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On environmental variability and limits to similarity
J. theor. Biol. (1980) 82,401-404
On Environmental G~RAN I.
Variability
AGREN
AND
and Limits to Similarity
TORBJ~RN FAGERSTR~~M
Swedish Coniferous Forest Project, The Swedish University of Agricultural Sciences, S-750 07 Uppsala, Sweden (Received 17 May 1979, and in revised form 14 August 1979) Different model approaches to the problem of environmental variability and limits to similarity are reviewed. It is shown that the assumption of a strictly flat resource spectrum leads to singular results. Hence, the result of May & MacArthur that a qualitative difference exists between a constant and a weakly varying environment is not generally valid. For small variations in the environment, several of the models reduce to a common form and the limits to similarity depend only weakly upon the magnitude of the environmental fluctuations. 1. Introduction In a classical work, May & MacArthur (1972) demonstrated that species segregating along a one-dimensional resource spectrum in a varying environment cannot be packed closer than by approximately one standard deviation of the utilization function. In contrast, they found that there is no such limit to the degree of niche overlap if the environment is strictly constant. This work has been challenged both from mathematical and biological standpoints (e.g. Feldman & Roughgarden, 1975; Abrams, 1975). In particular, the existence of a qualitative difference between a strictly constant environment, and one varying only very weakly, has recently been questioned by Turelli (1978) who found very small differences in the limits to similarity for the two cases. Also, other authors (MacArthur & Levins, 1967; Christiansen & Fenchel, 1977) have calculated finite limits to similarity in strictly constant environments. All of the referred studies are based on analyses of the Lotka-Volterra competition equations. We will here review the relevant parts of these studies with the aim of clarifying how so different conclusions can be reached from seemingly identical assumptions. 2. Constant Environments Two conditions have been used as criteria for maintained co-existence to prevail: (i) the equilibrium population densities must be positive, and 401
0022~5193/80/030401+04
$02.00/O
@ 1980 Academic
Press Inc. (London)
Ltd.
402
G.I.AGREN
ANDT.
FAGERSTRCIM
(ii) the system must be locally stable. Turelli adopts only condition (i) and argues that his difference equations should also satisfy condition (ii) because their continuous-time analogues do. May & MacArthur focus on condition (ii) when determining the limits to similarity, ensuring, a priori, that condition (i) is always satisfied by explicitly prescribing the equilibrium population densities to be equal. At a closer look, this latter prescription proves to be non-trivial and leads to the apparent paradox that a deterministic environment is qualitatively different from a stochastic one, even if the variation in the latter is infinitesimally small. May & MacArthur state (p. 1111) that “for a large number of species, n >>1, (equal population densities) means a flat resource spectrum”. This is not exactly true; however large the number of species is, the resource spectrum has to be slightly depressed at each end. With the assumption that the competition coefficients behave as (Y“‘*, where m is the distance between the two competing species, and assuming a typical (Y= 0.6, the carrying capacities would have to relate as 0.70,0.94, l., l., . . . , l., O-94,0.70, along the resource spectrum in order to give equal equilibrium populations. For n >>1 these small depressions at the ends look indeed quite innocent. However, changing the resource spectrum to a strictly flat one introduces drastic changes in the limits to similarity. The increased carrying capacities of the outermost species then allow their population densities to increase, which increases their competitive pressure on their neighbours to the extent of excluding them already at finite separations of the niches. Thus, the factor putting a limit to similarity in a constant environment is the asymmetric competition resulting from end effects. In the work by May & MacArthur this asymmetry was precisely balanced by varying the carrying capacities of the species in precisely the right way. Their finding of no limit to similarityshort of perfect congruence-in a strictly constant environment seems therefore to be a result of this very particular choice of the shape of the resource spectrum. For any other such choice, asymmetry in competition due to end effects would sooner or later show up. We have calculated that for n large the limiting value of LY is 0.619 corresponding to d/w = 1.39 for a flat resource spectrum. At this value of (Y the population densities along the spectrum are: 2.31, O., 1.67,0.58, 1.26, O-84, 1*10,0*94, 1.04,0.98, 1.02,0*99, l., l., . . . and symmetrically at the other end of the spectrum. We have also tried other shapes of the resource spectrum but for all shapes obtained limiting a’s corresponding to d/w = 1 although the species to be excluded may now be at a different position along the resource spectrum. It therefore seems, except under very special conditions, that also a strictly constant environment imposes a limit to similarity, similar to that found for a varying environment. Note that in a two-species
