- Email: [email protected]
A fuzzy description on some ecological concept
Ecological Modelling 169 (2003) 361–366
Short communication
A fuzzy description on some ecological concept Wang Wan-Xiong a,∗ , Li Yi-Min b , Li Zi-Zhen c , Yang Fengxiang d a
State Key Laboratory of Arid Agroecology, Lanzhou University, Lanzhou 730000, PR China b Jiangsu Science and Engineering University, Jiangsu, PR China c Department of Mathematics, Lanzhon University, Lanzhou 730000, PR China d Department of Basic Science, Gansu Agriculture University, Gansu 730070, PR China
Received 22 November 2002; received in revised form 11 June 2003; accepted 16 July 2003
Abstract From the view of fuzzy set theory, first, we discuss competition of species i and j for many type resources (basic resource, less important resource, occasional resource and replaceable resource). The membership functions of all resources were given by using fuzzy tone operator, the fuzzy competition coefficient of two species was also supplied here; second, by using fuzzy set theory, fuzzy niche was given. Based on that, fuzzy measurement of niche width and overlap were also discussed here. All these are helpful for study the competition and coexist of the species. © 2003 Published by Elsevier B.V. Keywords: Fuzzy set theory; Competition; Niche; Niche overlap and breadth
1. Introduction The concept of niches was first proposed by Grinnell in (1917), who considered a niche a fundamental distribution unit of species. Elton (1933) emphasized the function of the species, and described the niche as the place of the species in the bio-environment or the role the species played in the community. Both Grinnell and Elton regarded the niche as a qualitative concept, which was different to put into practical quantitative use. A quantitative concept of the niche, therefore, was proposed by Hutchinson in 1957. According to his definition, a niche was a hypovolume of n-dimensional resources in which a species lives; thus the niche can be described quantitatively by the coordinates of the species on n-dimensional resources axis (Hutchinson, ∗ Corresponding author. E-mail address: [email protected] (W. Wan-Xiong).
0304-3800/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/S0304-3800(03)00279-5
1957). Based on Hutchinson’s concept of niches, the problem of niche overlap can be solved, though there is still work to be done (Giller, 1982). The “competitive exclusion principle” may be illustrated more precisely in terms of Hutchinson’s niche concept: two species cannot exist within the same niche; two species will compete more intensely if their niche overlap is increased, consequently competition will allow one species to persist and the other to become extinct (Christiansen and Fenchel, 1977). Because the boundary of the niche is not distinct as Hutchinson’s concept suggested; in fact, its boundary has the characteristics of ambiguity in both distribution and resource utilization of the species. The fuzzy set theory was introduced by Zadeh (1965), a fuzzy set is a set which allows partial membership, it is defined in a universe of discourse by a grade of membership function defined on all elements of that universe. The value of this function for a given element describes the degree to which that
362
W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
element is a member of the associated fuzzy set. By convention, a value of 1 denotes total membership, a value of 0 denotes total non-membership, and values between 0 and 1 denote different degrees of partial membership. If a grade of membership is 0 or 1 for all elements in the universe, the associated fuzzy set is a classical (ordinary) set. This means that ordinary sets are special cases of fuzzy sets. Since fuzzy sets allow partial membership, they eliminate the needed to definitely classify an object, analogous to the way probability eliminates the need to definitely predict the outcome of an event. Fuzzy methods are non-statistical in nature, however, and apply to a different class of phenomena, or at least to the elimination of a different kind of detail. Fuzzy set theory, which describes the ambiguity and uncertainty in mathematics, adopted by other ecologists, such as Bosserman and Ragade (1982) for ecosystem analysis and Salski (1992) for modeling an ecological process containing uncertain ecological data. The fuzzy sets theory is also introduced here by us for defining some ecological concept.
2. Fuzzy competition coefficient of two species First, we consider species i and j compete for many types of resources. Besides taking basic resources, euryphagous species still take less important resources, occasional resources and replaceable resources. In natural environment, inter-species competition is mainly for basic resources, meanwhile, for important resources, occasional resources and replaceable resources. During a certain period, every euryphagous species has a stable resources and nutrition structure which can be denoted as a universe of discourse. U = {basic resources, less important resources, occasional resources, replaceable resources} where {basic resources}, {less important resources}, {occasional resources} and {replaceable resources} are fuzzy sets, denoted them as A1 , A2 , A3 , A4 , ˜ ˜ ˜ ˜ respectively. Two euryphagous species competed for resources are dependent on the similar degree of their menu (nutritional structure). Intuitively, if the menu of two species is very close, they will compete more in-
tensely; otherwise, if the menu (nutritional structure) of two species is not similar to a large degree, they will compete less intensely. Therefore, we may describe inter-species competition with similarity of two species menu. Assuming that the menu (nutrition structure) of every species is a fuzzy set U = {s1 , s2 , . . . , sn } = {A1 , A2 , A3 , A4 }.
