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Mathematical Biosciences 184 (2003) 115–135
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Periodic coexistence of four species competing
for three essential resources
Bingtuan Li
a,*,1
, Hal L. Smith
b,2
a
b
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
Received 3 September 2002; received in revised form 29 January 2003; accepted 28 March 2003
Abstract
We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by
resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a
combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution
of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed
which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic
solution adding one new species at a time so long as asymptotic stability can be assured at each step.
Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that
competition can generate coexistence of species in the form of periodic cycles, and that the number of
coexisting species can exceed the number of resources in a constant and homogeneous environment.
2003 Elsevier Science Inc. All rights reserved.
Keywords: Resource competition; Coexistence; Bifurcation; Periodic orbit
1. Introduction
Resource competition is common in nature. Early experimental and modeling work in ecology
[1] led to the principle of competitive exclusion [2–5]. This principle states that among several
species competing for a common resource, only the best competitor survives, and the number of
*
Corresponding author. Tel.: +1-502 852 6826; fax: +1-502 852 7132.
E-mail address: [email protected] (B. Li).
1
This authorÕs research was partially supported by an ORAU Ralph E. Powe Junior Faculty Enhancement Award.
2
This authorÕs research was supported in part by NSF grant DMS 0107160.
0025-5564/03/$ - see front matter 2003 Elsevier Science Inc. All rights reserved.
doi:10.1016/S0025-5564(03)00060-9
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
coexisting species does not exceed the number of available resources. The mathematical results for
the classical chemostat equations of competition for a single resource, are extensive and nearly
complete [6–11]. These results predict that only the species with lowest break-even concentration
survives. See [12] for a review. The case of two species competing for two essential resources has
been completely settled, first by Leon and Tumpson [13] and later with more mathematical
precision by Hsu et al. [14] and Butler and Wolkowicz [15]. The results show that outcomes of
two-species competition for two resources are the same as those for the classical Lotka–Volterra
model of competition: competitive exclusion, coexistence, and bistability. Recently we completely
settled the case of three species competing for two resources, proving that two resources cannot
support three species [16]. The existing mathematical results for one or two limiting resources
support the competitive exclusion principle. Many ecological communities, however, are limited
by more than two resources [17–19]. The above results cannot explain how rich ensembles of
species can coexist on a limited number of resources, as in aquatic ecosystems.
Recent papers by Huisman and Weissing [20,21] represent a milestone in the study of resource
competition, and have triggered further interest in this area. Huisman and Weissing show numerically
that competition models with three or more limiting resources can generate sustained oscillations or
even chaotic dynamics of species abundance, even under constant resource supply and constant
physical conditions. In particular, they show numerically that periodic oscillations occur if three species
compete for three resources, and chaotic oscillations occur if five species compete for five resources,
and up to nine species can seemingly be supported by three recourses and as many as 12 species can
coexist on five resources. The first author [22] established the existence of the limit cycle for the case of
three species competing for three resources case which is fundamental to many of the conclusions of
Huisman and Weissing. A review of known results and some new ones can be found in [23].
The numerical work of Huisman and Weissing [20,21] displays interesting features of diversity of
competing species. Field studies have shown that a temporally varying environment may favor
species coexistence [24,25]. Hence it is conceivable that oscillations generated by competition
promote species diversity. However, both mathematicians and ecologists still lack analytic understanding of mechanisms that generate biodiversity in the context of resource competition.
The present paper is motivated by the numerical simulations in [20] showing that beginning with the
periodic coexistence of three species on three resources, one new species after another could then be
added sequentially to the community, finally achieving an apparently coexisting community of nine
species on the three resources. We employ a standard bifurcation technique (see e.g. [26]) which in
principle will allow adding a new species to a community coexisting via an asymptotically stable periodic
orbit, yielding a perturbed periodic orbit in which the new species is present at small amplitude. If the
perturbed periodic orbit is asymptotically stable, then the technique can be iterated to add yet another
species. Using this technique, we establish periodic coexistence of four species on three resources.
2. The standard model
The equations for n species Ni competing for m resources Rj are
Ni0 ¼ Ni ½li ðRÞ D; 1 6 i 6 n;
X
cjk lk ðRÞNk ;
R0j ¼ D½Sj Rj k
1 6 j 6 m:
ð2:1Þ
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
117
Here, the removal rate of the ith species and the resource turnover rate is D > 0 and the supply of
the jth resource is Sj > 0. cjk > 0 is the content of resource j in species k. Write C for the m nmatrix ðcji Þ, N for the n-vector of the Ni , R for the m-vector of the Rj , and S for the m-vector of
supply concentrations. Model (2.1) links the population dynamics of competing species with the
dynamics of the resources that these species are competing for. An interesting feature of this
model is that it uses the biological traits of species to predict the time course of competition.
We restrict attention to essential resources R for which the Law of the Minimum applies
li ðRÞ ¼ minffji ðRj Þg;
ð2:2Þ
j
where fji is a Monod function,
fji ðRj Þ ¼
ri Rj
:
Kji þ Rj
ð2:3Þ
See [27] for a derivation of (2.2) as well as alternatives to the law of the minimum. Model (2.1)
with the law of the minimum has been tested and verified extensively using competition experiments with phytoplankton species [20,25,28–32]. The model also provides a conceptual framework for competitive interactions among terrestrial plants [33,34].
Adding equations, we have
ðR þ CN Þ0 ¼ D½S ðR þ CN Þ
ð2:4Þ
which leads to
ðR þ CN ÞðtÞ ¼ ðR þ CN Þð0ÞeDt þ Sð1 eDt Þ:
The above integration establishes the existence of an exponentially attracting affine sub-manifold
M fðR; N Þ 2 Rmþn
: R þ CN ¼ Sg:
þ
ð2:5Þ
The dynamics restricted to this manifold are given by
Ni0 ¼ Ni ½li ðRÞ D;
R ¼ S CN ;
1 6 i 6 n;
ð2:6Þ
on the positively invariant polygonal set
C fN 2 Rnþ : CN 6 Sg:
A further reduction of dimension by one to the Ôcarrying simplexÕ can be made; see [23] for details.
Inequalities between vectors are to be interpreted component-wise. We write v 6 w if vi 6 wi for all
i and v < w if vi < wi for all i.
