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Simulation of Competitive Mechanism for Limited Resources by Two-Species Meta-Populations in Two Patches
Available online at www.sciencedirect.com
Procedia Environmental Sciences 2 (2010) 1083–1106
International Society for Environmental Information Sciences 2010 Annual Conference (ISEIS)
Simulation of Competitive Mechanism for Limited Resources by Two-Species Meta-Populations in Two Patches Liu Kunkuna, Li Chunhuia,*, Pang Aipinga, Cai Yanpengb a
Ministry of Education Key Laboratory of water-sediment Science, School of Environment, Beijing Normal University, Beijing, 100875, China b Faculty of Engineering, Dalhousie University, Halifax, Nova Scotia, B3J 1Z1, Canada
Abstract In this study, according to the model of the meta-population of two species competition for two kinds of linear-growth resources which is developed by Liang R.J., STELLA was used to simulate the competition mechanisms of two consumer species in different patches, utilizing the limited resources. The dynamic process was simulated under different situations: When the two species’ migration rates are o, which is a particular case that population compete for resource in one patch, the results are accord with R* Principle; When the two species’ migration rates are different, the coexistence of the two populations is possible in the patch ; When the two species’ individual average resource consumption=n rates are different, there exists obvious competing exclusion effect; When the resource growth rates in two patches are different, population density is higher in the patch with a higher resource growth rate, and populations in both patches are apt to coexist with a higher value of q and the distribution of population is partially influenced by the difference between the patches. Because there are still many parameters related with different aspects, it is better to use STELLA to simulate the dynamicity of both the resource and the population by changing the value of any parameter in the model conveniently.
© 2010 Published by Elsevier Ltd. Keywords:meta-population, competing for resource, STELLA; simulation
1. Introduction Competition is one of the main factors to create biological morphology, life cycle, biological community structure and dynamics, and it is also one of the key ecological processes to determine the ecosystem structure and function [1] .Mechanisms of species coexistence of competition and exclusion are the important research topics in Ecology. Model is one of the effective means for human to understand and solve practical problems, and it is widely used in different fields of science. There into, STELLA, which is one of system dynamics modeling tools with powerful modeling environment and simple operation mode, is highly respected by foreign researchers but is extremely rare in the domestic ecology research and applications where differential (difference) equation (s) is a basic form for describing the dynamic behavior. Previous study mainly used STELLA only for simulation of population growth in
* Corresponding author. Tel.: +86-10-58802756; fax: +86-10-58802756. E-mail address: [email protected]
1878-0296 © 2010 Published by Elsevier doi:10.1016/j.proenv.2010.10.120
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ecology. So it is very important that STELLA is applied in other areas to address such issues especially involve the analysis of a complex dynamic system’s response to different conditions like the complex species competition mechanism. The model here is general adaptive, which provides a theoretical reference for the conservation of biological diversity and the improvement of the resource utilization efficiency. Some meaningful exploration has been taken by using Lotka-Volterra Model [2], which is very important in the development history in Ecology. According to the different initial conditions, the results are also different: competition exclusion and stable coexistence[3]. On the basis of the model, a lot of species competition models in the related literature have emerged since the 1920s to investigate the interaction among species according to the competition coefficient[4].. After 1970s, the resource consumption-based resource competition theory quietly raised. During the period, people have designed varies kinds of consumer-recourse models [3,5-13].The key point in the theory is that it considers the dynamicity of both the resource and the consumer population competing for resource. Therefore, it is a mechanistic model which is to define the intra-specific competition and the inter-species competition according how the resource available extent is influenced by the species consumers. The theory first developed on the basis of Monod and Droop Equation which is based on chemostat model [14].To the 1980s, after Tilman’s research [12,15-16], a new resource competing model-based method emerged to predict the result of species competing according to the resource demand of the competing species. The main theory is still the competing exclusion theory. The theory figures that when multiple species compete for only one kind of resource, the key parameter in the models is R* which is the critical concentration of resource when the population growth rate equals the mortality rate. That means in the equilibrium condition without disturbance, the species with low R* will compete and replace all the other species which is the named R*-Principle [3-4]. Recently, the landscape pattern fragmentation due to irrational economic activities by human has caused a large number of species inhabit patchy habitats. Therefore, the research in meta-population theory developed fast and a lot of mathematical models emerged [12,15,17-21]. Some focus on two kinds of species competing for one or two kinds of resources in a patch, and some focus on two kinds of species competing for the same resource in two patches. Most of these have more discussions in equilibrium boundary condition than the population dynamic predict. Liang has established a two species met-population’s competing model to two kinds of exponential growth resources [22]. But he didn’t utilize any ecological modeling tools but the numerical simulation. STELLA is a user-friendly and commercial software package for building a dynamic modeling system. It uses an iconographic interface to facilitate construction of dynamic system models. The key features of STELLA consist of four tools: Stocks, Flows, Converters, Connectors. STELLA offers a practical way to dynamically visualize and communicate how complex systems and ideas really work. It has been widely used in biological, ecological, and environmental sciences [23-26.].An elaborate description of STELLA package can be found in Isee System (2006). It is a modeling tool for building a dynamic modeling system by creating a pictorial diagram of a system and then assigning the appropriate values and mathematical functions to the system [27]. The objective of this study is to simulate the dynamic mechanism of the meta-population of two species competing for resource in two patches by using STELLA. Here, I consider that since the dynamicity of both the resource and the consumer population competing for resource is the key point of the model, that is exactly in line with the Stella’ essence. Besides, by using Stella, it is easy to adjust the parameter to analyze the change of the result and to improve the model itself.
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2. Methodology 2.1. Overview of the study case First, MacArthur established the famous resource competing model which describes that species are distributed in a patch and make use of the resource [5, 28]. In 2004, Abrams raised the two population-two patches resource competing model [29]. They assumed that resource growth accords with the exponential growth or Logistic growth, and the scattered individuals separate equally in other patches. They focused on the two population’s coexistence conditions, but the population dynamic prediction research seems inadequate. On the basis of it, here comes a new model by Liang[22], which simulates two-species meta-population’ competitive mechanism for linear growth resources in two patches. The two species here can be different types of population and resources can also include inorganic (such as various nutrients, etc.) and organic (such as microbes, plants and prey, etc.). 2.2. Model formulation 2.2.1. Model assumptions Here, we need to re-emphasize the conditions in the model: 1. Two kinds of species live on two patches to compete for resource. 2. Resource growth accords with the exponential growth. 3. Species growth is restricted by resource availability and their densities. 4. Consumers move from each patch in the speed which is proportional to their densities in that patch. 2.2.2. Model functions On the basis of the conditions and assumptions above, the model should be:
dRa ra [(1 c1 ) N1a c1 N1b ](k1a Ra ) [(1 c2 ) N 2 a c2 N 2b ](k 2 a Ra ) dt dN1a p1[(1 c1 ) N1a c1 N1b ](k1a Ra ) / v1a m1[(1 c1 ) N1a c1 N1b ] dt dN 2 a p 2 [(1 c2 ) N 2 a c2 N 2b ](k 2 a Ra ) / v2 a m2 [(1 c2 ) N 2 a c2 N 2b ] dt dRb rb [(1 c1 ) N1b c1 N1a ](k1b Rb ) [(1 c2 ) N 2b c2 N 2 a ](k 2b Rb ) dt dN1b p1 [(1 c1 ) N1b c1 N1a ](k1b Rb ) / v1b m1[(1 c1 ) N1b c1 N1a ] dt dN 2b p 2 [(1 c2 ) N 2b c2 N 2 a ](k 2b Rb ) / v2b m2 [(1 c2 ) N 2b c2 N 2 a ] dt
(1) (2) (3) (4) (5) (6)
where Rj (j=a, b): the resource density in patch. Nij (i=1, 2; j=a,b): the consumer species i density in patch . rj (j=a, b): the resource growth rate in patch . ci (i=1, 2):the population’s migration force. kij (i=1, 2; j=a, b): the individual average resource consumption rate or the resource’ support capacity to the population or the population’s invasion rate. vij (i=1, 2; j=a, b): the average resource demand for the individual to survive, or the individual’s transformation efficiency of resource(1/v). pi (i=1, 2):the population’s reproduction rate. mi (i=1, 2):the population’s death rate.