˜
˜
˜
˜
That is to say, n types of the resources are divided into four resources grades, which denote four fuzzy sets of basic resources, less important resources, occasional resources and replaceable resources, respectively. Supposed that A is an important resources, then its ˜ can be written as membership function K(s)W(s)/r(s) maxs K(s)W(s)/r(s)
A(s) =
˜
where K(s) is the carrying capacity of the resource environment s. r(s) is the intrinsic growth rate of the resource s. W(s) is the weight number of the resource s. So far as euryphagous species, K(s) and r(s) effect of nature increase ability on competition, under K(s) and r(s) given condition, W(s) is the weight number that the membership degree of resource s to Ai . ˜ Measurement of W(s) can be obtained by frequency that resource types appear in stomach of the animal. For example, the variability of resource component of mates zibellina (frequency appears in stomach) (see Table 1). We can calculate the grade weight numbers of mates zibeline for given resources during a certain period, such as in 1923–1924 year, Small rodent Pine seed
W(s) = 0.6 W(s) = 0.2
Competition of two species depends on common resource and important grade for them. The membership function of grade of every resource can be given by tone operator of fuzzy sets theory (Luo, 1989). Let tone operator be Hε : F(u) → F(u) (ε is positive real number); (Hε × A)(s) = [A(s)]ε
˜
˜
A ∈ F(u).
˜
When ε > 1, Hε , is called concentrated operator; when ε < 1, Hε , is called diffused operator.
W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
363
Table 1 Measurement of W(s) (from Sun, 1988) Food
Years
Small rodent Sciurus vulgarius Eutamias sibiricus Lepustimidus Sorex Birds Insect Pine seed Berry Number of mates zibellina
1923–1924
1924–1925
1925–1926
1926–1927
1927–1928
1928–1930
1930–1931
60.0 10.0 20.0 10.0 10.0 10.0 0 20.0 0 1.3
75.0 25.0 20.0 0 0 0 0 50.0 0 0.8
37.5 25.0 6.2 12.5 0 18.7 6.2 62.5 6.2 0.9
78.6 2.4 19.0 7.1 21.4 14.3 7.1 35.7 9.5 0.9
12.5 33.3 8.3 12.5 8.3 12.5 25.0 79.2 4.2 0.5
15.4 23.1 15.4 30.8 15.4 15.4 15.4 53.8 7.7 0.6
34.4 15.6 9.4 3.1 9.4 18.8 6.2 68.8 3.1 0.8
For example, assuming that the membership function of main (basic) resources is X1 (s), then (Hε × Ai )(s) = [A(s)]ε = [X1 (s)]ε .
˜
˜
The membership function of less important resources is A2 = X2 (s)/s. ˜ 1/2 K(s)W(s)/r(s) 1/2 . X2 (s) = [X1 (s)] = maxs K(s)W(s)/r(s) The membership function of the occasional resources is X3 (s) A3 = . s ˜ 1/4 K(s)W(s)/r(s) 1/4 . X3 (s) = [X1 (s)] = maxs K(s)W(s)/r(s) The membership function of the occasional resources is X4 (s) A4 = s ˜ 1/8 K(s)W(s)/r(s) . X4 (s) = [X1 (s)]1/8 = maxs K(s)W(s)/r(s) Therefore, we may also assume that the common resources set of the species i and j is Xij (s) Sij = . s ˜ Then their common resources on every grade are that ykij (s) F kij = Ak ε˙ Sij = , k = 1, 2, 3, 4. s ˜ ˜ ˜ where ε˙ is Einstain operator (Luo, 1989).
If Cij denotes the set of common share resources of species i and j, then Cij = F 1ij ∪ F 2ij ∪ F 3ij ∪ F 4ij =
˜
˜
˜
˜
Cij (s) s
s
.
Definition. Assuming that X takes value on the resources set U, which expresses common resources of species i and j on every grade. Let R(X) = Fkij (k = 1, 2, 3, 4), then the possible distribution related to X (Luo, 1989) is nkij = R(X) = Fkij Poss(X is s) = Π(s) = maxs (ykij (s)ΛΠkij (s)). Therefore, we can get the following conclusion: the competition coefficient of species i and j for resource is that αij =
maxs (ykij (s)ΛΠkij (s)) . maxs (ykij (s)ΛΠkij (s))
The competition coefficient given by the above equation is near reality and more practical. Πkij (s) expresses that any resources become the probability of the kth common resources, denoted as Πkij (s) = ykij (s), therefore, αij =
maxs ykij (s) . maxs ykii (s)
If the resources is richer, two species are only interested in main resources and take few the other
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resources, then competition only depends on the main common resources, at the same time, αij =
maxs y1ij (s) . maxs y1ii (s)
X1 (s)/s is normal, If the main resources A = namely, maxs X1 (s) = ˜1, the important of all resources within the main resource type is the same, that is to say, X1 (s) = 1 for all competitors. y1ij =
X1 (s) × Xij (s) = Xij (s). 1 + (1 − X1 (s))(1 − Xij (s))
Therefore, αij =
maxs Xij (s) maxs Xii (s)
is the competition coefficient of two species for the optimal common resources.
3. Fuzzy niche Niche concept of the most popular, now, is Hutchinson’s definition (1957): a niche was a supervolumer which includes all living condition needed by the species existing and reproduction; thus all variable (resources) related to the living of the species must be included, and these variables should be independent. Because the niche is important in ecological systems, many ecologists are interested in that (Abrams, 1980; Horn, 1966; Pielou, 1971; Yang and Ma, 1992). Assuming A is a fuzzy set of n-dimensional real space then ˜ Aλ = {X|µA(x) ≥ λ, 0 ≤ λ ≤ 1}
˜
˜
is called λ weak truncated set. Aλ = {X|µA(x) > λ, 0 ≤ λ ≤ 1}
˜
˜
is called λ strong truncated set (Luo, 1989). Where µA(x) is the membership function of the fuzzy set A. ˜ ˜ fuzzy sets may be considered as a fundamental Both niche and realized niche. Let H : [0, 1] → Q(z), Λ → H(Λ). Satisfied with AΛ ⊆ H(Λ) ⊆ AΛ,
˜
˜
∀ 0 ≤ Λ ≤ 1.
