another response Function has been proposed by H o h g (see [I5]. [KI). The
H o h g type-1. response h c t ion.
nrhere a and 6 are positive constants. is based on the obsenmtion that a predator-s
search t h e Ïs rednced by the act of handhg its p
. Equation (2.3) is &O referred
as the dhchaelis-iLIeenten finction because büchaelis and 4Ienten [24[ used it when
studying enzymatic reactions. Equation (2.3): as shown in Figure 2.3? is a concave
down curve passing throrrgh the ongin and bounded by o. The parameter b is the
Fi,pre 2.3: Holling type41 response Function.
halj-saturation constant: that is. bepnd a prey population d u e 6. the predator-s
at tack capabiiity begins to sat tuate.
Though ot her response b
c t ions have been suggesied (see [35].
[30].[91. [391. [8].[-21. p4]).
the H o h g type41 response h c t i o n is the most important and useftd one. In or-
der to make mathematicai and-mis possible. we wi.ü work mit h the Holling t y p o I I
response in t his t hesis.
Generalized Gause Models
Game (101 proposed the predator-prey mode1
where z ( t ) and y@) represent the prey and predator densities at time t respectiwly:
a > O is the growth rate of the prey without the predator: y
> O is the death rate of
the predator without the prey; c > O is the rate of conversion of consnmed prey into
predator; p(x) is the response fimction of the predator.
In this thesis, we d consider a generalization of the
Gause
model proposed
independently by Hsu [17] and Freedman [8]:
In this model. g ( x ) is the specific growth rate of the prey in the absence of predator
where
(i) g(0) = u > O.
(ii) g(r) is continuous. differentiable and g t ( r ) 5 O for x
(fi) there
e,uists
2 0.
a K such that g ( K ) = O and g'(K) < O . where
is the c a m g
capacity of the environment.
Aiso. it is assumeci that p ( x ) satisfies
(i) p ( 0 ) = O.
(ü) p ( z ) is continuous and Merentiable ûnd $(s)> O For x
Of course. ep, >
2 0.
since otherwise. the population w o d d be driven to extinction.
The logistic h c t i o n (1.3) satisfies the above requirernents and is generdy used as
a prototype.
In this thesis,
we wiU dways use the Iogistic function for g ( x ) when
considering a generalized Gause model. As mentioned previoudy. the H o h g type-II
response ftmction (2.3) will be used for p(x) in generaIized Ganse models.
System (2.4) has equilibria & = (0,O),
EK = (K.O), E* = (x*.y*)
d
~
e
Using the logistic growth (1.2) for g(z) mud
i the Hoiling type41 response (2.3) for
p(x), we obtain the -stem:
The global qualitative behavior of -stem
(3.5) c m be describeci by the FoiIowing
t heoremL:
Theorem 2 h system (2.5).
fi)
al1 so~utionsare positzm and bowded urdh r ( 0 ) > O and y(0) > 0:
(ii) denote X =
& if ac > 7. If ac < 7 or b a c < 7 and K 5 A.
the equilibriwm
EK = ( K û ) is as~ymptotzcallystabk:
(iii) if X < K 5 b + 2X. the equilib~umF = (xa.y') is aqmptotically stable:
(zu) if X
(v) if
< K < b + X. E ik ybbally stable:
K > 6 + A. G is unstable and there is a unique stable limit cycle amund F.
'The proof of this thwrem can be found in [311.
Generalized Gause Mode1 with Prey Harvest-
2.3
ing
Brauer and Soudack [3] added a hmesting constant to -tem
(2.5). which gives the
bwem
where
H > O is the hwesting rate constant: J > O
is the minim~miprey population
required for the pretlutor population to estüblished itseiE
& > 0 is the eq~uvirlent
of 7 in (2.5). The equilibrium G = (r'. y') is given by
Brauer and Soudack [3] analyzed -stem
In sys-
(2.6) and obtained the Foiioning:
tem (2.6).
Theorem 3
(il
if
K < 25. then E = (r'.
H 5 Hc luhere H, is the critacal
y') is asymptotically stable
harvesting
for O 5
rate: that is. Hc is the value of H
for y' = 0.
(il) if K > 23 + -4. F is unstabie for O 5 H 5 Hc.-4sH increases /rom O
the system goes fiom a lima cycle arovnd
F
to
Hct
to an wtstable spiral where the
prey speçies is driven to eztznction.
(ziz) If 25 < K < 25 f At Wen then zs
to
a
suitch in stabzlity as H inmases fiom O
El,,- T h e systern goes fiom a stable spiral tu a limit cycle around F to an
unstable spiml where the prey species is driven to extinction.
