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The mechanisms of interspecific and intraspecific competition for resources and the dynamics model of community
Ecological Modelling, 69 (1993) 303-309 Elsevier Science Publishers B.V., Amsterdam
303
The mechanisms of interspecific and intraspecific competition for resources and the dynamics model of community H a n Boping and Lin Peng Dept. of Biology, Xiamen University, Xiamen, China (Received 22 January 1992; accepted 24 October 1992)
ABSTRACT Han, B.P. and Lin, P., 1993. The mechanisms of interspecific and intraspecific competition for resources and the dynamics model of community. Ecol. Modelling, 69: 303-309. In this paper, the absorption of resources and the physiological consumption are considered as two elements of a species' competitive ability, and plant species' maintenance of the balance between absorption of resources and physiologicalconsumption is a measurement of the species' competitive ability. New models on the basis of the Riccati equation are developed to express the dynamical behaviour of a community.
1. INTRODUCTION Research on plant interspecific and intraspecific competition mechanisms is helpful to u n d e r s t a n d not only population dynamics but also community structure and its evolution behaviour. The earlier researchers were interested not in the elements of competition but in the description of dynamics of some special experiments; but theoretical models without general mechanism devalue sometimes. Some ecologists have realized the defects in traditional theory (Murary, 1986; Tilman, 1986). Tilman (1986) made a very important contribution to resolve the problem, and H a n and Lin (1991) put forth the concept of M S C R (Minimal Surviving Concentration of Resource) and defined species' competitive ability on the adsorption of resources and its physiological consumption.
Correspondence to: Han Boping, Dept. of Biology, Xiamen University, Xiamen 361005, China. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
304
B. HANANDP. LIN
2. THE DEFINITION OF COMPETITION
Competition was first introduced into biology in Darwin's work "On the Origin of Species" (1859). In general, competition is defined as the interaction between two species for survival when a resource needed by both of them is short. The definition of competition does not represent a competitive process, and the elements of competitive ability of a species naturally become the premise to build a quantitative model of competition. For an individual, its survival means to maintain its metabolism, and the essential condition for survival is that the resource absorption meets the individual physiological consumption. So, competitive ability of a species can be defined as the ability of this species to maintain the balance, which depends on the species' physiological behaviour of absorption and consumption. 3. A MATHEMATICAL MODEL OF COMPETITIVE MECHANISM
To express clearly the competitive mechanism between two species for two kinds of resource is discussed first. R i is the concentration of available resource i in the environment, A i j is the absorption rate of species i for resource j, E i is the provision rate of resource i in the environment. The total absorption rate and the total consumption rate of a species for two kinds of resource are vectors, / ~ = (R1, R2), A~ = ( a l l ,
a12),
/ ~ = ( E l , E2), J 2 = ( a 2 1 , A22),
and the competition mechanism between two species for two kinds of resource is as follows: A~.-~=O,
(1)
A~. =A~.(/~,, t),
(2)
/~=/~0
t(2
- °eft ~ N/'A~ -/~ 0 i=1
C~. = C-~min--~-4d ,
/=1,2.
)
dt,
(3) (4)
In the equations above, ~ is total physiological consumption rate of species i, N,. is the individual number of species i. Identity (3) reflects resource dynamics with population growth in the environment; identity (4) means that the physiological consumption of a species for a resource is
MECHANISMS OF C O M P E T I T I O N FOR R E S O U R C E S
305
composed of two parts, minimum survival consumption (~min) and regulable consumption (Cid); by applying the Talor's theorem the linear expandence of A i yields
Ot + --~R Ji,o(t-to),
(5)
where O~/Ot depends on the physiological rhythm law of species and is the ability of a species to adapt to competitive pressure in a long time. BAi/BR, the variability of absorption of species i for resource with R, reflects the adaptibility of species to changing resources and is a main regulation_ behaviour of species when resource is short; (OAi/~t + (~AffBR)R') can be considered as a self-regulation ability of species i for resource absorption. Similarly, the linear expandence of Ci yields ~ = ~ ( t 0 ) + (d~min/dt +
d~a/dt)l,o(t- to).