In this paper, we are not interested in cases where a species becomes extinct in the absence of
competition. Thus, we assume that D < ri for all i so there exists positive real numbers kji such
that
ðfji ðRj Þ DÞðRj kji Þ > 0;
Rj 6¼ kji :
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
kji is the break-even concentration of resource j for the growth of species i when only resource j is
growth limiting. We also assume that Sj > kji for all i; j; resource supply exceeds the demand for
growth.
3. Hopf bifurcation for three species on three resources
Several of the exciting conclusions of Huisman and Weissing [20,21] stem from their numerical
discovery of a limit cycle for the three-resource three-species model for certain parameters. The
existence of such a limit cycle was later established in [22] but its stability was not addressed. Our
method of achieving coexistence of four species on three resources in the following section requires the existence of a periodic solution of the three species–three resource system that is asymptotically stable in the linear approximation. Obviously, the lack of smoothness of the vector
field causes difficulties. Therefore, we resort to a Hopf bifurcation analysis to obtain a periodic
orbit, asymptotically stable in the linear approximation, near a positive steady state where we can
be sure of the smoothness of our vector field.
We need only consider the (reduced) differential equations in a neighborhood of the positive
equilibrium N . To simplify even further, we follow Huisman and Weissing by choosing parameters such that in a neighborhood of the positive equilibrium we have
li ðRÞ ¼ fii ðRi Þ ¼ f ðRi Þ ¼ mRi =ðK þ Ri Þ;
R ¼ S CN ;
ð3:1Þ
where f is the same for all i ¼ 1; 2; 3. In words, the growth of the ith population is limited by the
ith resource near equilibrium; the additional assumption that fii ¼ f is for mathematical simplicity. In this case, near the positive equilibrium, (2.6) takes the form
Ni0 ¼ Ni ½f ðRi Þ D;
1 6 i 6 3:
ð3:2Þ
The matrix C is taken to be
2
3
1þu lþv mþw
C ¼ 4 m þ v 1 þ w l þ u 5;
lþw mþu 1þv
and l, m, u, v, w are all positive constants. Note that the sum of entries in each row of C is
1 þ l þ m þ u þ v þ w. We comment further on this choice below.
Define k ¼ kii by f ðkÞ ¼ D (we assume D < m). We will ensure that the positive equilibrium of
T
(3.2) is given by N ¼ ðL; L; LÞ , for our choice of L > 0, by allowing S to be to contain a free
parameter L. Take S ¼ ðS0 ; S0 ; S0 ÞT where S0 is to be determined. As we must have k ¼ Ri ¼
ðS CN Þi ¼ ½S0 ð1 þ l þ m þ u þ v þ wÞL, we let S0 ¼ k þ ð1 þ l þ m þ u þ v þ wÞL > 0. We
have thus constructed a one-parameter family of systems parametrized by L > 0.
The form of C is so artificial that one may not find it Ôin the fieldÕ. However, it helps with
computations and displays interesting features of competition. In the case u ¼ v ¼ w ¼ 0, C is a
cyclic matrix. Cyclic matrices have been used by May and Leonard [35] for Lotka–Volterra
competition models, as well as by Huisman and Weissing [21] in their numerical simulations. In
both papers, it is one part of an assumption of a cyclic displacement of species. In particular, in
[21], the cyclicity of C and certain condition on kij Õs lead to the cyclic fashion of displacement:
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
119
species 1 is the better competitor for resource 1 but becomes limited by resource 2, species 2 is the
better competitor for this resource but becomes limited by resource 3, finally species 3 is the better
competitor for resource 3 but becomes limited by resource 1 and so on. Our purpose here is to
establish asymptotically stable periodic cycles through Hopf bifurcation. Unfortunately, if C is
cyclic then stability calculations seem to show that the bifurcating periodic cycle is subcritical,
unstable. Therefore, we have added a symmetric matrix, determined by u, v, w, to the cyclic matrix
determined by l, m. We will use l as a bifurcation parameter in our analysis.
The Jacobian matrix J ðN Þ is given by J ðN Þ ¼ rLC where r ¼ f 0 ðkÞ. Its eigenvalues are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lþm2 1
3ðm lÞ2 þ 4ðu2 þ v2 þ w2 uv uw vwÞ rL;
e1;2 ¼
2
2
e3 ¼ rLð1 þ l þ m þ u þ v þ wÞ:
The first two eigenvalues become a pair of conjugate complex numbers if
2
3ðm lÞ þ 4ðu2 þ v2 þ w2 uv uw vwÞ < 0:
ð3:3Þ
On the other hand, at the critical value l ¼ 2 m,
d
lþm2
rL
rL
¼ > 0:
dl
2
2
We therefore have shown that a Hopf bifurcation occurs at l ¼ 2 m. As Li [22] has already
proved the existence of large amplitude periodic orbits for a similar parameter set, this result adds
nothing really new except for the potential, realized below, for establishing the existence of an
orbitally stable periodic orbit.
Proposition 1. If (3.3) holds, then a Hopf bifurcation occurs at l ¼ 2 m for (3.2).
