MATHEMATICAL MODELS FOR PLANT COMPETITION
AND DISPERSAL
by
DAVID NORMAN ATKINSON, B.S.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
May, 1997
/
m
^..M
I û\ÛI'^
r ^^y
ACKNOWLEDGMENTS
I would like to thank each of my committee members for the time that they
invested in my research: Dr. Mark McGinley, whose explanations helped in comprehending the underlying biological questions; Dr. Ed AUen, whose programming
skills enabled the spatially heterogeneous simulations; and of course, Dr. Linda
Allen, whose guidance, support, and patience made all of this possible.
n
CONTENTS
ACKNOWLEDGMENTS
ii
LIST OF TABLES
iv
LIST OF FIGURES
v
I. INTRODUCTION
1.1 The Origins of Modeling
1.2 Alternative Modeling Formulation
II. THE TWO-SPECIES HOMOGENEOUS MODEL
2.1 History
2.2 Derivation
2.3 Steady-States
2.4 Stabihty of the Steady-State
2.5 Numerical Simulations
1
1
3
6
6
6
8
10
14
III. THE THREE-SPECIES SPATIALLY HOMOGENEOUS
MODEL
3.1 Introduction
3.2 Assumptions
3.3 Derivations
3.4 Steady-States
3.5 Analysis of the Steady-States
3.6 Special Case
3.7 Numerical Simulations
18
18
18
19
21
22
25
31
IV. THREE-SPECIES SPATIALLY HETEROGENEOUS MODEL .
4.1 Introduction
4.2 Dispersal-competition model
4.3 Relationship to the spatially homogeneous model
4.4 Numerical Simulations
4.5 Numerical Simulations with control
41
41
41
42
43
47
V. DISCUSSION AND CONCLUSIONS
REFERENCES
52
54
ni
LIST OF TABLES
2.1
Stability Conditions for Two-Species
l^
3.1
Stability Conditions for Single Species Steady-States
23
3.2
Stability Conditions for Two-Species Steady-States
24
3.3 Stability Conditions for the Special Case for the Single-Species
Steady-States
26
3.4 Stability Conditions for the Special Case for Two-Species
Steady-States
26
IV
LIST OF FIGURES
2.1
Steady-State No. 2
15
2.2
Steady-State No. 3
16
2.3
Outcome dependent upon initial conditions
17
3.1
Region of StabiUty
30
3.2
Single Species Domination
32
3.3
Two-Species Domination
33
3.4
Three-Species Co-existence
33
3.5
Convergence due to parameter values, a = .5 and b = A
34
3.6
Convergence due to parameter values, a = .5 and b = .9
35
3.7
Region of Stability, a = .75
36
3.8
Convergence, interior to region
36
3.9
Nonconvergent, exterior to region
37
3.10 Phase plane diagram of nonconvergence
37
3.11 Limit Cycle
38
3.12 Phase Plane Limit Cycle
38
3.13 Oscillation I
39
3.14 Phase Plane Oscillation I
39
3.15 Oscillationll
40
3.16 Phase Plane OsciUation II
40
4.1
Initial Spatial Distribution
43
4.2
Single-Species Domination
44
4.3
Two-Species Domination
45
4.4
Three-Species Co-Existence
46
4.5
Special Case
47
4.6
Initial Control Conditions
48
4.7
NoControl
49
4.8
Eífects of Control on Species Ai
50
4.9
Effects of Control on Species Ai and A^
VI
51
CHAPTERI
INTRODUCTION
1.1 The Origins of ModeUng
It is often dif icult for ecologists to apply experimental methods to answer their
questions and to predict the consequences of population growth or variations in
time and space. Mathematical modeling, which is based upon biologically reasonable assumptions, provides an alternative approach to some of these problems.
Modeling allows ecologists to study how variables interact and to make predictions
about what would happen if some of the variables changed. Moreover, models
generated from first principles can be tested against observations in experimental
systems; thus, tests for the hypothesis are possible.
The foundation for present day mathematical population models was built by
Malthus, Verhulst, Lotka and Volterra. In 1798, Malthus proposed the exponential
growth model [12]. This model is of the form dN/dt
= rN, where A^ is the
population size and r is the intrinsic growth rate. This is an unrealistic model, since
it implies unbounded population growth and that even the most slowly reproducing
organism would cover the earth in a short period of time. Malthus realized this
and he wrote:
Through the animal and vegetable kingdoms, nature has scattered the
seeds of life abroad with the most perfuse and liberal hand. She has
been comparatively sparing in the room and the nourishment necessary
to rear them. The germs of existence contained in this spot of earth,
with ample food, and ample room to expand in, would fill miUions of
worlds in the course of a few thousand of years. Necessity, that emperious all pervading law of nature, restrains them within the prescribed
bounds. The race of plants, and the race of animals shrink under this
great restrictive law.[9]
However, Malthus did not further his work, instead the next step was performed
by Verhulst in 1838 [5]. Verhulst derived the logistic equation, dN/dt
N/K),
= rN{l
-
which introduced the notion of a carrying capacity, i.e., the population
at which ^
= 0, and density dependence of the growth rate.
Unfortunately,
Verhulst's work was overshadowed by that of Darwin's, and thus the logistic model
was lost for over 70 years.
In the twentieth century the next step in the evolution of mathematical models
was performed by Lotka and VoIterra[5]. The Lotka-Volterra model for species
competition is given by the equations
dNi
^^ (Ki -Ni
-ai2N2\
= riiVi (dt
' ' V
Ki
dN2
,,
fK2-N2-a2iNi
dt
^ "K
K2
)
where K^ is the carrying capacity for species i, and aij are the interspecific competition coefíicients [5]. This model was revolutionary because it included the effects
of interspecific competition, in addition to the properties inherited from the logistic
growth model. Lotka-Volterra two-species competition model has been expanded
to n-species competition model and analytical stability conditions have been found.
Due to the models' simplicity and its biologically meaningful parameters this model
is popular among ecologists.
The popularity of mathematical models in theoretical ecology has grown greatly
in recent years; thus, the need for better modeling techniques is apparent. The
Lotka-Volterra competition model is an extension of the continuous logistic growth
model [5]. The logistic model was proposed after studying organisms with extremely fast reproductive capabihties, including prokariotic and simple eukariotic
organisms; however, population histories of phylogenetically more evolved organisms are not accurately described by this model. Hence, there is need for other
modehng techniques, especially when interested in populations with only annual
reproductive tendencies.
1.2 Alternative Modeling Formulation
Discrete-time models are better suited for organisms with seasonal reproductive
patterns. Pakes and Maller developed a discrete-time model based on de Wit's
formulation for modeling species in competition [14, 22].
In the early fifties, the Agricultural Extension of the Dutch Government conducted field experiments concerning mixed cultivation of barley (Hordeum vulgare)
and oats {Avena savita) [22]. The data collected from these experiments provided
long term results, which were used by de Wit to formulate a model. Since spatial
location is particularly important in plant communities, crowding coefficients were
used to describe interactions, instead of competition coefl&cients [22].
To formulate the de Wit competition model the following parameters are defined [22]: 6*1 and ^2 represent the two plant species, Ai and A^ represent the area
of the field available for each species, respectively, and Zi and Z^ represent the
number of seeds planted of each species. The variables Oi and O2 are the seed
yields of each species, and Mi and M^ are the monoculture yields of Si and 5^,
respectively. Then the following relations hold:
L Ai+^
=
H.
= biZi/b^Z^.
Ai/A2
1
(1.1)
(1.2)
The multiplication factors 61 and b^ are the crowding coefficients. The ratios of
these coeflacients are called the relative crowding coefficients, i.e., ku = bib^^ is
the relative crowding coefficient of Si with respect to S^ and k^i = b^b ^. Let
zi = Zi[Zi -I- Z2\~^ and z^ = Z^^Zi + Z^]'^ be the relative seed frequency of both
species. It follows from the equation (1.2) that
A,
= b,Zi[biZ,-{-b2Z2]-' = kuZi[knZi
A2
=
b2Z2[biZi-^b2Z2]-^
+ Z2]-'
= k2iZ2[k2iZ2 +
Zi]-\
The yields are proportional to the area available for each species:
Oi =
biZi[biZ^ + b^Z^]-^ Mi = ki2zi[kuzx + Z2]-^ Mi
O2
b2Z2[biZi + b2Z2]-^M2 = k2iZ2[k2iZ2 +
=
zi]-^M2.
At this point, the de Wit model provides a basis for Pakes and Maller [14] to
create their model. A few further assumptions are made in Chapter II. However,
it will be easy to see the relationship between the Pakes and Maller two-species
competition model and the de Wit model.
In Chapter II, the two-species competition model is derived, analyzed and
numerically simulated. In Chapter III, the two-species model is generalized to three
species. The three-species model is then analyzed and simulations are performed.
