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Ideal free distribution of unequal competitors can be determined by the form of competition
Z theor BioL (1989) 138, 347-352
Ideal Free Distribution of Unequal Competitors can be Determined by the Form of Competition RYSZARD KORONA
Institute of Environmental Biology, Jagiellonian University, M. Karasia 6, 30-060 Krakow, Poland (Received 5 October 1988, and accepted in revised form 7 February 1989) A simple mathematical model of the ideal free distribution with unequal competitors is developed. It is assumed that resource input to patches is continuous and that resources are consumed immediately. Animals compete for resources in interactions with randomly met opponents. A unique equilibrium distribution is predicted. This distribution yields the same average gains in every patch. General agreement between this prediction and the results of empirical studies is discussed.
Introduction It is easy to apply the theory of the "ideal free distribution", IFD, (Fretwell & Lucas, 1970) to situations when all individuals are identical and costs of travel between patches are negligible. In this case individual gains should be equal and numbers of animals in different patches should reflect the relative qualities of these patches. During the two last decades several empirical studies were carded out to test these predictions. In "continuous input experiments" (Parker & Sutherland, 1986) resource items enter patches continuously and are utilized immediately. In these situations average gains in different patches are equal as a rule, despite individual differences in rewards within patches. According to the model proposed by Parker & Sutherland (1986), the equality of average gains can appear in "continuous input" models, but only as the most probable of many unstable equilibrium distributions. However, Houston & McNamara (1988) in a general analysis of this model show that the most probable distributions o f unequal competitors cannot result in equal gains in patches. I think that the crucial question is how to define competition between individuals. As it will be shown this can produce quite different distribution effects. My own approach, called the "local competition" model is presented first. Its main feature is that it brings about a unique solution which is consistent with the "traditional" IFD predictions. In the Discussion it will be compared with the model of Parker & Sutherland (1986) and the relevance of these models to earlier done experiments will be considered.
The Model Assume that animals compete for limited and patchily distributed resources and that they can estimate the quality of every patch ideally and choose one freely. 347 0022-5193/89/110347 + 06 $03.00/0
(~) 1989 Academic Press Limited
348
R. KORONA
Here, a system of two patches, A and B, is considered. The quality of a patch is defined as the number of indivisible items of resources entering it in a time unit (rate of inflow). Resource particles are utilized immediately by individuals scrambling for them. Individuals differ and n phenotypes can be distinguished. To each phenotype i (i = 1, 2, 3 , . . . , n) a given "competitive value", C~ (C~ = Cj for i = j ) , can be attributed. The competitive value C~ is determined by characters of the individual of phenotype i and determines its chances o f winning in conflicts with other competitors (e.g. the bigger the animal the higher its competitive value). In this study it is assumed that C~ remains constant in time in all patches. The term "local competition" describes the way in which individuals compete and not the spatial structure of populations. The crucial assumption is that the competitive value of a given individual manifests itself in contacts with other, randomly encountered competitors. Competitive behaviour (i.e. pushing, threatening) occurs only when an individual that is about to take a piece of resource considers another one as a rival. Except for this, each individual is equally free, i.e. it can get close to a resource particle and repeat an unsuccessful attempt to catch it. Let us calculate the average total gain of an individual of phenotype i in patch A. The average number of resource items per individual in this patch is given by the equation LA =
Q~ Ep,~
(j = 1, 2 , . . . , n),
(1)
J
where N~ denotes the total number of individuals of phenotype j in both patches and pj is the proportion of such individuals in patch A, and QA is the quality of patch A. An individual of any phenotype meets an individual of phenotype k ( k = 1, 2 , . . . , n) in patch A with the probability
p~ M~A--
ypj~
(2)
J
The chances of getting a particle are considered to be proportional to the competitive values of both rivals. Thus, when a given particle is taken in a conflict between i and k, then the probability that an individual of phenotype i succeeds is equal to Ci
Ci,k -- Ci + Ck"
(3)
The total reward Ri.A of a competitor of phenotype i in A depends on how much resources per competitor there are (LA) and whom a competitor encounters (Mk~A for every phenotype), that determines its chances in conflicts (q.k). Therefore Ri.A = 2LA ~, Mk, ACi.k. k
(4)
IDEAL
FREE
349
DISTRIBUTION
Note that the right side is multiplied by two because two individuals are directly engaged when one particle is taken. This can be seen by assuming that all competitors are identical, in which case R~.A= LA. Equation (4), after substituting eqns (1), (2) and (3), yields
2QA ( piN, p2N2 p,N, ) R'A-~,p~N i " \Y. pjN~ c,.,+ ~,PJN c,2+....+ ~,p.ilVj c,.,, . J
J
J
J
Calculations of the reward of an individual of phenotype i in patch B, R~.n, are made analogically. When individuals of all phenotypes can be found in different patches in equilibrium distribution, then it follows from the IFD theory that
Ri.A = Ri.8.