H is called a nested sets of fuzzy set A, when Λ1 < ˜ Λ2 ⇒ H(Λ1 ) ⊇ H(Λ2 ). Definition. Assuming Λ is a vector of n-dimensional real space, it expresses a state of a species living space, Λ = (λ1 , λ2 , . . . , λn ) is a living environment that the species can be tolerant, λ1 , λ2 , . . . , λn express coordinate of all ecological factors (biotic and abiotic factors) of the species living and reproduction, respectively. Then the nested sets H(Λ∗ ) are a niche occupied by the species X, where Λ∗ ||Λ||. (1) If H(Λ∗ ) is a niche occupied by the species X, then X ∈ H(Λ∗ ). It expresses the ecological space (including climate, resources, etc.) that the species can be tolerant. Meanwhile, there is µA(x) ≥ Λ∗ ⇒ ||X|| ≤ R(Λ∗ ).
˜
In geometry, R(Λ∗ ) is radius of n-dimensional supervolume, the space boundary of niche occupied by species X is a n-dimensional supervolume taken R(Λ∗ ) as radius, all ecological factors are included in, such as temperature, humid, altitude, PH, resources, and competition etc., related to the species living and reproduction. (2) If Λ∗1 < Λ∗2 , particular, |λi1 | < |λi2 |, i = 1, 2, . . . , n, then H(Λ∗1 ) ⊇ H(Λ∗2 ). These show that: more great the absolute value of the ecological factors λi (i = 1, 2, . . . , n), the less the niche H(Λ∗ ) occupied by the species is correspondingly. (3) According to the expression theorem of fuzzy sets theory (Luo, 1989), A = ∪Λ∈[0,1] ΛH(Λ) expresses all populations˜ occupied a given area. Those are called a community in ecology, namely, a community can be expressed as a fuzzy set A.
˜
More general, Λ = (λ1 , λ2 , . . . , λn ), is a n-dimensional vector. Assuming that the species X has an “optimal tolerated range” which can be expressed as fuzzy number λi (i = 1, 2, . . . , n) on axis of every ecological factor. ˜λi (i = 1, 2, . . . , n) is obviously a ˜ fuzzy number, its membership bounded and closed function is α ∈ [m, n] = φ 1, λi (α) = L(α), α < n ˜ R(α), α > n
W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
where L(α) is an increasing and left continuous function, and limα→−∞ L(α) = 0, namely, 0 ≤ L(α) < 1. R(α) is a decreasing and right continuous function, and limα→+∞ R(α) = 0.0 ≤ R(α) ≤ 1. Therefore, λi = {[mλi , nλi ], Lλi (α), Rλi (α)},
˜
(i = 1, 2, . . . , n)
expresses an optimal tolerated range of the species on an ecological factor axis λi . According to extensional theorem of fuzzy set, we can assume that H : (λ1 , λ2 , . . . , λn ) = Λ∗ → H (Λ∗ ) = V
˜ ˜
˜
˜
˜
365
a fuzzy number, Λ∗ = (λ1 , λ2 , . . . , λn ) expresses the ˜ ˜ living space. ˜ optional state of the species Therefore, ∗ H (Λ ) expresses the optional niche occupied by the ˜ species X. According to decomposition theorem, there is α = 0 or 1 expresses the minimum and maximum niche of the species X. 3.1. Fuzzy niche width and overlap In the past decade, research on niche form and relation is one of the most popular subject in community ecology. Abrams (1980) and Slobodkichoff and Schulz (1980) had in detail evaluated measurement method on niche width and overlap. Here we will be discussed that by fuzzy method.