We next discuss the stability of the bifurcating periodic orbit. We use v1 and v3 to denote the
eigenvectors corresponding to e1 and e3 , respectively. Here we consider only a special case: u ¼ 1,
v ¼ 0:25, w ¼ 0:5, and m ¼ 1:5. This choice ensures that (3.3) holds for all l 0:5. Now we follow
the recipe on pages 86 of [36]. Real and imaginary parts of eigenvectors are
Rv1 ¼ ð1; 0; 1ÞT ;
pffiffiffi
pffiffiffi
Iv1 ¼ ð0; 5=5; 5=5ÞT ;
v3 ¼ ð1; 1; 1ÞT :
We change coordinates by
N ¼ ðL; L; LÞT þ Py;
where
2
1
P ¼4 0
1
p0ffiffiffi
pffiffiffi5=5
5=5
ð3:4Þ
3
1
15
1
is obtained from eigenvectors. Then
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
2
P 1
p2ffiffiffi
¼ 1=34 5
1
1
pffiffiffi
2 5
1
3
1ffiffiffi
p
5 5:
1
We only require the differential equation for y, y 0 ¼ F ðyÞ, at the critical value of the Hopf
parameter l ¼ 1=2. Its components are given by
F1 ¼ 1=3½2N1 ðf ðR1 Þ DÞ N2 ðf ðR2 Þ DÞ N3 ðf ðR3 Þ DÞ;
pffiffiffi
F2 ¼ 5=3½N1 ðf ðR1 Þ DÞ 2N2 ðf ðR2 Þ DÞ þ N3 ðf ðR3 Þ DÞ;
F3 ¼ 1=3½N1 ðf ðR1 Þ DÞ þ N2 ðf ðR2 Þ DÞ þ N3 ðf ðR3 Þ DÞ;
where N is given by (3.4) and R ¼ S CN becomes
pffiffiffi
R1 ¼ k ð 5=4Þy2 ð19=4Þy3 ;
R2 ¼ k ð1=4Þy1 ð19=4Þy3 ;
pffiffiffi
R3 ¼ k þ ð1=4Þy1 þ ð 5=4Þy2 ð19=4Þy3 :
Following Guckenheimer and Holmes [37, Eq. (3.4.9), p. 152], we need only determine the
coefficient a in the normal form for the system y 0 ¼ F ðyÞ, restricted to the center manifold,
y3 ¼ a11 y12 þ a12 y1 y2 þ a13 y22 þ h:o:t:
where Ôh.o.t.Õ stands for higher order terms (which need not be computed). One can use Maple or
other computational software to show that
a¼
3Km
5ð4k þ 4KÞ3
ð3:5Þ
< 0:
In this case, a supercritical, stable Hopf bifurcation occurs, as stated in the following proposition.
Proposition 2. Assume that u ¼ 1, v ¼ 0:25, w ¼ 0:5, m ¼ 1:5, and S0 ¼ k þ ð1 þ l þ m þ u þ
v þ wÞL where k ¼ D=ðm DÞ, L > 0 and m > D > 0. Then there is a supercritical, asymptotically
stable periodic orbit of (3.2) bifurcating from the steady state N ¼ ðL; L; LÞ at the critical value
l0 ¼ 1=2.
Remark 1. The biological meaning of our assumptions and parameter choices in Proposition 2 are
as follows. Eq. (3.1) says that population i is growth limited by resource i for i ¼ 1; 2; 3. Our
choice u ¼ 1, v ¼ 0:25, w ¼ 0:5, m ¼ 1:5, and l 0:5 translates to
c11 ¼ c13 ¼ 2 > c12 ¼ 0:75 þ ðl l0 Þ;
c21 ¼ 1:75 > c23 ¼ 1:5 þ ðl l0 Þ;
c22 ¼ 1:5;
ð3:6Þ
c32 ¼ 2:5 > c33 ¼ 1:25 > c31 ¼ 1 þ ðl l0 Þ:
This means that population 1 and 3 consume the same amount of resource 1 and more than
population 2 consumes; that population 1 consumes the most of resource 2, followed by population 2 and 3 which consume nearly the same amount; finally, population 2 consumes twice as
much of resource 3 as does population 3, which in turn consumes more than population 1.
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
121
Remark 2. There are other parameter values of l, m, u, v, w which also yield supercritical Hopf
bifurcations. For example, in the case u ¼ 1, v ¼ 0:25, w ¼ 0:5, and m ¼ 0:6, it can be shown that a
supercritical, stable Hopf bifurcation occurs at the critical value l0 ¼ 1:4.
Remark 3. The stability of Hopf bifurcation shown in Proposition 2 and Remark 2 is robust. In
fact, if one changes the parameters slightly, then a < 0 is still valid and the purturbed system
undergoes a supercrital Hopf bifurcation.
Remark 4. In the case that C is cyclic, i.e., u ¼ v ¼ w ¼ 0, one can find that the coefficient a in the
normal form restricted to the center manifold is determined by
2
a¼
3Kmð1 mÞ
3
4ðk þ KÞ
> 0:
This shows that a subcritical, unstable Hopf bifurcation occurs at the critical value l0 ¼ 1 m for
m 6¼ 1.
We next demonstrate numerically the periodic cycle in Proposition 2 with L ¼ 2, S0 ¼ 12,
m ¼ 1, D ¼ 1562=2187, K ¼ 1 and l ¼ 0:5004, the latter being very close to the critical value
l0 ¼ 0:5. Fig. 1 show the periodic oscillations of the system due to Hopf bifurcation. In all cases,
we chose the initial values N1 ð0Þ ¼ 2:14, N2 ð0Þ ¼ 1:8, and N3 ð0Þ ¼ 1:75.
Fig. 1 displays the small-amplitude oscillations of the simplified (3.2) in a neighborhood of N where (3.1) is satisfied. This same solution is also a solution of the system in the form (2.6) with
li ðRÞ given by (2.2), fji given by (2.3), ri ¼ 1, and on choosing K ¼ ðKji Þ as
2
3
1
0:75 0:25
K ¼ 4 0:25
1
0:75 5:
0:75 0:25
1
In a neighborhood of N , this system agrees with (3.2). This choice of Kji implies relationships
among the requirements for growth kji ¼ kKji . Namely,
k11 > k12 > k13 ;
k22 > k23 > k21 ;
ð3:7Þ
k33 > k31 > k32 :
Huisman and Weissing [38] predict that Ôif each species consumes most of the resource for which it
has the intermediate requirement, the system generates species oscillationÕ. However, our oscillations are produced under different circumstances. In our case, (3.6) and (3.7) imply that species 3
consumes the most of resource 1 (actually, species 1 consumes the same amount) but has the
lowest requirement for it; species 2 consumes the most of resource 3 but has the lowest requirement for it; species 1 consumes the most resource 2 but has the lowest requirement for it.
There are some further differences between the periodic solutions discovered numerically in
[20,22] and the solution described in Fig. 1. In [20], the simulations show that the species are
oscillating with phase shifts due to the use of a cyclic matrix. Clearly, this is not the case in Fig. 1
since our matrix C is a combination of a cyclic matrix and a symmetric matrix.
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
Fig. 1. Oscillations of three species on three resources due to Hopf bifurcation: (a) time course of the abundances of
three species competing for three resources; (b) the corresponding limit cycle; (c) time course of the abundances of the
corresponding three resources.