ENVIRONMENTAL
VARIABILITY
403
community the competition is necessarily perfectly symmetric so that there will be no limit to similarity due to end-effects, qualitatively distinguishing this community from communities of three or more species. The limit to similarity for a two-species community in a constant environment is d/w > 0, the case d/w = 0 being only neutrally stable. 3. Varying Environments Another important question is how rapidly the limit to similarity will change when a small variability is introduced in the environment, and to what extent it is dependent upon the particular choice of model for the resource spectrum, a curved one (May & MacArthur, 1972) or a flat one Turelli, 1978). Turelli calculates his limits from the condition that an intruder should be able to increase its population density. This criterion makes it difficult to extend the analysis beyond a few (3 resident) species because thereafter the number of possible positions of the intruder becomes too large to make the calculations possible. In addition, Turelli considers two different models for introducing the stochasticity into the equations as well as three models of interaction between the resident species; the intruder competes equally with all the residents. However, in all six cases his results in the limit of small variations can be reduced to (Y” = a, (1 - &*) where LX,and (Y, are the limiting a-values in a varying and a constant environment, respectively, c.I* is either the variance or the squared coefficient of variation of the stochastic term in the difference equation and b is a parameter that depends upon the choice of model, number of resident species and their intrinsic growth rates but is of the order of 1. The criterion used by May & MacArthur that the minimum eigenvalue of the competition matrix Amin > ~*/6 = q*, where a2 is the variance and 6 the mean of the (equal) carrying capacities will give the very similar condition cyU= CY,(1 -s*) for two of the competition models used by Turelli (the residents do not compete whereas the intruder experiences a competition cy from all the residents, and all the species compete with competition coefficient CX). The third of Turelli’s models, which is similar to the original model of May & MacArthur, has two residents with competition coefficient a4 and one intruder in-between with competition coefficient cy. Application of the eigenvalue criterion to this case and approximation to small CJgives (Y” = (Y, (1-q) which is a stronger (4 < 1) dependence on the environmental variability than that obtained with Turelli’s criterion. Yet, when 4 has moderate values, LYEand CY,do not differ significantly in numerical terms, the most important difference between May & MacArthur and Turelli being that assuming a flat resource spectrum means (Y== 0.544 while with equal equilibrium populations LY,= 1.
404
G. I. AGREN
AND
T. FAGERSTRbM
4. Conclusions In conclusion, the prescription of equal equilibrium populations leads to peculiar results in a strictly constant environment while the choice of model has little influence on how a, depends upon the environmental variability. However, what seems to be crucial is the microscopic interpretation given to a. May & MacArthur suggest that a should be related to niche separation, d, and niche width, w, as a =exp (-d2/4w2), in which case all that really matters is whether a = 1 giving d/w = 0 or a < 1 giving d/w = 1. Other interpretations of a (Abrams, 1975) give a much smoother relation between a and d/ w, a = 1 - bd/ w, where b is of the order of 1. In these cases there is no sharp transition at a particular value of d/w between regions of coexistence and non-coexistence. REFERENCES ABRAMS, P. (1975). Theor. pop. Biol. 8,356. CHRISTIANSEN, F. B. & FENCHEL, T. M. (1977).
Theories of Populations in Biological New York: Springer-Verlag. FELDMAN,M.W.& ROUGHGARDEN, J. (1975). Theor. pop.Biol. 7,197. MACARTHUR,R.&LEVINS, R.(1967). Am.Nat. 101,377. MAY, R. M. & MACARTHUR, R. H. (1972). Proc. nam. Acad. Sci. U.S.A. 69,1109. TURELLI, M. (1978). Proc. nam. Acad. Sci. U.S.A. 755085. Communities.