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Kij * Rj /vij: the resource’s bearing capacity of the population.<1. According to the assumption by Abrams, the intrinsic growth rate of R is r, q is for the patch parameter .The resource intrinsic growth rate in patch a and b is:
ra ® ¯rb
r (q 1) / q r (q 1) / q
(7) (8)
Among the parameters, the state variables contain: Rj (j=a, b): the resource density in plague. Nij (i=1, 2; j=a, b): the density of the consumers species i in patch j. They change over time and can be expressed by:
Ra (t ) Rb (t )
Ra (t dt ) \ R a J R a Rb (t dt ) \ R b J R b
N1a (t )
N1a (t dt ) \ N 1a J N1a1
N 2 a (t )
N 2a (t dt ) \ N 2a J N 2 a
(9) (10) (11) (12)
N1b (t )
N1b (t dt ) \ N 1b J N1b
(13)
N 2b (t )
N 2b (t dt ) \ N 2b J N 2b
(14)
2.3. Model structure To establish a simulation model with STELLA, the first step is to develop a conceptual model, a basic structure that can capture the processes described before. As shown in Fig.1, the rectangles are stocks that graphically represent the resource density in each patch, or the consumer species density in each patch. The flow symbols (represented by double lines with arrows and switches) represent the rates of flow into or out of the stocks. The other variables are converters (represented by empty circles) that denote the rules or conditions controlling the stocks and flows through the use of connectors (represented by single lines with arrows).
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(a) The original one
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(b) The modified one(more easily understandable) Where the cap represents Kij * Rj /vij: the resource’s bearing capacity of the population. Fig.1. Conceptual model of meta-population of two species competing for resource in two patches
In the figure of resource density, the blue and red curves represent the density of Ra and Rb, and in the figure of community density, the blue, red, purple, green curves represent the density of N1a, N1b, N2a, N2b respectively. The following figures are the same. The growth makes the increase of the resource density in each patch, and the decrease of the density is decided by the amount eaten by the two consumer species that contain the species in this patch and those moved from the other patch and not contain those moved out of this patch. As to the species density, similarly, the birth makes the increase of the consumer density, and the decrease is decided by the death rate. Besides, each element is related with many factors such as the resource’s bearing capacity of the population, the population’s migration force and so on. The time step is one month, and the simulation time is 700 months. According to Liang’s chosen in parameters, the values are fixed in Table 1: Table 1. The values of the parameters in the model parameter
parameter's meaning
value
parameter
parameter's meaning
value
c1
population 1’s migration force
0.05
c2
population 2’s migration force
0.05
k1a
population 1's individual average resource consumption rate in patch a
0.3
k2a
population 2's individual average resource consumption rate in patch a
0.3
k1b
population 1's individual average resource consumption rate in patch b
0.3
k2b
population 2's individual average resource consumption rate in patch b
0.3
v1a
population 1's average resource demand for the individual to survive in patch a
0.5
v2a
population 2' s average resource demand for the individual to survive in patch a
1
v1b
population 1's average resource demand for the individual to survive in patch b
0.5
v2b
population 2's average resource demand for the individual to survive in patch b
1
p1
population 1's reproduction rate
0.1
p2
population 2's reproduction rate
0.1
m1
population 1's death rate
0.02
m2
population 2's death rate
0.02 0.036
q
the patch parameter
2
r
the resource' intrinsic growth rate of
ra
the resource growth rate in patch a
0.054
rb
the resource growth rate in patch b
0.018
initial conditions parameter
parameter's meaning
value
parameter
parameter's meaning
value
Ra (0)
initial resource density in patch a
0.2
Rb(0)
initial resource density in patch b
0.2
N1a(0)
the consumer species 1's initial density in patch a
0.05
N2a (0)
the consumer species 2's initial density in patch a
0.2
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
the consumer species 1's initial density in patch b
N1b(0)
0.05
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the consumer species 2's initial density in patch b
N2b (0)
0.2
Remark: The values here can be changed rationally to examine the response of the model in different conditions or for the different purposes.
2.4. Model calibration Because the validation of the resource competing model is limited to zooplankton, and it is carried under the stable conditions in laboratory, besides, it doesn’t have the abundant historical data like the climate prediction; it can not be validated by the data on the past. At the same time, it also can not be tested objectively by the fact in the short-term future like the weather and the earthquake. Therefore, the model here is not easy to be validated by the real data, but the related experiment results can be considered for reference, like R*-principle or Liang’s results by numerical simulation. 2.5. Model verification 2.5.1. Long-term stability: Run the model with values given before and set the simulation time 600 months and 1200 months separately. The simulation results are shown in Fig.2 and Fig.3.
Resource density
1: Ra
2: Rb
1
0.75 1
0.5
1
1
2
0.25
150
0
1: N1a 1 Population density
2 2
1
300 Time (months)
2: N1b
2
450
3: N2a
600
4: N2b
0.75 0.5
1
1 1
0.25
3 1
0
2
2
4
150
2
2 3
4
300 Time (months)
3
4
3
450
4
600
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Fig.2. Dynamic changes of resource and population in both patches in 600 months
1: Ra
2: Rb
Resource density
1 0.75 1
0.5 2 1
1
2
1
2
0.25
0
1: N1a
Population density
2
300
600 Time (months)
2: N1b
1
900
3: N2a
1200
4: N2b
0.75 1
1
0.5
1
0.25 1
0
2
2
2
3 4
3
300
4
600 Time (months)
2 3
4
3
900
4
1200
Fig.3. Dynamic changes of resource and population in both patches in 1200 months
Compare the simulative results in Fig.2 and Fig.3, we can figure that the number of resource and population in different time series both reach to a stable level finally. So the model has a relatively good stabilization, namely the long-term stability. 2.5.2. Model rationality: In general, when the parameters are set, patch a and b are different in the resource growth rate, which accords with that in patch a, resource reaches to a higher level than in patch b. Besides, in each patch, population 1 gets the upper hand at last and excludes population 2, which accords with that the individual in population 1 has a higher transformation efficiency of resource than population 2. That is to say, at a basic level, the model reacts as what we
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expect. But of course, there is something else in the model or its reaction to disturbance that we can’t comprehend. Further experiments and theoretical validation are needed. 2.5.3. Sensitive analysis: The model contains 6 state variables and a lot of parameters. The sensitive analysis has figured that the change of every parameter will result the change of at least one state variable, various in sensitive. Take the sensitive analysis of q for an example here. q stands for the difference between two patches. Take sensitive analysis with STELLA by choosing 4 values in its threshold value˄can’t be attained because of the material limitation, but can be estimated approximately through the process of sensitive analysis. According to the value Liang has chosen, the values of q here and the corresponding values of N1a and the results are shown in Fig.4.
Population 1 density in patch a
N1a: 1 - 2 - 3 - 4 1 0.75 1
1
0.5
1
0.25 3 1
2 2
3
4
2 3
4
4
4
2
175
0
350
525
700
Time (months) Fig.4. Population 1 density in patch a when q=1.5, 2.67, 3.83, 5.
According to the sensitive analysis equation: S [wN1a / N1a ] /[ w q / q] , a mended equation is given 3 as: S [( N1ai 1 N1ai ) / N1ai ] /[( qi 1 qi ) / qi ] /3
¦ i 1
Here, take the average value of N1a during the last 100 months as the value of N1a in the equation. Table 2. The average values of N1a during the last 100 months q
1.50
2.67
3.83
5.00
N1a
0.62
0.51
0.46
0.44
S= (0.51-0.62)/0.62/[(2.67-1.50)/1.50]+(0.46-0.51)/0.51/[(3.83-2.67)/2.67]+(0.44-0.46)/0.46/[(5.00-3.83)/3.83]= -0.198 Likewise, as to N1b, S=0.836.