(6)
For a species, its minimum survival consumption rate is the product of long-term evolution, dCid/d t is the species' self-regulation ability of physiological consumption for a resource. Any plant species maintains the balance between the absorption of resource and its physiological consumption with the two kinds of self-regulation. The critical survival equation of species 1 is
A~(g, /) - ~min = 0
(7)
at the critical point, t = t~', R = R~, the concentration of resource 1 or 2 reaches the MSCR of species 1. The critical survival equation of species 2 is
A~(/{, t ) - - ( 2 m i n = 0
(8)
at the critical point, t = t~', / ~ = / ~ ' , the concentration of resource 1 or 2 reaches the MSCR of species 2. The competitive result depends on the time length used by the species to reach their critical points.
4. SPECIES' COMPETITIVE STRATEGIES For a plant species, the competitive results relies on its maintenance to minimum physiological consumption. A plant species has developed some strategies to adapt competitive pressure in its evolution history; t h e physiological consumption of species i for a resource, C i =Cimin--[-Cid, a plant species first decreases the regulable consumption (C/d) to maintain its minimum physiological consumption when resource is short of maintaining its optimum growth, and the self-regulation method is a kind of short-term
306
B. HAN AND P. LIN
strategy for a species to adapt to resource pressure, for instance, by decreasing the offspring number. In the process of evolution, the main self-regulation behaviour is to increase the efficiency of metabolism and to decrease the minimum survival consumption rate. On the other hand, a species' self-regulation mechanism to the absorption of resource runs when resource concentration is lower than its optimum value. There R' < 0 when resource is short, the smaller OAi/OR is, the stronger the self-regulation ability for a plant species to maintain the optimum absorption rate of resource is. The self-regulation abilities to the absorption and the physiological consumption of resource compose the two elementary strategies to survive under competitive pressure, and the two strategies work on some special physiological and morphological characters; many evidences have proved the view (Chapin III, 1991). 5. A NEW DYNAMICS MODEL OF A SINGLE-SPECIES POPULATION U N D E R COMPETITION
In the Verhulst-Pearl logistic model of population (May, 1974), there are two defects; first, a population can reach its intrinsic rate of natural increase only when the population size is zero, but in the actual environments with rich resources, any population can reach its intrinsic rate of natural increase; second, the actual increase rate of a population is considered to relate to its individual number N in the Verhulst-Pearl logistic model, but in many experiments (Mooney et al., 1987), the actual growth rate of a population also depends on the resource provision in the environment (Trugill, 1977). On the basis of the competitive mechanism and the Shelford's tolerance law (Odum, 1971), Roo t is set for the optimum resource concentration of the species, Rmi n the MSCR of the species, there exists r(t) = rm(R - R m i n ) / ( R o p t - Rmin).
(9)
The Verhulst-Pearl logistic model is advanced as follows, d N / d t = rm(R - Rmin)/(Rop t - Rmin)g R =R o-a(N'A
(10)
-E)t
where A is the absorption rate of a species of a resource, E the provision rate of a resource in the environment, both A and E are constants. Further we set B = r m R min//(Rop t - Rmi n), C = rmE/(
Ropt - R m i n ) ,
O=Fmh//(Ropt-Rmin).
307
MECHANISMSOF COMPETITIONFOR RESOURCES Equation (10-1) can be rewritten as dN(t) at - (B + C t ) N ( t ) + D t X 2 ( t ) .
(11)
This equation is called the Riccati equation (Kamoker, 1977; Schafe and Schmidt, 1982); its initial condition is that N = N o at t = 0. The solution of this equation is
N0
f'
N(t) =
(12)
1 + D N o t e m+ct~/2 dt e - m - c t : / 2 to
6. COMMUNITY DYNAMICS MODELS UNDER COMPETITION FOR MANY KINDS OF RESOURCES As shown in Section 3, the dynamical model of two species for many i kinds of resources is first, discussed. Roptj is the optimum concentration of i resource j for species i. Rminj is the minimum survival concentration of resource j for species i. On the basis of a single population dynamics model, the dynamical model of two species u n d e r competition for two kinds of resource, there exists
dN1/dt=rml(Rl_Rminl)(R2 1
--
1 1 Rmin2)//(Roptl
--
1 1 Rminl)//(Ropt2
--
1 Rmin2)N1
d g 2 / / d t = rm2 (R1 - R 2 i n l ) ( R 2 - R2in2)//(g2ptl - R2inX)//(R2pt2- R2in2) N2
R 2 = R2o -
N~(t) "Aia - E2 t
i= (13) where rm/ is the intrinsic rate of natural increase of species i. The result of competition can be obtained through analysing the stable condition of population dynamics, that is, d N 1 / d t -- 0 and d N 2 / d t -- 0, there exist four possible states,