Berlin,
Heidelberg,
On Environmental G~RAN I.
Variability
AGREN
AND
and Limits to Similarity
TORBJ~RN FAGERSTR~~M
Swedish Coniferous Forest Project, The Swedish University of Agricultural Sciences, S-750 07 Uppsala, Sweden (Received 17 May 1979, and in revised form 14 August 1979) Different model approaches to the problem of environmental variability and limits to similarity are reviewed. It is shown that the assumption of a strictly flat resource spectrum leads to singular results. Hence, the result of May & MacArthur that a qualitative difference exists between a constant and a weakly varying environment is not generally valid. For small variations in the environment, several of the models reduce to a common form and the limits to similarity depend only weakly upon the magnitude of the environmental fluctuations. 1. Introduction In a classical work, May & MacArthur (1972) demonstrated that species segregating along a one-dimensional resource spectrum in a varying environment cannot be packed closer than by approximately one standard deviation of the utilization function. In contrast, they found that there is no such limit to the degree of niche overlap if the environment is strictly constant. This work has been challenged both from mathematical and biological standpoints (e.g. Feldman & Roughgarden, 1975; Abrams, 1975). In particular, the existence of a qualitative difference between a strictly constant environment, and one varying only very weakly, has recently been questioned by Turelli (1978) who found very small differences in the limits to similarity for the two cases. Also, other authors (MacArthur & Levins, 1967; Christiansen & Fenchel, 1977) have calculated finite limits to similarity in strictly constant environments. All of the referred studies are based on analyses of the Lotka-Volterra competition equations. We will here review the relevant parts of these studies with the aim of clarifying how so different conclusions can be reached from seemingly identical assumptions. 2. Constant Environments Two conditions have been used as criteria for maintained co-existence to prevail: (i) the equilibrium population densities must be positive, and 401
0022~5193/80/030401+04
$02.00/O
@ 1980 Academic
Press Inc. (London)
Ltd.
402
G.I.AGREN
ANDT.
FAGERSTRCIM
(ii) the system must be locally stable. Turelli adopts only condition (i) and argues that his difference equations should also satisfy condition (ii) because their continuous-time analogues do. May & MacArthur focus on condition (ii) when determining the limits to similarity, ensuring, a priori, that condition (i) is always satisfied by explicitly prescribing the equilibrium population densities to be equal. At a closer look, this latter prescription proves to be non-trivial and leads to the apparent paradox that a deterministic environment is qualitatively different from a stochastic one, even if the variation in the latter is infinitesimally small. May & MacArthur state (p. 1111) that “for a large number of species, n >>1, (equal population densities) means a flat resource spectrum”. This is not exactly true; however large the number of species is, the resource spectrum has to be slightly depressed at each end. With the assumption that the competition coefficients behave as (Y“‘*, where m is the distance between the two competing species, and assuming a typical (Y= 0.6, the carrying capacities would have to relate as 0.70,0.94, l., l., . . . , l., O-94,0.70, along the resource spectrum in order to give equal equilibrium populations. For n >>1 these small depressions at the ends look indeed quite innocent. However, changing the resource spectrum to a strictly flat one introduces drastic changes in the limits to similarity. The increased carrying capacities of the outermost species then allow their population densities to increase, which increases their competitive pressure on their neighbours to the extent of excluding them already at finite separations of the niches. Thus, the factor putting a limit to similarity in a constant environment is the asymmetric competition resulting from end effects. In the work by May & MacArthur this asymmetry was precisely balanced by varying the carrying capacities of the species in precisely the right way. Their finding of no limit to similarityshort of perfect congruence-in a strictly constant environment seems therefore to be a result of this very particular choice of the shape of the resource spectrum. For any other such choice, asymmetry in competition due to end effects would sooner or later show up. We have calculated that for n large the limiting value of LY is 0.619 corresponding to d/w = 1.39 for a flat resource spectrum. At this value of (Y the population densities along the spectrum are: 2.31, O., 1.67,0.58, 1.26, O-84, 1*10,0*94, 1.04,0.98, 1.02,0*99, l., l., . . . and symmetrically at the other end of the spectrum. We have also tried other shapes of the resource spectrum but for all shapes obtained limiting a’s corresponding to d/w = 1 although the species to be excluded may now be at a different position along the resource spectrum. It therefore seems, except under very special conditions, that also a strictly constant environment imposes a limit to similarity, similar to that found for a varying environment. Note that in a two-species
