CHIN. PHYS. LETT. Vol. 27, No. 1 (2010) 018701
Competitive Exclusion Principle Revised by Noise
*
LIU Yong-Jiang(刘永姜)** , WAN GAi-Ling(王爱玲), WANG Biao(王彪), LIU Zhao-Hua(刘兆华)
Key Laboratory for AMT Shanxi, North University of China, Taiyuan 030051
(Received 7 September 2009)
A fundamental tenet in theoretical ecology is the competitive exclusion principle. Two competitive species for a
limited resource cannot coexist and thus one of the species will be driven to extinction. However, we show that
noise can revise this principle in a resonance-like manner, which makes coherence resonance in the system. Our
obtained results well enrich the findings in the interaction of populations in ecosystems, which may explain some
filed observations in the real world.
PACS: 87. 23. Cc, 82. 40. Ck, 05. 45. Pq
DOI: 10.1088/0256-307X/27/1/018701
Competitive exclusion principle is one of most fundamental rules in ecosystems. More specifically, two
species competing for a resource cannot coexist and
one of the species may disappear. This principle is
supported by many mathematical models, in which
the Lotka–Volterra model for two competing species
is the most famous.
On the other hand, laboratory experiments play an
important role in supporting the competitive exclusion
principle in ecosystems. One series of laboratory studies was conducted by Park and Mertz using two species
of flour beetles (of the genus Tribolium).[1−5] Park and
his collaborators used a difference equation model in
these studies, rather than the Lotka–Volterra differential equations.[6] They did not give a mathematical
analysis of their model in detail, but they worked under the assumption that the model possesses the same
dynamic scenarios as the Lotka–Volterra model.
However, noise has long been considered an important factor that should be included in the modeling of ecological systems. Holling emphasized the
importance of environmental variability in ecological
dynamics and resilience.[7] Inherent uncertainties in an
ecological system (e.g. varying rainfall or the nutrient
inputs) widely exist. Additionally, there are considerable anthropogenic disturbances that exacerbate the
uncertainty in the way an ecosystem responds.[8] Besides random fluctuations, there are also seasonal variations in different ecological parameters.[9,10] Some
people consider the effects of seasonality on the phytoplankton enrichment in lakes.[9] And some groups have
investigated the effect of noise in ecologically relevant
models. The influence of noise in the input levels to a
lake, determined by maximizing benefits or utility on
optimal policy choices, has been well investigated.[10]
Thus, when studying the interactions of different populations, noise can not be ignored.
We firstly consider a classical Lotka–Volterra
model of two-species competition[11,12] defined by the
equations
𝑑𝑢
= 𝑟𝑢(1 − 𝑢 − 𝛼𝑣),
(1a)
𝑑𝑡
𝑑𝑣
= 𝑟𝑣(1 − 𝛼𝑢 − 𝑣),
(1b)
𝑑𝑡
where 𝑢 and 𝑣 are the population densities, 𝑟 is the
growth rate of 𝑢 and 𝑣, and 𝛼 accounts for the interactions among the species. This Lotka–Volterra model
has been widely studied, and it is well known that, for
𝛼 < 1, both species are present, but for 𝛼 > 1, exclusion takes place through a symmetry-breaking bifurcation and one of the species is eliminated.[12]
Combining with noise term, we have the system as
follows:
𝑑𝑢
= 𝑟𝑢(1 − 𝑢 − 𝛼𝑣),
(2a)
𝑑𝑡
𝑑𝑣
= 𝑟𝑣(1 − 𝛼𝑢 − 𝑣) + 𝜂(𝑟, 𝑡).
(2b)
𝑑𝑡
In Eq. (2), the stochastic factors are taken into account as the term, 𝜂(𝑟, 𝑡), which is obtained from microscopic interaction in the space[13−15] where the typical white noise will emerge. Recently, colored noise
and white noise have both been used in describing
ecological evolution.[16−18] White noise is the limiting case of colored noise, so we consider the more
general case-colored noise. Due to the coupling, the
noise in Eq. (2b) will have an influence on Eq. (2a)
as well. Hence, we only need consider the noise presented in one of Eqs. (2a) and (2b) (here, we consider
noise presented in Eq. (2b)). The noise term 𝜂(𝑟, 𝑡) is
introduced additively in space and time, which is the
Ornstein–Uhlenbech process that obeys the stochastic
partial differential equation[19,20]
1
1
𝜕𝜂(𝑟, 𝑡)
= − 𝜂(𝑟, 𝑡) + 𝜉(𝑟, 𝑡),
𝜕𝑡
𝜏
𝜏
(3)
where 𝜉(𝑟, 𝑡) is a Gaussian white noise with zero mean
and correlation,
⟨𝜉(𝑟, 𝑡)⟩ = 0,
(4a)
* Supported by the PhD Research Fund of the Ministry of Education of China under Grant No 20050204JJ, and the Natural
Science Fund of the Shanxi Province (20040202JJ).