(5)
To see intuitively the meaning of this condition, suppose that only animals of one phenotype are to have equal rewards in both patches. For many distributions of this phenotype it would be possible to find such distributions of others that would satisfy this condition. But the rewards of other phenotypes have to be balanced also. All of them play the same "mixed strategy" (Maynard Smith, 1982). Thus, our aim is to find the solution for every phenotype simultaneously, namely the solution of a system of n equations, tt can be shown (see the Appendix) that this system simplifies to the following condition for each phenotype i
Q~
Q~
(6)
Note that at equilibrium the proportion of individuals which stay in patch A is the same for each phenotype. Let us denote this frequency as p* and solve eqn (6)
Q~
p* - - -
QA + Qn"
(7)
Thus, competitors have to be distributed in a unique way. The equilibrium distribution has three features: (i) the total number of animals is divided between patches in a proportion reflecting the quality of the patches, (ii) each phenotype distributes itself between different patches in the same proportions and, as a result, (iii) the average gain for all individuals is equal within both patches. (It was assumed only that rewards were equal for each particular phenotype.) Such predictions are very clear and can be verified unequivocally. It is also possible to relax two previous restrictions. Firstly, the chance of winning need not be linearly proportional to the competitive value [eqn (3)]. This chance can increase for the "stronger" individual 2 2 [e.g. Ci.k= Ci/(C~ + C2)] or decrease, but it would not change the outcome of the model (see matrices X and Y in the Appendix). Secondly, I assumed that there
350
R. KORONA
are two patches only. But the above model can be applied to describe a process between two neighbouring patches in a system of many patches.
Discussion
In order to introduce the approach of Parker & Sutherland (1986) let us consider a hypothetical example. Imagine a pond inhabited by a certain species of filtrating animals. There are suitable patches for these animals on the bottom of the pond. Within a given patch the individual gains of competitors are linearly proportional to their rates of water filtration. According to our intuition there may be many equilibrium distributions of unequal animals. To justify this expectation it is sufficient to imagine that in an equilibrium state one animal is interchanged with two smaller ones from another patch, and that the sum of their filtration rates is equivalent to that of the replaced animal. The rewards of other competitors in both patches will remain unchanged, but the average gain per individual is lowered in the first one and increased in the second. We can ask if different distributions are equally probable and if the distributions resulting in equal average rewards in both patches are the most probable. Houston & McNamara (1988) in their general model find that for systems with two phenotypes the stronger competitors more frequently occupy the better patch and this yields higher average gain in this patch. They argue that when the total number of competitors is small then the most probable distributions bring about equal rewards. This does not mean, however, that the average expected rewards are equal. They are not as one can see in fig. l(i) in the paper of Houston & McNamara (1988). Generally, in the systems described above, each individual influences all other competitors' gains immediately, because they directly divide a common stream of resources. In a sense each individual weighs its strength or efficiency against the whole group. There can be no doubt that the model described above does apply in numerous situations, but I believe that already existing experimental studies of animal distribution, when the input of resources is continuous, generally fit the assumptions of the "local competition" model better. The reason is that individuals actively tried to get as many items as possible, and engaged in conflicts with particular competitors. The experiments involving sticklebacks (Milinski, 1979, 1984), mallards (Harper, 1982), and cichlid fish (Godin & Keenleyside, 1984) can be classified as "local competition" studies. Here all the animals were actively scrambling for distinct food particles and were free to move within a patch. This freedom was proved by Godin & Keenleyside (1984) who could not detect despotism or monopolization. In all these experiments animals were able to change patches and they did so, hence different phenotypes were present in both patches. Patches were qualitatively identical and differed only in the rate of resource input. Thus, the assumptions of the model were relevant. The results reveal that both the total numbers of competitors in patches and the average gains in patches were consistent with ideal free predictions. Moreover, the data from these experiments confirm the prediction about the similarity of distributions of different phenotypes between patches. Explicitly it was
IDEAL
FREE
DISTRIBUTION
351
f o r m u l a t e d by Milinski (1984): "the g o o d competitors as well as the p o o r ones were distributed between the patches in the ratio o f patch profitabilities". It must be stressed that the presented m o d e l is relevant for specified situations. In a sense, its a s s u m p t i o n that the competitive abilities o f animals manifests itself in interactions with other individually met competitors is an opposite extreme o f the a s s u m p t i o n m a d e by Parker & Sutherland (1986) that an individual weighs its strength against the whole group. M a n y other systems are possible and can be studied both theoretically a n d experimentally. After all, the most attractive feature o f the I F D theory is that it predicts the p o p u l a t i o n p h e n o m e n a from individual behaviour. I thank A. Lomnicki, J. Kozlowski and D. Padley for their advice and helpful criticism. This study was supported by a grant of the Polish Academy of Sciences CPBP 04.03-1/1.