expresses the fittest niche of the species X. All conditions of physical and chemical environment of the 3.1.1. Niche width species within the niche V is optimum. The member˜ From above statement, when interval number ship function, which the species is subordinated to the λ (i = 1, 2, . . . , n) is a fuzzy number, Λ∗ = i niche V , is ˜ ˜ 0, if H −1 (v) = 0 µV (u) = n ∨(∧i=1 )µλi (ui ), otherwise, (u1 , . . . , un ) ∈ H −1 (v) ˜
˜
where H −1 (v) = {(u1 , u2 , . . . , un )|H(u1 , u2 , . . . , un ) = v}. Therefore, every species is corresponding to an optimal niche. It is a n-dimensional fuzzy set, in geometry, a n-dimensional fuzzy hypovolume. In particular, assuming that λi is a special fuzzy number, namely, ˜ denoted as λi = [mλ , nλ ], (i = interval number, i i 1, 2, . . . , n), which express ˜ecological amplitude of the species. According to Shelford’s “law of tolerance”, every species has a tolerant range to every environment, that is to say, there is an ecological minimum and maximum. The range between ecological minimum and maximum is called ecological amplitude. Thus, Λ∗ = (λ1 , λ2 , . . . , λn )
˜ ˜
˜
= ([mλ1 , nλ1 ], [mλ2 ,nλ2 ], . . . , [mλn ,nλn ]). is a vector consisted of n-dimensional closed intervals. H(Λ∗ )α = H(λ1 , λ2 , . . . , λn )α
˜ ˜
˜
= H(λ1,α , λ2,α , . . . , λn,α ),
α ∈ [0, 1]
is called truncated set of the fuzzy set H (Λ∗ ). When n-dimensional ecological factor λi (i = 1, 2, . . . , n) is
˜
(λ1 , λ2 , . . . , λn ) is the optional state of the species liv˜ space, ˜ ˜ ) represents the niche of the species X. ing H (Λ ˜ H (Λ) occupied by a species X must Therefore, ˜niche ˜ ˜i ) on every resources axis, H(λi ) have a projection H(λ ˜ niche on the ith dimensional ˜ expresses a projection axis, its width is, w(w(H(λi )) = nλi − mλi the width ˜ of niche occupied by the species X, therefore may be denoted as B(H (Λ)) = w(w(H(λ1 )), w(H(λ2 )), . . . , w(H(λn ))
1/2 n ˜ ˜ ˜ ˜ ˜ 2 (w(H(λi ))) . i=1
˜
The equation reflects resource diversity used by the species, namely, the tolerant (or used) range of the species X for n ecological factors. If a small part within whole resources is really used by the species, then there must be some w(H(λi )) are very small or equal zero. Therefore, B(H (Λ))˜ is small. In this case, we ˜ ˜ have a narrower niche. will say that the species 3.1.2. Niche overlap Assuming that H (Λ1 ) and H (Λ2 ) are the niche of ˜ ˜ ˜ The formula of the the species X and ˜Y, respectively.
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W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
niche overlap can be written as the following H XY (Λ) = H (Λ1 ) ∩ H (Λ2 )
˜
˜
where
˜ ˜ ˜
˜ ˜ ˜ ˜ ∩(H(λ21 ), H(λ22 ), . . . , H(λ2n )) ˜ ˜ ˜ (HXY (λ1 ), HXY (λ2 ), . . . , HXY (λn )). ˜ ˜ ˜
= (H(λ11 ), H(λ12 ), . . . , H(λ1n ))
HXY (λi ) = HX (λ1i ) ∩ HY (λ2i )
˜
˜
˜
[max(mλ1i , mλ2i ), max(nπ1i , nλ2i )], i = 1, 2, . . . , n. λ1i = mλ1i , nλ1i ,
λ2i = [mλ2i , nλ2i ].
˜
˜
Therefore, H XY (Λ) is a n-dimensional closed fuzzy vector, the ˜niche˜ overlap is a overlap of the niche HX (λ1i ) and HY (λ2i ) projected on each environment ˜ λ1i and λ2i .˜ factor ˜ above˜ statement we know when HXY (λi ) is From more great, the niche overlap of the species X and Y is much more. This results in a very similar niche in which the species X and Y live, and inter-species competition will be intensify, Finally, one of the species could be eliminated or niche could be separated. According to the niche overlap defined by the paper, we know that it is a comprehensive index. If niche overlap happens only on some environment factors λi , the species may not be competition, certainly, species may be competition when their overlap niches are some decisive factors. The relation of niche overlap and competition may be described by the formula HXY (·). 3.2. Further consideration Fuzzy niche concept of the species is similar to Hutchinson’s concept in some ways, however, it has
some improvements over the classical concept. For making the fuzzy concept more sound and easy for use, still much work to be done.
Acknowledgements This research was supported by the National Nature science Foundation of China (Nos. 39970135, 30070139). References Abrams, P., 1980. Some comments on measuring niche overlap. Ecology 61 (1), 44–49. Bosserman, R.W., Ragade, R.K., 1982. Ecosystem analysis using fuzzy set theory. Ecol. Model. 16, 191–208. Christiansen, F.B., Fenchel, T.M., 1977. Theories of Population in Biological Communities. Springer, Berlin. Elton, C., 1933. The Ecology of the Animals. Méthuen, London. Giller, P.S., 1982. Community Structure and the Niche. Chapman and Hall, London. Grinnell, J., 1917. The niche relationships of the California thrasher. Auk. 21, 364–382. Horn, H.S., 1966. Measurement of overlap in comparative ecological studies. Am. Nat. 100, 419–424. Hutchinson, G.E., 1957. Concluding remarks. Cold Spring Harbor Synp. Quant. Biol. 22, 415–427. Luo, C., 1989. Introduction Fuzzy Sets. Beijing Normal University Press (in Chinese). Pielou, E.C., 1971. Niche width and overlap: a method for measuring them. Ecology. 53 (4), 687–692. Salski, A., 1992. Fuzzy knowledge-based models in ecological research. Ecol. Model. 63, 103–112. Slobodkichoff, C.N., Schulz, W.C., 1980. Measures of niche overlap. Ecology 61, 1051–1055. Sun R., 1988. The Principle of Animal Ecology. Beijing Normal University Press (in Chinese). Yang, X., Ma, J., 1992. A review on some terms related to niche and their measurement. Chin. J. Ecol. 11 (2), 44–49. Zadeh, L.A., 1965. Fuzzy sets. Inf. Control 8, 338–353.