Chapter IV extends the time-dependent model to a time and spatially dependent
model, discrete in time but continuous in space. This latter model is related to
models developed by E. Allen et al. [1] and L. AUen et al. [2]. However, they did
not apply the model to three-species.
The two and three-species time and spatially-dependent model has application
to weed control. One species may represent a desirable plant species, such as
an agricultural crop, and the remaining species are weeds which may outcompete
the desirable plant.
To prevent the weed species from replacing the desirable
plant, controls are appUed to the weed species to reduce their density. Numerical
simulations are performed in Chapter IV to iUustrate weed control in some cases.
The final chapter presents a discussion of the results.
CHAPTER II
THE TWO-SPECIES HOMOGENEOUS MODEL
2.1 History
In Australia, there has been great interest in modeUng the spread of clover
strains. Since most clover are a food source for livestock, there is an aim of substituting new improved strains for less desirable ones [16]. R. C. Rossiter has been a
keystone individual in the long term field studies of binary clover cultures. With
collected data, Rossiter and M. J. Palmer calculated the de Wit competition coefficients for some clover strains [17]. The competition coefficients were then used
by Anthony Pakes and Ross Maller [14] to create a two-dimensional difference
equation for the adult plant density of species one and two in year t, Ai{t) and
A2(t).
2.2 Derivation
To derive the Pakes and Maller competition model, we begin by defining several
state variable and parameters that are needed in the model description [2, 3, 14, 19].
Let
rt, St
Xt, yt
Pw, Qw
PG, QG
PEI QE
PA, QA
=
=
=
=
=
=
residual seed density in the year t for species 1 and 2.
new seed density produced in year t for species 1 and 2.
probabiUty that a seed survives over winter.
probability that seeds germinate.
probabiUty that plants become estabUshed.
probability that established plants become adults.
In general, the parameters Pw, PG, PE, and PA depend on temperature, soil
conditions, and other environmental variables [3]. The probabiUties with P stand
6
for species 1 and Q for species 2.
The residual seeds in the year t consist of seeds from previous years that have
survived the winter and remained dormant, as well as new seeds that have not
germinated after surviving the winter. Thus, r^ and St satisfy the equations,
n = Pw{i - PG)[xt-i + n-i],
st =
Qw{i-QG)[yt-i-^st-i].
Applying the de Wit model, the maximum seed pool for species one in year t
is
Pt
=
Xt + rt
kuM.PEPAPGPwPt-i
kuPEPAPGPwPt-l
+ QEQAQGQwqt
- 1
+ Ps{l -
PG)Pt-V
and for species two the maximum seed pool at time t is
qt
=
yt + St
k^iM^QEQAQGQwqt-i
PEPAPGPWP^-I
+
k2lQEQAQGQwqt-l
+
Qs{l-QG)qt-i.
Adult plants are formed from seeds pt and qt from year t that survive the winter,
enter year t-\-l, germinate, become established, and finally reach adulthood. Let
adult plants of species i in year t be denoted by A^, i = 1,2. Then from the above
equations it foUows that
aiMiÅ:i2^4irø
ki2Ai{t)
. .,
+ A2(t)
^^(^+^) = ^21^(t)+^(t)-^'^^'^(^)'
8
where a, =
PAPEPGPW,
h = Pw{l -
PG),
«2 -
QAQEQGQW,
and 62 = Qw{l
-
QG).
The number of parameters in the above model can be reduced to five by replacing Ai{t) by cÃi{t), where c = {1 - bi)/aiMi.
The simplified system has the
form:
Mt-^l) = PÍl-0)J^;^^fl^^^^^PMt),
(2.2)
where the tilde notation has been dropped for simpUcity, a = bi, (3 = b^,
p = «2^2(1 - 6i)/[aiMi(l - 62)]- Model (2.1) and (2.2) wiU be analyzed in the
remainder of this chapter.
2.3 Steady-States
The methods used to solve for the steady-states in discrete models (difference
equations) are comparable to those used in solving continuous models (differential
equations). For example, when solving for the steady-states of the Lotka-Volterra
model, it is assumed that the population growth rate is zero (e.g., ^
= 0). In
discrete models, it is assumed that Ai{t + 1) = Ai{t), i.e., AAi = 0.
In the two-species model (2.1) and (2.2), there are three steady-states of interest. Two of these steady-states are where one species is at carrying capacity
and the other is locally extinct. Depending on certain relationships, these steadystates are either globally or locally stable. The third steady-state occurs when
co-existence is possible. Solving for population one at carrying capacity and pop-
ulation two extinct, (^i,0), gives the value Ai = 1. Solving for population two
at carrying capacity and population one locally extinct, (0,^2), gives the value
A2 = p. Solving for a positive steady-state, (^i,.^^), and applying the difference
equation for Ai we obtain the foUowing:
kuAi + A2
1
_
^12
kuAi + A2
ku
= kuAi + A2
A2 = ku{l-Ar)
(2.3)
A,
(2.4)
= 1 - ^ .
'«^12
Applying the difference equation for A^ we obtain:
A;2i^2 + Ai
^21
T
1 = p-k2iA2 + Ai
pk2i
= k^iA^ + Ai
Ai
A2 = p-T^
k2i
Ai = k2i{p-A2).
(2.5)
(2.6)
Substituting (2.3) into (2.5) and solving for Ai.
ku{l-Ai)
=
p-
Ai
^21
Ai =
k2i{p-ku)
1 - kuk2i
Substituting (2.4) into (2.6) and solving for A^:
k2i{p-A2)
= 1
A2
k 12
10
^
^
^12(1 -
pk2i)
1 — kuk^i
Thus, the three steady-states are:
(1,0), (o,p), (^^^(P-M M i - P / . . )
\
1 — /C12/C21
1 —
Kuk^i
2.4 StabiUty of the Steady-State
To test the local stabiUty of the steady states, we use the Jacobian matrix. The
Jacobian matrix is a matrix consisting of the partial derivatives of the equations
with respect to each variable:
J ^ -
/
\
I
\
where Ai{t + 1) = F{Ai{t),A2{t))
dF{A^,A2)
dAi
dGjAuAj)
dAi
dFjAiM
dA2
dG{AuA2)
dA2
\
I '
/
and Ai{t + 1) = G{Ai{t),A2{t)).
Matrix J is
evaluated at the steady-states.
The linearized system is X{t + 1) = JX{t), where X{t -\-1) = {Ai{t + 1) Ai,A2{t + 1) — A^)^ and matrix J is evaluated at the steady-state {Ai^A^)'^. The
linearized system converges to the origin if and only iî Ai{t -\-1) converges to the
steady-state value, i = 1,2. However, X{t + 1) converges to the origin if and
only if the eigenvalues of J satisfy |AÎ| < 1, 2 = 1, 2. It is welI-known[5] that the
eigenvalues of a 2 x 2 matrix satisfy |Ai| < 1 if and only ii\tr{J)\
< l-\-det{J) < 2,
where tr{J)is the trace of J and det{J) is the determinant of J.
The Jacobian matrix for the system (2.1) and (2.2) Ai{t-\-l) = F{Ai{t), A^^t))
is
"' ~
//1
\k^9.(k^2Ai+A2)-ki2{ki2Ai)
.
/ ( I - a j — ^
^k,2Ai+A2y'
^ ^
'
.(Q.
n
fc2lA2
P\P
^^(fc21>Í2+Al)2
(ry-\)
^12-^1
^^
^Hki2Ai+A2)^
u(l _ 0\k2l{k2lA2
+ A^)-k2l{k2lAi)
/^V^
^f
(fc2lA2 + .4i)2
n
^P
11
To test for stabiUty of the single-species equiUbrium, (1,0), the Jacobian matrix is
evaluated at (1,0):
\^ 0 p{l-
/3)k2i + P
Since this matrix is upper triangular (aU entries below the main diagonal are zeros),
the eigenvalues are equal to the main diagonal entries.
Ai = a
and
A^ = p{l — (3)k2i + P
Clearly a < 1; thus, stability depends on A^. Note that A^ < 1 if and only if
P<
l/k2i.
At the steady-state {0,p), the Jacobian matrix has the form
^^ / (l-a)^ + a 0
l^ -{1-Í3)k2i
(3
Since this matrix is lower triangular (all entries above the main diagonal are zeros),
the eigenvalues are equal to the main diagonal entries.
k
Ai = (1 — a)
\-a
and
A^ =/?
P
Clearly j3 < 1; thus, stability depends on A^. Note that Ai < 1 if and only if
p > kuAt the positive steady-state {Ai^A^), the Jacobian matrix simplifies to
/ ( l - « ) ( ;âÍTfeF + "
1, p ( / 5 - l ) ( ; ^ t T ^
Recall that 1 = ^J^^.A^
(" - Dí l í í í í ^
p ( l - / î ) ( S i ^ + /3
and 1 = p^^Jf.^,- The matrix simphfies to
12
^=f(i-")â+"
("-i)â
Thus, stabiUty is determined by the trace and determinant of J. Note that tr{J)
(•^ ~ '^^át + '^ + (^ - ^ ) ^ ^ + /^' so that tr{J) > 0. The determinant of J is
éet(J)=(il-a)^^a)Ul-P)AL^A(i-^)MÍ(l^^^
V
ku
J\
pk2i
J
\
ku
J\
pk2i
After simplification,
detiJ) = ( ^ l M + "(1 - /^)-4^ + «^.