It should be noted thet dthough a switch in stability fkom stable to unstable is
possible in case (fi). the inverse cannot o c c w for system (2.6): that is. t here is no
mitch From umtabiiit.y to stability.
Varying the harvesting constant H affects the stability of the -stem (2.6).which
makes this system different From system (2.5) where this possibiüty was absent. In
chapters 3 and 4. we wiU add delas in systern (2.6). and study how this dfects the
model.
It is weil-known that the lot ka-Volterra mode1 (2.1) is not accurate since it assiunes
that members of poptdations c m react instantaneously to ÿny environmental change.
In reality. delays shotdd be present in both predator and prey popttlations. For
simpücit. however. LVangersky ilnd Cunningham [381 proposed the mocLel
which has deInys in the predator population oniy (see [38] for detds).
The results drawn by Wangersky and Cunningham are as foiIows:
vaqhg the
deiay or the equilibrium d u e for the prey population in the absence of predation to
the eqdibrium value with predation can remit in different types of solutions. These
solutions can range fkorn stabIe nodes to stabIe spird to Lmit cycle. The analysis
dune by Wangersky and Crinningham has been criticized by Goel et aL
[El,
but
their anaiysis is also incomplete. An analysis of this mode1 wiU be done in Chapter
5 where we will look at this mode1 modiûed by üdding a harvesting constant in the
prey equation.
Chapter 3
Generalized Gause Model with
Prey Harvesting and Delay in the
Prey Specific Growth
3.1
The Model
Let x ( t ) and y(t) denote the population density of the prey and predator population
respectiveiy In this section. we coosider the system:
where p > O is a constant. f (L)is the specsc growth nte of the prey in the absence of
predators. xh(x) is the response function, J is the minimum prey popdation required
for the predator population to estabkh itself. and H is the constant-rate harvesting
of the prey species x. Also. f (O) 2 O and f (x) is continuous and decreasing in x. The
dehy T > O is a coostant representing the assumption that in the absence of predators.
the prey's growth iç affected by population density only after a Lued period of time.
The system (3-2) satisfies the condit ions:
where g(z) = xh(c)is the resporise function. The equilibrium point is @en by sv = J
md
r 0 [ f(r')- g*h(x*)]- H = O
(3.4)
if (3.4) has non-negative real solution for y*. LVe c m see thst as H increases. y'
decreases continuotisly until it resches zero at
H
= x8f ( Y )which $ves us the critical
hmest rate
H = r'f (Y) = J f ( J ) .
We now iinearize the systern about the equilibriurn point ( Y .y'). Let X = r -s*.Y =
y - y'. We then obtain the lînearized qstern
X r ( t ) = Y / ' ( x 8 ) X ( t- T ) + ( f (x*)- y B g r ( x B ) ) X (-t )g ( r W ) Y ( t )
Y t ( t ) = pf gr( x W ) X ( t ) .
From the Iinearized çystem we obtain the characteristic equation:
(3.5)
p = [f (z*) - y*g'(x')jr.
Now. Ietting r = Ar. we h d that the zeros of
must be a.iI in the left haif plane for the equiiibriurn point to be a s ~ p t o t i c d ystable.
This predator-prey system hüs been andyzed by Bntier
(11.
1 wiü outiine his
resdts in the next section anci I wiU Iater add to his resttits. looking for bifurcations
in the system.
3.2
Criteria of stabiiity: Brauer
We can find e?rpIicit stability criteria using the two following theorems proved in [II.
Theorem 4 The zeros of the function
urhere p > O. q 5 0-O 5 a 5 r2/-1,are al[ in the left hal'f plane if and only if
P C L 1
p f q
< 0.
Jc+
-
< y a/ïj,ï j = arcos(-pl*).
Theorem 5 For p = O. the zeros of the fvnction
K ( r ) = (2'
urhel-e O
5 n2/4.q 5 0.
are ail
+ a ) e Z - qz,
in the left half plane if
-q
< */2 - 2a/n
Looking st (3.4) we c m see that
and since hr(s') 5 O . we obtain that p = O iff hr(r') = O astl H = O. which rneûns
that there is no hmesting and the predator-prey interaction is h e u .
For p # O. we let p =
UT.
q =
-h.n
= cr2. where a = f(r*)- y'g'(xS).
b = -X'~'(X*)~ c = py*g(r*)gr(o*).
Using Theorem 1. we o b t a h that the equilibrium
(r'.y*) is aspptotically stable if
which means that
T
must be smaller than the Iarger root of the quadratic equation
-2
+
Y(b'
- a') t p r - -9y- =O.
According to Theorem 4 6 > a and
T
< l / a are also required for the eqtuübrium
to be aspptotically stable.