1 R 1 - Rminl = 0 (I) (III)
g 1 -R2minl = 0
1 R 1 - Rminl = 0 2 g 2 - Rmin2 = 0
(II) ( R2 - R1in22
=
0
R 2 -- Rmin2 = 0 (IV) { Re - Rlin22 = 0 R 1 - Rminl = 0
308
B. H A N A N D P. L I N
In state (I), the factor limiting the increase of two populations is resource 1, two populations can stay at stable state and co-exist for a long time only 1 2 when Rminl = Rminl , or a species will be excluded by another species with a lower MSCR. In state (II), the factor limiting the increase of two populations is resource 2, two populations co-exist at a stable state only when 1 _ 2 Rmin2- Rmin2. In states (III) and (IV), the factors limiting the increase of two populations are different and two species can co-exist for a long time due to avoiding competition. When the dynamics model of a two-species population is expanded to n species, the dynamics model of community with n species yields easily m
i H (Rj-Rm,nj) j=l
dN/ =rmi
m
~i
1~ ( i -- R i p t ) j = 1 R°ptj
(14)
Rj=Rio-a
Ni'Ai~(t)-E s t i
i = l, 2 , . . . , n ; j = l, 2 . . . . , m . Although the dynamics model of community with n species under competition for m kinds of resource has more-complex competitive behaviour between species, the competitive result can be obtained by discussing the stable condition of community dynamics. When the factors limiting the increase of every population are same, only the species with lower MSCR can survive for a long time; when the factors limiting the increase of population are different from each other, different species can co-exist due to avoiding the direct competition. 7. C O M P E T I T I O N
AND NICHE SEPARATION
A resource can be considered as a dimension of a niche; a niche overlaps yields when different species need the same resources. For example, we still discuss Eqs. (13) representing competition between two species with overlap. At d N 1 = d N 2 = 0, the four possible states (I)-(IV) show the four kinds of niche relationship between two species. In state (I), two species have a large range of niche overlap due to resource I limiting both of them; the competition is so intensive that one of the species is excluded by the one with the lower MSCR. In state (II), similarly, a large range of niche overlap due to resource 2 limiting both them results in intensive competition. In states (III) and (IV), since the resource limiting the two species is different, that is, the resource niches between them separate them and the
MECHANISMS OF C O M P E T I T I O N FOR R E S O U R C E S
309
two species avoid direct competition for resource and coexist. In a community, we easily obtain a similar result. 8. CONCLUSION
The definition of competitive ability on the absorption of a resource and the physiological consumption is the basis which the new community dynamics models are built on. Emphasizing the intrinsic mechanism of community dynamical behaviour is the most important property of the new models. The dynamics model on the basis of the Riccati equation eliminates the defects of the Verhulst-Pearl logistic model because absorption o f r e s o u r c e a n d its physiological c o n s u m p t i o n are c o n s i d e r e d . T h e species' competitive strategies derived from mathematical model of competitive m e c h a n i s m c o i n c i d e with physiology. T h a t species with d i f f e r e n t limiting r e s o u r c e s can avoid d i r e c t c o m p e t i t i o n shows t h e n i c h e s e p a r a t i o n and t h e f o r m i n g c o n d i t i o n o f a stable c o m m u n i t y s t r u c t u r e . REFERENCES Chapin III, F.S., 1991. Integrated responses of plants to stress. BioScience, 41(1): 29-36. Han, B.P. and Lin, P., 1991. Plant interspecific and intraspecific competition mechanism. In: Advances in Ecology. China Science and Technology Press, pp. 135-136. Kamoker, E., 1977. A Handbook of Differential Equations. Science Press, pp. 28-31. May, R.M., 1974. Stability and Complexity in Model Ecosystems, 2nd ed. Princeton University Press, Princeton, NJ, pp. 13-6. Mooney, H.A., Pearcy, R.W. and Ehleringer, J., 1987. Plant physiological ecology today. BioScience, 37(1): 18-21. Murary Jr., B.G., 1986. The structure of theory and the role of competition in community dynamics. Oikos, 46: 145-158. Odum, E.P., 1971. The Fundamentals of Ecology, 3rd ed. W.B. Saunders, Philadelphia, PA, pp. 106-138. Schafe, F.W. and Schmidt, D., 1982. The Fundamentals of Differential Equations. People Education Press, pp. 15-21. Tilman, D., 1986. In: N.J. Crawley (Editor), Plant Ecology. Blackwell Scientific Publications, pp. 51-75. Trugill, S.T., 1977. Soil and Vegetation System. Clarendon Press, Oxford, pp. 20-24.