ENVIRONMENTAL
VARIABILITY
403
community the competition is necessarily perfectly symmetric so that there will be no limit to similarity due to end-effects, qualitatively distinguishing this community from communities of three or more species. The limit to similarity for a two-species community in a constant environment is d/w > 0, the case d/w = 0 being only neutrally stable. 3. Varying Environments Another important question is how rapidly the limit to similarity will change when a small variability is introduced in the environment, and to what extent it is dependent upon the particular choice of model for the resource spectrum, a curved one (May & MacArthur, 1972) or a flat one Turelli, 1978). Turelli calculates his limits from the condition that an intruder should be able to increase its population density. This criterion makes it difficult to extend the analysis beyond a few (3 resident) species because thereafter the number of possible positions of the intruder becomes too large to make the calculations possible. In addition, Turelli considers two different models for introducing the stochasticity into the equations as well as three models of interaction between the resident species; the intruder competes equally with all the residents. However, in all six cases his results in the limit of small variations can be reduced to (Y” = a, (1 - &*) where LX,and (Y, are the limiting a-values in a varying and a constant environment, respectively, c.I* is either the variance or the squared coefficient of variation of the stochastic term in the difference equation and b is a parameter that depends upon the choice of model, number of resident species and their intrinsic growth rates but is of the order of 1. The criterion used by May & MacArthur that the minimum eigenvalue of the competition matrix Amin > ~*/6 = q*, where a2 is the variance and 6 the mean of the (equal) carrying capacities will give the very similar condition cyU= CY,(1 -s*) for two of the competition models used by Turelli (the residents do not compete whereas the intruder experiences a competition cy from all the residents, and all the species compete with competition coefficient CX). The third of Turelli’s models, which is similar to the original model of May & MacArthur, has two residents with competition coefficient a4 and one intruder in-between with competition coefficient cy. Application of the eigenvalue criterion to this case and approximation to small CJgives (Y” = (Y, (1-q) which is a stronger (4 < 1) dependence on the environmental variability than that obtained with Turelli’s criterion. Yet, when 4 has moderate values, LYEand CY,do not differ significantly in numerical terms, the most important difference between May & MacArthur and Turelli being that assuming a flat resource spectrum means (Y== 0.544 while with equal equilibrium populations LY,= 1.
404
G. I. AGREN
AND
T. FAGERSTRbM
4. Conclusions In conclusion, the prescription of equal equilibrium populations leads to peculiar results in a strictly constant environment while the choice of model has little influence on how a, depends upon the environmental variability. However, what seems to be crucial is the microscopic interpretation given to a. May & MacArthur suggest that a should be related to niche separation, d, and niche width, w, as a =exp (-d2/4w2), in which case all that really matters is whether a = 1 giving d/w = 0 or a < 1 giving d/w = 1. Other interpretations of a (Abrams, 1975) give a much smoother relation between a and d/ w, a = 1 - bd/ w, where b is of the order of 1. In these cases there is no sharp transition at a particular value of d/w between regions of coexistence and non-coexistence. REFERENCES ABRAMS, P. (1975). Theor. pop. Biol. 8,356. CHRISTIANSEN, F. B. & FENCHEL, T. M. (1977).
Theories of Populations in Biological New York: Springer-Verlag. FELDMAN,M.W.& ROUGHGARDEN, J. (1975). Theor. pop.Biol. 7,197. MACARTHUR,R.&LEVINS, R.(1967). Am.Nat. 101,377. MAY, R. M. & MACARTHUR, R. H. (1972). Proc. nam. Acad. Sci. U.S.A. 69,1109. TURELLI, M. (1978). Proc. nam. Acad. Sci. U.S.A. 755085. Communities.
Berlin,
Heidelberg,