Thus, for stability we need to show, tr{J) < 1 + det{J) < 2. First consider
tr{J) <
(1
l-\-det{J).
^ ) ^ , ^ ^ ^ (1 .pA. ^ ^ < , ^ (1 - » ) ^ / ? ^ »(1 - 0)A. ^ ^ ^ .
A:i2
pk2i
ku
pk2i
Simplification leads to
A2
Ai
+ - 7 ^ <1
ku
pk2i
Application of (2.3) yields
1- Ai-\- — ^ < 1
or
1/A:2i < p.
pk2i
Now, the second inequaUty of the stabiUty criteria, 1 + det{J) < 2, requires
^^il-a)A.P_^ail-m.^^^^,
ku
pk2i
AppUcation of (2.3) yields
(1 - «)(i - AO/? + í ^ ^ ^ ^ +«/? < 1
13
or
/? - A,/3 + aØA, + " ( ^ : ^ ) - ^ ' < 1
pk 21
or
-A,f3{l-a)<{l-(3)(l
aAi
pk 21
Now, the left side of the above inequality is negative. The right side wiU be shown
to be positive, which means the inequaUty l-\-det{J)
< 2 is automatically satisfied.
From (2.6) aAi = ak^i^p - A^) < k^ip or 1 - g ^ > 0.
Note also that the steady-state values must be positive, so that the inequality
P > -j^ impUes kuk^i
> 1 and ku > P- So the steady-state (^1,^2) is locally
stable if and only if Å:i2 > p > ^ . The foUowing table gives the conditions for
each steady-state to be stable such that no other steady-state is stable.
Table 2.1: Stability Conditions for Two-Species
Condition 1 Condition 2
Steady-State
(1.0)
(0,P)
Ík2i{p-ki2)
\
1-A;i2fc2i '
ki2{l-pk2i)\
1-^:12^21
/
p<^
^
k2i
P> ku
ku > P
P< ku
p>ú:
P>Ú-i
It has been shown that the local stabiUty results are global stabiUty results.
Pakes and MaUer[14] proved that if the positive steady-state exists and is locally
stable (Å:i2 > p > 1/^21)7 then it is globally stable for positive initial conditions.
14
Pakes and MaUer[14]used a Liapunov function of the form:
V{A,{t),A2{t))
=
c,{A,{t)-A,-A,\n{A,{t)/A,))-^C2{A2{t)-A2-A2lii{A2/A2{t)),
where Ci and c^ are appropriately chosen constants and (^1,^2) is the positive
steady-state. They showed that dV/dt < 0 for aU positive values {Ai{t),A2{t))
7^
('"^1,^2)- In addition, they showed using geometric arguments in the phase plane
that (1,0)^ is globally stable (with respect to positive initial conditions) ií p <
1/^21, ^12 and {0,p) is globally stable if p > A;i2,1/^:21 • In the remaining case,
ku < p < l/k^i, there is no globaUy stable steady-state. Depending on the initial
conditions, solutions either converge to (1,0)^ or (0,p)^. Thus, the two-species
discrete time model behaves in a manner similar to the continuous-time LotkaVolterra model (see Table 2.1).
Numerical simulations are performed in the next section and the simulations are
graphed in the phase plane. In this manner, it is easy to see the close relationship
between our discrete model and the continuous-time Lotka-Volterra model.
2.5 Numerical Simulations
In this section, numerical simulations show the outcome of experiments for
specific parameter values. These parameter values along with initial conditions
determine the outcome of competition. In Figures 2.1 through 2.3, we see the
results of the numerical simulations. The soUd line represents species one zerogrowth isocUne and the dash-dotted line represents the zero-growth isocline of
species two. The open circles correspond to initial population densities, while the
dotted lines represent the change in population with respect to time.
15
The first figure (Figure 2.1) represents stabiUty of the steady-state (0, p). Parameter values were chosen such that p > ku and p > -^, this coincides with
steady-state (0, p) being stable. The parameter values for this simulation are as
foUows: a = .8, /3 = .9, ku = -8, k^i = 1.2, and p=l.
The results of the numer-
ical simulation agree with the analytically predicted results, i.e., the steady-state
solution (0,p) is stable for all positive initial conditions..
species 1
Figure 2.1: Steady-State No. 2
16
The second figure (Figure 2.2) coincides with the third steady state, the positive
steady-state. Parameter values where chosen to satisfy the inequality l/k^i < p <
ku- The values chosen for this simulation were a = .6, (3 = .8, ku = 1-4, A:2i = 1.3,
and p = 1. The simulation outcome agrees with the analytical outcome; the steadystate solution ( ^ ^ ^ ^ , '^jB^^^)
is stable for aU positive initial conditions.
1.5
1
p
. .o
1 --" V .
\
-
~ -^
v^
CN
(/)
Q>
' \
"o
\
CD
Q.
W
0.5-
\>
" "X
V
O'
0
0
' \
\
1
0.5
species 1
Figure 2.2: Steady-State No. 3
1.5
17
The final figure (Figure 2.3) is the case where either species may dominate the
competition; the outcome is based upon initial conditions. Parameter values were
chosen to faU aU analytical conditions, i.e., to satisfy the inequaUty ku < P <
l/k^i-
Parameter values used in this simulation are a = .8, (3 = .9, ku = -9,
^21 = -5, and p = 1.2. The results of the numerical simulation coincide with
the analytical prediction, i.e., the steady-state solution is dependent upon initial
conditions, either solutions converge to (1,0) or (0,p).
species 1
Figure 2.3: Outcome dependent upon initial conditions
CHAPTER III
THE THREE-SPECIES SPATIALLY HOMOGENEOUS MODEL
3.1 Introduction
The three-species model can be derived by making some assumptions about
the relationship between seeds and adult plants. These assumptions are similar to
those used for the two-species model. The derivation can be extended very easily
to n species, where n > 3. Two basic assumptions are needed. These assumptions
are stated in the next section. The model is derived in section 3.3 and analyzed in
the foUowing section. A special case is considered in section 3.6.
3.2 Assumptions
Assume that the seeds produced in one generation from the adult plants satisfy
the foUowing relationships:
I
i ^ Í L + ÍL+^3
mi
si/mi _
S2/m2
777-2
^ 3
Ax_ si/mi _ A^ s^/m^ _
A2 sz/m^
^As' Sz/m^,
A^
^Aj,
where m^ is the seed yield for each of the three strains in mono-culture; si, s^,
and 53 are the new seeds produced and Ai, A2, and A^ are the densities of the
adult plants for the three strains. Assumption I states that the sum of the relative
yields for all seeds grown together should be approximately one. This is related to
de Wit's assumption about the sum of occupied area being one, assumption (1.1).
The formula is correct, when for example Si = 0 = 52, since without competition,
seed yield for A^ wiU reach m^; it is also correct for the other two cases, i.e.,
18
19
when 5i = 0 = S3 and S2 = 0 = S3. Assumption II states that the ratios for the
density of new seeds produced are proportional to the ratios of the densities of the
corresponding adult plants, which is de Wit's second assumption (1.2).
3.3 Derivations
From assumptions I and II, we have
1
=
Si/mi
l
^2/^2
^3/^3
Si/mi
1
=
5i/mi
si/mi
^3
1+ J1^+
Ci^i
C2^i
mi^li
si
=
Ai+A2/ci
+ A^/c2
Now using the relation between new seeds si, residual seeds r^, and adults ,4i for
strain 1:
Ai{t + l) = PAPEPGPw[si{t) + ri{t)] and n{t) = Pw{l-PG)[si{t-l)
+
r^{t-l)],
where Pw is the probability that seeds survive the winter, PG is the probability
that seeds germinate, PE is the probability that seedlings become established, and
PA is the probability that established plants reach the adult stage. The above two
equations show that ri{t) can be written in terms of ^i(^):
uit) =
^^Ait),
^A^E^G
where 1 — P G is the probability that the seeds of ^ i ( t ) do not germinate. Substituting the above expressions for ri{t) and Si{t) into the equation for .4i(í + 1):
^ i ( t + l)
=
PAPEPGPwSi{t) +
{l-PG)PwAi{t)
PAPEPGPwmiAi{t)
Ai{t) + A2{t)/ci + As{t)/c2
+
{l-PG)PwA,{t)
20
PiAi{t)
+
Mt)-^Mt)/ci+As{t)/c2
where Pi = PAPEPGPwmi
and a =
aAi{t),
SimUar equations can be derived
{1-PG)PW
for .42(t + 1) and As{t + 1), where the probabiUties depend on the particular
strain. Thus, the system of three difference equations for adult plant densities has
the foUowing form:
Ai{t + 1)
=
A2{t + 1) =
A3{t+1)
=
PiAi{t)
+ aAi{t)
A,{t) + A2{t)/ci + As{t)/c2
P2A2{t)
+ (3A2{t)
A2{t) + CiAi{t) + A3{t)/c3
PzA^{t)
+
A^{t) + C2Ai{t) + c^A2{t)
lA^{t).