REFERENCES S. D. & LUCAS, H. L. (1970). On territorial behaviour and other factors influencing habitat distribution in birds. Aeta biotheor. 19, 16. G O D I N , J.-G. J. & KEENLEYSIDE,M. H. A. (1984). Foraging on patchily distributed prey by a cichlid fish (Teleostei, Cichlidae): a test of the ideal free distribution theory. Anita. Behav. 32, 120. HARPER, D. G. C. (1982). Competitive foraging in mallards: "ideal free" ducks. Anim. Behav. 30, 575. H O U S T O N , A. I. & MCNAMARAJ. M. (1988). The ideal free distributions when competitive abilities differ: an approach based on statistical mechanics. Anita. Behav. 36, 166. MAYNARDS M I T H , J. (1982). Evolution and the Theory of Games. Cambridge: Cambridge University Press. MILINSKI, M. (1979). An evolutionarily stable feeding strategy in sticklebacks. Z. Tierpsychol. 51, 36. MILINSKI, M. (1984). Competitive resource sharing: an experimental test of a learning rule for ESS. Anita. Behav. 32, 233. PARKER, G. A. & SUTHERLAND, W. J. (1986). Ideal free distributions when individuals differ in competitive ability: phenotype limited ideal free models. Anim. Behav. 34, 1222. FRETWELL,
APPENDIX E q u a t i o n (5) for n p h e n o t y p e s in the matrix n o t a t i o n can be re-written as follows:
= [ c l l... . . . .cl2 ..... ...
c,n lxl/(I[(1-p')N'-p2)N2"'"
C21
C22
•••
C2n
Lc,1
c,2
...
c,,
(8) l(1-p,)N,
Let X be the square matrix in eqn (8). To eliminate X, both sides o f the e q u a t i o n s h o u l d be multiplied f r o m the left by the inverse matrix X -1. To prove the existence o f X -~ it is sufficient to s h o w that the det X is different f r o m zero. By eqn (3),
352
R.
det X =
KORONA
C,
C1
Ci
C I+ C I
C 1 "JI-C 2
C 1 "~ C n
C2 C2 + Cl
C2 C2+C2
C2 C2+C.
c.
c.
c.
c.+c~
c.+c2
C.+n
= fi C~ det Y i=1
where
y=
1
1
1
C l ~- C I
C 1 "~- C2
C I -~- Cn
1
1
1
C2+C~
C2+C2
C2+C.
1
1
1
C.+C,
C.+C2
C.+C.
Matrix Y is symmetric, so everything that is true for rows of Y can be applied to columns as well. Let us assume that Ci is positive and Ci are so arranged that C~ < C~+,, which does not limit the generality of the model. Some features of Y can be easily detected• (i) Elements in every row are positive and monotonically decreasing and the rate of decrease is different for each row. (ii) Let us choose two rows g and h (g < h). The ratio of corresponding elements (from the column i) is: 1
C ~+ C, Ch+C, r(g°h°i) 1 - C~ + Ci • =
ch + c, From the above equation we can conclude that for each pair of rows (g and h), the ratios of two corresponding elements r are always positive and monotonically decreasing and the rate of decrease is different for each pair o f rows. By (i) and (ii) we see that no two rows can be identical or proportional and that no linear combination of rows can result in a row in which any two elements are equal. Therefore a row consisting of zeros cannot be produced. Consequently, det Y ~ 0, hence det X ~ 0 and X -I exists• After multiplying two sides of eqn (8) by X -~ we get FP1Nl]
...
[(1-pl)N~ ]
-[
Since two vectors are equal if their corresponding elements are equal the last equality implies eqn (6).