Criteria of St ability and Bifurcation
Xow. it is not gunranteeci that condition (3.13) wiii be satisfied.
if it is not. we c m tr-y
to find the d ~ i of
e ro ilt which a bifurcilt ion occurs: t hat is. for r < 7-0 the etluilibritun
(Y.y') is s%obleand for r
> T , it is unstable. The met hod hiis two main steps. First.
for T = O. we mtm show that under certain conditions irlI eigenvalues X have negative
real parts. Second. for r # O. we must show that there is a pure imae@mry root. This
împlies t hat t here is a b i h c a t ion at a vainlue 7-0 meaning that the eqtdibntm (x'. y ')
changes fkom stability for r
< O to instability for r > 0.
Consider the characteristic equation
where
For r = O the characteristic equation becomes
which hm the roots
Looking at equation (3.16).
p
Rie
c m see that X has negative red roots if and o d y if
+ q > O and a > O or equivdentiy.
Since p > O. g'(x*) > O. g(rU)> O. m d g' > 0. the Iast condition is always true ÿnd
we are lefi with (3.17). Now for r
have:
+ O. if X = iw is a root ofequation (3.14).then we
Lemma 1 If p
+ q > O and p-'- q'
negative real parts for ail r
stable f o r alt
T
1 O;
- Za > O . then all roots of equation (3.14) houe
that is. the eqwi6b~rium(2'. y*) is asymptoticall~
3 O.
From (3.21) we can see that Q-e obtain a unique positive sohtion
we obtain two i m a g i n q ~ o l uions.
t
A,
LJ:
if
= iw, . C V e can now find the \dalue of r;
C
sttbstituting d z and solving for r in eqttations (3.L8) ÿnd (4.19). We obtain
From the above andysis. we have the following restdt.
Lemma 2 Let p
+q
> O. If $ - p'.
then the equation (3.14) with
q" - p'
+ 2a > O and ($
-'.p
T
+ 2a
> O and
(qy
-# +
= 4a4 hdd.
= rr has a pair of pure imaginaw roots
+ 212)'
&i.iw.
I/
> -ka2 hold and T = $ (r = r; respectizrehj).
then the equatzon (3.14) has a pair of pure imaginary mots I i w , (+tu- respectively).
To see if a blfurc'ation occurs. we need to ver@ the transversality conditions:
Dserentiating equation (3.14) with respect to
T
we obtain:
Unless p = q = O. we cm see t hat dl purely irnaginary roots are simple. -Aiso.
We therefore obtain (ushg equation (3.20) for the Iast step):
From (3.21) (3.24). anci the last resdt . we c m see t h i ~ tthe transversality conditions
are satisfied. Therefore.
the FolIowing t heorem:
r;'
are
bifurcation values. Regrotiping our restdts we have
Theorem 6 Let r be defined by epatzoion (3.25).
(i) I f p + q > O a n d p b 2 - i ' -
2a > O , t h e n the equilibrium (x*:y') of systern (3.2)
is asyrnptoticully stable for all r
(k)
If p + q > O. i2- p'.
+20 > O
> 0.
and (q' - $ + 20)'
= 4a2.then there is one
+ 20)'
> 402,then there ezists a
bifirciation at ri.
(ki) f'jp + q > O. q' -p.'
+ 2 0 > O and (q'
-'.p
positive integer k such that there are k svitches /rom stabilitg to instabdity. In
other words. when
the equdiblrium ( x ' . y') of syste*m(3.2) is stable. and when
(o*.
y*) is unstable. Therefore. there are bi/îLrcations at (Y.9') for al/
3.4 Examples
Let
Figure 3.1: The equilibriiun (Y.y') = (20.15) is aqmptotically stable when r ( 0 ) =
40. y(0) = 16.
where f(c)is the logistic growth und xh(r) is the response function. We then obtain
the system
Let r = 2. p = 1. -4 = 10. .J = 20. k = 40 d H = 10 and r = O . CVe want to
determine the stability of the mode1
where the equiübrium (z'. y') = (20.15). In that case. me can see in Figure 3.1 that
the predator and prey populations wilI spiral toward the equilibrium (20.15). We c m
also Iook at predators and prey separately to study their behaviors in tirne. From
Figures 3.2 and 3.3. we can see that bot6 the prey and the predator populations
Figure 3.2: The prey popdation converges to its equilibrium vd~ie.
Figure 3.3: The predator population converges to its eq~dibriumvalue.
converge in h i t e time to their equilibrium 'x = 20 and y* = 15 respectiveLy.