Because of assumption 1 the isoclines have the form:
(f) = A,{t) + A2{t)/ci + A^{t)/c2,
p = A2{t) +
a
where (f) = Pi/{1 -a),
CiAi{t)+A3{t)/c3,
= A3{t) + C2Ai{t) + C3A2{t),
p = P^/^l - (3), and a = Pz/{1 - l). The coefficient matrix
A = {ttij) has the particular form:
/ ttii ai2 ai3 \
A =
a2i
022
Íi23
V ^31 a32 a33 /
/ 1
=
Ci
V C2
1/ci
l/c2 ^
1
1/C3
C3
1
,
/
where the symmetric terms satisfy aijttji = 1. We wiU consider the more general
system:
Ai{t + 1)
A2{t + 1)
As{t + 1)
PiAi{t)
Ai{t) + kuA2{t) +
kuAs{t)
+ «-4i(t).
P2A2{t)
+ PA2Ít),
A2{t) + k2iAi{t) + k^sAs^t)
PsAs^t)
+ 7.-l3(i),
As{t) + ksiAi{t) + k32A2{t)
(3.1)
(3.2)
(3.3)
21
where the coefficient m a t r i x A satisfies:
(
1
ku
k2l
1
V ^31 A:32
ki3 ^
k23
1
/
Note t h a t we have used the reciprocal of the crowding coefficients % .
3.4 Steady-States
In the three-species model there are seven steady-states of interest. To simplify
the derivation of these steady-states, recall the relations Pi = (^(1 - a ) , P^ =
p{l — j3), and P^ = a{l — 7 ) .
Steady-states one through three are where one
species is at carrying capacity and the other two are extinct. They are given by
((/>, 0,0), (0,p, 0), and (0,0, cr). The steady-states four through six are when two
species co-exist and the third is extinct. They are given by
(f) - kup
P - k^i^
,0
1 - k^okoi
- 1^121^21
kiok
12/^21 ' -1
L—
1 - ^13^31
5
p - ^230-
^l
1 - ^13^31
(T - k32p
1 — ^23^32 1 ~ ^23^32
Finally steady-state seven occurs when co-existence between all three species
is possible. It is given by
. _
(^(1 - A:23A;32) - ^(^^12 - hsh^)
1 - kuk^i
_
p{l-
- kuk^i
- A:23^32 + hzk^^k^i
+
ki3k3i) - (f){k2i - A:23A:3i) - ^'(A:^^ -
—
^^n
- (^{ku -
— -
kuk^^)
kuk^shi
k^k^i)
—
.
1 - Å:i2A:21 - ^13^31 - ^23^32 + ^13^32^21 + ^12^23^31
_
a{l-
kuk^i)
- </>(^3i - ^21^32) - p{k32 -
kuk^i)
1 - Å:i2A:21 - ^13^31 - ^23^32 + ^13^32^21 + A^i^/c^^/^^l
22
3.5 Analysis of the Steady-States
SimUar to the work done on the two-dimensional model, conditions for local
stability are determined from the linearized system. StabiUty of a steady- state
( ^ i , ^ 2 , A3) is determined by the eigenvalues of the Jacobian matrix evaluated at
the steady-state. For the three-dimensional model the Jacobian matrix is of the
form:
/
^
-\-a
i-<l>){l-a)ki2Ai
(-p)(l-/3)fc2iA2
J{Ai,A2,A3)
^ 1 3
62
l
^
{-a){l-j)k3iA3
62
^
{-a){l--f)k32Az
/2
where di = {1 - a)(l){kuA2 + k^A^),
{-(t>){l-a)ki3Ai
/2
\
{-p){l-hk23A2
62
h. ^ ^
/
/2 ^
/
^
d^ = {Ai + kuA^ + ki^A^)^,
P)p{k2iAi + k23A3),e2 = {A^ + k^iAi + /^23^^3)2, /1 = (1 - j)a{k3iAi
e^ = (1 -
+ k^^A^), and
/2 = (^3 + k^iAi + /1:32^2)^. Note that the diagonal elements are positive and that
the off-diagonal elements are nonpositive.
Again local stability of the steady-state is dependent upon the magnitude of
the eigenvalues from the evaluated Jacobian matrix; the steady-state is stable if
all eigenvalues have magnitude less than one. This analysis is similar to the twodimensional case. In addition each nonzero component of the steady-state must
be positive e.g., Ai > 0, ^2 > 0, A3 > 0 when all the species co-exist.
The analysis of the first three steady-states, namely the single-species steadystates, is similar to that of the first two in the two-species case. Recall that the
three single-species steady-states are (ø, 0, 0), (0, p, 0), and (0, 0, a), where the first
component is Ai, second component is A2, and the third component is .43. By
reordering the states, the evaluated Jacobian matrix can be put into an upper
triangular matrix form and the eigenvalues are then the elements on the main
diagonal. Conditions for stabiUty are given in Table 3.1.
23
Table 3.1: Stability Conditions for Single Species Steady-States
Steady-State
(^,0,0)
Condition 1
Condition 2
P < k2i(/)
a < Å;3iø
{0,p,0)
(f) < kup
a < k32p
(0,0,(7)
^ < ki3a
p < k23(^
The next three steady-states are with two species coexisting and the third extinct. The evaluated Jacobian matrix can be put in block upper triangular matrix
form, meaning the eigenvalues associated with the n x n matrix are equivalent to
those of the smaller block matrices. For example, the steady-state {Ai^A^, 0), has
the Jacobian matrix:
/
di
, ^
{-p){l-l3)k2iA2
J,( ^ 1 , ^ 2 , 0 )
62
V
0
{-<p){l-a)ki2Ai
^Xô
{-ct>){l-a)ki3Ai
i-pKl-hk23A2
'^
62
0
\
62
f+ 7
/2
/
'
The matrix J{AI,A2,O) is a block upper triangular matrix; hence, the eigenvalues
of this matrix are equal to the eigenvalues of the 2 x 2 and 1 x 1 matrices. The
analysis of the 2 x 2 matrix is identical to that in the two-dimensional case. An
additional condition is that the eigenvalue of the 1 x 1 matrix (which is identical to
the single element of that matrix) has to have magnitude less than one. Thus, the
conditions for local stability at this steady-state and the steady-states (.4i,0,.43)
and (0,^2,^3) are given in Table 3.2.
Necessary conditions for the stability of the final steady-state, where all three
species coexist, (^1,^2, ^3), are not easily found. In situations like this, typicaUy
Jury conditions wiU be appUed[5]. For a 3 x 3 matrix, the Jury conditions are as
foUows:
24
Table 3.2: StabiUty Conditions for Two-Species Steady-States
Steady-State
(î
4>-ki2P
p-k2i4)
-fci^fc^i ' l—ki^k-}.
-Æ ^21^
<P<-t
•^Ll^
- A- - ky,
k3i(l>
p-k23a
(0,
a-kz2P
-fc2.sfc.'H2 '
Condition 2
Condition 1
l-fc2.'^fc.'^?
)
<(y<-t
k3l{(p-ki2p)+k32{p-k2l4>)
l-k-í9.k9.^
^ fc2i {(t>-ki3a)+k23 (cr-fc3i <p)
P ^
l-fci.'.fc..i
a <
fcl2(p-fc23q-)-ffcl3(o'-fc32P)
0<
k32P
r^
'32P <<a
^ <<fcf^
l-fc:>.3fc.'^2
1 + ai + a2 + a3 > 0, 1 — ai + a^ — a^ > 0
la^l < 1, 1 - {a^Y > 1^2 - a^ail,
where A^ + a^X^ + a^X + a^ = 0 is the characteristic equation of the matrix in
question. However, due to the large number of parameters in the three-dimensional
model neither eigenvalue analysis nor analysis of the Jury conditions yield simple
relationships between parameter values. For reference purposes, the coefficients a^,
2 = 1, 2,3, in the characteristic equation are given below.
_di_
«1
(f)^
diei
ei_
f ]:-(X-(3
P'
a'
, difi
ci/i
.
\^i , /
,
\^i , ^i/i
(12
, /
, /3\/i
a'
9i-
92- 9i + Oí(3 + a-f + /3j
áiei
as = - 7
{(f>p)'
difi
^
"(0^)'
Cl/l
a-
"(P^)'
d,
Ør,2
(f>'
Cl
«7— - a(3—
p'
+ api + 092 + 193 - (^h
Where í^i = (1 - ?){1 -1)^23^32^2^3/{p(7),
and P3 = (1 - Oí){l -
0)kuk2iAiA2/{(f)p).