Ideal Free Distribution of Unequal Competitors can be Determined by the Form of Competition RYSZARD KORONA
Institute of Environmental Biology, Jagiellonian University, M. Karasia 6, 30-060 Krakow, Poland (Received 5 October 1988, and accepted in revised form 7 February 1989) A simple mathematical model of the ideal free distribution with unequal competitors is developed. It is assumed that resource input to patches is continuous and that resources are consumed immediately. Animals compete for resources in interactions with randomly met opponents. A unique equilibrium distribution is predicted. This distribution yields the same average gains in every patch. General agreement between this prediction and the results of empirical studies is discussed.
Introduction It is easy to apply the theory of the "ideal free distribution", IFD, (Fretwell & Lucas, 1970) to situations when all individuals are identical and costs of travel between patches are negligible. In this case individual gains should be equal and numbers of animals in different patches should reflect the relative qualities of these patches. During the two last decades several empirical studies were carded out to test these predictions. In "continuous input experiments" (Parker & Sutherland, 1986) resource items enter patches continuously and are utilized immediately. In these situations average gains in different patches are equal as a rule, despite individual differences in rewards within patches. According to the model proposed by Parker & Sutherland (1986), the equality of average gains can appear in "continuous input" models, but only as the most probable of many unstable equilibrium distributions. However, Houston & McNamara (1988) in a general analysis of this model show that the most probable distributions o f unequal competitors cannot result in equal gains in patches. I think that the crucial question is how to define competition between individuals. As it will be shown this can produce quite different distribution effects. My own approach, called the "local competition" model is presented first. Its main feature is that it brings about a unique solution which is consistent with the "traditional" IFD predictions. In the Discussion it will be compared with the model of Parker & Sutherland (1986) and the relevance of these models to earlier done experiments will be considered.
The Model Assume that animals compete for limited and patchily distributed resources and that they can estimate the quality of every patch ideally and choose one freely. 347 0022-5193/89/110347 + 06 $03.00/0
(~) 1989 Academic Press Limited
348
R. KORONA
Here, a system of two patches, A and B, is considered. The quality of a patch is defined as the number of indivisible items of resources entering it in a time unit (rate of inflow). Resource particles are utilized immediately by individuals scrambling for them. Individuals differ and n phenotypes can be distinguished. To each phenotype i (i = 1, 2, 3 , . . . , n) a given "competitive value", C~ (C~ = Cj for i = j ) , can be attributed. The competitive value C~ is determined by characters of the individual of phenotype i and determines its chances o f winning in conflicts with other competitors (e.g. the bigger the animal the higher its competitive value). In this study it is assumed that C~ remains constant in time in all patches. The term "local competition" describes the way in which individuals compete and not the spatial structure of populations. The crucial assumption is that the competitive value of a given individual manifests itself in contacts with other, randomly encountered competitors. Competitive behaviour (i.e. pushing, threatening) occurs only when an individual that is about to take a piece of resource considers another one as a rival. Except for this, each individual is equally free, i.e. it can get close to a resource particle and repeat an unsuccessful attempt to catch it. Let us calculate the average total gain of an individual of phenotype i in patch A. The average number of resource items per individual in this patch is given by the equation LA =
Q~ Ep,~
(j = 1, 2 , . . . , n),
(1)
J
where N~ denotes the total number of individuals of phenotype j in both patches and pj is the proportion of such individuals in patch A, and QA is the quality of patch A. An individual of any phenotype meets an individual of phenotype k ( k = 1, 2 , . . . , n) in patch A with the probability
p~ M~A--
ypj~
(2)
J
The chances of getting a particle are considered to be proportional to the competitive values of both rivals. Thus, when a given particle is taken in a conflict between i and k, then the probability that an individual of phenotype i succeeds is equal to Ci
Ci,k -- Ci + Ck"
(3)
The total reward Ri.A of a competitor of phenotype i in A depends on how much resources per competitor there are (LA) and whom a competitor encounters (Mk~A for every phenotype), that determines its chances in conflicts (q.k). Therefore Ri.A = 2LA ~, Mk, ACi.k. k
(4)
IDEAL
FREE
349
DISTRIBUTION
Note that the right side is multiplied by two because two individuals are directly engaged when one particle is taken. This can be seen by assuming that all competitors are identical, in which case R~.A= LA. Equation (4), after substituting eqns (1), (2) and (3), yields
2QA ( piN, p2N2 p,N, ) R'A-~,p~N i " \Y. pjN~ c,.,+ ~,PJN c,2+....+ ~,p.ilVj c,.,, . J
J
J
J
Calculations of the reward of an individual of phenotype i in patch B, R~.n, are made analogically. When individuals of all phenotypes can be found in different patches in equilibrium distribution, then it follows from the IFD theory that
Ri.A = Ri.8.