303
The mechanisms of interspecific and intraspecific competition for resources and the dynamics model of community H a n Boping and Lin Peng Dept. of Biology, Xiamen University, Xiamen, China (Received 22 January 1992; accepted 24 October 1992)
ABSTRACT Han, B.P. and Lin, P., 1993. The mechanisms of interspecific and intraspecific competition for resources and the dynamics model of community. Ecol. Modelling, 69: 303-309. In this paper, the absorption of resources and the physiological consumption are considered as two elements of a species' competitive ability, and plant species' maintenance of the balance between absorption of resources and physiologicalconsumption is a measurement of the species' competitive ability. New models on the basis of the Riccati equation are developed to express the dynamical behaviour of a community.
1. INTRODUCTION Research on plant interspecific and intraspecific competition mechanisms is helpful to u n d e r s t a n d not only population dynamics but also community structure and its evolution behaviour. The earlier researchers were interested not in the elements of competition but in the description of dynamics of some special experiments; but theoretical models without general mechanism devalue sometimes. Some ecologists have realized the defects in traditional theory (Murary, 1986; Tilman, 1986). Tilman (1986) made a very important contribution to resolve the problem, and H a n and Lin (1991) put forth the concept of M S C R (Minimal Surviving Concentration of Resource) and defined species' competitive ability on the adsorption of resources and its physiological consumption.
Correspondence to: Han Boping, Dept. of Biology, Xiamen University, Xiamen 361005, China. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
304
B. HANANDP. LIN
2. THE DEFINITION OF COMPETITION
Competition was first introduced into biology in Darwin's work "On the Origin of Species" (1859). In general, competition is defined as the interaction between two species for survival when a resource needed by both of them is short. The definition of competition does not represent a competitive process, and the elements of competitive ability of a species naturally become the premise to build a quantitative model of competition. For an individual, its survival means to maintain its metabolism, and the essential condition for survival is that the resource absorption meets the individual physiological consumption. So, competitive ability of a species can be defined as the ability of this species to maintain the balance, which depends on the species' physiological behaviour of absorption and consumption. 3. A MATHEMATICAL MODEL OF COMPETITIVE MECHANISM
To express clearly the competitive mechanism between two species for two kinds of resource is discussed first. R i is the concentration of available resource i in the environment, A i j is the absorption rate of species i for resource j, E i is the provision rate of resource i in the environment. The total absorption rate and the total consumption rate of a species for two kinds of resource are vectors, / ~ = (R1, R2), A~ = ( a l l ,
a12),
/ ~ = ( E l , E2), J 2 = ( a 2 1 , A22),
and the competition mechanism between two species for two kinds of resource is as follows: A~.-~=O,
(1)
A~. =A~.(/~,, t),
(2)
/~=/~0
t(2
- °eft ~ N/'A~ -/~ 0 i=1
C~. = C-~min--~-4d ,
/=1,2.
)
dt,
(3) (4)
In the equations above, ~ is total physiological consumption rate of species i, N,. is the individual number of species i. Identity (3) reflects resource dynamics with population growth in the environment; identity (4) means that the physiological consumption of a species for a resource is
MECHANISMS OF C O M P E T I T I O N FOR R E S O U R C E S
305
composed of two parts, minimum survival consumption (~min) and regulable consumption (Cid); by applying the Talor's theorem the linear expandence of A i yields
Ot + --~R Ji,o(t-to),
(5)
where O~/Ot depends on the physiological rhythm law of species and is the ability of a species to adapt to competitive pressure in a long time. BAi/BR, the variability of absorption of species i for resource with R, reflects the adaptibility of species to changing resources and is a main regulation_ behaviour of species when resource is short; (OAi/~t + (~AffBR)R') can be considered as a self-regulation ability of species i for resource absorption. Similarly, the linear expandence of Ci yields ~ = ~ ( t 0 ) + (d~min/dt +
d~a/dt)l,o(t- to).
(6)
For a species, its minimum survival consumption rate is the product of long-term evolution, dCid/d t is the species' self-regulation ability of physiological consumption for a resource. Any plant species maintains the balance between the absorption of resource and its physiological consumption with the two kinds of self-regulation. The critical survival equation of species 1 is
A~(g, /) - ~min = 0
(7)
at the critical point, t = t~', R = R~, the concentration of resource 1 or 2 reaches the MSCR of species 1. The critical survival equation of species 2 is
A~(/{, t ) - - ( 2 m i n = 0
(8)
at the critical point, t = t~', / ~ = / ~ ' , the concentration of resource 1 or 2 reaches the MSCR of species 2. The competitive result depends on the time length used by the species to reach their critical points.