í/2 = (1 - a ) ( l - 7)^i3A;3i--li.43/((/)(7).
25
3.6 Special Case
Next, a special case of competition is considered, where information about
the eigenvalues can be obtained. May and Leonard[10] investigated the special
case of non-transitive competition in the three-species Lotka-Volterra competition
equations. In one case they assumed that Ai out competes A^, A^ out competes
A3, and A3 out competes A^. For certain parameter values there are non-periodic
solutions of bounded amplitude but ever increasing cycle time. For this special
case, the model takes the form:
. .
X
(1 -a)Ai{t)
. ..
^-(*+^)=^uo+a^;;.43(^)+"^-<^)
^3(^^^)-.43(.)l\7w+U(0^^-^-(^)
For this model, there are three single-species steady-states given by (1,0,0), (0,1,0),
and (0,0,1); three two-species steady-states given by ci(l - a, 1 - 6,0), Ci(0,1 a^l-b),
and c i ( l - 6 , 0 , 1 - a ) , where ci = 1/(1-a6); one three-species steady-state
given by C2(l, 1,1), where c^ = 1/(1 + a + b).
The local stabiUty of these steady-states can be checked easily from the conditions derived in the previous section. For example, the single-species steady-state
(1,0,0) is locally stable provided a, b > 1. The same conditions hold for the other
single-species steady-states; the conditions are given in Table 3.3.
Each of the two-species steady-states is stable provided a, 6 < 1 and one additional condition is satisfied. The stability conditions are given in the foUowing
table.
26
Table 3.3: Stability Conditions for the Special Case for the Single-Species
Steady-States.
Steady-State
Stability Conditions
(1,0,0)
a, & > 1
(0,1,0)
a, b> 1
(0,0,1)
a, b> 1
Table 3.4: Stability Conditions for the Special Case for Two-Species Steady-States
Steady-States
Stability Conditions
ci{0,l-a,l-b)
a, 6 < 1 and 1 — ab < a{l — a) + 6(1 --b)
a, 6 < 1 and 1 — ab < a{l — a) + 6(1 -b)
-
ci(l
a, 6 < 1 and 1 — a6 < a(l — a) + 6(1-b)
-
ci(l
-a,l-b,0)
-b,0,l-a)
The non-transitive case considered by May and Leonard[10] is the particular
case a < 1 and 6 > 1. In this case there are no stable one or two-species steadystates, Ai out competes A^, A^ out competes A3, and A3 out competes .4i. A
non-transitive case occurs also when a > 1 and 6 < 1. When a > 1 or 6 > 1 the
two-species steady-states are locally unstable and the single-species steady-states
are stable.
The conditions for stability of the three-species steady-state are still complicated even in this special case because of the large number of parameters
{a, (3, 7, a, and 6). However, the Jacobian matrix has a relatively simple form:
27
/ a + 6 + a — (1 — a)a —{1 — a)b
-{l-(3)b
Ji = C2
a + b + (3
-{l-(3)a
V
-{l-l)a - ( 1 - 7 ) 6 a + 6 + 7
where c^ = l / ( l + a + 6). Some sufficient conditions for local stability of the
three-dimensional steady-state can be derived from Gerschgorin's theorem [13]. It
foUows from Gerschgorin's theorem that the eigenvalues A^ must satisfy
|c2(a + b + k) - Xi\ < C2(l - k){a + 6),
where k is either a, (3, or 7. It foUows that |AÎ| < 1 if
C2[(a + 6 + A;) + (1 - k){a + 6)] < 1.
Simplifying this inequality we obtain a sufficient condition for local stability of the
three-species steady-state:
a + 6 < 1.
Some stronger sufficient conditions can be obtained for the case that o; = /? = 7.
When a = /? = 7, the matrix Ji has the form of a circulant matrix [13]:
Ji = C3
í 1 A B\
B 1 A
[A B 1 J
where C3 = C2(a + 6 + a ) , A = -{l-a)a/{a
+ b+a), B = -{l-a)b/{a
+ b+a), and
C2 = l / ( l + a + 6). This case appears to be closely related to the continuous case
analyzed by May and Leonard [10]; they obtained a circulant matrix as well for
the Jacobian matrix. The eigenvalues for this circulant matrix can be calculated
explicitly [13].
28
The eigenvalues for the matrix Ji are
Ai =
C3{l + A + B)
X2 =
C3{l-l/2A-l/2B)
X3 =
+
l/2{A-B)iV2>
C3{l-l/2A-l/2B)-l/2{A-B)iy/2>.
Substitution of the values for C3, A, and B give the explicit values for the eigenvalues
of the matrix Ji:
Ai =
A2
=
A3
a
3/26 - l/2ab - l/2aa + a + 3/2a + l/2i(6 -a-ab
l + a+ 6
3/26 - l/2ab - l/2aa + a + 3/2a - l/2z(6 -a-ab
=
+ aa)V^
+ aa)y/?>
1+a+b
For stability the magnitude of the eigenvalues must be less than one. Clearly
|Ai| = a < 1. For A2 and A3, we require jA^I =
XÍXÍ=
9fíA? + ^Af < 1. Since A2 and
A3 are complex conjugates, it is sufficient to show A2A3 < 1 or c^^A^A^ — 1) < 0.
The expression A2A3 equals,
a^a^ - a^ab + 3aa + 3aí6 + 3a^ + 3a6 + 36^ - 3^6^ + a^ + b'^a^ - ba^ - Zaa^ - a^a
(l + a + 6)2
Grouping like terms, the expression c^^A^A^ — 1) simplifies to
{a^ - l)(a2 - a6 + 1 + 6^ - 6 - a) + 3(a - l)(a - a' + 6 - 6^).
However, this latter expression is negative if
{a + l){a^ - a6 + 1 + 6^ - 6 - a) + 3(a - a^ + 6 - 6^) > 0
since a < 1. Simplification once again leads to the required inequaUty for local
stability:
a{a^ - a6 + 1 + 6^ - 6 - a) + 2(a - a^ + 6 - 6^) + 1 - a6 > 0.
29
Letting f{a, 6) = (a^ - a6 + 1 + 6^ - 6 - a), 9{a, 6) = (a - o^ + 6 - 6^), and
h{a, 6) = 1 - a6, we obtain the following inequaUty,
af{a, b) + 2g{a, 6) + h{a, 6) > 0.
(3.4)
To find sufficient conditions when the expression (3.4) is minimized, we can
consider the functions f, 9, and h separately. Consider the function f = a^ — ab +
6^ — a — 6 + 1, it has a minimum at a = 1 and 6 = 1 . It is straightforward to show
that a = 1 and 6 = 1 is the only critical value of / ; i.e., df /da = 0 = df /db at
a = 1 and 6 = 1 . Also, checking the second derivatives we find that since
d^f
''
da^
d^f
^
, <92/ d^f
= 2 and "^ "^
db^
da^ db^
12'--.2
d'f
dadb
= 3
at (1,1) there is a local minimum[21]. Since it is the only minimum, it must be
a global minimum. AIso, /(1,1) = 0. So for (a, 6) ^ (1,1), f{a,i>) > 0. Hence
/ is positive Va, 6 ^ 1. The functions 9{a,b) and h{a,b) are strictly positive for
a, 6 < 1. Since a G [0,1) this system, where a = (3 = j , is locally stable at the
three-species steady-state when a, 6 < 1 and a G [0,1).
Figure 3.1 shows the graphs of A2A3 = 1 for varying values oî a {a = 0.1,
0.25, 0.5, 0.75, 0.9). The region below the curves are the a and 6 parameter
values where the three-species steady-state is stable, A2A3 < 1. The square region
0 < a < l , 0 < 6 < l , which was analytically shown to be stable, is contained in
each region.
This special case of the Lotka-Volterra competition has received much attention
because of its unusual mathematical behavior [10, 11, 20]. The osciUatory behavior
with increasing cycle length that occurs in the continuous-time Lotka-Volterra
30
2
\0.9 '
'
1.8 - 0 . 7 5 \
1.6
0 . 5 ^ \ \ .
0.25^"---.^\xX
1.4
0.1
~~^^^^^^^^^\
1.2
03
^^^w
1
OJ
1
> 0.8
1
0.6
1
0.4
1
1
1
1
0.2
0.4
0.6
0.8
1
1
1.2
values of a
1
1.4
'
0
0
1
0.2
1.6
1.8
2
Figure 3.1: Region of Stability
system occurs in more general systems, e.g., where a and 6 are replaced by a^ and
bi, = 1, 2, 3, 0 < a^ < 1 < 6i and a^ + 6^ > 2 [20].