(5)
To see intuitively the meaning of this condition, suppose that only animals of one phenotype are to have equal rewards in both patches. For many distributions of this phenotype it would be possible to find such distributions of others that would satisfy this condition. But the rewards of other phenotypes have to be balanced also. All of them play the same "mixed strategy" (Maynard Smith, 1982). Thus, our aim is to find the solution for every phenotype simultaneously, namely the solution of a system of n equations, tt can be shown (see the Appendix) that this system simplifies to the following condition for each phenotype i
Q~
Q~
(6)
Note that at equilibrium the proportion of individuals which stay in patch A is the same for each phenotype. Let us denote this frequency as p* and solve eqn (6)
Q~
p* - - -
QA + Qn"
(7)
Thus, competitors have to be distributed in a unique way. The equilibrium distribution has three features: (i) the total number of animals is divided between patches in a proportion reflecting the quality of the patches, (ii) each phenotype distributes itself between different patches in the same proportions and, as a result, (iii) the average gain for all individuals is equal within both patches. (It was assumed only that rewards were equal for each particular phenotype.) Such predictions are very clear and can be verified unequivocally. It is also possible to relax two previous restrictions. Firstly, the chance of winning need not be linearly proportional to the competitive value [eqn (3)]. This chance can increase for the "stronger" individual 2 2 [e.g. Ci.k= Ci/(C~ + C2)] or decrease, but it would not change the outcome of the model (see matrices X and Y in the Appendix). Secondly, I assumed that there
350
R. KORONA
are two patches only. But the above model can be applied to describe a process between two neighbouring patches in a system of many patches.
Discussion
In order to introduce the approach of Parker & Sutherland (1986) let us consider a hypothetical example. Imagine a pond inhabited by a certain species of filtrating animals. There are suitable patches for these animals on the bottom of the pond. Within a given patch the individual gains of competitors are linearly proportional to their rates of water filtration. According to our intuition there may be many equilibrium distributions of unequal animals. To justify this expectation it is sufficient to imagine that in an equilibrium state one animal is interchanged with two smaller ones from another patch, and that the sum of their filtration rates is equivalent to that of the replaced animal. The rewards of other competitors in both patches will remain unchanged, but the average gain per individual is lowered in the first one and increased in the second. We can ask if different distributions are equally probable and if the distributions resulting in equal average rewards in both patches are the most probable. Houston & McNamara (1988) in their general model find that for systems with two phenotypes the stronger competitors more frequently occupy the better patch and this yields higher average gain in this patch. They argue that when the total number of competitors is small then the most probable distributions bring about equal rewards. This does not mean, however, that the average expected rewards are equal. They are not as one can see in fig. l(i) in the paper of Houston & McNamara (1988). Generally, in the systems described above, each individual influences all other competitors' gains immediately, because they directly divide a common stream of resources. In a sense each individual weighs its strength or efficiency against the whole group. There can be no doubt that the model described above does apply in numerous situations, but I believe that already existing experimental studies of animal distribution, when the input of resources is continuous, generally fit the assumptions of the "local competition" model better. The reason is that individuals actively tried to get as many items as possible, and engaged in conflicts with particular competitors. The experiments involving sticklebacks (Milinski, 1979, 1984), mallards (Harper, 1982), and cichlid fish (Godin & Keenleyside, 1984) can be classified as "local competition" studies. Here all the animals were actively scrambling for distinct food particles and were free to move within a patch. This freedom was proved by Godin & Keenleyside (1984) who could not detect despotism or monopolization. In all these experiments animals were able to change patches and they did so, hence different phenotypes were present in both patches. Patches were qualitatively identical and differed only in the rate of resource input. Thus, the assumptions of the model were relevant. The results reveal that both the total numbers of competitors in patches and the average gains in patches were consistent with ideal free predictions. Moreover, the data from these experiments confirm the prediction about the similarity of distributions of different phenotypes between patches. Explicitly it was
IDEAL
FREE
DISTRIBUTION
351
f o r m u l a t e d by Milinski (1984): "the g o o d competitors as well as the p o o r ones were distributed between the patches in the ratio o f patch profitabilities". It must be stressed that the presented m o d e l is relevant for specified situations. In a sense, its a s s u m p t i o n that the competitive abilities o f animals manifests itself in interactions with other individually met competitors is an opposite extreme o f the a s s u m p t i o n m a d e by Parker & Sutherland (1986) that an individual weighs its strength against the whole group. M a n y other systems are possible and can be studied both theoretically a n d experimentally. After all, the most attractive feature o f the I F D theory is that it predicts the p o p u l a t i o n p h e n o m e n a from individual behaviour. I thank A. Lomnicki, J. Kozlowski and D. Padley for their advice and helpful criticism. This study was supported by a grant of the Polish Academy of Sciences CPBP 04.03-1/1.