FVe c m va- the harvesting comant H to see how it affects the dynamics. From
the ntmerical andysis. we concIude that a variation in H does not change the stabiiity
of the model (3.29). We
&O
notice that vaq-ing the hiirvesting constant H for
r = O modifies the d u e of y*. The more a prey population is harvested. the lower
is the number of predators at the equilibrirmi. and the less a prey population is
harvested. the higher is the number of predators y'. R e c d that the critical hawest
rate H = Jf ( J ) = 20 in this case. Therefore, for H < 20 the equilibrium is stable
and positive. but for H 2 70 the predator species is driwn to extinction and the
system colIspses. We aIso notice that the smder H is the faster the prey and the
Figure 3.4: Behavior of the prey popttiation in tirne for
r(0)= 40. y(0) = 16.
H
= 5 and
H = 10 üt
y-prcdutor
3s
i
tS 30
-*
*-..
s
O
-
O
50
1SC>
100
20t>
25<)
t-t~rnc
Figure 3.5: Behavior of the predator population in t h e for H = 5 mcl H = 10 at
r(0)= 40. y(0) = 16.
predator populations go to the equilibrïum (r*.y'). An example of these behaviors
is shown in Figures 3.4 and 3.5.
Wë are dso interested in studying the effect of the del- r on the dynamics of
the model. From the niunerÏcai andysis. we notice that increasing the d u e of T wiü
change the stability of the eqwlibrium (x*.y').
The eqwübrium wiu be stable for
s m d T but a limit cycle wiII be created for bigger T. Increasing r even more Ni1Ilead
to an unstable systern. We can see
dready present .
in Figure 3.6 that for r = 0.526 a Iimit cycle is
y-predator
17
18
Figure 3.6: There iJ
il
'
t
19
20
'
21
x-prey
I
22
24
23
bifurcating periodic solution for
T
= 0.826.
Figure 3.7: The oscillationsof the prey and predator popdationç in t h e for r = 0.826
y-predator
Figure 3.8: Behavior of the predator poptilation for diaerent values of
and H = 15 (bottom) for r = 0.826.
H
= 10 (top)
As with r = O. we c m observe the behaviors of the prey and the predator io
tirne. Looking at the esample of Figure 3.7. we notice that the predütor and prey
populations reach a periodic oscillation arouncl the eqtdibrium (r'.g * ) = (20.15) in
finit e t ime.
From these results. we c m now compare the numerical iuidysis and the mathematical mdysis with Bratiers results. Using the same purmeters as Brauer. but
using a difFerent mothematicd analysis. we obtah that the equilibritm (30.15) has
a bifurcation d u e at r = -8256.Brauer had thet the equilibrium (30.15) was stable
For T < 0.826. Therefore. euen though the signification of
T
mers we can appreciate
the similarity of the quantitative r e d t s . We can &O note h-orn Figures 3.6 and 3.7
t hat
the numencd resdts support the mat hematicd analysis.
As i
n the case for r = O. we can verify numerÏcally if varyhg H wilI affect the
dynLtmics of the model. Looking at Figure 3.8? me c m see that for T = -826.mqing
the value of the harvesting constant H changes the y* value of the equilibrium point
( * * )
As in the
case for
T
= 0' Uicreasing
H decreases y' and decreasing H
increases y*. The graph of x in time has been ornitteci. but vitrying the value of H
does not change its behavior in t h e . Unlüce when r = O, varying H for r = .Y26
does not change the speed at which the b i t cycle is fomed.
This example corresponds to part (Ki) of Theorem 6. So even though we have
seen graphs of the h t bifurcation. we should keep in mind that it is the ûrst one of
a series of bifurcations.
Chapter 4
Generalized Gause Model wit h
Prey Harvesting and Delay in the
Predator Response Function
4.1 The Model
Let x ( t ) and y ( t ) denote the population density of the prey and predator populations
respectively. In this chapter. we consider the . i t e m :
where c > O is the rate of conversion of consumed prey to predator. d > O is the death
rate of the predator in the absence of the prey, H is the constant-rate hitrvesting of
the prey species z. Ah. f (x)Ïs the specific growth rate of the prey in the absence of
predators where f (O)
3 O and f (x) is continuous and decreasing in x. The capture
rate of prey per predator. that is the bnctional response is given by xh(x) = g(x)
where h(x) > O and h'(x) 5 O and g'(x) > O- The delay
T
> 0 is a constant
repreçenting the ossumption that in the absence of prey. the predatorDsgrowth is
affected by population density ody d e r a Lued period of t h e .
(2'.
The equilibrium
y *) is given by
y'
nrhere
I
'
=
r'f (x*)- H
x* h ( x a )
and
is a non-negative real d u e . The y' value implies that r'f (r') >
now lineurize the system about the eqttiiibrium
(1'.y').