** Email: [email protected]
c 2010 Chinese Physical Society and IOP Publishing Ltd
○
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CHIN. PHYS. LETT. Vol. 27, No. 1 (2010) 018701
The colored noise 𝜂(𝑟, 𝑡), which is temporally correlated and white in space, satisfies
⟨𝜂(𝑟, 𝑡)𝜂(𝑟′ , 𝑡′ )⟩ =
(︁ |𝑡 − 𝑡′ | )︁
𝜀
𝛿(𝑟 − 𝑟′ ),
exp −
𝜏
𝜏
(5)
where 𝜏 controls the temporal correlation, and 𝜀 measures the noise intensity.
We will systematically analyze effects of nonzero
𝜎 and 𝜏 on the competitive model for 𝛼 > 1, with
the aim of reporting noise-induced transitions in a
resonance-like manner depending on 𝜎 and 𝜏 , thus evidencing coherence resonance in the studied system
and revising the competitive exclusion principle.
The Runge–Kutta method is used for the deterministic part in Eq. (2). The stochastic partial differential Eq. (3) is integrated numerically by applying
the Euler method. Several different discrete methods
(simple Euler and Runge–Kutta) have been checked,
and the results are almost the same. The code is implemented in Matlab 7.3 and the 𝑤𝑔𝑛 function is used
for the main numerical integration.
In Figs. 1 and 2, we show the time series of the two
populations without noise. One can see from that the
population 𝑢 always tends to extinction and 𝑣 persist.
That is to say, if the initial value of the population
is smaller, it will disappear regardless of the parameter values, which is called the competitive exclusion
principle.
1.0
by nonzero values of 𝜎 in a resonant manner. More
specifically, small 𝜎 are able to sustain only small density of 𝑢. As 𝜎 increases, the density of 𝑢 will reach the
maximal value at a certain value of 𝜎. For the larger
𝜎, the intensity of 𝑢 is small. In other words, results
presented in Fig. 3 thus indicate a typical coherence
resonance scenario.[21−24]
1.0
(a)
0.8
u
(4b)
0.6
0.4
0.2
0.0
1.0
(b)
0.8
v
⟨𝜉(𝑟, 𝑡)𝜉(𝑟′ , 𝑡′ )⟩ = 2𝜀𝛿(𝑟 − 𝑟′ )𝛿(𝑡 − 𝑡′ ).
0.6
0.4
0.2
0
5
10
15
Time t
20
25
30
Fig. 2. Time series of two populations. Parameter values
are taken as 𝑟 = 0.5 and 𝛼 = 5, and the initial conditions
are 𝑢(0) = 0.9 and 𝑣(0) = 1.
0.8
(a)
0.6
0.8
0.4
u
u
0.6
0.4
0.2
0.2
0
10
0
0
(b)
0.4
0.6
0.8
Fig. 3. An illustration of intensity of 𝑢 for different values of noise intensity. Note that, for the no-noise model,
the intensity of population 𝑢 will tends to zero. Coherence
resonance occurs when combined with noise. Here 𝑟 = 0.5,
𝛼 = 1, 𝜏 = 6, and the initial conditions are 𝑢(0) = 1 and
𝑣(0) = 10.
v
6
4
2
0
0
0.2
Noise intensity σ
8
5
10
Time t
15
20
Fig. 1. Time series of two populations. Parameter values
are taken as 𝑟 = 0.5 and 𝛼 = 1, and the initial conditions
are 𝑢(0) = 1 and 𝑣(0) = 10.
Now, we consider the effect of noise on the intensity of 𝑢. We run the simulations until the intensity
of 𝑢 reaches a fixed values or until it shows a behavior
that does not seem to change its characteristics any
longer. The results presented in the panels of Fig. 3
clearly show that the density of 𝑢 will larger than zero
The role of temporal correlation 𝜏 of the colored
noise is also significant including its intensity in controlling the intensity of the population 𝑢. Now, it is
natural to ask what the consequence of the temporal
correlation of the colored noise is. In order to well understand that mechanism, we also perform a series of
simulations for fixed noise intensity 𝜀, which is shown
in Fig. 4. We choose a set of 25 parameters for numerical simulations. We find that, when 𝜏 < 3.5 or
𝜏 > 10, noise has no effect on the intensity of 𝑢. How-
018701-2
CHIN. PHYS. LETT. Vol. 27, No. 1 (2010) 018701
ever, when 𝜏 ∈ (3.5, 10), the intensity of 𝑢 is more
than zero, which is induced by noise, and reaches the
maximum value at 𝜏 ≈ 6.
meaningful suggestions given.
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Fig. 4. An illustration of intensity of 𝑢 for different values
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In summary, we have shown that color noise introduced in the system of a competitive model can revise
the competitive exclusion principle in a resonant manner depending on the intensity of noise and temporal correlation. The reported phenomenon is identical
as coherence resonance reported previously in temporal and spatially extended dynamical systems.[21,22]
Ecologically speaking, noise effect should not be neglected when considering the interaction of the populations, which may have an effect on the dynamics
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We are grateful to the anonymous referees for the
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