REFERENCES S. D. & LUCAS, H. L. (1970). On territorial behaviour and other factors influencing habitat distribution in birds. Aeta biotheor. 19, 16. G O D I N , J.-G. J. & KEENLEYSIDE,M. H. A. (1984). Foraging on patchily distributed prey by a cichlid fish (Teleostei, Cichlidae): a test of the ideal free distribution theory. Anita. Behav. 32, 120. HARPER, D. G. C. (1982). Competitive foraging in mallards: "ideal free" ducks. Anim. Behav. 30, 575. H O U S T O N , A. I. & MCNAMARAJ. M. (1988). The ideal free distributions when competitive abilities differ: an approach based on statistical mechanics. Anita. Behav. 36, 166. MAYNARDS M I T H , J. (1982). Evolution and the Theory of Games. Cambridge: Cambridge University Press. MILINSKI, M. (1979). An evolutionarily stable feeding strategy in sticklebacks. Z. Tierpsychol. 51, 36. MILINSKI, M. (1984). Competitive resource sharing: an experimental test of a learning rule for ESS. Anita. Behav. 32, 233. PARKER, G. A. & SUTHERLAND, W. J. (1986). Ideal free distributions when individuals differ in competitive ability: phenotype limited ideal free models. Anim. Behav. 34, 1222. FRETWELL,
APPENDIX E q u a t i o n (5) for n p h e n o t y p e s in the matrix n o t a t i o n can be re-written as follows:
= [ c l l... . . . .cl2 ..... ...
c,n lxl/(I[(1-p')N'-p2)N2"'"
C21
C22
•••
C2n
Lc,1
c,2
...
c,,
(8) l(1-p,)N,
Let X be the square matrix in eqn (8). To eliminate X, both sides o f the e q u a t i o n s h o u l d be multiplied f r o m the left by the inverse matrix X -1. To prove the existence o f X -~ it is sufficient to s h o w that the det X is different f r o m zero. By eqn (3),
352
R.
det X =
KORONA
C,
C1
Ci
C I+ C I
C 1 "JI-C 2
C 1 "~ C n
C2 C2 + Cl
C2 C2+C2
C2 C2+C.
c.
c.
c.
c.+c~
c.+c2
C.+n
= fi C~ det Y i=1
where
y=
1
1
1
C l ~- C I
C 1 "~- C2
C I -~- Cn
1
1
1
C2+C~
C2+C2
C2+C.
1
1
1
C.+C,
C.+C2
C.+C.
Matrix Y is symmetric, so everything that is true for rows of Y can be applied to columns as well. Let us assume that Ci is positive and Ci are so arranged that C~ < C~+,, which does not limit the generality of the model. Some features of Y can be easily detected• (i) Elements in every row are positive and monotonically decreasing and the rate of decrease is different for each row. (ii) Let us choose two rows g and h (g < h). The ratio of corresponding elements (from the column i) is: 1
C ~+ C, Ch+C, r(g°h°i) 1 - C~ + Ci • =
ch + c, From the above equation we can conclude that for each pair of rows (g and h), the ratios of two corresponding elements r are always positive and monotonically decreasing and the rate of decrease is different for each pair o f rows. By (i) and (ii) we see that no two rows can be identical or proportional and that no linear combination of rows can result in a row in which any two elements are equal. Therefore a row consisting of zeros cannot be produced. Consequently, det Y ~ 0, hence det X ~ 0 and X -I exists• After multiplying two sides of eqn (8) by X -~ we get FP1Nl]
...
[(1-pl)N~ ]
-[
Since two vectors are equal if their corresponding elements are equal the last equality implies eqn (6).