4. SPECIES' COMPETITIVE STRATEGIES For a plant species, the competitive results relies on its maintenance to minimum physiological consumption. A plant species has developed some strategies to adapt competitive pressure in its evolution history; t h e physiological consumption of species i for a resource, C i =Cimin--[-Cid, a plant species first decreases the regulable consumption (C/d) to maintain its minimum physiological consumption when resource is short of maintaining its optimum growth, and the self-regulation method is a kind of short-term
306
B. HAN AND P. LIN
strategy for a species to adapt to resource pressure, for instance, by decreasing the offspring number. In the process of evolution, the main self-regulation behaviour is to increase the efficiency of metabolism and to decrease the minimum survival consumption rate. On the other hand, a species' self-regulation mechanism to the absorption of resource runs when resource concentration is lower than its optimum value. There R' < 0 when resource is short, the smaller OAi/OR is, the stronger the self-regulation ability for a plant species to maintain the optimum absorption rate of resource is. The self-regulation abilities to the absorption and the physiological consumption of resource compose the two elementary strategies to survive under competitive pressure, and the two strategies work on some special physiological and morphological characters; many evidences have proved the view (Chapin III, 1991). 5. A NEW DYNAMICS MODEL OF A SINGLE-SPECIES POPULATION U N D E R COMPETITION
In the Verhulst-Pearl logistic model of population (May, 1974), there are two defects; first, a population can reach its intrinsic rate of natural increase only when the population size is zero, but in the actual environments with rich resources, any population can reach its intrinsic rate of natural increase; second, the actual increase rate of a population is considered to relate to its individual number N in the Verhulst-Pearl logistic model, but in many experiments (Mooney et al., 1987), the actual growth rate of a population also depends on the resource provision in the environment (Trugill, 1977). On the basis of the competitive mechanism and the Shelford's tolerance law (Odum, 1971), Roo t is set for the optimum resource concentration of the species, Rmi n the MSCR of the species, there exists r(t) = rm(R - R m i n ) / ( R o p t - Rmin).
(9)
The Verhulst-Pearl logistic model is advanced as follows, d N / d t = rm(R - Rmin)/(Rop t - Rmin)g R =R o-a(N'A
(10)
-E)t
where A is the absorption rate of a species of a resource, E the provision rate of a resource in the environment, both A and E are constants. Further we set B = r m R min//(Rop t - Rmi n), C = rmE/(
Ropt - R m i n ) ,
O=Fmh//(Ropt-Rmin).
307
MECHANISMSOF COMPETITIONFOR RESOURCES Equation (10-1) can be rewritten as dN(t) at - (B + C t ) N ( t ) + D t X 2 ( t ) .
(11)
This equation is called the Riccati equation (Kamoker, 1977; Schafe and Schmidt, 1982); its initial condition is that N = N o at t = 0. The solution of this equation is
N0
f'
N(t) =
(12)
1 + D N o t e m+ct~/2 dt e - m - c t : / 2 to
6. COMMUNITY DYNAMICS MODELS UNDER COMPETITION FOR MANY KINDS OF RESOURCES As shown in Section 3, the dynamical model of two species for many i kinds of resources is first, discussed. Roptj is the optimum concentration of i resource j for species i. Rminj is the minimum survival concentration of resource j for species i. On the basis of a single population dynamics model, the dynamical model of two species u n d e r competition for two kinds of resource, there exists
dN1/dt=rml(Rl_Rminl)(R2 1
--
1 1 Rmin2)//(Roptl
--
1 1 Rminl)//(Ropt2
--
1 Rmin2)N1
d g 2 / / d t = rm2 (R1 - R 2 i n l ) ( R 2 - R2in2)//(g2ptl - R2inX)//(R2pt2- R2in2) N2
R 2 = R2o -
N~(t) "Aia - E2 t
i= (13) where rm/ is the intrinsic rate of natural increase of species i. The result of competition can be obtained through analysing the stable condition of population dynamics, that is, d N 1 / d t -- 0 and d N 2 / d t -- 0, there exist four possible states,
1 R 1 - Rminl = 0 (I) (III)
g 1 -R2minl = 0
1 R 1 - Rminl = 0 2 g 2 - Rmin2 = 0
(II) ( R2 - R1in22
=
0
R 2 -- Rmin2 = 0 (IV) { Re - Rlin22 = 0 R 1 - Rminl = 0
308
B. H A N A N D P. L I N