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3. Results and discussion 3.1 A particular case—resource competing in one patch Adjust the values of the parameters: c1=c2=0; Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.5; k1a=k1b=k2a=k2b=0.3; m1=m2=0.2; p1=p2=0.33; ra = rb=0.145(ra and rb are used here for convenience to control the resource growth rate to be equal in both patches.); v1a=v1b=0.345, v2a=v2b=0.72. ra and rb are used here for convenience to control the resource growth rate to be equal in both patches. The simulation results are shown in Fig.5.
Resource and population density
1: Ra
2: Rb
1
3: N1a
0.75
1
2
2
3
4: N1b
1
4
2
3
5: N2a
1
4
2
3
4
0.5 4 1
0.25
3
5
0
25
5
50 Time (months)
5
75
5
100
Fig.5. Resource competing in one patch (c1=c2=0)
c1=c2=0, which means that the two patches are independent, and the values of parameters except v in both patches are equal, so the resource and species’ dynamic in two patches are completely the same. (N2b is not in the figure, but it goes the same as N2a) There is a periodical vibration damp movement between resource and population 1. Because the consumer has a time lag effect to resource, the phase of species density is behind the resource density, presenting plus correlation because of the feed back effect between them. The situation here is a little like the prey model by Lotka-Volterra. Because the population density can be adjusted by itself at some level, and population 2 should be considered because it is also the consumer for the same resource, the density vibration amplitude decreases rapidly and the densities of resource and population 1 reach to a balance quickly. The cycle time changes when the values of parameters change. Population 2 die out over period of time despite whether the change of parameters values because its transformation efficiency of resource (v) is lower than population 1, which accords with R*-Principle that species with the lowest R* will eliminate other species in balanced state and coexisting is impossible. Even the original population quantity of species 2 is larger than 1, the results are the same, which is shown in Fig. 6.
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Population density
1: N1a
2: N1b
1
1093
4: N2b
3: N2a 1 2
0.75
1
2
1
2
0.5 0.25 1
2
3
4 3
0
25
4
50 Time (months)
3
4
3
75
4
100
Fig.6. Resource competing in one patch (c1=c2=0, N1a=N1b
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Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
2 1
2
1
2
0.25 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1
0.5
1
0.25
1 2
3
0
3
175
2
2
2 4
4
350 Time (months) (a) c1=c2=0
3
4
3
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
1095
2: Rb
1
0.75 0.5
1 1
1
2
1
2
2
0.25 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1 1
0.5 0.25
1 2
3 4
0
2
175
3
4
2
350 Time (months) (b) c1=c2=0.15
3
2
4
525
3
4
700
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Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
0.25
2
2
0
Population density
1: N1a
1 2
175
2
350 Time (months)
2: N1b
1
1
525
3: N2a
4: N2b 1
1
0.75
700
1
0.5 0.25
1 2
3 3
4
0
2
175
4
2
350 Time (months)
3
4
2
525
3
4
700
(c) c1=c2=0.3 Fig.7. Resource competing in two patches (v1
We can figure that, because ra>rb, the resource quantity Ra grows quickly at first, reaches to the maximum value ,and then decreases because of the increase of populations before a new balance through dynamical evolution. Population 1 density increases fast and reaches to a stable state because of the higher transformation efficiency, becoming the dominant species. In contrast, Population 2 density decreases because of population 1’s competing exclusion effect and dies out at last. In patch b, the resource and population densities are both lower because the lower resource growth rate. When population migration rate is low(c=0), the situations in patch 1 and 2 are similar. Population 1 is the dominant species while population 2 dies out, though they are not in the same level. But as the value of c increases, the population 1 density increases to the saturation fast, and in patch b, population 1 is no longer the dominant species and dies out, too. This is not only because the resource growth rate is lower, but also because population density in patch a is higher and more individuals migrate to patch b, which has a strong competing exclusion effect on all the species in patch 2, reducing the species diversity. As to resource density, because of the invasion of species in patch a, resource density in patch b reaches to a smaller value in the balanced state.
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(2). c1=0.01, c2=0.45 When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.2; k1a=k1b=k2a=k2b=0.3; m1=m2=0.02; p1=p2=0.1; q=1.5; r=0.036,let v1=v2, and as the valve of v changes, the simulation results are:
Resource density
1: Ra
2: Rb
1
0.75 0.5
1
0.25
1 2
0
Population density
1: N1a
1 2
175
350 Time (months)
2: N1b
1
0.75
2
525
700
4: N2b
3: N2a
1
1
1
3
3
0.5
1
2
3
3
1
0.25
2 2
4
0
175
4
2
350 Time (months) (a) v1=v2=0.3
2
4
525
4
700
1098
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Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
0.25
1
1 2
2
2 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5 1
1 3
0.25
1
3
3
3
1 2 2
4
0
175
4
2
350 Time (months) (b) v1=v2=0.5
4
2
525
4
700
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Resource density
1: Ra
1099
2: Rb
1
0.75 0.5
1
1
1
2
2
2
0.25
1
2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5 1
0.25 1
1
3
3
1
3
3
2 2
4
0
175
4
2
350 Time (months)
4
2
525
4
700
(c) v1=v2=0.7 Fig.8. Resource competing in two patches (c1
We can figure that, coexistence of the two populations is possible in patch a. Population 1 density is higher than 2 possibly because of the following reasons: Population in patch a migrate to patch b and vise versa, but the net effect is that population migrate to patch b because of the higher population density (the higher resource growth rate). In addition, Population 2’s net migration is larger than population 1 because of its higher migration rate. Similarly in patch b, population 2’s density is smaller, but they all die out at last because of the resource limitation. As the transformation efficiency (1/v) decreases, population density decreases, too, causing the resource and population density reach at different levels. Besides, the difference between population 1 and 2 decreases as the value of v increases. As to resource density, it increases, but the extent in patch a is bigger than b possibly because of the net migration from patch a to patch b. It seems that population with lower migration rate has a stronger competitive power here, which doesn’t contradict with Xu’s conclusions [30]. The background is different. We assume that the initial values of N1a and N2a are equal, and there are two different patches in a habitat. But when there are many patches and population with
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higher migration rate first enters a patch in a habitat, it will eliminate the others, which is different to some extent in two patches under the conditions here. 3.2.2 Resource growth rate influence---resource characteristic (1). ra=0.055, rb=0.05 When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.5; k1a=k1b=k2a=k2b=0.3; m1=m2=0.02; p1=p2=0.1; v1a=v1b=0.5; v2a=v2b=1; c1=c2=0.1,the simulation results are:
Resource density
1: Ra
2: Rb
1
0.75 0.5 1
0.25
Population density
:
1
2
1
2
1
0
1: N1a
2
2
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1
0.5
1
1
2
2 2
2
0.25
3 4 3
0
175
Fig.9. Resource competing in two patches (ra≠rb, v1≠v2)
4
350 Time (months)
3
4
3
525
4
700
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Even with a little difference between the value of ra and rb, the population density in patch a is obvious higher, and as there is a larger gap between the values of ra and rb, the gap of population densities between patch a and patch b is larger. That is to say, population density is higher in the patch with a higher resource growth rate. (2). ra≠rb When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.2; k1a=k1b=k2a=k2b=0.3; v2a=v2b=1;c1=c2=0.1, let q=1,3,5 separately, the simulation results are:
Resource density
1: Ra
m1=m2=0.02;
p1=p2=0.1;
v1a=v1b=0.5;
2: Rb
1
0.75 1
0.5
1
1
1
0.25 2
2
0
Population density
1: N1a
2
175
2: N1b
1
2
350 Time (months)
525
4: N2b
3: N2a
1
1
0.75
700
1
0.5 0.25
1 3 2 4
0
2
175
3
4
2
350 Time (months) (a) q=1
3
4
2
525
3
4
700
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Resource density
1: Ra
2: Rb
1
0.75 0.5 1
1
0.25
0
1: N1a
2
1
2
1
2
2
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
Population density
:
0.75 1
1
0.5
1
0.25
1
2
3 4
0
2
2
2 3
175
4
350 Time (months) (b) q=3
3
4
3
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
2: Rb
1
0.75 0.5 1
1
2
2
1
2
1
2
0.25
0
1: N1a
Population density
1103
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5
1
1 1
0.25
1
2 3
0
2
2
2 4
3
175
4
350 Time (months)
3
4
3
525
4
700
(c) q=5 Fig.10. Resource competing in two patches (v1≠v2)
We can figure that, when q=1, ra=2r, rb=0, that is to say, resource grows well in patch a, but can’t grow in patch b. The growth situation in patch a accords with R*- Principle, but the two species can’t survive in patch b and die out at last because of the resource limitation. As the value of patch parameter (q) increases, the resource growth rate on both patches gradually approaches equilibrium. Population can survive in both patches and each accords with R*-Principle. Besides, the densities of both populations also approach equilibrium gradually. For better observation, we collate the simulation results when q=1, 2,3,4,5, which are shown in table 3:
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Table 3. The values of resource and populations of different situations q 1 2 3 4 5
Ra 0.34 0.33 0.33 0.33 0.33
Rb 0 0.33 0.33 0.33 0.33
N1a 0.79 0.58 0.51 0.47 0.45
N1b 0 0.14 0.21 0.25 0.27
N2a 0 0 0 0 0
N2b 0 0 0 0 0