Let -Y= r - 'r . Y = y - .'y
We then obtaui the iineiuized system
From the Linearized systern we obtain the chusacteristic equat ion:
where
H. CVe
4.2
Criteria of Stability and Bifurcation
As in the mode1 with prey delay. we c m try to h d the value of ro a t which bifurcation
occurs: that is, for T < TO the equilibrium (x*,y*) is stable and for T > T* it is unstable.
The method used is explained in the prerious chapter.
Consider the characteristic equat ion
where p and r are defined in the previotti section. For
T
= O the characteristic
equation b ecornes
which has the roots
Looking at (4.5). we can see t hat X hns negat ive r d roots if and o d y if
p > O and r
> 0.
but r is aiwatVilys
positive so we only need p > O. Xow for r # O. if X = iur is a root of
equation (4.3): we t hen have:
where
Squarîng both sides gives:
Adding both equütions aad regrouphg by the pomr of
ir.
. we obtsin the foollowing
fourth degree polynomid:
Erom which we obtaîn
Rom (4.10). since
solution
+ 4f'
>
we c m see that we obtain one and ody one positive
4.
Thus. equûtion (4.3) has one pair of pure-
can now find the value of r,f substituthg
i m a g i n q roots iziw+. We
J:and soivirtg for r in equations (4.7)
and (4.8). We obtain
From the above andysis. we have the foUowiug resuit.
Lemma 3 ï f p > O and r = r& then eqvatzon (4.3) has a pazr of pure imaginary
rooh fiw,.
To see if a bifurcation occurs. we need to verify the tranwersality conditions:
Ditferentiating equation (4.3) with respect to
T
we obtiun:
Unless p = q = O. we can see that dl pure- imaginÿ- roots are simple. .Uso.
(4.13)
We therefore obtain ((using equation (4.9) for the last step):
From (4.10). (4.12). and the Iast result. we can see t hat the trans~ersalityconditions
are satxed.
Therefore.
7f
are bifurcation values. Regrouping our results we have
the foIIowing theorem:
Theorem 7 Let r be defined bg equation (4.11). Lf p > O and r =
TC. then
the
epztzbnum (x' ,y*) of system (4.2) 6as a pair 01pwely imaginary mots &iu+ for
r = r: and is stable tir T < r,f and unstable for r > r,f. The system undergoes a
biJvrcation at r,f = $ardan
(5).
Looking at Theorem 6 and Theorem 7. we c m see that delays have a major impact
on the dynamics of models. Indeed. the systerns (3.2) and (4.2) are the same except
that t he deIay was in the logistic function for the &st one and in the fimctiond
response of the predator equation for the second one. LVe cm see that the dp*mics
of the second mode1 is mitch simpler thnn the Ekst one.
4.3
Examples
Let
f )= r (1
-)
i
and h ( r ) = x + -4.
where f (r)is the logistic growth and rh(t) is the response hinction.
L W t hen obtain
the system
Let r = 2. -4 = 40. k = 50 and
H
determine the stability of the mode1
= 20, c = 6. d = 3 and r = O , bVe wmt to
y-predotor
F i e 4.1: The equilibrium (r*.y*) = (40.12) is on asymptoticdy stable node for
Figure 4.2: The prey popdation converges to its eqidibrium value.
where the equilibrittm (x'. y') = (40.12). In this case. we c a s see in Fibpe 4.1 t hat
for r = O the predator and prey populations do not spiral toward the eqttiiibntun
(40.17)as in the previous chapter: the ecpfibrium (40.12) Ïs a stable node.
We can
tirne.
&O
look at predetors and prey sepwately to study their behaviors in
Rom Figures 4.2 and 4.3- we can see that both the prey and the predator
converge in finite time to their eqnilibrium values x* = -10 and y' = 12 respective-
We wish to study the effect of the delay T on the dpamics of the model. From the
Figure 4.3: The predator popdation converges to its equilibrium d u e .
Figure 4.4: The equilibrium (r*.y') = (40.12) is an iuymptorically stable foctis for
7 = 7.
numerical andysis. we notice t hat increusing the value of r
change the dq-nilmics
of the equilibrium (x*.y*). The equilibrium wilI be a stable focus for maiI T . and a
Mt cycle wiil be created for bigger T. Increasing r even more will lead to an tmstable
system. For example. we c m see in Figure 1.1 and Figure 4.5 that the equilibrium
point (40. L2) is a stable Focus for r = 7 and that for r = 9 a Limit cycle is present.