The numerical example of three species coexisting on three resources via a periodic solution
which appear in [20,22] has a Z3 symmetry which we comment on here. The matrix C is taken to be
cyclic, e.g., u ¼ v ¼ w ¼ 0, which commutes with the cyclic permutation matrix
2
3
0 1 0
Q ¼ 4 0 0 1 5;
1 0 0
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
123
QC ¼ CQ. Q generates the cyclic Lie group C ¼ fQl : l ¼ 0; 1; 2g acting on R3 . The supply vector S
is taken to be constant so QS ¼ S and the Monod functions ½fji ðRj Þ are also chosen in a cyclic
fashion so that QlðRÞ ¼ lðQRÞ. It follows that if F denotes the vector field (2.6), then
QF ðN Þ ¼ F ðQN Þ which in turn implies that QN ðtÞ is a solution whenever N ðtÞ is a solution of
(2.6). Assuming that there is a unique periodic solution N ðtÞ of minimal period T , which numerical
simulations strongly suggest, then it follows that there exists h 2 ð0; T Þ such that QN ðtÞ ¼ N ðt þ hÞ
for all t : N2 ðtÞ ¼ N1 ðt þ hÞ; N3 ðtÞ ¼ N2 ðt þ hÞ ¼ N1 ðt þ 2hÞ; N1 ðtÞ ¼ N3 ðt þ hÞ ¼ N1 ðt þ 3hÞ. It follows that h ¼ T =3 of h ¼ 2T =3. Consequently, as it appears in the figures in [20], N2 and N3 are
merely phase shifted N1 by a third of a cycle. Such solutions are often called Ôrotating wave solutionsÕ [39]. Alternatively, a Hopf bifurcation analysis in this case establishes the existence of
periodic rotating wave solutions. See e.g. [39, Theorem 8.2, Chapter XVII].
4. Perturbation of the Hopf bifurcation by a fourth species
Our goal in this section is to show that we can add a suitable fourth species to an asymptotically
stable three-species periodic orbit obtained by Hopf bifurcation, yielding a periodic orbit with
four species coexisting on the same three resources. We start with the system of three species
competing for three resources
Ni0 ¼ Ni ½li ðRÞ D;
1 6 i 6 3;
ð4:1Þ
where
R ¼ SðlÞ CðlÞN
and CðlÞ, SðlÞ depend on parameter l for l l0 . The matrix C in the previous section can be
viewed as a special case of CðlÞ. We assume that
li ðRÞ ¼ f ðRi Þ;
i ¼ 1; 2; 3;
ð4:2Þ
where f ðRi Þ ¼ rRi =ðK þ Ri Þ, holds in a neighborhood of a positive steady state. Matrix CðlÞ and
supply vector SðlÞ are chosen such that for some N ¼ ðL; L; LÞT with L > 0, N ¼ N is a steady
T
state, for all l l0 . Thus, if f ðkÞ ¼ D, R ðk; k; kÞ ¼ SðlÞ CðlÞN . By continuity, there exists
d > 0 such that if kR R k < d, then (4.2) holds.
A new species N4 is defined by a choice of cj4 > 0, r4 > 0, Kj4 > 0 for j ¼ 1; 2; 3, determining
r 4 Rj
:
ð4:3Þ
l4 ðRÞ min
16j63
Kj4 þ Rj
The resulting four-species system is given by
ð4:4Þ
Ni0 ¼ Ni ðli ðRÞ DÞ; i ¼ 1; 2; 3; 4;
e N, C
e is the 3 · 4 matrix obtained from CðlÞ by adding the fourth column
where R ¼ SðlÞ C
T
ðc14 ; c24 ; c34 Þ . System (4.4) is built on (4.1), describing competition among four species for three
resources.
We will choose species 4 to be growth limited by resource 1 near the Hopf periodic solution.
This choice, a completely arbitrary one, suggests using K14 as a (secondary) bifurcation parameter.
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
The following theorem is our main result. It requires no assumptions regarding the consumption
profile (the cj4 ) of species 4.
Theorem 1. Suppose that a supercritical Hopf bifurcation occurs from N at l ¼ l0 for system (4.1),
e ðt; lÞ ¼ ð N
e 1 ðt; lÞ; N
e 2 ðt; lÞ; N
e ðt; lÞÞT
that is, there exists an asymptotically stable periodic orbit N
with minimal period T > 0 for l > l0 and l very close to l0 . Then, for a fixed l > l0 near l0 , and
and K24 ; K34 > 0 such that for some values of K14 with
cj4 > 0 (j ¼ 1; 2; 3), r4 > D, there exists K14
jK14 K14 j suciently small, (4.4) has a periodic orbit with the property that all four components are
positive.
Proof. We have
e ðt; lÞ N N
e ðtÞ ¼ N
e ðt; lÞ for the asymptotically stable periodic
for l 2 ðl0 ; þ l0 Þ. Fix such a l and write N
e
e
solution of periodic T > 0. Set R ðtÞ ¼ SðlÞ CðlÞ N ðtÞ.
> 0 such that f14 ðR1 Þ ¼ r4 R1 =ðK14 þ R1 Þ satisfies
Fix K14
#
Z "
e 1 ðtÞ
1 T
r4 R
D dt ¼ 0
ð4:5Þ
aðK14 Þ ¼
e 1 ðtÞ
T 0
K14 þ R
when K14 ¼ K14
. As r4 > D, it is easy to see that such a value of K14 is uniquely defined. Hereafter,
.
we restrict K14 to be near K14
e ðtÞ we choose K24 , K34 such that
In order to ensure that l4 ðRÞ ¼ f14 ðR1 Þ for R R
r 4 R2
r4 R3
r4 R1
;
>
K24 þ R2 K34 þ R3 K14 þ R1
whenever R ¼ ðR1 ; R2 ; R3 ÞT is near R . We just need to take K24 , K34 sufficiently small, since by
continuity, if K24 ¼ 0 we have
e2
r4 R
r4 R1
¼ r4 >
:
e2
K14 þ R1
K24 þ R
We have defined our new species N4 in such a way that it is R1 -limited near R .