From the values of the balanced state, we can figure that, when q=1, there is no population in patch b although there is a high population density of N1a in patch a, which causes the total population density in patch a is the highest in all the situations but poor in species diversity. As the value of q increases, the difference between population densities of both species diminishes, and the total density stays the same (0.72). It seems that population is distributed in two patches according to the value of patch parameter. In conclusion, populations in both patches are apt to coexist with a higher value of q. Furthermore, the total densities of both resource and population have little relationship with the original value of population density and the value of c, but are related with the values of m and v which represent the population characteristics. That is to say, the distribution of populations is partially influenced by the difference between the patches and the populations’ migration ability. 4. Conclusions x First point, the model of meta-population of two species competing for resource in two patches is a simplification of the meta-population of multi-population competing for resource in multi-patches. In this study, STELLA was used to simulate the competition mechanisms. The dynamic process was simulated under different situations: the two species’ migration rates are o; the two species’ migration rates are different; the two species’ individual average resource consumption rates are different; the resource growth rates in two patches are different. Because there are still many parameters related with different aspects, it is better to use STELLA to simulate the dynamicity of both the resource and the population by changing the value of any parameter in the model conveniently. x Second point, the results of simulation by adjusting some of the parameters are as follows. (1) Parameters related with the population When c1=c2=0, this is a particular case that population compete for resource in one patch. The results are accord with R* Principle. When v1≠v2, c1=c2, there exists obvious competing exclusion effect. When c1≠c2, v1=v2, the coexistence between population 1 and 2 is possible in patch a. (2) Parameters related with the resource Population density is higher in the patch with a higher resource growth rate. Populations in both patches are apt to coexist with a higher value of q and the distribution of population is partially influenced by the difference between the patches.
5. Research prospect (1). The model contains many complex parameters, and each influences the simulation results. That causes a difficult problem-how to analyze the simulation results by changing the values of the parameters and to find out the some laws. For example, if the value of parameter related to the same population in different patches is not equal, what the results should be? (2). Because the validation of the resource competing model is limited to zooplankton, and it is carried under the stable conditions in laboratory, besides, it can not be validated by the data on the past, and can not be tested objectively by the fact in the short-term future, the model here is not easy to be validated by the real data. Therefore,
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
more experiment data are needed to validate the model in order to better predict the population and resource development. (3). The model of the meta-population of multi-population competing for resource in multi-patches is more practical and should be developed. The key point is that the model should be simple but can reflect the actual situation. Acknowledgements This research was supported by the National Science Foundation (50709002) and National Water Pollution Control and Treatment Project of China (2008ZX07209-009).Thanks to Zhang Lixiao for his encouragement to study, innovate and valuable opinions, support. At last, thanks to all the co-supervisors who gave valuable opinions and support to the successful completion of this study. References [1]Fan, J.W., Zhong, H P.,et al. (2004). Study of plant competition. Grassland Science.13, 1-8 [2]Hanski l. (1999). Metapopulation Ecology. Oxford University Press [3]Zhang D Y, et al. (2000). Researches on Theoretical Ecology. Beijing: Higher Education Press and Springer Press. 151-191 [4]Brock W A, Xepapadeas A. (2002). Optimal ecosystem management when species compete for limiting resources. Journal of Environmental Economics and Management. 44, 189-220 [5]MacArther R H. (1972). Geographical Ecology. Harper and Row, New York [6]Steward F M, Levin B R. (1973). Partitioning of resources and the outcome of interspecific: a model and some general considerations. American Naturalist. 107, 171-196 [7]Greeney W J., Bella D A., Curt H C. , 1973. A theoretical approach to interspecific competition in phytoplankton communities. American Naturalist. 107, 405-425 [8]O’Brien N J. (1974). The dynamics of nutrient limitation of Phytoplankton algae, a model reconsidered. Ecology. 55, 134-141 [9]Leon J, Tumpson D. (1975). Competition between two species for two complementary or substitutable resources. Journal Theoretical Biolog. 50, 185-201 [10]Taylor P, Williams P. (1975). Theoretical studies on the coexistence of competing species under continuous flow conditions. Canadian Journal of Microbiology. 21, 96-98 [11]Tilman D. (1977). Resource competition between planktonic algae: an experimental and theoretical approach. Ecology. 58, 338-348 [12]Tilman D. (1982). Resource Competition and Community Structure [M]. Princeton University Press, Princeton, N J [13]Zhang D Y. (1991). Exploitation competition and coexistence of two plant populations on a number of growth-limiting resources. Ecological Modeling. 53, 263-279 [14]Monod J. (1950). La technique de culture continue: theories et applications. Ann. Inst. Pasteur. 79, 390-410 [15]Tilman D. (1988). Plant strategies and the dynamics and structure of plant communities. Princeton University Press, Princeton, N J [16]Pacala S, Tilman D. (1994). Limiting similarity in mechanistic and spatial models of plant competition in heterogeneous environments. Am. Nat. 143, 222-257 [17]Tilman D, Pacala S. (1993). The maintenance of species richness in plant communities. In: Ricklefs, R. E.& Schluter, D. eds. Species diversity in ecological communities; Historical and Geographic Perspectives University of Chicago Press. 13-25 [18]Amarasekare P. (2003). Competitive coexistence in spatially structured environments: a synthesis. Ecol Lett. 6, 1109-1122 [19]Takeuchi Y. (1989). Diffusion-mediated persistence in two-species competition. Lotka-Volterra model. Math Biosci. 95, 65-83 [20]Kishimoto K. (1990). Coexistence of any number of species in Lotka-Volterra competitive system over two patches. Theor. Popul Biol. 38. 149-158 [21]Amarasekare P, Nisbet R.(2001). Spatial heterogeneity, source-sink dynamics and the local coexistence of competing species. Am. Nat. 158, 572-584 [22]Liang, R J. (2007). Dynamic mechanism of metapopulation of two species competing for resources. Journal of Linyi Normal University. 6, 71-75 [23]Hannon and M. Ruth. (1994). Dynamic Modeling. Springer-Verlag, New York [24]Peterson and B. Richmond (1996). STELLA Research Technical Documentation, High Performance Systems, Hanover, NH [25]Costanza A., Voinov, R.et al.(2002). Integrated ecological economic modeling of the Patuxent river watershed, Maryland, Ecol. Monogr. 72, 203-231
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[26]Aassine and El Jai (2002). Aassine and M.C. El Jai, Vegetation dynamics modelling: a method for coupling local and space dynamics. Ecol. Model. 154,237–249 [27]Isee System. (2006). Technical document for the iThink and STELLA software [28]MacArther R H. (1970). Species packing and competitive equilibrium for many species. Theor. Popul Biol. 1, 1-11 [29]Abrams P A Wilson W G., 2004. Coexistence of competitors in metacommunities due to spatial variation in resource growth rates: does R* predict the outcome of competition? Ecology Letters.7, 929-940 [30]Xu Cailin, Li Zizhen et al. (2001). Escape and co-existence in patchy habitats and computer simulation study. Journal of Desert research, 21, 296-298
Procedia Environmental Sciences 2 (2010) 1083–1106
International Society for Environmental Information Sciences 2010 Annual Conference (ISEIS)
Simulation of Competitive Mechanism for Limited Resources by Two-Species Meta-Populations in Two Patches Liu Kunkuna, Li Chunhuia,*, Pang Aipinga, Cai Yanpengb a
Ministry of Education Key Laboratory of water-sediment Science, School of Environment, Beijing Normal University, Beijing, 100875, China b Faculty of Engineering, Dalhousie University, Halifax, Nova Scotia, B3J 1Z1, Canada
Abstract In this study, according to the model of the meta-population of two species competition for two kinds of linear-growth resources which is developed by Liang R.J., STELLA was used to simulate the competition mechanisms of two consumer species in different patches, utilizing the limited resources. The dynamic process was simulated under different situations: When the two species’ migration rates are o, which is a particular case that population compete for resource in one patch, the results are accord with R* Principle; When the two species’ migration rates are different, the coexistence of the two populations is possible in the patch ; When the two species’ individual average resource consumption=n rates are different, there exists obvious competing exclusion effect; When the resource growth rates in two patches are different, population density is higher in the patch with a higher resource growth rate, and populations in both patches are apt to coexist with a higher value of q and the distribution of population is partially influenced by the difference between the patches. Because there are still many parameters related with different aspects, it is better to use STELLA to simulate the dynamicity of both the resource and the population by changing the value of any parameter in the model conveniently.