A more practical type of stability behavior than the local stability criteria
that we have considered has been examined in some continuous and discrete-time
competition models [6, 7, 8]. This more practical stability behavior is referred to as
permanence or persistence. A competitive system is said to be permanent if there
exists a compact set K in the interior of R^ such that all solutions beginning in
the interior of R i end up in K. The system is said to be persistent if
limsupAi{t)
> 0, 2 = 1,2,3
í-^oo
for aU solutions in the interior of R^. In the case that the positive equiUbrium
is globaUy stable, then the system is permanent and persistent. However, in the
special case, where solutions have ever increasing cycle length and get close to the
31
single-species steady-states, the system is persistent but not permanent. There
has been some recent articles related to persistence and permanence in discrete
competitive systems [6, 7]. Unfortunately, the results discussed in these articles
do not apply to our discrete-time models. However, persistence and permanence
in our discrete-time system is a problem that needs further investigation.
In the next section, some numerical simulations of the three-species competition
model are performed.
3.7 Numerical Simulations
In this section solutions to this model are studied numerically. Parameter values
are chosen to test the analytical results determined in section 3.5, and the special
case studied in section 3.6. Since a, (3 and 7 do not analytically eífect the system
outcome, they wiU be assumed constant, a = 0.8, (3 = 0.9, and 7 = 0.6 for the
general case. Ai{t), A^^t) and A^^t) wiU be represented by a solid, dashed, and
dotted line, respectively.
Figure 3.2 is representative of the first three steady-states, single species domination. Parameter values were chosen to satisfy the necessary conditions determined analytically, i.e., p < k2i(f> and a < k^i^p. Since these conditions are necessary for local asymptotic stability, it is important to make certain that the parameter values or initial conditions do not satisfy conditions for a different steady-state.
The parameter values used in this simulation are A;i2 = 0.5, Å:^^ = 0.8, A:3i = 1.4,
ki3 = 0.5, A:32 = 0.4, A;2i = 1.1, (f) = 1, p = 0.8, and a = 0.9. The numerical results agree with the analytical prediction, i.e., the steady-state solution is
((^,0,0) = (1,0,0).
32
150
200
250
time
Figure 3.2: Single Species Domination
Figure 3.3 iUustrates the next three steady states, i.e., two species domination.
The parameter values were chosen to satisfy the necessary conditions for stability
of (^1,^2,0), but not of (^i, 0, ^3) nor (0, A2, A3). The parameter values selected
for this simulation are ku = 0.5, A:^^ = 1.5, k^i = 1.1, Å:i3 = 1, k^^ = 1.2, A:2i = 0.7,
(p = l^ p = 0.8, and a = 0.9. The results from the numerical simulation supported
the analytical results; solutions converge to (^1,^2,0) ^ (0.92,0.15,0).
Figure 3.4 is an example of three-species co-existence. The parameter values
must fail all of the necessary conditions in Tables 3.1 and 3.2 for a one or two-species
steady-state to exist. The parameter values used in this simulation are ku = 0.5,
^23 = 0.6, Å;3i = 0.55, Å:i3 = 1.45, Å:^^ = 1.5, ^21 = 1.6, (f) = 1, p = .S, and a =
.9. Solutions converge to the steady-state (^1,^2,^3) = (0.1886,0.2049,0.4889);
however the convergence is osciUatory.
33
1
1
[
-
0.8
A /
:\\y
0.6
-
' \
\ \
\
\
s
0.4
_
N
-- - - -
0.2-
-
1
[
1
1
50
100
150
200
250
tinne
Figure 3.3: Two-Species Domination
1
r
1
r
—r
I
1
1
r
1
-
0.8
-
t-0.6
g
.•
1 \
3
Q.
O
1
' 1
•/» 1
; M
1 '\
1- • . + • 1
:
^^0.4
' •
'' 1 f-iM\ >•' \\'
• t
\
0.2
.'
\ '
1 \ V /
\ A
1
\/
A
'v'\
'A\
1 1\
'i
'A '^A -A -A
\ ' í \ ''A
1 \ \ ' í \ \ • // \ \\ ; ' /A\ \\ M A• \\ . /• /AM\ • ' / A\ \\
/
•V / ' / '• V / M' / A / \J
/•.
l' \
/ A'
i /• \ / A
0 1
0
200
1
'
1
400
600
800
r
1
1000
time
1200
. . .
.1.
1400
1 . .
1600
Figure 3.4: Three-Species Co-existence
1
1800
2000
34
The next set of figures illustrate the special case discussed in section 3.6. In
this section we made the assumptions that a = (3 = ^ and (f) = p = a. For the
foUowing graphs we wiU refer to these parameters as a and (f).
In Figures 3.5 and 3.6 the parameters a and b are both less than one. The
parameter values for Figure 3.5 are a = 0.6, a = 0.5, b = 0.4, ^ = 1, and
a + b < 1. The values for Figure 3.6 are a = 0.8, a = 0.5, b = 0.9 and ø = 1.
The results of both simulations agree with the analytical prediction, i.e., a stable
positive steady-state is reached. In the case of Figure No. 3.5 the steady-state is
(0.526,0.526,0.526) and in Figure 3.6 the steady-state is (0.417,0.417,0.417).
Figure 3.5: Convergence due to parameter values, a = .5 and b = .4.
Consider the special case where a < 1 and b > 1. Let a = .75, then from
analytical work values of a and b that satisfy the inequality (3.5), wiU converge
to a stable positive steady-state. Figure 3.7 shows the region of stabiUty bounded
by a solid Une. May and Leonard proved solutions containing parameter values
satisfying the inequality a + b < 2 (the region bounded by a dotted Une) wiU
converge to a stable positive steady-state, in the continuous time Lotka-\blterra
35
100
150
200
250
time
Figure 3.6: Convergence due to parameter values, a = .5 and b = .9.
model[10]. This model is of the form
dNi
~dt' = Ni[l - Ni - aN2 - bN^],
dN2
= N2[l - bN^ - N2 - aN^],
dt
dN3
=
N3[l-aNi-bN2-N3].
dt
Three pairs of a and b values are chosen to compare the simulated results. These
three pairs are given in Figure 3.7.
Figures 3.8, shows the results of the numerical simulation for (a, b) = (0.5,1.2).
It supports the analytical results that for a given value of a, values of parameters
a and b which satisfy inequality (3.5), converge to the stable positive steady-state,
in this case (^1,^2, ^3) = (0.37,0.37,0.37).
Figures 3.9 and 3.10, iUustrate the dynamics when (a, b) = (0.5,1.7). In Figure
3.9 the system does not converge to a stable steady state. In the phase plane
figure (Fig. 3.10), there appears to be a limit cycle encircling the steady-state
(0.3125,0.3125,0.3125).
36
'
p
I
' "'•T
r
1
r
1
• (0.1, 1.85)
1.8
s.
1.6
•.,
^<;^
1.4
o
1.2
X.
(0.5,1.2)
B
0)
D
o
(0.5,1.7)
\.^^
N^
r
> 0.8
>,^
0.6
N:
0.4
\"
0.2
n
1
0.2
0.4
_...!
0.6
1
0.8
]
1
1
1.2
values of a
J
1
1.4
1.6
\
1
1.8
Figure 3.7: Region of Stability, a = .75
Figure 3.8: Convergence, interior to region.
Figures 3.11 and 3.12 present an interesting case. The point (a, b) = (0.1,1.85)
fails the inequality (3.5), but satisfies the conditions estabUshed by May and
Leonard [10] for stabiUty in the three-dimensional continuous Lotka-Volterra model.
The numerical results of this simulation support our analytical findings, i.e., the
system does not converge to a stable steady-state. However, the system appears to
37
200
400
600
800
1000
time
1200
1400
1600
1800
2000
Figure 3.9: Nonconvergent, exterior to region
species 1
species 2
Figure 3.10: Phase plane diagram of nonconvergence
38
converge to a stable limit cycle. This also appears to be the case for a+6 = 2, which
for the latter case May and Leonard analytically proved the solution converged to
a stable limit cycle [10].
1
r-
0.8
200
400
600
800
1000
time
1200
1400
1600
1800
2000
Figure 3.11: Limit Cycle
species 1
species 2
Figure 3.12: Phase Plane Limit Cycle
39
The final case that May and Leonard[10] considered was a < 1 and b > 2.
Their results showed a limit cycle of ever increasing cycle time. The remaining
figures in this section iUustrate this case. Let a = .75 and (f> = l. For Figures 3.13
and 3.14, a = 0.5 and b = 2.5.
200
1000
time
1200
1400
1600
1800
2000
Figure 3.13: OsciUation I
specles 1
species 2
Figure 3.14: Phase Plane OsciUation I
40
As the difference between the parameters a and b increases, the nonperiodicity
of the osciUations becomes more apparent. In Figures 3.15 and 3.16, let a = 0.5
and 6 = 5.
T
1 -
:' 1 : ':[ 1
'
••1
'••
'
:i r
!