e ðtÞ to be a 4-vector by adding a zero fourth component, and note
Next, we redefine N
e
e
R ðtÞ ¼ S C N ðtÞ (still a 3 vector). Hereafter, N will denote a vector in R4 . Then, so long as
e ðtÞk is small
e ðtÞk and jK14 K j are sufficiently small then R ¼ S CN is such that kR R
kN N
14
enough that
r 4 R1
l4 ðRÞ ¼
:
ð4:6Þ
li ðRÞ ¼ f ðRi Þ; 1 6 i 6 3;
K14 þ R1
e ðtÞ is a T -periodic solution of
We conclude that N
Ni0 ¼ Ni ½li ðS CN Þ D;
1 6 i 6 4;
and furthermore that (4.6) holds provided N remains near the periodic orbit
e ðtÞ : t 2 Rg
c fN
j is sufficiently small.
and jK14 K14
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
125
Because we assume that the Hopf bifurcation gives a family of asymptotically stable periodic
e ðtÞ consist of 1, q2 , q3 and
orbits, the Floquet multipliers of the periodic solution N
q4 expðaðKÞÞ;
where jqi j < 1 for i ¼ 2; 3 and K ¼ K14 . By construction (4.5), q4 satisfies
q4 ðK Þ ¼ 1;
and
where K ¼ K14
dq r4
ðK Þ ¼ dK
T
Z
0
T
e 1 ðtÞ
R
dt < 0:
e 1 ðtÞÞ2
ðK þ R
ð4:7Þ
e ðtÞ changes from unstable to stable as K
Hence, the stability property of the periodic solution N
increases through the value K .
We may now proceed exactly as in the analysis of the food chain model in Chapter 3 of [12] to
e ð0Þ but with all components positive.
determine the initial conditions of a periodic orbit near N
0
e
e
Choose a point P0 ¼ N ðt0 Þ 2 c and let V0 ¼ N ðt0 Þ. Then a neighborhood U of P0 in P0 þ V0? gives
a transversal to the flow. We can choose an orthonormal basis fu1 ; u2 ; u3 g for V0? where
u3 ¼ ð0; 0; 0; 1Þ and u1 , u2 have vanishing 4-th components such that the corresponding coordinates ðX ; Y ; ZÞ 2 U vanish at P0 and where Z ¼ N4 . Then the Poincare map P : U ! U , satisfies
P ð0; 0; 0; KÞ ¼ ð0; 0; 0Þ for K near K and P3 ðX ; Y ; 0; KÞ 0 since Z ¼ N4 ¼ 0 is invariant. Note
that P ¼ ðP1 ; P2 ; P3 Þ and only P3 depends on K. Its Jacobian is given by
2
3
P1X P1Y P1Z
DP ð0; 0; 0; KÞ ¼ 4 P2X P2Y P2Z 5:
0
0 P3Z
The 2 · 2 sub-matrix B on the upper left represents the Jacobian of the Poincare map
ðX ; Y Þ ! ðP1 ðX ; Y ; 0Þ; P2 ðX ; Y ; 0ÞÞ of the three dimensional sub-system (N4 ¼ 0) corresponding to
e 2 ðtÞ; N
e 3 ðtÞÞ so it has eigenvalues q2 , q3 of modulus less than one. The
e 1 ðtÞ; N
the periodic solution ð N
third eigenvalue of DP is the Floquet multiplier q4 ðKÞ ¼ P3Z . When K ¼ K , q4 ðK Þ ¼ 1 so the
corresponding eigenvector is v ¼ ða; b; 1ÞT where
ðB IÞða; bÞT ¼ ðP1Z ; P2Z ÞT ;
where I is the identity matrix. As the eigenvalues of B lie inside the unit circle, B I is invertible so
T
ða; b; 1Þ is well-defined.
According to Theorem 6.3 of Smith and Waltman, Chapter 3, in [12], the Poincare map P has a
branch of fixed points
T
N ¼ P0 þ s½au1 þ bu2 þ ð0; 0; 0; 1Þ þ oðsÞ;
jsj < v;
for K14 ¼ K þ jðsÞ where oðsÞ 2 V0? satisfies oðsÞ=s ! 0 as s ! 0 and jð0Þ ¼ 0 and j is continuously differentiable. Obviously, this fixed point corresponds to initial data N ð0Þ of a periodic
solution of our system with K14 ¼ K þ jðsÞ and period near T . Furthermore, as N4 ð0Þ ¼ s þ
oðsÞ > 0, the solution N ðtÞ has all components positive. The proof is complete. 126
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
Two possible bifurcation diagrams are depicted below, assuming that j0 ð0Þ 6¼ 0. Each depicts
two branches of solutions, the trivial one with N4 ð0Þ 0 representing the three-species periodic
e and the transversal branch where N4 ð0Þ 6¼ 0. In the left bifurcation diagram, where
orbit N
0
j ð0Þ > 0, an unstable periodic solution exists for K > K . In the right one, where j0 ð0Þ < 0, a
stable periodic orbit exists for K < K .
That the stability of the bifurcating branch of periodic orbits depends on the sign of j0 ð0Þ is
well-known (e.g. [40, Theorem 13.8]).
We recall that Theorem 1 was obtained with virtually no assumptions regarding species 4 other
than the arbitrary one that it is growth limited by resource 1. In order that the secondary bifurcation from the Hopf solution is asymptotically stable so that the four species can coexist
stably, we expect that assumptions regarding the consumption profiles cj4 of species 4 will be
required. Indeed, we may also need to amend our arbitrary assumption that species 4 is growth
limited by resource 1. In Appendix A, we develop a formula for the direction of bifurcation (see
(A.10)), and consequently, for the stability of the bifurcating branch. This formula is affine in the
cj4 and involves aspects of the three species ecosystem consisting of species 1–3 through solutions
of the non-homogeneous linear system obtained from the Jacobian matrix evaluated along the
Hopf periodic solution. Unfortunately, it is sufficiently complicated that we are unable to determine the sign of j0 ð0Þ by analytic means.
Proposition 2 in the previous section provides a family of asymptotically stable periodic orbits
due to Hopf bifurcation. Applying Theorem 1, we have the following corollary.
Corollary 1. Assume that the hypotheses of Proposition 2 are satisfied. Then, for a fixed
and K24 ; K34 > 0 such that for
l > l0 ¼ 1=2 near l0 , and cj4 > 0 (j ¼ 1; 2; 3), r4 > D, there exists K14
some values of K14 with jK14 K14 j suciently small, (4.4) has a periodic orbit with the property that
all four components are positive.