© 2010 Published by Elsevier Ltd. Keywords:meta-population, competing for resource, STELLA; simulation
1. Introduction Competition is one of the main factors to create biological morphology, life cycle, biological community structure and dynamics, and it is also one of the key ecological processes to determine the ecosystem structure and function [1] .Mechanisms of species coexistence of competition and exclusion are the important research topics in Ecology. Model is one of the effective means for human to understand and solve practical problems, and it is widely used in different fields of science. There into, STELLA, which is one of system dynamics modeling tools with powerful modeling environment and simple operation mode, is highly respected by foreign researchers but is extremely rare in the domestic ecology research and applications where differential (difference) equation (s) is a basic form for describing the dynamic behavior. Previous study mainly used STELLA only for simulation of population growth in
* Corresponding author. Tel.: +86-10-58802756; fax: +86-10-58802756. E-mail address: [email protected]
1878-0296 © 2010 Published by Elsevier doi:10.1016/j.proenv.2010.10.120
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ecology. So it is very important that STELLA is applied in other areas to address such issues especially involve the analysis of a complex dynamic system’s response to different conditions like the complex species competition mechanism. The model here is general adaptive, which provides a theoretical reference for the conservation of biological diversity and the improvement of the resource utilization efficiency. Some meaningful exploration has been taken by using Lotka-Volterra Model [2], which is very important in the development history in Ecology. According to the different initial conditions, the results are also different: competition exclusion and stable coexistence[3]. On the basis of the model, a lot of species competition models in the related literature have emerged since the 1920s to investigate the interaction among species according to the competition coefficient[4].. After 1970s, the resource consumption-based resource competition theory quietly raised. During the period, people have designed varies kinds of consumer-recourse models [3,5-13].The key point in the theory is that it considers the dynamicity of both the resource and the consumer population competing for resource. Therefore, it is a mechanistic model which is to define the intra-specific competition and the inter-species competition according how the resource available extent is influenced by the species consumers. The theory first developed on the basis of Monod and Droop Equation which is based on chemostat model [14].To the 1980s, after Tilman’s research [12,15-16], a new resource competing model-based method emerged to predict the result of species competing according to the resource demand of the competing species. The main theory is still the competing exclusion theory. The theory figures that when multiple species compete for only one kind of resource, the key parameter in the models is R* which is the critical concentration of resource when the population growth rate equals the mortality rate. That means in the equilibrium condition without disturbance, the species with low R* will compete and replace all the other species which is the named R*-Principle [3-4]. Recently, the landscape pattern fragmentation due to irrational economic activities by human has caused a large number of species inhabit patchy habitats. Therefore, the research in meta-population theory developed fast and a lot of mathematical models emerged [12,15,17-21]. Some focus on two kinds of species competing for one or two kinds of resources in a patch, and some focus on two kinds of species competing for the same resource in two patches. Most of these have more discussions in equilibrium boundary condition than the population dynamic predict. Liang has established a two species met-population’s competing model to two kinds of exponential growth resources [22]. But he didn’t utilize any ecological modeling tools but the numerical simulation. STELLA is a user-friendly and commercial software package for building a dynamic modeling system. It uses an iconographic interface to facilitate construction of dynamic system models. The key features of STELLA consist of four tools: Stocks, Flows, Converters, Connectors. STELLA offers a practical way to dynamically visualize and communicate how complex systems and ideas really work. It has been widely used in biological, ecological, and environmental sciences [23-26.].An elaborate description of STELLA package can be found in Isee System (2006). It is a modeling tool for building a dynamic modeling system by creating a pictorial diagram of a system and then assigning the appropriate values and mathematical functions to the system [27]. The objective of this study is to simulate the dynamic mechanism of the meta-population of two species competing for resource in two patches by using STELLA. Here, I consider that since the dynamicity of both the resource and the consumer population competing for resource is the key point of the model, that is exactly in line with the Stella’ essence. Besides, by using Stella, it is easy to adjust the parameter to analyze the change of the result and to improve the model itself.
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2. Methodology 2.1. Overview of the study case First, MacArthur established the famous resource competing model which describes that species are distributed in a patch and make use of the resource [5, 28]. In 2004, Abrams raised the two population-two patches resource competing model [29]. They assumed that resource growth accords with the exponential growth or Logistic growth, and the scattered individuals separate equally in other patches. They focused on the two population’s coexistence conditions, but the population dynamic prediction research seems inadequate. On the basis of it, here comes a new model by Liang[22], which simulates two-species meta-population’ competitive mechanism for linear growth resources in two patches. The two species here can be different types of population and resources can also include inorganic (such as various nutrients, etc.) and organic (such as microbes, plants and prey, etc.). 2.2. Model formulation 2.2.1. Model assumptions Here, we need to re-emphasize the conditions in the model: 1. Two kinds of species live on two patches to compete for resource. 2. Resource growth accords with the exponential growth. 3. Species growth is restricted by resource availability and their densities. 4. Consumers move from each patch in the speed which is proportional to their densities in that patch. 2.2.2. Model functions On the basis of the conditions and assumptions above, the model should be:
dRa ra [(1 c1 ) N1a c1 N1b ](k1a Ra ) [(1 c2 ) N 2 a c2 N 2b ](k 2 a Ra ) dt dN1a p1[(1 c1 ) N1a c1 N1b ](k1a Ra ) / v1a m1[(1 c1 ) N1a c1 N1b ] dt dN 2 a p 2 [(1 c2 ) N 2 a c2 N 2b ](k 2 a Ra ) / v2 a m2 [(1 c2 ) N 2 a c2 N 2b ] dt dRb rb [(1 c1 ) N1b c1 N1a ](k1b Rb ) [(1 c2 ) N 2b c2 N 2 a ](k 2b Rb ) dt dN1b p1 [(1 c1 ) N1b c1 N1a ](k1b Rb ) / v1b m1[(1 c1 ) N1b c1 N1a ] dt dN 2b p 2 [(1 c2 ) N 2b c2 N 2 a ](k 2b Rb ) / v2b m2 [(1 c2 ) N 2b c2 N 2 a ] dt
(1) (2) (3) (4) (5) (6)
where Rj (j=a, b): the resource density in patch. Nij (i=1, 2; j=a,b): the consumer species i density in patch . rj (j=a, b): the resource growth rate in patch . ci (i=1, 2):the population’s migration force. kij (i=1, 2; j=a, b): the individual average resource consumption rate or the resource’ support capacity to the population or the population’s invasion rate. vij (i=1, 2; j=a, b): the average resource demand for the individual to survive, or the individual’s transformation efficiency of resource(1/v). pi (i=1, 2):the population’s reproduction rate. mi (i=1, 2):the population’s death rate.