0.8-" ; ;r i
• i' '
:
: -r 1
j
i
(
r^
I
i; 1
li; i: i
'í 'i '
'n!j\
200
^
r
n
•. (
1
• i
r
;i
n
•
i
1
)
r
r
:
• • " •
'
\
• 1
1
• '
1
'r
r
r
1
•1
r
r
.1
:)
i
i
)
)
i
;r
r^
i
•
(
•(
(
i
)
)
i
i
i
(
r
"
1
-1
j
i
i
-
:(
(
:'
:i
(
:r
r
r
I
0.2::,:
1
1
)
í
•
r
1 i
0.4 t
••'
r
.0.6
i
r
n
1
5
)
1
i
i
r
r
i
i
i
i
-
\l 1 l í ,
1
600
800
r
1000
time
1200
:
l
1
1400
;:
1600
,
1800
2000
Figure 3.15: OsciUation II
0.8--
species 1
specres 2
Figure 3.16: Phase Plane OsciUation II
CHAPTER IV
THREE-SPECIES SPATIALLY HETEROGENEOUS MODEL
4.1 Introduction
The spatially-independent three-species model for competing annual plants discussed in Chapter II, is generalized to a spatially-dependent integrodiíference equation. This expansion was formulated and analyzed by E. J. AUen et al. [1] and L.
AUen and E. AUen [3] for one and two-species. In this chapter we wiU develop and
numerically simulate a spatially-dependent model for three-species.
4.2 Dispersal-competition model
A plant model with seed dispersal can be derived in a manner similar to the
model without dispersal. Thus, the derivation of the three-species competition
model with dispersal is similar to the derivation of the three-dimensional model in
Chapter III. The expression m^ is replaced with
mi{t,x) = min lMi,di j
fi{x,z)Ai{t,z)dz\,
where for species i, di is the average number of seeds produced per adult in a unit
area, fi{t,z) is the dispersal kernel (e.g., normal density), Mi is maximum seed
density, Ai{t,x) is the adult plant density at position x, and Si is the seed density.
Let
Ni=
í
JR
f^{x,z)Ai{t,z)d:
Assume, as in Chapter III,
41
42
l « A + f^+^3
mi
m2
ma
si/mi
Ni si/mi
~CiT7-5 — 1 —
S2M2
A^2' ss/ms
II
=
Ni s^/m^
C2 — , — - , —
N3' 83/^3
=
N^
C3
N3'
After a slight generalization, one obtains a system of three integrodifí'erence equations for adult plant densities:
w^
-,
4 /. , 1
\
Pimi{t,x)Ni{t,x)
X
P2m2{t,x)N2{t,x)
, ,
,
, /o. ., X
-"^^*-*•''") = iv.(M) + ;c.jv,(.,x) + fe.3ÍV3(^,x)+^^^'^'"^'
. ,
^ .
•^^^' + ^ ' " ) =
P3m3{t,x)N3{t,x)
iV3(M)+;^3(iV((í,x) +
^ . .
fc3.iV.(í,x)+^^^(*'^)-
Note that any of the equations above can be written as
Ai{t, x) = Ci{t, x) min lMi,di
fi{x, z)Ai{t, z)dz > + XiAi{t, x),
where the coefficient function Ci represents the effects of competition, 0 < Ci{t, x) <
1, and xi = Oi^ X2 = /^, Xs = 1- When Ci = 1, then there is only a single species
present and the model reduces to a single species case considered in AUen et al.[2].
4.3 Relationship to the spatially homogeneous model
Without spatial variations this model reduces to the spatiaUy homogeneous
model of Chapter III. Let Ai be constant in space, then Ni = A, and mi{t)
min{Mi,diAi{t)}.
=
However, mi{t) = diAi{t) implies that (0,0,0) is our solution;
thus, mi{t) = Mi. This model has the same homogeneous steady-states. as the
model in Chapter III.
43
4.4 Numerical Simulations
In this section we conduct numerical simulations with the spatially heterogeneous model. These simulations are performed with the parameter values from the
Chapter III. The results of the simulations are consistent with previous results,
i.e., in many cases the heterogeneous model behaves like the homogeneous model
as time increases.
In the foUowing figures, several parameter are kept constant. Unless otherwise
stated, assume a = 0.8, /? = 0.9, 7 = 0.6, (f)=l,p
D3 = 1, di = 30, d2 = 20, d3 = 10, and fi{x,z)
= 0.8, a = 0.9, D^ = 3, i:>2 = 2,
= exp(-(a: -
zf/Di)/y/D^
is the normal density function. RecaU that Pi = (t){l - a), P^ = p{l - P), and
P3 = (j(l - 7). Figure 4.1 represents the initial species distribution, for aU but the
last example in this section. The soUd, dashed, and dotted lines represent Ai, A2,
and A3, respectively.
Figure 4.1: Initial Spatial Distribution
44
Figure 4.2 represents the dynamics of a spatially heterogeneous system with
homogeneous (spatially independent) parameter values equal to those of the system
in Figure 3.2. Let ku = 0.5, k^^ = 0.8, k^i = 1.4, Å;i3 = 0.5, k^^ = 0.4 and k^i = 1.1.
The results of this simulation is consistent with our results in Chapter III. i.e., this
system approaches a single-species steady-state. The numerical solutions show
domination by species A^. Parts a, b, and c of Figure 4.2 represent the system at
t = 20,t = 50, and t = 70, respectively. Figure 4.2d shows the dispersal of species
Ai, where the dashed line is at t = 20, the dotted lines are at t = 30, 50, 70, and
the solid line is at t = 100.
Figure No. 4.2b
Flgure No. 4.2a
1
1
.
.-^0.8
w
c0)
Q0.6
\
^
'
•
X
c
g
'^ 0.4
/
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.•
\
••
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c
>•
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(/)
c
o
Q0.6
^
g
^ 0.4
.
'•
•
3
Q.
13
Q.
i2o.2
íf 0.2
.
- -
•
"
'
y '
. -^ '
0_
()
1
—
—
^''*^'-^^
* ^
0
•
10
5
• ''
()
X
Figure No. 4.2c
Figure No. 4.2d
—
.
•
'
^
1
5
X
•
• '
\
,: ^'.
1
—
\ .
• . . .
_
•^ • .
.^0.8
c
•^0.8
0)
0
Q0.6
c
g
Q0.6
•co 0 . 4
co 0.4
" • • • • • . . .
•
\
c
UOj
Q.
3
Q.
í£o.2
£0.2
• .
•
\
\
•
\
\
•
s
•s
_ _ .
0
)
(
—
" ~
• ' ' '
5
•
—
•
^
10
0
(}
X
•^
5
1
X
Figure 4.2: Single-Species Domination
Figure 4.3 illustrates two-species domination. Further discussion of this system
45
wiU be in the foUowing section. The parameter values were selected to agree with
the system in Figure 3.3; thus, ku = 0.5, k^^ = 1.5, k^i = 1.1, ki3 = 1, A:^^ = 1-2
and k^i = 0.7. The results from this simulation are consistent with our results
from the previous chapter. The system approaches the two-species steady-state
(^1,^2,0) = (0.92,0.15,0). Figure 4.3 parts a through d represent the system at
í = 20, 50,150,300.
Figure No. 4.3a
Figure No. 4.3b
1
.•^0.8
^~~~~~-—.-^^^^
.•^0.8
<n
c
c
CD
1
Q0.6
c
g
« 0.4
0)
•^
^
•'N.
'^
I
Q 0.6
c
g
;ro 0.4
\
\
^
\
>.
23
Q.
Q.
£o.2
/
\.
••^
'
"
~
'
\
.'
\
\v
•s .•
\ .
. \
x"
^
•
~ ^ .
/ ^
t£o.2
'i
\
^
y'
\
'^ .
^
• - .
5
10
5
X
X
Figure No. 4.3c
Figure No. 4.3d
10
Figure 4.3: Two-Species Domination
Figure 4.4 is an example of three-species coexistence. The parameter values
are selected to agree with those in Figure 3.4, i.e., kx2 = 0.5, k^^ = 0.6, k^x = 0.55,
ki3 = 1.45, k32 = 1.5 and k^i = 1-6. Figure 4.4 parts a through d represent
the system at í = 50,150,300, 500, respectively. These resuUs are converging
46
to the steady-state solution, (^i,.^^,^^) = (0.1886.0.2049.0.4889); however, this
convergence is osciUatory.
Figure No. 4.4a
Figure No. 4.4b
1
1
.^0.8
.•^0.8
w
c
cu
w
c
CD
Q0.6
Q0.6
^ 0.4
rt 0.4
c
g
•
c
g
3
' '
3
Q.
a.
2 0.2
^
•
••
• '
^
£0.2
_
.
-
•
"
.
'
•
_y
0
()
•
•
5
0
0
10
. . ^
•
5
X
1
X
Figure No. 4.4d
Figure No. 4.4c
0.45
ity
0.6
^
V)
w
SO.4
.
•
•
•
0.4
c
cu
'
Q
c
o
Q0.35
•
ati
c
g
ÍS 0.3
" " • • - . . . . . .
=»0.2
Q.
^
•
•
ZJ
Q.