In order to demonstrate that our analytical result, Corollary 1, can provide observable (stable)
small-amplitude oscillations we must resort to numerical simulations. Our goal in these simulations is only to provide such evidence. As a consequence, the solutions so obtained will necessarily
be characterized by very low abundance of species 4. Simulations showing apparently stable
coexistence of four species on three resources with more biologically plausible species abundances
are inferred from Fig. 1(c) in [20].
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
127
Fig. 2. Periodic coexistence of four species illustrating Corollary 1: (a) time course of the (very low) abundance of the
invading species N4 ; (b) time course of the abundances of N1 , N2 , and N3 .
Fig. 2 shows coexistence of four species obtained by adding species 4 to the asymptotically
stable three-species periodic orbit shown in Fig. 1. As in Fig. 1, D ¼ 1562=2187 0:7142203932,
and k ¼ 2:5. Species 4 is characterized by r4 ¼ 0:8, c14 ¼ 0:36, c24 ¼ 0:3, c34 ¼ 0:24, K24 ¼ 0:1,
0:300256 obtained from
K34 ¼ 0:15, and K14 ¼ 0:3. The latter is very near K14
K14
ðr4 =D 1Þk
e ðtÞ R ¼ ðk; k; kÞ. Besides being growth limited by resource 1,
which follows from (4.5) with R
our choice of relatively small values of cj4 imply that species 4 consumes the least of each of the
three resources. Resource requirements for growth of species 4, given by kj4 ¼ ½D=ðr4 DÞKj4 ,
imply that it requires essentially the same level of resource 1 as species 1 does (the highest among
the others) but is intermediate between the two species having the lowest requirements for resource 2 and 3. Initial data for species 4 is taken to be N4 ð0Þ ¼ 0:001, while initial values of other
species remain the same as in Fig. 1.
5. Discussion
In this paper we established coexistence of four species on three essential resources using bifurcation theory. Our method requires the existence of a periodic solution representing coexistence of
128
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
three species on three resources which is asymptotically stable in the linear approximation. In
order to satisfy this stringent requirement, we first showed that for certain parameters, smallamplitude stable periodic oscillations of three species competing for three resources occur due to a
super-critical Hopf bifurcation. We next built the periodic coexistence of four species on the
periodic oscillations of three species through a secondary bifurcation. Unfortunately, the formula
for the stability of the secondary branch is so intractable that we were unable to determine the
characteristics of species 4 which make it capable of successfully invading the three-species
community yet not displacing any of the original competitors. However, our numerical simulations suggest that the bifurcation mechanism can produce stable coexistence of four species on
three resources even though we cannot explain precisely why. Our results add to the growing
awareness that resource competition can generate oscillations which create an opportunity to
increase species diversity. Furthermore, we provide some rigorous mathematical analysis leading
to the conclusion, apparent form the numerical simulations reported in [20,21], that the number of
coexisting species can exceed the number of limiting resources in a constant and homogeneous
environment. The remainder of this section is devoted to suggesting some extensions of our work
which can be carried out.
We assumed that in the case of three species competing for three resources, at N species i is
limited by resource i for each i and the matrix of contents of resources in species is a sum of a
cyclic matrix and a symmetric matrix to ensure a supercritical, stable Hopf bifurcation to occur. A
similar technique can be used to discuss Hopf bifurcation of the model of four species competing
for four resources that takes a form of (3.2) with (3.1) for 1 6 i 6 4. In fact, if matrix ðcij Þ44 takes
the cyclic form
3
2
1 m l c
6 c 1 m l7
7
6
4 l c 1 m 5;
m l c 1
one can see at the positive equilibrium ðL; L; L; LÞ, the eigenvalues of the Jacobian matrix are given
by rLð1 þ l þ m þ cÞ, rLð1 þ l m cÞ, 1 l þ iðm cÞ, and 1 l iðm cÞ. If we use l as a
bifurcation parameter and assume 1 þ l > m þ c and m 6¼ c, then a Hopf bifurcation occurs at
l ¼ 1. If one assumes that ðcij Þ44 is a sum of a cyclic matrix and a symmetric matrix
3
1þu mþv lþw cþx
6 c þ v 1 þ u m þ x l þ w7
7
6
4 l þ w c þ x 1 þ u m þ v 5;
mþx lþw cþv 1þu
2
then the eigenvalues of Jacobian
are rLð1 þ l þ m þ c þ v þ w þ xÞ, q
rLð1
þ l þ u þ w m
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c v xÞ, 1 þ u l w þ ðm cÞ2 þ ðv xÞ2 , and 1 þ u l w ðm cÞ2 þ ðv xÞ2 . It
follows that Hopf bifurcation occurs at l ¼ 1 þ u w, if jm cj > jv xj and 1 þ l þ u þ w >
m þ c þ v þ x. In the case of n species competing for n resources, one can use ðcij Þnn of the cyclic
form
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
2
1
6 ln1
6
4 l1
l1
1
l2
l2
l2
l3
129
3
ln1
ln2 7
7:
5
1
As shown in [41], the eigenvalues of this matrix are
1þ
n1
X
lj kjk ;
k ¼ 0; . . . ; n 1;
j¼1
where k ¼ expð2pi=nÞ. One can then discuss when Hopf bifurcation occurs. Alternatively, one can
use the sum of a cyclic matrix and a symmetric matrix for general n to analyze Hopf bifurcation.
In Section 4, we employed a bifurcation technique to add a new species to an asymptotically
stable three-species periodic orbit due to Hopf bifurcation. This technique can be iterated to add
more species into the system. In general, if one has an asymptotically stable n-species periodic
cycle with small amplitudes, then one can show periodic coexistence of n þ 1 species. The amplitudes of periodic oscillations obtained by Hopf bifurcation are so small that oscillations would
probably go unnoticed behind the noise of any real-world data set. Yet even these small-amplitude oscillations are apparently sufficient for adding more species into the community. It is interesting to note that competition itself can generate diversity of competing species in the form of
periodic cycles, without involving external factors.
An alternative analysis to that given here would be to begin with the positive steady state N of
the three-species, three resource model for parameter values l near l0 at which a Hopf bifurcation
occurs. By adding a suitable fourth species with parameter K14 , we could determine a critical value
, such that the new species is neutrally stable to invasion of the three-species steady state N .