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Kij * Rj /vij: the resource’s bearing capacity of the population.<1. According to the assumption by Abrams, the intrinsic growth rate of R is r, q is for the patch parameter .The resource intrinsic growth rate in patch a and b is:
ra ® ¯rb
r (q 1) / q r (q 1) / q
(7) (8)
Among the parameters, the state variables contain: Rj (j=a, b): the resource density in plague. Nij (i=1, 2; j=a, b): the density of the consumers species i in patch j. They change over time and can be expressed by:
Ra (t ) Rb (t )
Ra (t dt ) \ R a J R a Rb (t dt ) \ R b J R b
N1a (t )
N1a (t dt ) \ N 1a J N1a1
N 2 a (t )
N 2a (t dt ) \ N 2a J N 2 a
(9) (10) (11) (12)
N1b (t )
N1b (t dt ) \ N 1b J N1b
(13)
N 2b (t )
N 2b (t dt ) \ N 2b J N 2b
(14)
2.3. Model structure To establish a simulation model with STELLA, the first step is to develop a conceptual model, a basic structure that can capture the processes described before. As shown in Fig.1, the rectangles are stocks that graphically represent the resource density in each patch, or the consumer species density in each patch. The flow symbols (represented by double lines with arrows and switches) represent the rates of flow into or out of the stocks. The other variables are converters (represented by empty circles) that denote the rules or conditions controlling the stocks and flows through the use of connectors (represented by single lines with arrows).
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(a) The original one
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(b) The modified one(more easily understandable) Where the cap represents Kij * Rj /vij: the resource’s bearing capacity of the population. Fig.1. Conceptual model of meta-population of two species competing for resource in two patches
In the figure of resource density, the blue and red curves represent the density of Ra and Rb, and in the figure of community density, the blue, red, purple, green curves represent the density of N1a, N1b, N2a, N2b respectively. The following figures are the same. The growth makes the increase of the resource density in each patch, and the decrease of the density is decided by the amount eaten by the two consumer species that contain the species in this patch and those moved from the other patch and not contain those moved out of this patch. As to the species density, similarly, the birth makes the increase of the consumer density, and the decrease is decided by the death rate. Besides, each element is related with many factors such as the resource’s bearing capacity of the population, the population’s migration force and so on. The time step is one month, and the simulation time is 700 months. According to Liang’s chosen in parameters, the values are fixed in Table 1: Table 1. The values of the parameters in the model parameter
parameter's meaning
value
parameter
parameter's meaning
value
c1
population 1’s migration force
0.05
c2
population 2’s migration force
0.05
k1a
population 1's individual average resource consumption rate in patch a
0.3
k2a
population 2's individual average resource consumption rate in patch a
0.3
k1b
population 1's individual average resource consumption rate in patch b
0.3
k2b
population 2's individual average resource consumption rate in patch b
0.3
v1a
population 1's average resource demand for the individual to survive in patch a
0.5
v2a
population 2' s average resource demand for the individual to survive in patch a
1
v1b
population 1's average resource demand for the individual to survive in patch b
0.5
v2b
population 2's average resource demand for the individual to survive in patch b
1
p1
population 1's reproduction rate
0.1
p2
population 2's reproduction rate
0.1
m1
population 1's death rate
0.02
m2
population 2's death rate
0.02 0.036
q
the patch parameter
2
r
the resource' intrinsic growth rate of
ra
the resource growth rate in patch a
0.054
rb
the resource growth rate in patch b
0.018
initial conditions parameter
parameter's meaning
value
parameter
parameter's meaning
value
Ra (0)
initial resource density in patch a
0.2
Rb(0)
initial resource density in patch b
0.2
N1a(0)
the consumer species 1's initial density in patch a
0.05
N2a (0)
the consumer species 2's initial density in patch a
0.2
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
the consumer species 1's initial density in patch b
N1b(0)
0.05
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the consumer species 2's initial density in patch b
N2b (0)
0.2
Remark: The values here can be changed rationally to examine the response of the model in different conditions or for the different purposes.
2.4. Model calibration Because the validation of the resource competing model is limited to zooplankton, and it is carried under the stable conditions in laboratory, besides, it doesn’t have the abundant historical data like the climate prediction; it can not be validated by the data on the past. At the same time, it also can not be tested objectively by the fact in the short-term future like the weather and the earthquake. Therefore, the model here is not easy to be validated by the real data, but the related experiment results can be considered for reference, like R*-principle or Liang’s results by numerical simulation. 2.5. Model verification 2.5.1. Long-term stability: Run the model with values given before and set the simulation time 600 months and 1200 months separately. The simulation results are shown in Fig.2 and Fig.3.
Resource density
1: Ra
2: Rb
1
0.75 1
0.5
1
1
2
0.25
150
0
1: N1a 1 Population density
2 2
1
300 Time (months)
2: N1b
2
450
3: N2a
600
4: N2b
0.75 0.5
1
1 1
0.25
3 1
0
2
2
4
150
2
2 3
4
300 Time (months)
3
4
3
450
4
600
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Fig.2. Dynamic changes of resource and population in both patches in 600 months
1: Ra
2: Rb
Resource density
1 0.75 1
0.5 2 1
1
2
1
2
0.25
0
1: N1a
Population density
2
300
600 Time (months)
2: N1b
1
900
3: N2a
1200
4: N2b
0.75 1
1
0.5
1
0.25 1
0
2
2
2
3 4
3
300
4
600 Time (months)
2 3
4
3
900
4
1200
Fig.3. Dynamic changes of resource and population in both patches in 1200 months
Compare the simulative results in Fig.2 and Fig.3, we can figure that the number of resource and population in different time series both reach to a stable level finally. So the model has a relatively good stabilization, namely the long-term stability. 2.5.2. Model rationality: In general, when the parameters are set, patch a and b are different in the resource growth rate, which accords with that in patch a, resource reaches to a higher level than in patch b. Besides, in each patch, population 1 gets the upper hand at last and excludes population 2, which accords with that the individual in population 1 has a higher transformation efficiency of resource than population 2. That is to say, at a basic level, the model reacts as what we
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expect. But of course, there is something else in the model or its reaction to disturbance that we can’t comprehend. Further experiments and theoretical validation are needed. 2.5.3. Sensitive analysis: The model contains 6 state variables and a lot of parameters. The sensitive analysis has figured that the change of every parameter will result the change of at least one state variable, various in sensitive. Take the sensitive analysis of q for an example here. q stands for the difference between two patches. Take sensitive analysis with STELLA by choosing 4 values in its threshold value˄can’t be attained because of the material limitation, but can be estimated approximately through the process of sensitive analysis. According to the value Liang has chosen, the values of q here and the corresponding values of N1a and the results are shown in Fig.4.
Population 1 density in patch a
N1a: 1 - 2 - 3 - 4 1 0.75 1
1
0.5
1
0.25 3 1
2 2
3
4
2 3
4
4
4
2
175
0
350
525
700
Time (months) Fig.4. Population 1 density in patch a when q=1.5, 2.67, 3.83, 5.
According to the sensitive analysis equation: S [wN1a / N1a ] /[ w q / q] , a mended equation is given 3 as: S [( N1ai 1 N1ai ) / N1ai ] /[( qi 1 qi ) / qi ] /3
¦ i 1
Here, take the average value of N1a during the last 100 months as the value of N1a in the equation. Table 2. The average values of N1a during the last 100 months q
1.50
2.67
3.83
5.00
N1a
0.62
0.51
0.46
0.44
S= (0.51-0.62)/0.62/[(2.67-1.50)/1.50]+(0.46-0.51)/0.51/[(3.83-2.67)/2.67]+(0.44-0.46)/0.46/[(5.00-3.83)/3.83]= -0.198 Likewise, as to N1b, S=0.836.
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3. Results and discussion 3.1 A particular case—resource competing in one patch Adjust the values of the parameters: c1=c2=0; Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.5; k1a=k1b=k2a=k2b=0.3; m1=m2=0.2; p1=p2=0.33; ra = rb=0.145(ra and rb are used here for convenience to control the resource growth rate to be equal in both patches.); v1a=v1b=0.345, v2a=v2b=0.72. ra and rb are used here for convenience to control the resource growth rate to be equal in both patches. The simulation results are shown in Fig.5.