As with r
= O. we can observe the behaviors of the prey and the predator in
time. Looking at the exampIe of Figme 4.6. we notice that for r = 7. the predator
population reaches the e q f i b ~ u m(40,12) in finite t h e . However. the t h e required
to do so, Ïs much iarger than when T = O. h o , the predator population reaches a
Figure 4.5: There is a biFurcating penodic solution for r = 9.
periodic oscillation around the equiübrium (x*.y*) in h i t e t ime. The süme resultJ are
obtained doing a sirnilu iuidysis of the prey population. From these residts. we c a s
now compare the numericd iuidysis and the mat hemüt icul analysis. h[st hemüt icdiy.
we obtain t hat the eqtdibrium (40.12) hm a bifurcation value at r
= 8.205. From
the numericd mdysis. we have ümit cycle at r = 9 (see Figure 4.5). which meiuis
that the bifurcation occurs before r = 9. Therefore. the numerical irnalysis stipports
the mat hematical restrlts.
We can &O vary the hmesting constant H to see how it ÿffects the dymuics.
From the numerical anaiysis. we notice thnt the harvesting constant H for different r
changes the equüibrium in the predator coordlliate y*. The more a prey popdation is
harvested. the lower is the number of predators at the equilibrium. and the less a prey
population is harvested. the higher is the number of predators y'.
In this euample.
the critical harvesting rate Ïs H = x f (x) = 16. Therefore. For H < L6 the equiiibriurn
is stable and positive. but for
H 2 16 the prey species is driven to extinction and
the system colIapses. Unlike the example in the previous chapter. a variation in H
can change the stabiIity of the mode1 (4.15). For euample. we can see in Figures 4.7
and 4 8 chat for r = 9 and H = 25 (a value close to the criticaI hwersting rate) the
Figure &6: The predator population in tirne for r = 7 (above) and
x- Wcy
50
.
-4s
-
T
= 9 (below).
C t t O
30
,
O
50
LOO
c-rimc:
150
200
2SO
Figure C7: Behavior of the prey population for different values of H for r = 9.
Figure 4.8: Behavior of the predator population for different values of H for T = 9.
prey and predator populations go to the equiiïbrium (x'. y') = (40.2) in finite tirne:
therefore. changing the value of H c m change the stübiiity of the mode1 (4.15).
Chapter 5
Wangersky-Cunningham Mode1
with Prey Harvesting
5.1
The Mode1
Let r(t)and y ( t ) denote the population density of the prey and predator popdations
respectively. In this chapter. we consider the system:
where
rl Îs
the rate of increase of the prey population.
r2
is the death rate of the
predator popdation. 6 is the coeEcient of effect of predation on x. c is the coefficient
of effect of predation on g7 H is the constant-rate harvesting of the prey species x.
h o ,a = r L / K ,where Kr, a density-dependent tenn. representç the ümitation upon
the growth of the prey other than by predation. The delay r > O is a constant based
on the assumption that the number of predators dependç on the number of prey
and predetors present at some previous tirne. The equiübrium ( z * ' y * ) is given by
z* = r 2 / cand
if crlrz - a$ - Hc? 3 O. PVe can see that as H increases.
y' decreases continuousiy
until it reaches zero at the criticd h m e s t rate
We can simplifjr the Wtem by LneanUng about the eqtfibrium (x'. p.). Let X =
r
- r'. Y = y - y'. CVe then obtirh the Iinezlrized bystem
X r ( t ) = (rI - Zax8 - b y e ) X ( t )- &z8Y(t).
Y f ( t ) = q 8 X ( t- 7)- -Y(t) + cx*Y(t - r).
From the linexized system we o b t a h the characteristic equiition:
where
5.2
Condition for S t ability and Bifurcation
As in the two previous chapters, we want to determine whether there exist roots with
negative r d parts' which Ïncrease to zero and eventudy become positive as T varies.
For r = O the characteristic equation becomes
which has the roots
X=
-(P + S ) &
+ J)'
- 4 ( q+ r )
3
-.
(5.6)
Looking at equation (5.6). we can see thut X has negütive real roots if and ooIy if
p+s>O
Yow for r # O. if X
(5.7)
=i
u is a root of eqtmtion (5.4) we have:
-J+ qe-w
+ph
where
and q + r > 0 .
+ r + iswe-lm
= 0.
Simplifying equations (5.8) and (5.9). we obtain the fourth order pol-yuornial
W4
+ (3- iS- 2T)W- f r- - q- = 0.