K14
Consequently, the Jacobian matrix of the four-species system at ðN ; 0Þ, at parameter values
, would have a zero eigenvalue and a purely imaginary conjugate pair. This
l ¼ l0 and K14 ¼ K14
. However, as no steady
degeneracy could then be analyzed for parameters l l0 and K14 K14
state can exist for four species on three resources, we expect a degeneracy in the standard normal
form for this co-dimension two bifurcation.
Appendix A
In this brief appendix we give an alternative analysis to that derived in [12] and used in the
previous section. Rather than using the Poincare map, we use a perturbation argument and the
Fredholm alternative which leads to a formula for the direction of bifurcation (of course, one may
also derive such a formula using the Poincare map approach). We carry out this program in some
generality since it may be useful in other applications.
Consider the following system of differential equations:
x0 ¼ F ðx; yÞ;
y 0 ¼ yGðx; y; kÞ;
where x 2 Rn and y; k 2 R, and F , G are smooth. We assume that there is a T -periodic solution
x ¼ pðtÞ, y 0 and that p is asymptotically stable solution of
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B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
x0 ¼ F ðx; 0Þ:
If fact, we assume that the Floquet multipliers of the variational equation
X 0 ¼ AðtÞX ;
AðtÞ Fx ðpðtÞ; 0Þ
ðA:1Þ
lie inside the unit circle except for the simple multiplier l ¼ 1 corresponding to the T -periodic
solution X ¼ p0 ðtÞ.
The stability of ðpðtÞ; 0Þ as a solution of the full system is determined by the variational
equation
Z 0 ¼ Bðt; kÞZ;
where Z ¼ ðX ; Y Þ and
Fx ðpðtÞ; 0Þ Fy ðpðtÞ; 0Þ
Bðt; kÞ ¼
:
0
GðpðtÞ; 0; kÞ
It is easy to see that its fundamental matrix UðtÞ, satisfying Uð0Þ ¼ I is partitioned as
U11 ðtÞ
U
ðtÞ
12
Rt
UðtÞ ¼
;
0
exp½ 0 GðpðsÞ; 0; kÞ ds
where U11 ðtÞ is the fundamental matrix for (A.1) and
Z s
Z t
1
U11 ðtÞU11 ðsÞFy ðpðsÞ; 0Þ exp
GðpðrÞ; 0; kÞ dr ds:
U12 ðtÞ ¼
0
0
It follows that ðpðtÞ; 0Þ is stable if
Z T
GðpðtÞ; 0; kÞ dt < 0
gðkÞ 0
and unstable if the reversed inequality holds. We assume there exists k0 such that gðk0 Þ ¼ 0 and
that
Z T
dg
Gk ðpðtÞ; 0; kÞ dt 6¼ 0:
ðA:2Þ
ðk0 Þ ¼
dk
0
For example, if the derivative is negative, as in our application, then ðpðtÞ; 0Þ is stable for k > k0
and unstable for k < k0 , at least for k k0 .
We seek a periodic solution nearby ðx; yÞ ¼ ðpðtÞ; 0Þ for k k0 , but with y > 0. Our method
follows that described in [42, Chapter 1]. Of course, the period of this periodic solution will vary
so it is useful to scale the time variable s ¼ xt so the system becomes
x_x ¼ F ðx; yÞ;
x_y ¼ yGðx; y; kÞ;
ðA:3Þ
where x_ denotes the derivative with respect to s. Now we seek a periodic solution of fixed period T
in the s variable
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
131
xðs; Þ ¼ pðsÞ þ p1 ðsÞ þ ;
yðs; Þ ¼ y1 ðsÞ þ ;
xðÞ ¼ 1 þ x1 þ ;
kðÞ ¼ k0 þ k1 þ ;
where is a small parameter. We demand that yi , pi are T -periodic satisfying
0 ¼ pi ð0Þ p_ ð0Þ;
1 ¼ y1 ð0Þ;
yi ð0Þ ¼ 0; i P 2:
ðA:4Þ
These will be motivated further below. The second set of conditions imply that yð0Þ ¼ y1 ð0Þ ¼ .
Inserting this expansion into (A.3), we find, after some computation, that equality of the order
1 terms on both sides requires that
p_ 1 AðsÞp1 ¼ Fy ðp; 0Þy1 x1 p_ ;
y_ 1 GðsÞy1 ¼ 0
ðA:5Þ
hold. We
R T abuse notation slightly by setting GðsÞ GðpðsÞ; 0; k0 Þ.
As 0 GðsÞ ds ¼ 0, the second equation has a subspace of T -periodic solutions spanned by
Z s
GðsÞ ds :
y1 ¼ exp
0
In order to solve the first equation we recall the Fredholm Alternative (see e.g. Lemma 1.1 of
[43, Chapter IV]). The equation
u_ AðsÞu ¼ f ðsÞ
has a T -periodic solution corresponding to continuous, T -periodic f if and only if
Z T
f ðsÞ vðsÞ ds
0¼
ðA:6Þ
ðA:7Þ
0
holds, where v is the unique (up to scalar multiple) non-trivial T -periodic solution of the adjoint
equation
v_ þ AðsÞT v ¼ 0:
RT
It is known that 0 p_ v ds 6¼ 0. In fact, since p_ v is independent of t, we may choose v such that
p_ v 1. Furthermore, if (A.7) holds, then there is a one-parameter family of T -periodic solutions
of (A.6) given by
u ¼ u0 þ cp_ ;
where u0 is a particular T -periodic solution and c is an arbitrary constant. Conditions (A.4) are
chosen to fix constant c uniquely such that uð0Þ p_ ð0Þ ¼ 0 ¼ u0 ð0Þ p_ ð0Þ þ cjp_ ð0Þj2 .
Hence, the first equation is solvable for T -periodic p1 if and only if
0¼
Z
T
½Fy ðp; 0Þy1 x1 p_ v ds:
0
132
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
This determines x1 ,
RT
Fy ðp; 0Þy1 v ds
x1 ¼ 0 R T
p_ v ds
0
and T -periodic p1 satisfying (A.4).