Short communication
A fuzzy description on some ecological concept Wang Wan-Xiong a,∗ , Li Yi-Min b , Li Zi-Zhen c , Yang Fengxiang d a
State Key Laboratory of Arid Agroecology, Lanzhou University, Lanzhou 730000, PR China b Jiangsu Science and Engineering University, Jiangsu, PR China c Department of Mathematics, Lanzhon University, Lanzhou 730000, PR China d Department of Basic Science, Gansu Agriculture University, Gansu 730070, PR China
Received 22 November 2002; received in revised form 11 June 2003; accepted 16 July 2003
Abstract From the view of fuzzy set theory, first, we discuss competition of species i and j for many type resources (basic resource, less important resource, occasional resource and replaceable resource). The membership functions of all resources were given by using fuzzy tone operator, the fuzzy competition coefficient of two species was also supplied here; second, by using fuzzy set theory, fuzzy niche was given. Based on that, fuzzy measurement of niche width and overlap were also discussed here. All these are helpful for study the competition and coexist of the species. © 2003 Published by Elsevier B.V. Keywords: Fuzzy set theory; Competition; Niche; Niche overlap and breadth
1. Introduction The concept of niches was first proposed by Grinnell in (1917), who considered a niche a fundamental distribution unit of species. Elton (1933) emphasized the function of the species, and described the niche as the place of the species in the bio-environment or the role the species played in the community. Both Grinnell and Elton regarded the niche as a qualitative concept, which was different to put into practical quantitative use. A quantitative concept of the niche, therefore, was proposed by Hutchinson in 1957. According to his definition, a niche was a hypovolume of n-dimensional resources in which a species lives; thus the niche can be described quantitatively by the coordinates of the species on n-dimensional resources axis (Hutchinson, ∗ Corresponding author. E-mail address: [email protected] (W. Wan-Xiong).
0304-3800/$ – see front matter © 2003 Published by Elsevier B.V. doi:10.1016/S0304-3800(03)00279-5
1957). Based on Hutchinson’s concept of niches, the problem of niche overlap can be solved, though there is still work to be done (Giller, 1982). The “competitive exclusion principle” may be illustrated more precisely in terms of Hutchinson’s niche concept: two species cannot exist within the same niche; two species will compete more intensely if their niche overlap is increased, consequently competition will allow one species to persist and the other to become extinct (Christiansen and Fenchel, 1977). Because the boundary of the niche is not distinct as Hutchinson’s concept suggested; in fact, its boundary has the characteristics of ambiguity in both distribution and resource utilization of the species. The fuzzy set theory was introduced by Zadeh (1965), a fuzzy set is a set which allows partial membership, it is defined in a universe of discourse by a grade of membership function defined on all elements of that universe. The value of this function for a given element describes the degree to which that
362
W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
element is a member of the associated fuzzy set. By convention, a value of 1 denotes total membership, a value of 0 denotes total non-membership, and values between 0 and 1 denote different degrees of partial membership. If a grade of membership is 0 or 1 for all elements in the universe, the associated fuzzy set is a classical (ordinary) set. This means that ordinary sets are special cases of fuzzy sets. Since fuzzy sets allow partial membership, they eliminate the needed to definitely classify an object, analogous to the way probability eliminates the need to definitely predict the outcome of an event. Fuzzy methods are non-statistical in nature, however, and apply to a different class of phenomena, or at least to the elimination of a different kind of detail. Fuzzy set theory, which describes the ambiguity and uncertainty in mathematics, adopted by other ecologists, such as Bosserman and Ragade (1982) for ecosystem analysis and Salski (1992) for modeling an ecological process containing uncertain ecological data. The fuzzy sets theory is also introduced here by us for defining some ecological concept.
2. Fuzzy competition coefficient of two species First, we consider species i and j compete for many types of resources. Besides taking basic resources, euryphagous species still take less important resources, occasional resources and replaceable resources. In natural environment, inter-species competition is mainly for basic resources, meanwhile, for important resources, occasional resources and replaceable resources. During a certain period, every euryphagous species has a stable resources and nutrition structure which can be denoted as a universe of discourse. U = {basic resources, less important resources, occasional resources, replaceable resources} where {basic resources}, {less important resources}, {occasional resources} and {replaceable resources} are fuzzy sets, denoted them as A1 , A2 , A3 , A4 , ˜ ˜ ˜ ˜ respectively. Two euryphagous species competed for resources are dependent on the similar degree of their menu (nutritional structure). Intuitively, if the menu of two species is very close, they will compete more in-
tensely; otherwise, if the menu (nutritional structure) of two species is not similar to a large degree, they will compete less intensely. Therefore, we may describe inter-species competition with similarity of two species menu. Assuming that the menu (nutrition structure) of every species is a fuzzy set U = {s1 , s2 , . . . , sn } = {A1 , A2 , A3 , A4 }.