^ ~~~~-~~J~ ~"'"~' ~
O
CL
''
2 0.25
0
^
0.2
()
5
10
()
X
5
1
X
Figure 4.4: Three-Species Co-Existence
Figure 4.5 iUustrates the conditions of the example which May and Leonard
investigated[10], where a < 1 and b > 2. The parameter values were selected
corresponding to Figure 3.13; hence, a = 0.5, b = 2.5, a = /? = 7 = 0.75, and
^ = p = a = 1. The results of the numerical solution were consistent with
those in Chapter III. The simulation approached no steady-state and osciUations
were present. Figure 4.5 consists of 12 graphs, the time for each graph is í =
0,10,50,100,150,200,250,300,350,400,450,500.
4'
Figure 4.5: Special Case
4.5 Numerical Simulations with control
In this section, we discuss the numerical simulations for a three-species heterogeneous model with an eífective control to prevent weed species from spreading.
In this section, assume Ai and A^ are the weed species, while A^ represents a
desirable plant, such as an agricultural crop. The control is applied only to the
weed species in the region occupied by the desirable plant species. This model is
appUcable to rangeland management, where control methods are appUed only to
a particular area to eliminate weed species.
Figure 4.6 iUustrates the initial conditions for aU experiments in this section,
where a soUd, dashed, and dotted Une represent species .4i, .^2, and .43. respec-
48
tively. The space is partitioned into two regions, the first consisting of both weed
species, and the second containing our desirable plant species. Parameter values
for these simulations are identical to those in Figure 4.3.
Figure 4.6: Initial Control Conditions
Figure 4.7 represents the outcome of competition with no control. Parts a
through d represent the system at t = 20, 70,120,150. The results of this simulation is identical to the results in Figure 4.3, which impUes that initial spatial
distribution does not eífect the outcome of competition. Here, with no control, the
two weed species Ai and .42 dominate the competition, eliminating the species .43.
Figure 4.8 iUustrates the effect of control on species .4i. A control is appUed
that reduces the density of species Ai or A2 by a proportional amount pi or p^ on
49
Figure No. 4.7a
Figure No. 4.7b
1
.^0.8
c
(U
^
Q
c 0.6
o
'^0.4
, • • • • '
*^ \
3
Q.
1 .'^
i2o.2
>.
\
• \
Nv
^•
n
0
^""^"^^
5
10
X
Figure No. 4.7c
Figure No. 4.7d
Figure 4.7: No Control
the region [5, oo). For each of these figures the system is at í = 500 and the species
A2 is not controUed. Figure 4.8a represents 100% control of Ai, i.e., all of species
Ai is removed from the region containing the desired plant species. Figures 4.8b,
4.8c, and 4.8d illustrate 95%, 92.5%, and 90% control of species Ai, respectively.
Figure 4.9 illustrates the eífects of control on species Ai and A^. The system
is at í = 500 and the control on species A2 is held constant at 50%. In parts a
through d, species Ai is controUed at the same levels as in Figure 4.8.
There are two major results from these particular simulations. First, the desired
plant yield increases with higher levels of control on species .4i, and is optimized
at 100% control of species A^. Second, additional control of species .4^ has little
50
Figure No. 4.8a
Flgure No. 4.8b
1
1
.^0.8
•
.-^0.8
co
c
\
. • • •
co
c
CD
CD
Q0.6
uo!
Q0.6
c
0
^ 0.4
^0.4
3
.
• \
u
Q.
Q.
£0.2
£0.2
•
0
()
- » -
5
^
' ^
—
0
10
()
5
X
X
Figure No. 4.8c
Figure No. 4.8d
1
.-^0.8
\
K
1
,
..^0.8
to
Ui
c
cu
c
Q0.6
0)
-
Q0.6
c
•
c
0
0
•coO.4 •
^0.4
u
13
Q.
Q.
.0.2
•
û-0.2
• - . ^.
• ~
•
0
C)
•
-
~
0
5
10
C)
X
5
1C
X
Figure 4.8: Effects of Control on Species Ai
impact on the relative fitness of the desired crop. Although both weed species Ai
and A2 would dominate A3 with no control, it is control of only one of the weed
species Ai that is need for the desired crop to survive.
51
Figure No. 4.9a
Figure No. 4.9b
1
1
.-^0.8
(0
c
.^0.8
c
Q0.6
c
o
"cs 0.4
Q
c 0.6
(D
'
\
.
•
•
(D
.0
•
•'
\
Q.
"co 0.4
u
•
Q.
£0.2
S.0.2
0
0
•
c)
5
X
10
C)
Figure No. 4.9c
1
.-^0.8
•
<
1
5
X
Figure No. 4.9d
1
____—'-—"
'•
•
.-^0.8
•
(fi
(0
c
c
CD
(D
Q
c 0.6
Q
c 0.6
"co 0.4
1c0.4
Q.
Z5
Q.
0
0
£0.2
0
•
•
£0.2
•^
c)
5
X
v^ _
10
0)
c
5
X
Figure 4.9: Effects of Control on Species Ai and .4^
1
CHAPTER V
DISCUSSION AND CONCLUSIONS
The underlying foundation for our research is the Pakes and MaUer [14] twospecies competition model (Chapter II). Their model is based upon the de Wit [22]
replacement series model, which was used to predict seed yields of adult plants in
mixture [14, 19]. The model was formulated and tested against data coUected by
R. C. Rossiter [16, 17, 18, 19]. The data of Rossiter was clover species in binary
mixtures and applied to the two-species model. In their 1985 paper [19] they
stated,
Even now, many established pastures, . . . , consist of three or four
strains even though only one or two of these may predominate. Modeling the population dynamics of such mixtures should not be an insurmountable challenge; our model could be extended to mixtures of
more than two strains. (141)
However, Pakes and Maller only considered two strains. Via local and global
stability analyses, they were able to determine the qualitative behavior of solutions
to a very general two-species competition model. The behavior of their model is
very similar to that of the Lotka-Volterra continuous-time model [4, 5, 12, 15].
In this research the Pakes and Maller model has been generalized to a threespecies model (Chapter III). Analytically, we have solved for the steady-states
and performed a local stability analysis for the one and two-species steady-states
(Chapter III), which accounts for six of the possible seven steady-states. Conditions for the stability of the positive steady-state are given by the Jury conditions.
Furthermore, in the analysis of the three-species model in Chapter III, we
considered a special case related to the continuous time Lotka-\'olterra model of
52
53
May and Leonard [10]. This case iUustrates the unusual behavior found in nonUnear
systems. Our analysis and simulations revealed results simUar to that of May and
Leonard. We proved that when a < 1 and b < 1, where a = Å;i2 = k^^ = k^i and
b = ki3 = k32 = k^i, the system is asymptotically stable. The stability can be
extended to other values of a, b, and a which satisfy the specific inequaUty (3.5).
AIso worth noting is the osciUatory behavior of the solutions graphed in Figures
3.13 through 3.16.
In Chapter IV, we derived and simulated the spatially heterogeneous model,
where the population spreads seeds in one spatial dimension. The basis for this
work can be found in AUen et al. [1, 2, 3]. We demonstrated that the constant
steady-states are the same in both models. In many cases, numerical solutions of
this model converged to the steady-state solutions of the spatially homogeneous
model; however, there may be a lot of spatial variation prior to convergence. In
section 4.5, we discussed an application of this model, which iUustrates the use of
control over weeds.
There exists much future work to be done on the spatially homogeneous (Chapter III) and heterogeneous (Chapter IV) models. Further work includes finding
global stability conditions for all of the steady-states, the existence of spatially
heterogeneous steady-states, and investigating questions related to permanence or
persistence. Some future objectives are to apply these models to weed control, to
determine rates of spread with uniform and nonuniform controls, and the amount
of control necessary to prevent weed infestations.
REFERENCES
[1] AUen, E.J., L.J.S. AUen, and X. GilUam. 1996. Dispersal and competition
models for plants. J. Math. Biol. 34: 455-481.
[2] AUen, L.J.S., E.J. AUen, and S. Ponweera. 1996. A mathematical model for
weed dispersal and control. Bull. Math. Biol. To appear.
[3] AUen, L. J.S. and E. J. AUen. 1996. Mathematical Models for the Dispersal
and Control of Undesirable Plants on Rangeland. Proc. ofthe Fifth Int'l Conf.
on Desert Developement. To appear.
[4] Campbell, N. A. 1990, Biology. The Benjamin/Cummings Publishing Company, I n c , Muelo Park, CA.
[5] Edelstein-Keshet, L. 1988. Mathematical
I n c , New York.
Models in Biology. McGraw-HiU,
[6] Franke, J.E. and A-A. Yakubu. 1996. Extinction and persistence of species
in discrete competitive systems with a safe refuge. J. Math. Anal. Appl. 203:
746-761.
[7] Franke, J.E. and A-A. Yakubu. 1996. Extinction of species in in agestructured, discrete noncooperative systems. J. Math. Biol. 34: 422-454.
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