3
3
(5.10)
from whch nre c m obtain
Et follows that if
are satisfled. t hen eqtiation (5.10) does not have positive solut ions: t bat is. the chi~riicteristic equation (5.4) does not have piireiy imaginary roots. Ecpation (5.7) gzurantees that ail rootü of equation (5.5) have negative real parts. Lking Rouché's t heorem.
we can conclude the FoIIowing:
Lemma 4 If conditions (5.7) and (5.12) aare satisfied. then al1 mots of equation (5.4)
have negutzve real parts for al1 r
T
2 0:
that is. the epiiibri.um (2'. y') ks stable /or al1
> 0-
From (5.11) we can see that we obtain one positive solution :
w if
hold. we obtain two imaginary solutions. AL = iu*. We can now End the d u e of r,'
by substituting wg and solving for
in eqttations (5.8) and (5.9). Solving for
<.
we obtain
Rom the above analysis. nre have the FoUowing remit.
Lemma 5 1' conditions (5.7) and (5.13) hold. then e p a t i o n (5.4) ur&thr = r - has
a pair
of pure imaginary roots &i.iw+.
If conditions (5.7) and (5.14) hold and r = r;
-
(r = r, respectively). then equation (5.4) has a paw of pure imuginary roots *i~*,
(3ziu- respect aue19).
To see if a bifurcÿt ion occurs. we need
to
verify the transversality coriditions:
Differentiating ecluation (5.4) with respect to
T
we obtÿin:
Udess p = q = s = O. we can see that all pure- h a g i n q roots are simple.
We therefore obtain (using equation (5.10) for the last step):
=
{..
(a-'}
.-
A*
.&o.
Rom (5.11). (5.14).and the 1 s t result. we can see t hat the tranmersality condit ions
are satisfied. Therefore.
are bihircation values. Regrouping our rendts we have
the foiIowing t heorem:
Theorem 8 Let
TF be defined by equation (5.15).
Then.
fi) If (5.7) and (5.12) hold. then the equilibrium (r*.y*) of sptem
f o r ail
(ii)
(5.2) is stable
r 2 O.
If (5.7)and (5.13) hold.
then there is one bifiit~cationat ro.
(iii) If (5.7) and (5.14) hold. then there exists a positive inteyer k such that theare
are 12 sun'tches /rom stabiliby to instability. In other wolrds. when
the equllibrium (z*:y') of system (5.2) is stable. and urhen
(f.
)'y.'t
is unsta6k. Thereforet the* are bifvzations at (x*.g*) for all
y-predator
8
6
3
.tprey
IO
Figure 5.1: The equilibri~un(L'. y') = (5.6815) is asymptoticülly stable when r(0)=
2. y(0) = 10.
Let
rl
= 30.
Q
= 15. a = 1. 6 = 1. c = 3.
H
= T and r = O. which satisdies part (ü)
of Theorem 8. We want to determine the stability of the mode1
cix
dt
= x(t)[lO - c(t)-
-
- r.
where the equilibrium (r*.
y*) = (5.68/5). In this case. we can see in Figure 5.1 that
the predator and prey populations d spiral toward the eqtdibrium (5.68/5.15).
We can also bok
at predators and prey separately to study their
behaviors in t h e .
From Figme 5.2. we can see that the predator popdation converges in finite time to
its equilibrium d u e y* = 68/5. The prey population has the sazne behavior. We
&O
notice that the Wmgersky-Citnnfngham mode1 reaches its equilibrium (z*.y*)
y -prcduror
f
t-tirnc
1O
Figure 5.2: The predator population converges to its equilibrîum value y ' = 6815.
Figure 5.3: Behuvior of the predator population in tirne for H = 1 and H = 7 when
~ ( 0=
) 2. y(0) = 10.
much fuster that the preiious models. which might occur because the stntcture of the
Wangers-Cunnigham mode1 is much simpler.
LVe can vary the harvesting constant H to see how it d e c t s the dpamics. From
the numerical analysis. we conclude that a d a t i o n in H does not change the stability
of the mode1 (5.19). LVe also notice that varying the harvesting constant H for r = O
m o ~ e the
s d u e of y*. The more a prey population is hamesteci, the lower is the
number of predators at the equiIibnum. and the less a prey popuIation is harvested.
the higher is the number of predators y*. The criticd harvest rate is H = J f (J)= 75
in thÏs case. Therefore, for H < 75 the equilibrium is stable and positive. but for
y-predator
O
2
r
1
4
6
P
.
I
8
LO
x-prey
14
12
16
Figure 5.4: There is a bifurcating periodic solution for r = 0.05.
H 2 75 the prey species ù clriven to extinction and the qrstem co!Iapses. However.
for H 2 25. condition (5.7) does not hold. so H < 25 and the system cloes not
collapse. We aiso notice that the smder H is the f'ter the prey and the prediitor
populations go to the eq~ulibritun(x'. y*). An example of these behaviors are shom
in Figures 5.3.