Resource and population density
1: Ra
2: Rb
1
3: N1a
0.75
1
2
2
3
4: N1b
1
4
2
3
5: N2a
1
4
2
3
4
0.5 4 1
0.25
3
5
0
25
5
50 Time (months)
5
75
5
100
Fig.5. Resource competing in one patch (c1=c2=0)
c1=c2=0, which means that the two patches are independent, and the values of parameters except v in both patches are equal, so the resource and species’ dynamic in two patches are completely the same. (N2b is not in the figure, but it goes the same as N2a) There is a periodical vibration damp movement between resource and population 1. Because the consumer has a time lag effect to resource, the phase of species density is behind the resource density, presenting plus correlation because of the feed back effect between them. The situation here is a little like the prey model by Lotka-Volterra. Because the population density can be adjusted by itself at some level, and population 2 should be considered because it is also the consumer for the same resource, the density vibration amplitude decreases rapidly and the densities of resource and population 1 reach to a balance quickly. The cycle time changes when the values of parameters change. Population 2 die out over period of time despite whether the change of parameters values because its transformation efficiency of resource (v) is lower than population 1, which accords with R*-Principle that species with the lowest R* will eliminate other species in balanced state and coexisting is impossible. Even the original population quantity of species 2 is larger than 1, the results are the same, which is shown in Fig. 6.
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Population density
1: N1a
2: N1b
1
1093
4: N2b
3: N2a 1 2
0.75
1
2
1
2
0.5 0.25 1
2
3
4 3
0
25
4
50 Time (months)
3
4
3
75
4
100
Fig.6. Resource competing in one patch (c1=c2=0, N1a=N1b
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Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
2 1
2
1
2
0.25 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1
0.5
1
0.25
1 2
3
0
3
175
2
2
2 4
4
350 Time (months) (a) c1=c2=0
3
4
3
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
1095
2: Rb
1
0.75 0.5
1 1
1
2
1
2
2
0.25 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1 1
0.5 0.25
1 2
3 4
0
2
175
3
4
2
350 Time (months) (b) c1=c2=0.15
3
2
4
525
3
4
700
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Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
0.25
2
2
0
Population density
1: N1a
1 2
175
2
350 Time (months)
2: N1b
1
1
525
3: N2a
4: N2b 1
1
0.75
700
1
0.5 0.25
1 2
3 3
4
0
2
175
4
2
350 Time (months)
3
4
2
525
3
4
700
(c) c1=c2=0.3 Fig.7. Resource competing in two patches (v1
We can figure that, because ra>rb, the resource quantity Ra grows quickly at first, reaches to the maximum value ,and then decreases because of the increase of populations before a new balance through dynamical evolution. Population 1 density increases fast and reaches to a stable state because of the higher transformation efficiency, becoming the dominant species. In contrast, Population 2 density decreases because of population 1’s competing exclusion effect and dies out at last. In patch b, the resource and population densities are both lower because the lower resource growth rate. When population migration rate is low(c=0), the situations in patch 1 and 2 are similar. Population 1 is the dominant species while population 2 dies out, though they are not in the same level. But as the value of c increases, the population 1 density increases to the saturation fast, and in patch b, population 1 is no longer the dominant species and dies out, too. This is not only because the resource growth rate is lower, but also because population density in patch a is higher and more individuals migrate to patch b, which has a strong competing exclusion effect on all the species in patch 2, reducing the species diversity. As to resource density, because of the invasion of species in patch a, resource density in patch b reaches to a smaller value in the balanced state.
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
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(2). c1=0.01, c2=0.45 When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.2; k1a=k1b=k2a=k2b=0.3; m1=m2=0.02; p1=p2=0.1; q=1.5; r=0.036,let v1=v2, and as the valve of v changes, the simulation results are:
Resource density
1: Ra
2: Rb
1
0.75 0.5
1
0.25
1 2
0
Population density
1: N1a
1 2
175
350 Time (months)
2: N1b
1
0.75
2
525
700
4: N2b
3: N2a
1
1
1
3
3
0.5
1
2
3
3
1
0.25
2 2
4
0
175
4
2
350 Time (months) (a) v1=v2=0.3
2
4
525
4
700
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Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
2: Rb
1
0.75 0.5
1 1
0.25
1
1 2
2
2 2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5 1
1 3
0.25
1
3
3
3
1 2 2
4
0
175
4
2
350 Time (months) (b) v1=v2=0.5
4
2
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
1099
2: Rb
1
0.75 0.5
1
1
1
2
2
2
0.25
1
2
0
Population density
1: N1a
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5 1
0.25 1
1
3
3
1
3
3
2 2
4
0
175
4
2
350 Time (months)
4
2
525
4
700
(c) v1=v2=0.7 Fig.8. Resource competing in two patches (c1
We can figure that, coexistence of the two populations is possible in patch a. Population 1 density is higher than 2 possibly because of the following reasons: Population in patch a migrate to patch b and vise versa, but the net effect is that population migrate to patch b because of the higher population density (the higher resource growth rate). In addition, Population 2’s net migration is larger than population 1 because of its higher migration rate. Similarly in patch b, population 2’s density is smaller, but they all die out at last because of the resource limitation. As the transformation efficiency (1/v) decreases, population density decreases, too, causing the resource and population density reach at different levels. Besides, the difference between population 1 and 2 decreases as the value of v increases. As to resource density, it increases, but the extent in patch a is bigger than b possibly because of the net migration from patch a to patch b. It seems that population with lower migration rate has a stronger competitive power here, which doesn’t contradict with Xu’s conclusions [30]. The background is different. We assume that the initial values of N1a and N2a are equal, and there are two different patches in a habitat. But when there are many patches and population with
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higher migration rate first enters a patch in a habitat, it will eliminate the others, which is different to some extent in two patches under the conditions here. 3.2.2 Resource growth rate influence---resource characteristic (1). ra=0.055, rb=0.05 When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.5; k1a=k1b=k2a=k2b=0.3; m1=m2=0.02; p1=p2=0.1; v1a=v1b=0.5; v2a=v2b=1; c1=c2=0.1,the simulation results are:
Resource density
1: Ra
2: Rb
1
0.75 0.5 1
0.25
Population density
:
1
2
1
2
1
0
1: N1a
2
2
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 1
1
0.5
1
1
2
2 2
2
0.25
3 4 3
0
175
Fig.9. Resource competing in two patches (ra≠rb, v1≠v2)
4
350 Time (months)
3
4
3
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
1101
Even with a little difference between the value of ra and rb, the population density in patch a is obvious higher, and as there is a larger gap between the values of ra and rb, the gap of population densities between patch a and patch b is larger. That is to say, population density is higher in the patch with a higher resource growth rate. (2). ra≠rb When Ra=Rb=0.2; N1a=N1b=N2a=N2b=0.2; k1a=k1b=k2a=k2b=0.3; v2a=v2b=1;c1=c2=0.1, let q=1,3,5 separately, the simulation results are:
Resource density
1: Ra
m1=m2=0.02;
p1=p2=0.1;
v1a=v1b=0.5;
2: Rb
1
0.75 1
0.5
1
1
1
0.25 2
2
0
Population density
1: N1a
2
175
2: N1b
1
2
350 Time (months)
525
4: N2b
3: N2a
1
1
0.75
700
1
0.5 0.25
1 3 2 4
0
2
175
3
4
2
350 Time (months) (a) q=1
3
4
2
525
3
4
700
1102
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Resource density
1: Ra
2: Rb
1
0.75 0.5 1
1
0.25
0
1: N1a
2
1
2
1
2
2
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
Population density
:
0.75 1
1
0.5
1
0.25
1
2
3 4
0
2
2
2 3
175
4
350 Time (months) (b) q=3
3
4
3
525
4
700
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
Resource density
1: Ra
2: Rb
1
0.75 0.5 1
1
2
2
1
2
1
2
0.25
0
1: N1a
Population density
1103
175
350 Time (months)
2: N1b
1
525
700
4: N2b
3: N2a
0.75 0.5
1
1 1
0.25
1
2 3
0
2
2
2 4
3
175
4
350 Time (months)
3
4
3
525
4
700
(c) q=5 Fig.10. Resource competing in two patches (v1≠v2)