Equality of the order 2 terms requires that
p_ 2 AðsÞp2 ¼ Fy ðpðtÞ; 0Þy2 x1 p_ 1 x2 p_ þ Qðp1 ; y1 Þ;
y_ 2 GðsÞy2 ¼ x1 y_ 1 þ y1 ½Gx ðp; 0; k0 Þp1 þ Gy ðp; 0; k0 Þy1 þ Gk ðp; 0; k0 Þk1 hold. The term Qðp1 ; y1 Þ is a quadratic form whose exact nature we do not require.
As our primary interest is in determining k1 , we need only consider the second of these
equations. By the Fredholm alternative, now with 1=y1 ðsÞ as the periodic solution of the adjoint
equation to the second equation of (A.5), the second equation is solvable for T -periodic y2 if and
only if
Z T
½x1 GðsÞ þ Gx ðp; 0; k0 Þp1 þ Gy ðp; 0; k0 Þy1 þ Gk ðp; 0; k0 Þk1 ds:
0¼
0
This determines k1 as
RT
½Gx ðp; 0; k0 Þp1 þ Gy ðp; 0; k0 Þy1 ds
;
k1 ¼ 0
RT
Gk ðp; 0; k0 Þ ds
0
ðA:8Þ
where we have used again that G has zero mean value. The denominator does not vanish by (A.2).
Solution y2 is determined up to an arbitrary multiple of y1 which spans the null space. Obviously,
this arbitrary constant is determined by our normalization (A.4) that y2 ð0Þ ¼ 0.
In principle, the first equation may also be solved for T -periodic p2 for a unique choice of x2 . In
fact, the expansion can be continued indefinitely with pi , yi , xi , ki uniquely determined by (A.4).
It is well-known (e.g. [40, Theorem 13.8]) that the stability of the bifurcating periodic solution
depends on the sign of k1 and the sign of dg=dkðk0 Þ in (A.2). For simplicity, we describe the
relation when the latter is negative as is the case in our application. In that case, if k1 < 0, the
periodic solution satisfies y > 0 for > 0, implying that k < k0 and it is asymptotically stable. If
k1 > 0, the periodic solution satisfies y > 0 for k > k0 and is unstable.
Observe that the numerator in (A.8) can be written as
Z T
Z T
d Gðxðs; Þ; yðs; Þ; k0 Þ ds ¼
½Gx ðp; 0; k0 Þp1 þ Gy ðp; 0; k0 Þy1 ds:
d ¼0 0
0
In biological terms, it measures the change in the specific growth rate of the introduced species y
in the direction of the bifurcating branch, when the parameter k is fixed at threshold k0 .
In our particular problem, x ¼ ðN1 ; N2 ; N3 Þ, y ¼ N4 , k ¼ K14 , and
Gðx; y; kÞ ¼ f14 ðR1 Þ D;
T
F ðx; yÞ ¼ ðx1 ½f ðR1 Þ D; x2 ½f ðR2 Þ D; x3 ½f ðR3 Þ DÞ ;
P
where Rj ¼ S0 3i¼1 cji xi cj4 y. Thus,
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
133
Gx ¼ f140 ðR1 Þðc11 ; c12 ; c13 Þ;
Gy ¼ f140 ðR1 Þc14 ;
R1
;
Gk ¼ r4
ðK þ R1 Þ2
Fy ðp; 0Þ ¼ ðp1 f 0 ðR1 Þc14 ; p2 f 0 ðR2 Þc24 ; p3 f 0 ðR3 Þc34 ÞT :
Consequently,
RT 0
f ðR1 Þ½c14 y1 þ ðc11 ; c12 ; c13 Þp1 ds
k1 ¼ 0 14
;
dg
ðk0 Þ
dk
Z T
y1 ½p1 v1 f 0 ðR1 Þc14 þ p2 v2 f 0 ðR2 Þc24 þ p3 v3 f 0 ðR3 Þc34 ds;
x1 ¼ ðA:9Þ
0
and
2
3
p1 f 0 ðR1 Þc14
p_ 1 ¼ AðsÞp1 y1 4 p2 f 0 ðR2 Þc24 5 x1 p_ ;
p3 f 0 ðR3 Þc34
P
e 1; N
e 2; N
e 3 Þ is our periodic
where Rj ¼ S0 3i¼1 cji pi and A ¼ Fx ðp; 0Þ. Recall that ðp1 ; p2 ; p3 Þ ¼ ð N
solution given by Proposition 2. As
Z T
dg
dq d R1 ðs; 0Þ
r4
D ds < 0
ðk0 Þ ¼ T
ðK Þ ¼
dk
dK
dK14 K14 ¼K 0
K14 þ R1 ðs; 0Þ
by (4.7), the denominator of (A.9) is negative. The numerator can be written as
Z T
d R1 ðs; Þ
r
D
ds;
4 d ¼0 0
K þ R1 ðs; Þ
P3
where R1 ðs; Þ ¼ S0 i¼1 c1i xi ðs; Þ c14 yðs; Þ.
As k1 is precisely j0 ð0Þ from the previous section, its sign is critical for stability of our bifurcating T -periodic solution. We had like to show that k1 < 0, implying stability for the positive
branch where K14 < K . We have cj4 , j ¼ 1; 2; 3, at our disposal. The equations for x1 and p1 imply
that
p1 ¼ c14 q1 þ c24 q2 þ c34 q3 ;
where q1 is the unique T -periodic solution of
2 1 0
3
p f ðR1 Þ
5 x1 p_
q_ 1 ¼ AðsÞq1 y1 4
0
1
0
RT
_ ð0Þ ¼ 0 and x11 ¼ 0 y 1 p1 v1 f 0 ðR1 Þ ds. Terms qj and xj1 satisfy similar equations
satisfying qP
1 ð0Þ p
and x1 ¼ 3j¼1 cj4 xj1 . It is evident then that we may as well take c14 ¼ 1 for then (A.9) takes the
form
134
B. Li, H.L. Smith / Mathematical Biosciences 184 (2003) 115–135
RT
k1 ¼
0
f140 ðR1 Þðy1 þ c1 ½q1 þ c24 q2 þ c34 q3 Þ ds
dg
ðk0 Þ
dk
;
ðA:10Þ
where c1 ¼ ðc11 ; c12 ; c13 Þ. Thus, k1 is an affine function of the positive parameters c24 and c34 . It is
easy to see that the qj cannot vanish identically.
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