LVe are also interested in studying the effect of the del-
T
on the d ~ u m i c sof
the model. From the numerical malysis. we notice that increasing the vahe of r
change the stability of the equiübrium ( Y ,y').
smd
T
The equiiibrium will be stable for
but a limit cycle will be created for bigger r. hcreaskg r even more wiil
ledd to an unstable -stem.
We c m see in Figure 5.4 that for T = 0.05 a limit cycle
is dready present.
As with r = O. we can observe the behaviors of the prey and the predators in
tirne. Lookuig at the exampile of Figure 5.5? we notice that the predator population
reaches a periodic oscillation around the equiübritun (x*:y*) = (5,68/5) in h i t e
Figure 5.5: The oscillations of the prey iuid predator populations in t h e for r = 0.05
t h e . The prey population shows the m n e behilvior. We sshould &O note that again
the Wmgers-Cunningham mode1 reaches the limit cycle state much Faster t h m the
models st udied before.
From these results. we cm now compare the numerical mdysis a s d the mat hemi~ticdanalysis. Mat hematically. we obtain t hüt the eqtdibrium (r'.y ') = (5.6815)
hiis u bifurcation value at r zz 0.0385. From the numericd aneiysis. we have
a
limit
cycle at r = 0.05. which means that the bûurcation occurs before T = 0.05. Therefore.
the numerical analysis supports the mat hematicd results.
As in the case for r = O. we can ve*
numericdly whether varying H d d e c t
the dpamics of the model. Looking at Fieme 5.6. it is not obvious that for r = 0.05.
varyuig the value of the hmesting constant H changes the y' value of the equilibrium
point (z*, y*). We calculate that Cor H = L. y' = 7.115. cornpiireci to y' = 6815 for
H = 7. Therefore. varying H also varies y*. As in the case for T = O. Uirreasing H
decreases y' m d decredsing H increases y*. The graph of z in tirne has been omitted.
but vafying the value of H does not change its behavior in t h e .
y-predator
30
25
20
1s
10
5
Figure 5.6: Behavior of the predator population for different d u e s of H for r = 0.05.
Chapter 6
Conclusions
We have seen throughout this thesis that incorporuting a harvesting constant in a
model and varying it cûn change the stability of a -tem.
although it does uot
seem to be a generd d e . From the mathematicai maiysis. change in stability is not
supposed to happen when altering H as long as it respects the conditions and is below
the criticai harvesting rate Hc.However. we have seen boom computer simulations iu
Chapter 4 that increasiising H stficiently codd change a Limit cycle into a stable spirai.
More research wodd need to be done to determine if this behavior is a particuluity
of systern (5.1) or if it also occurs in other biologicai models.
We have also seen that üdding deiays into a model c m change its stabilit. A
particdarly interesthg case has been seen in Chapter 3. Indeed. nre have shonm that
adding a delay in the logistic growth term of the prey equation cretited k switches
from stability to instability as r increases. This means that the -stem can go back
to stability when unstable. This phenornenon H e r s h-orn the generalized Gause
model (2.5) where once the system has become unstable it couId not go back to a
stable state.
There is still a tremendous m o u n t of work to do in this area. For example.
it would be interesting to see whiit wodd be the behaviors of systemç (3.2).(4.2)
and (5.2) when the hmesting constant is in the predator equation- CVe codd expect
some Merences boom the examples seen in this thesis. at l e s t for systerns (3.2)
and (4.2) where Brauer asd Soudack (see [3]?[2]) noticed different m e s of solutions
whether the harvesting n r i in the prey or in the predator equation when the dehy
was nil. Idedy. we would be interesteci in studying systems (3.2). (-4.2) wud
i (5.2)
with both predator and prey hmesting constants since we iisuaily harvest. or woidd
Like to h m e s t . both populations.
It wodd di0 be interesting to study. üt Ieast
with cornputer simulations. the kVangemky-Cunninghm mode1 wit h dela-ys in bot h
the predator and prey eqttntions. We codd dso üdd hiwesting constiints to cieliq
rnodels like
M a y [21]
and
Volterra [37] and BreIot [4]
It is important to understand rnodels with dehy and hanresting and dthough
much as been done with models with either delay o r hamesting. Iitth research hss
been done toward models h*Wlg both components. It is particularly important to
have good hmesting models since the need to manage properly our ecosystem ancl
the economy depending on it is immediate. As the global population increiws at
an e-qonentid rate. we put our environment under dangerotts pressure. Therefore.
having the tools to avoid over-exploiting the ecosystem is a small but important step
in keeping the globd ecosystern in a viable state.
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