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BEHAVIOURAL SENSITIVITY OF THE
LOTKA-VOLTERRA MODEL
Johan Swart
Department of Mathematics & Applied Mathematics
University of Natal
Pietermaritzburg
South Africa
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ABSTRACT
The classic model for a deterministic,
continuous growth, one-predator-one-prey
system is that of Lotka and Volterra. It is well
known that this model predicts neutral
stability in which the constant amplitudes of
the oscillations are determined by the initial
conditions. Without changing the underlying
model assumptions and by altering only the
predator functional response to prey density,
it is shown that damped oscillations towards
stable equilibrium or explosive oscillations
or a stable limit cycle can be generated as
model output.
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Introduction
The classic Lotka-Volterra model for a
deterministic predator-prey system with
continuous growth is the simplest
representation of non-linear predator prey
interaction. The model predicts oscillatory
behaviour of the populations with constant
amplitudes strongly dependent on the initial
conditions. Thus the model predicts that an
observed oscillation of many periods duration
is due to an event that occurred earlier on ,
rather than being inherent to the system. In
most texts the model is abandoned as it is felt
that its unnatural behaviour makes further
study unprofitable (Pielou 1977,p91 '). In
order to obtain different behavioural modes
the underlying basic model is then changed
by assuming additional relationships amongst
model variables- usually these comprise self
regulating terms or a predator-prey response
that takes into account the likely effect on
per capita growth rates of the relative sizes of
the interacting populations.
1.Pielou,E.C.(1977). Mathematical Ecology. Wiley-Interscience, New
York.
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More realistic models resulting in damped
oscillations towards stable equilibrium or
explosive oscillations and eventual extinction
of predator-prey populations or stable limit
cycles are available in the literature (Leslie
& Gower 1960, Freedman & Waltman1975,
Nicholson & Bailey1935, Samuelson! 9672 )
We show that the Lotka-Volterra model is
behaviourally sensitive to the predator
functional response to prey density and that
all the behavioural modes mentioned above
are possible for this model. We follow a
system dynamics approach.
2. Freedman,H.l.,and Waltman,P.(1975). Perturbation of two dimen-
sional predator-prey equations. SIAM J. Appl. Math. 28, 1-10.
Leslie,P.H., and Gower,J.C.(1960). The properties of a stochastic
model for the predator-prey type of interaction between two
species. Biometrika 47, 219-234.
Nicholson,AJ., and Bailey, V.A.(1935). The balance of animal
populations. Proc. Zool. Soc. Lond. 1935, 551-598.
Samuelson,P.A.(1967). A universal cycle? Operations
Research-Verfahren 3, 307-320.
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The Model
Consider a simple predator-prey model in
which the following are assumed:
(1)The prey has an unlimited food supply so
the net prey growth rate is proportional to
prey density;
(2)The birth rate of the predator is
proportional to predator density hut
is modified according to prey abundance;
(3)The death rate of the predator is
proportional to predator density;
(4)The predation rate of prey is proportional
to predator density and is modified
according to prey density.
Growth + = Predation
Rate | _ PREY Rate /!
4 Feel LP . we
! , Say ot ‘
@ ae ~~ PM . ;
a2 Predation /
BM
. oo Multiplier”
Birth Multiplier Zz
Z ’
y -
3 2
Birth b ~~ Death
Rate PREDATOR F— pt
—_ Ss a 7 4
oe ane
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Model Formulation
d(PRED)/dt = Birth Rate - Death Rate
= PRED*BM(PREY) - PRED*DN
d(PREY)/dt - Growth Rate- Predation Rate
= PREY*GF - PRED*PM(PREY)
The functions BM and PM are increasing
functions of PREY density - assume that BM
varies between 0 and 0.4 and PM between 0
and 8 respectively, as PREY density increases
from 0 to 20. BM is taken to be linear.
Predator functional response graphs, PM,
used in the model are shown below.
Predator Functional Responses
= 34
a 8
concave
\ <— linear
<— sigmoidal
9) 4 8 12 16 20
Prey density
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The Lotka-Voiterra Model
BM and PM are both assumed linear.
BM(PREY)= (0.4/20)*PREY
PM(PREY)= (8/20)*PREY
yielding the following model equations:
d(PRED)/dt- PRED*(0.02*PREY - DN)
d(PREY)/dt=- PREY*(GF - 0.4*PRED)
sesenaee Lotka-Volterra equations
Both species oscillate periodically around
their equilibrium values PRED-GF/0.4 and
PREY-DN/0.02 with average population values
equal to their equilibrium values. For GF= 0.4
and DN=0.2 these values are respectively
PRED=1 , PREY=10. The constant amplitudes of
the oscillations are detemined by the initial
values.
N=
Ne
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Predator linear response
1 Prey 2 Predator
500
16.250 J ~
1.250
\
12500 4
1.000
= \\ \ 2
e750 6 64
0.750
3.058 7 .
0.500" ig’ 12.500 25.000 37.500 50.000
Time
Prey vs Predator
16.250 34
Prey On,
Pa
12.500 4
8750 4
5.000
osoo 0750 1000 1250 1.500
Predator
Ne
Ne
Ne
Ne
N=
Predator sigmoidal response
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1 Prey 2 Predator
20.000 _
1.500
16.250 4
1.250
"3388
2 i a 2
i | 1 Venere
azo | ya _——
0.750
1
3-908 T T T T 1
9.500 go 20.000 40.000 60.000 80.000
Time
Prey vs Predator
16.250 945
Prey a,
12500 4+ .
8750 4 ee ant!
5.000 T u T 7
0.500 0.750 1.000 T2568 1.500
Predator
n= N= N=
ey
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Predator concave response
1 Prey 2 Predator
30.000 _
2500
1
38 4 ,
15.900
7:250 VA,
WN J/ J
aS W
1
00 1 i
99 oo mT ""20.000 40.000 sano ss
Time
30.000 4 Prey vs Predator
Prey A te
22500 | 2 -
1s000 4° = hank ‘
: 3 ‘ ey
ar | £ ‘
: 4 ;
7s00 4: LY r; .
3 na
i y
00 oes 61250 «+1875 ~~ 2500
Predator
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Stable limit cycle
15.000
Prey
12.750 4
10.500 4
8.250 4
7 7 T 6.000 T T T
0.625 0.850 1.075 0.400 0.625 0.850 1.075,
Predator
Predator functional response
8.000 | _ Input Output
0.0 0.0
2.000 0.440
4.000 2.080
6.000 2.320
8.000 4.080
z 10.000 4.360
12.000 5.920
14.000 6.440
16.000 6.560
18.000 7.640
20.000 8.000
Become Algebra
pee,
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Conclusion
The simple predator-prey model investigated
here is behaviourally sensitive to the
predator functional response to prey density.
At least four distinct behavioural modes are
possible- neutral Stability, damped
oscillations towards stable equilibrium ,
explosive oscillations and stable limit
cycles.