˜
˜
˜
˜
That is to say, n types of the resources are divided into four resources grades, which denote four fuzzy sets of basic resources, less important resources, occasional resources and replaceable resources, respectively. Supposed that A is an important resources, then its ˜ can be written as membership function K(s)W(s)/r(s) maxs K(s)W(s)/r(s)
A(s) =
˜
where K(s) is the carrying capacity of the resource environment s. r(s) is the intrinsic growth rate of the resource s. W(s) is the weight number of the resource s. So far as euryphagous species, K(s) and r(s) effect of nature increase ability on competition, under K(s) and r(s) given condition, W(s) is the weight number that the membership degree of resource s to Ai . ˜ Measurement of W(s) can be obtained by frequency that resource types appear in stomach of the animal. For example, the variability of resource component of mates zibellina (frequency appears in stomach) (see Table 1). We can calculate the grade weight numbers of mates zibeline for given resources during a certain period, such as in 1923–1924 year, Small rodent Pine seed
W(s) = 0.6 W(s) = 0.2
Competition of two species depends on common resource and important grade for them. The membership function of grade of every resource can be given by tone operator of fuzzy sets theory (Luo, 1989). Let tone operator be Hε : F(u) → F(u) (ε is positive real number); (Hε × A)(s) = [A(s)]ε
˜
˜
A ∈ F(u).
˜
When ε > 1, Hε , is called concentrated operator; when ε < 1, Hε , is called diffused operator.
W. Wan-Xiong et al. / Ecological Modelling 169 (2003) 361–366
363
Table 1 Measurement of W(s) (from Sun, 1988) Food
Years
Small rodent Sciurus vulgarius Eutamias sibiricus Lepustimidus Sorex Birds Insect Pine seed Berry Number of mates zibellina
1923–1924
1924–1925
1925–1926
1926–1927
1927–1928
1928–1930
1930–1931
60.0 10.0 20.0 10.0 10.0 10.0 0 20.0 0 1.3
75.0 25.0 20.0 0 0 0 0 50.0 0 0.8
37.5 25.0 6.2 12.5 0 18.7 6.2 62.5 6.2 0.9
78.6 2.4 19.0 7.1 21.4 14.3 7.1 35.7 9.5 0.9
12.5 33.3 8.3 12.5 8.3 12.5 25.0 79.2 4.2 0.5
15.4 23.1 15.4 30.8 15.4 15.4 15.4 53.8 7.7 0.6
34.4 15.6 9.4 3.1 9.4 18.8 6.2 68.8 3.1 0.8
For example, assuming that the membership function of main (basic) resources is X1 (s), then (Hε × Ai )(s) = [A(s)]ε = [X1 (s)]ε .
˜
˜
The membership function of less important resources is A2 = X2 (s)/s. ˜ 1/2 K(s)W(s)/r(s) 1/2 . X2 (s) = [X1 (s)] = maxs K(s)W(s)/r(s) The membership function of the occasional resources is X3 (s) A3 = . s ˜ 1/4 K(s)W(s)/r(s) 1/4 . X3 (s) = [X1 (s)] = maxs K(s)W(s)/r(s) The membership function of the occasional resources is X4 (s) A4 = s ˜ 1/8 K(s)W(s)/r(s) . X4 (s) = [X1 (s)]1/8 = maxs K(s)W(s)/r(s) Therefore, we may also assume that the common resources set of the species i and j is Xij (s) Sij = . s ˜ Then their common resources on every grade are that ykij (s) F kij = Ak ε˙ Sij = , k = 1, 2, 3, 4. s ˜ ˜ ˜ where ε˙ is Einstain operator (Luo, 1989).
If Cij denotes the set of common share resources of species i and j, then Cij = F 1ij ∪ F 2ij ∪ F 3ij ∪ F 4ij =
˜
˜
˜
˜
Cij (s) s
s
.
Definition. Assuming that X takes value on the resources set U, which expresses common resources of species i and j on every grade. Let R(X) = Fkij (k = 1, 2, 3, 4), then the possible distribution related to X (Luo, 1989) is nkij = R(X) = Fkij Poss(X is s) = Π(s) = maxs (ykij (s)ΛΠkij (s)). Therefore, we can get the following conclusion: the competition coefficient of species i and j for resource is that αij =
maxs (ykij (s)ΛΠkij (s)) . maxs (ykij (s)ΛΠkij (s))
The competition coefficient given by the above equation is near reality and more practical. Πkij (s) expresses that any resources become the probability of the kth common resources, denoted as Πkij (s) = ykij (s), therefore, αij =
maxs ykij (s) . maxs ykii (s)
If the resources is richer, two species are only interested in main resources and take few the other
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resources, then competition only depends on the main common resources, at the same time, αij =
maxs y1ij (s) . maxs y1ii (s)
X1 (s)/s is normal, If the main resources A = namely, maxs X1 (s) = ˜1, the important of all resources within the main resource type is the same, that is to say, X1 (s) = 1 for all competitors. y1ij =
X1 (s) × Xij (s) = Xij (s). 1 + (1 − X1 (s))(1 − Xij (s))
Therefore, αij =
maxs Xij (s) maxs Xii (s)
is the competition coefficient of two species for the optimal common resources.
3. Fuzzy niche Niche concept of the most popular, now, is Hutchinson’s definition (1957): a niche was a supervolumer which includes all living condition needed by the species existing and reproduction; thus all variable (resources) related to the living of the species must be included, and these variables should be independent. Because the niche is important in ecological systems, many ecologists are interested in that (Abrams, 1980; Horn, 1966; Pielou, 1971; Yang and Ma, 1992). Assuming A is a fuzzy set of n-dimensional real space then ˜ Aλ = {X|µA(x) ≥ λ, 0 ≤ λ ≤ 1}
˜
˜
is called λ weak truncated set. Aλ = {X|µA(x) > λ, 0 ≤ λ ≤ 1}
˜
˜
is called λ strong truncated set (Luo, 1989). Where µA(x) is the membership function of the fuzzy set A. ˜ ˜ fuzzy sets may be considered as a fundamental Both niche and realized niche. Let H : [0, 1] → Q(z), Λ → H(Λ). Satisfied with AΛ ⊆ H(Λ) ⊆ AΛ,
˜
˜
∀ 0 ≤ Λ ≤ 1.