We can figure that, when q=1, ra=2r, rb=0, that is to say, resource grows well in patch a, but can’t grow in patch b. The growth situation in patch a accords with R*- Principle, but the two species can’t survive in patch b and die out at last because of the resource limitation. As the value of patch parameter (q) increases, the resource growth rate on both patches gradually approaches equilibrium. Population can survive in both patches and each accords with R*-Principle. Besides, the densities of both populations also approach equilibrium gradually. For better observation, we collate the simulation results when q=1, 2,3,4,5, which are shown in table 3:
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Table 3. The values of resource and populations of different situations q 1 2 3 4 5
Ra 0.34 0.33 0.33 0.33 0.33
Rb 0 0.33 0.33 0.33 0.33
N1a 0.79 0.58 0.51 0.47 0.45
N1b 0 0.14 0.21 0.25 0.27
N2a 0 0 0 0 0
N2b 0 0 0 0 0
From the values of the balanced state, we can figure that, when q=1, there is no population in patch b although there is a high population density of N1a in patch a, which causes the total population density in patch a is the highest in all the situations but poor in species diversity. As the value of q increases, the difference between population densities of both species diminishes, and the total density stays the same (0.72). It seems that population is distributed in two patches according to the value of patch parameter. In conclusion, populations in both patches are apt to coexist with a higher value of q. Furthermore, the total densities of both resource and population have little relationship with the original value of population density and the value of c, but are related with the values of m and v which represent the population characteristics. That is to say, the distribution of populations is partially influenced by the difference between the patches and the populations’ migration ability. 4. Conclusions x First point, the model of meta-population of two species competing for resource in two patches is a simplification of the meta-population of multi-population competing for resource in multi-patches. In this study, STELLA was used to simulate the competition mechanisms. The dynamic process was simulated under different situations: the two species’ migration rates are o; the two species’ migration rates are different; the two species’ individual average resource consumption rates are different; the resource growth rates in two patches are different. Because there are still many parameters related with different aspects, it is better to use STELLA to simulate the dynamicity of both the resource and the population by changing the value of any parameter in the model conveniently. x Second point, the results of simulation by adjusting some of the parameters are as follows. (1) Parameters related with the population When c1=c2=0, this is a particular case that population compete for resource in one patch. The results are accord with R* Principle. When v1≠v2, c1=c2, there exists obvious competing exclusion effect. When c1≠c2, v1=v2, the coexistence between population 1 and 2 is possible in patch a. (2) Parameters related with the resource Population density is higher in the patch with a higher resource growth rate. Populations in both patches are apt to coexist with a higher value of q and the distribution of population is partially influenced by the difference between the patches.
5. Research prospect (1). The model contains many complex parameters, and each influences the simulation results. That causes a difficult problem-how to analyze the simulation results by changing the values of the parameters and to find out the some laws. For example, if the value of parameter related to the same population in different patches is not equal, what the results should be? (2). Because the validation of the resource competing model is limited to zooplankton, and it is carried under the stable conditions in laboratory, besides, it can not be validated by the data on the past, and can not be tested objectively by the fact in the short-term future, the model here is not easy to be validated by the real data. Therefore,
Liu Kunkun et al. / Procedia Environmental Sciences 2 (2010) 1083–1106
more experiment data are needed to validate the model in order to better predict the population and resource development. (3). The model of the meta-population of multi-population competing for resource in multi-patches is more practical and should be developed. The key point is that the model should be simple but can reflect the actual situation. Acknowledgements This research was supported by the National Science Foundation (50709002) and National Water Pollution Control and Treatment Project of China (2008ZX07209-009).Thanks to Zhang Lixiao for his encouragement to study, innovate and valuable opinions, support. At last, thanks to all the co-supervisors who gave valuable opinions and support to the successful completion of this study. References [1]Fan, J.W., Zhong, H P.,et al. (2004). Study of plant competition. Grassland Science.13, 1-8 [2]Hanski l. (1999). Metapopulation Ecology. Oxford University Press [3]Zhang D Y, et al. (2000). Researches on Theoretical Ecology. Beijing: Higher Education Press and Springer Press. 151-191 [4]Brock W A, Xepapadeas A. (2002). Optimal ecosystem management when species compete for limiting resources. Journal of Environmental Economics and Management. 44, 189-220 [5]MacArther R H. (1972). Geographical Ecology. Harper and Row, New York [6]Steward F M, Levin B R. (1973). Partitioning of resources and the outcome of interspecific: a model and some general considerations. American Naturalist. 107, 171-196 [7]Greeney W J., Bella D A., Curt H C. , 1973. A theoretical approach to interspecific competition in phytoplankton communities. American Naturalist. 107, 405-425 [8]O’Brien N J. (1974). The dynamics of nutrient limitation of Phytoplankton algae, a model reconsidered. Ecology. 55, 134-141 [9]Leon J, Tumpson D. (1975). Competition between two species for two complementary or substitutable resources. Journal Theoretical Biolog. 50, 185-201 [10]Taylor P, Williams P. (1975). Theoretical studies on the coexistence of competing species under continuous flow conditions. Canadian Journal of Microbiology. 21, 96-98 [11]Tilman D. (1977). Resource competition between planktonic algae: an experimental and theoretical approach. Ecology. 58, 338-348 [12]Tilman D. (1982). Resource Competition and Community Structure [M]. Princeton University Press, Princeton, N J [13]Zhang D Y. (1991). Exploitation competition and coexistence of two plant populations on a number of growth-limiting resources. Ecological Modeling. 53, 263-279 [14]Monod J. (1950). La technique de culture continue: theories et applications. Ann. Inst. Pasteur. 79, 390-410 [15]Tilman D. (1988). Plant strategies and the dynamics and structure of plant communities. Princeton University Press, Princeton, N J [16]Pacala S, Tilman D. (1994). Limiting similarity in mechanistic and spatial models of plant competition in heterogeneous environments. Am. Nat. 143, 222-257 [17]Tilman D, Pacala S. (1993). The maintenance of species richness in plant communities. In: Ricklefs, R. E.& Schluter, D. eds. Species diversity in ecological communities; Historical and Geographic Perspectives University of Chicago Press. 13-25 [18]Amarasekare P. (2003). Competitive coexistence in spatially structured environments: a synthesis. Ecol Lett. 6, 1109-1122 [19]Takeuchi Y. (1989). Diffusion-mediated persistence in two-species competition. Lotka-Volterra model. Math Biosci. 95, 65-83 [20]Kishimoto K. (1990). Coexistence of any number of species in Lotka-Volterra competitive system over two patches. Theor. Popul Biol. 38. 149-158 [21]Amarasekare P, Nisbet R.(2001). Spatial heterogeneity, source-sink dynamics and the local coexistence of competing species. Am. Nat. 158, 572-584 [22]Liang, R J. (2007). Dynamic mechanism of metapopulation of two species competing for resources. Journal of Linyi Normal University. 6, 71-75 [23]Hannon and M. Ruth. (1994). Dynamic Modeling. Springer-Verlag, New York [24]Peterson and B. Richmond (1996). STELLA Research Technical Documentation, High Performance Systems, Hanover, NH [25]Costanza A., Voinov, R.et al.(2002). Integrated ecological economic modeling of the Patuxent river watershed, Maryland, Ecol. Monogr. 72, 203-231
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[26]Aassine and El Jai (2002). Aassine and M.C. El Jai, Vegetation dynamics modelling: a method for coupling local and space dynamics. Ecol. Model. 154,237–249 [27]Isee System. (2006). Technical document for the iThink and STELLA software [28]MacArther R H. (1970). Species packing and competitive equilibrium for many species. Theor. Popul Biol. 1, 1-11 [29]Abrams P A Wilson W G., 2004. Coexistence of competitors in metacommunities due to spatial variation in resource growth rates: does R* predict the outcome of competition? Ecology Letters.7, 929-940 [30]Xu Cailin, Li Zizhen et al. (2001). Escape and co-existence in patchy habitats and computer simulation study. Journal of Desert research, 21, 296-298