In state (I), the factor limiting the increase of two populations is resource 1, two populations can stay at stable state and co-exist for a long time only 1 2 when Rminl = Rminl , or a species will be excluded by another species with a lower MSCR. In state (II), the factor limiting the increase of two populations is resource 2, two populations co-exist at a stable state only when 1 _ 2 Rmin2- Rmin2. In states (III) and (IV), the factors limiting the increase of two populations are different and two species can co-exist for a long time due to avoiding competition. When the dynamics model of a two-species population is expanded to n species, the dynamics model of community with n species yields easily m
i H (Rj-Rm,nj) j=l
dN/ =rmi
m
~i
1~ ( i -- R i p t ) j = 1 R°ptj
(14)
Rj=Rio-a
Ni'Ai~(t)-E s t i
i = l, 2 , . . . , n ; j = l, 2 . . . . , m . Although the dynamics model of community with n species under competition for m kinds of resource has more-complex competitive behaviour between species, the competitive result can be obtained by discussing the stable condition of community dynamics. When the factors limiting the increase of every population are same, only the species with lower MSCR can survive for a long time; when the factors limiting the increase of population are different from each other, different species can co-exist due to avoiding the direct competition. 7. C O M P E T I T I O N
AND NICHE SEPARATION
A resource can be considered as a dimension of a niche; a niche overlaps yields when different species need the same resources. For example, we still discuss Eqs. (13) representing competition between two species with overlap. At d N 1 = d N 2 = 0, the four possible states (I)-(IV) show the four kinds of niche relationship between two species. In state (I), two species have a large range of niche overlap due to resource I limiting both of them; the competition is so intensive that one of the species is excluded by the one with the lower MSCR. In state (II), similarly, a large range of niche overlap due to resource 2 limiting both them results in intensive competition. In states (III) and (IV), since the resource limiting the two species is different, that is, the resource niches between them separate them and the
MECHANISMS OF C O M P E T I T I O N FOR R E S O U R C E S
309
two species avoid direct competition for resource and coexist. In a community, we easily obtain a similar result. 8. CONCLUSION
The definition of competitive ability on the absorption of a resource and the physiological consumption is the basis which the new community dynamics models are built on. Emphasizing the intrinsic mechanism of community dynamical behaviour is the most important property of the new models. The dynamics model on the basis of the Riccati equation eliminates the defects of the Verhulst-Pearl logistic model because absorption o f r e s o u r c e a n d its physiological c o n s u m p t i o n are c o n s i d e r e d . T h e species' competitive strategies derived from mathematical model of competitive m e c h a n i s m c o i n c i d e with physiology. T h a t species with d i f f e r e n t limiting r e s o u r c e s can avoid d i r e c t c o m p e t i t i o n shows t h e n i c h e s e p a r a t i o n and t h e f o r m i n g c o n d i t i o n o f a stable c o m m u n i t y s t r u c t u r e . REFERENCES Chapin III, F.S., 1991. Integrated responses of plants to stress. BioScience, 41(1): 29-36. Han, B.P. and Lin, P., 1991. Plant interspecific and intraspecific competition mechanism. In: Advances in Ecology. China Science and Technology Press, pp. 135-136. Kamoker, E., 1977. A Handbook of Differential Equations. Science Press, pp. 28-31. May, R.M., 1974. Stability and Complexity in Model Ecosystems, 2nd ed. Princeton University Press, Princeton, NJ, pp. 13-6. Mooney, H.A., Pearcy, R.W. and Ehleringer, J., 1987. Plant physiological ecology today. BioScience, 37(1): 18-21. Murary Jr., B.G., 1986. The structure of theory and the role of competition in community dynamics. Oikos, 46: 145-158. Odum, E.P., 1971. The Fundamentals of Ecology, 3rd ed. W.B. Saunders, Philadelphia, PA, pp. 106-138. Schafe, F.W. and Schmidt, D., 1982. The Fundamentals of Differential Equations. People Education Press, pp. 15-21. Tilman, D., 1986. In: N.J. Crawley (Editor), Plant Ecology. Blackwell Scientific Publications, pp. 51-75. Trugill, S.T., 1977. Soil and Vegetation System. Clarendon Press, Oxford, pp. 20-24.