H is called a nested sets of fuzzy set A, when Λ1 < ˜ Λ2 ⇒ H(Λ1 ) ⊇ H(Λ2 ). Definition. Assuming Λ is a vector of n-dimensional real space, it expresses a state of a species living space, Λ = (λ1 , λ2 , . . . , λn ) is a living environment that the species can be tolerant, λ1 , λ2 , . . . , λn express coordinate of all ecological factors (biotic and abiotic factors) of the species living and reproduction, respectively. Then the nested sets H(Λ∗ ) are a niche occupied by the species X, where Λ∗ ||Λ||. (1) If H(Λ∗ ) is a niche occupied by the species X, then X ∈ H(Λ∗ ). It expresses the ecological space (including climate, resources, etc.) that the species can be tolerant. Meanwhile, there is µA(x) ≥ Λ∗ ⇒ ||X|| ≤ R(Λ∗ ).
˜
In geometry, R(Λ∗ ) is radius of n-dimensional supervolume, the space boundary of niche occupied by species X is a n-dimensional supervolume taken R(Λ∗ ) as radius, all ecological factors are included in, such as temperature, humid, altitude, PH, resources, and competition etc., related to the species living and reproduction. (2) If Λ∗1 < Λ∗2 , particular, |λi1 | < |λi2 |, i = 1, 2, . . . , n, then H(Λ∗1 ) ⊇ H(Λ∗2 ). These show that: more great the absolute value of the ecological factors λi (i = 1, 2, . . . , n), the less the niche H(Λ∗ ) occupied by the species is correspondingly. (3) According to the expression theorem of fuzzy sets theory (Luo, 1989), A = ∪Λ∈[0,1] ΛH(Λ) expresses all populations˜ occupied a given area. Those are called a community in ecology, namely, a community can be expressed as a fuzzy set A.
˜
More general, Λ = (λ1 , λ2 , . . . , λn ), is a n-dimensional vector. Assuming that the species X has an “optimal tolerated range” which can be expressed as fuzzy number λi (i = 1, 2, . . . , n) on axis of every ecological factor. ˜λi (i = 1, 2, . . . , n) is obviously a ˜ fuzzy number, its membership bounded and closed function is α ∈ [m, n] = φ 1, λi (α) = L(α), α < n ˜ R(α), α > n
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where L(α) is an increasing and left continuous function, and limα→−∞ L(α) = 0, namely, 0 ≤ L(α) < 1. R(α) is a decreasing and right continuous function, and limα→+∞ R(α) = 0.0 ≤ R(α) ≤ 1. Therefore, λi = {[mλi , nλi ], Lλi (α), Rλi (α)},
˜
(i = 1, 2, . . . , n)
expresses an optimal tolerated range of the species on an ecological factor axis λi . According to extensional theorem of fuzzy set, we can assume that H : (λ1 , λ2 , . . . , λn ) = Λ∗ → H (Λ∗ ) = V
˜ ˜
˜
˜
˜
365
a fuzzy number, Λ∗ = (λ1 , λ2 , . . . , λn ) expresses the ˜ ˜ living space. ˜ optional state of the species Therefore, ∗ H (Λ ) expresses the optional niche occupied by the ˜ species X. According to decomposition theorem, there is α = 0 or 1 expresses the minimum and maximum niche of the species X. 3.1. Fuzzy niche width and overlap In the past decade, research on niche form and relation is one of the most popular subject in community ecology. Abrams (1980) and Slobodkichoff and Schulz (1980) had in detail evaluated measurement method on niche width and overlap. Here we will be discussed that by fuzzy method.
expresses the fittest niche of the species X. All conditions of physical and chemical environment of the 3.1.1. Niche width species within the niche V is optimum. The member˜ From above statement, when interval number ship function, which the species is subordinated to the λ (i = 1, 2, . . . , n) is a fuzzy number, Λ∗ = i niche V , is ˜ ˜ 0, if H −1 (v) = 0 µV (u) = n ∨(∧i=1 )µλi (ui ), otherwise, (u1 , . . . , un ) ∈ H −1 (v) ˜
˜
where H −1 (v) = {(u1 , u2 , . . . , un )|H(u1 , u2 , . . . , un ) = v}. Therefore, every species is corresponding to an optimal niche. It is a n-dimensional fuzzy set, in geometry, a n-dimensional fuzzy hypovolume. In particular, assuming that λi is a special fuzzy number, namely, ˜ denoted as λi = [mλ , nλ ], (i = interval number, i i 1, 2, . . . , n), which express ˜ecological amplitude of the species. According to Shelford’s “law of tolerance”, every species has a tolerant range to every environment, that is to say, there is an ecological minimum and maximum. The range between ecological minimum and maximum is called ecological amplitude. Thus, Λ∗ = (λ1 , λ2 , . . . , λn )
˜ ˜
˜
= ([mλ1 , nλ1 ], [mλ2 ,nλ2 ], . . . , [mλn ,nλn ]). is a vector consisted of n-dimensional closed intervals. H(Λ∗ )α = H(λ1 , λ2 , . . . , λn )α
˜ ˜
˜
= H(λ1,α , λ2,α , . . . , λn,α ),
α ∈ [0, 1]
is called truncated set of the fuzzy set H (Λ∗ ). When n-dimensional ecological factor λi (i = 1, 2, . . . , n) is
˜
(λ1 , λ2 , . . . , λn ) is the optional state of the species liv˜ space, ˜ ˜ ) represents the niche of the species X. ing H (Λ ˜ H (Λ) occupied by a species X must Therefore, ˜niche ˜ ˜i ) on every resources axis, H(λi ) have a projection H(λ ˜ niche on the ith dimensional ˜ expresses a projection axis, its width is, w(w(H(λi )) = nλi − mλi the width ˜ of niche occupied by the species X, therefore may be denoted as B(H (Λ)) = w(w(H(λ1 )), w(H(λ2 )), . . . , w(H(λn ))
1/2 n ˜ ˜ ˜ ˜ ˜ 2 (w(H(λi ))) . i=1
˜
The equation reflects resource diversity used by the species, namely, the tolerant (or used) range of the species X for n ecological factors. If a small part within whole resources is really used by the species, then there must be some w(H(λi )) are very small or equal zero. Therefore, B(H (Λ))˜ is small. In this case, we ˜ ˜ have a narrower niche. will say that the species 3.1.2. Niche overlap Assuming that H (Λ1 ) and H (Λ2 ) are the niche of ˜ ˜ ˜ The formula of the the species X and ˜Y, respectively.
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niche overlap can be written as the following H XY (Λ) = H (Λ1 ) ∩ H (Λ2 )
˜
˜
where
˜ ˜ ˜
˜ ˜ ˜ ˜ ∩(H(λ21 ), H(λ22 ), . . . , H(λ2n )) ˜ ˜ ˜ (HXY (λ1 ), HXY (λ2 ), . . . , HXY (λn )). ˜ ˜ ˜
= (H(λ11 ), H(λ12 ), . . . , H(λ1n ))
HXY (λi ) = HX (λ1i ) ∩ HY (λ2i )
˜
˜
˜
[max(mλ1i , mλ2i ), max(nπ1i , nλ2i )], i = 1, 2, . . . , n. λ1i = mλ1i , nλ1i ,
λ2i = [mλ2i , nλ2i ].
˜
˜
Therefore, H XY (Λ) is a n-dimensional closed fuzzy vector, the ˜niche˜ overlap is a overlap of the niche HX (λ1i ) and HY (λ2i ) projected on each environment ˜ λ1i and λ2i .˜ factor ˜ above˜ statement we know when HXY (λi ) is From more great, the niche overlap of the species X and Y is much more. This results in a very similar niche in which the species X and Y live, and inter-species competition will be intensify, Finally, one of the species could be eliminated or niche could be separated. According to the niche overlap defined by the paper, we know that it is a comprehensive index. If niche overlap happens only on some environment factors λi , the species may not be competition, certainly, species may be competition when their overlap niches are some decisive factors. The relation of niche overlap and competition may be described by the formula HXY (·). 3.2. Further consideration Fuzzy niche concept of the species is similar to Hutchinson’s concept in some ways, however, it has
some improvements over the classical concept. For making the fuzzy concept more sound and easy for use, still much work to be done.
Acknowledgements This research was supported by the National Nature science Foundation of China (Nos. 39970135, 30070139). References Abrams, P., 1980. Some comments on measuring niche overlap. Ecology 61 (1), 44–49. Bosserman, R.W., Ragade, R.K., 1982. Ecosystem analysis using fuzzy set theory. Ecol. Model. 16, 191–208. Christiansen, F.B., Fenchel, T.M., 1977. Theories of Population in Biological Communities. Springer, Berlin. Elton, C., 1933. The Ecology of the Animals. Méthuen, London. Giller, P.S., 1982. Community Structure and the Niche. Chapman and Hall, London. Grinnell, J., 1917. The niche relationships of the California thrasher. Auk. 21, 364–382. Horn, H.S., 1966. Measurement of overlap in comparative ecological studies. Am. Nat. 100, 419–424. Hutchinson, G.E., 1957. Concluding remarks. Cold Spring Harbor Synp. Quant. Biol. 22, 415–427. Luo, C., 1989. Introduction Fuzzy Sets. Beijing Normal University Press (in Chinese). Pielou, E.C., 1971. Niche width and overlap: a method for measuring them. Ecology. 53 (4), 687–692. Salski, A., 1992. Fuzzy knowledge-based models in ecological research. Ecol. Model. 63, 103–112. Slobodkichoff, C.N., Schulz, W.C., 1980. Measures of niche overlap. Ecology 61, 1051–1055. Sun R., 1988. The Principle of Animal Ecology. Beijing Normal University Press (in Chinese). Yang, X., Ma, J., 1992. A review on some terms related to niche and their measurement. Chin. J. Ecol. 11 (2), 44–49. Zadeh, L.A., 1965. Fuzzy sets. Inf. Control 8, 338–353.