Behavioral Ecology

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Behavioral Ecology Vol. 14 No. 1: 109-115
© 2003 International Society for Behavioral Ecology

The ideal free distribution when the resource is variable

Hiroshi Hakoyama

Hokkaido National Fisheries Research Institute, Katsurakoi 116, Kushiro 085-0802, Japan

Address correspondence to H. Hakoyama. E-mail: hako{at}affrc.go.jp.

Received 17 December 2001; revised 14 May 2002; accepted 14 May 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
On the basis of the ideal free distribution (IFD) model, two stochastic models that incorporate the uncertainty of the information used for decision making were considered to investigate the effects of the variability in the resource supply rate on the IFD under continuous input conditions. In the uncertain-information model, competitors cannot trace the variation of the supply rate and use the expectation of the supply rate or previous payoffs for decision making. Both submodels predict matching of means, in which the average number of competitors for each patch is proportional to the average supply rate in the patch. In the perfect-information model, competitors continuously know and trace the environment conditions. Numerical predictions depend on the relative size of the resource variance between patches. When the resource variance in the good patch is sufficiently larger than that in the poor patch, it predicts undermatching of means; when the variance of the supply rate for each patch is small and proportional to the average of the supply rate in the patch, it predicts matching of means; and when the resource variance in the poor patch is larger than (or equal to) that in the good patch, it predicts overmatching of means. These results indicate the importance of clarifying the assumption on the uncertainty in information for decision making and the type of the resource variance for the test of the IFD under conditions where the resource supply rate is stochastic.

Key words: continuous input condition, ideal free distribution, resource variability, uncertainty.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
The ideal free distribution (IFD) theory predicts the distribution of animals that compete for resources that are distributed in patches (Fretwell, 1972; Fretwell and Lucas, 1970). If all competitors are equal in food acquisition ability, ifthey can move between patches without cost, and if they have perfect information of the resource supply and compe titors' distribution, then each competitor is expected to go to the patch where its gain is the highest (an evolutionarilystable strategy; Maynard Smith, 1982). As a result, atequilibrium theintake rate is equal both across patches and between competitors. Because the resource input rate remains constant through time in the original IFD model, no individual could improve its intake rate by moving at equilibrium. Under continuous input conditions in which resource items are used as soon as they are input into the patches (Parker and Sutherland, 1986; Tregenza, 1994), a competitor's intake rate is equal to the supply rate divided by the number of competitors present. In this case, the IFD theory predicts that the number of competitors for each patch is proportional to the resource input rate in the patch (input-matching rule).

However, as shown by Recer et al. (1987) and Earn and Johnstone (1997), when the resource supply rate is a random variable, continuous input models do not always predict matching of means in which the average number of competitors for each patch is proportional to the average resource supply rate in the patch. I show here that numerical predictions depend on the uncertainty of information used for the decision making and the type of the resource variance. "Uncertainty" means lack of knowledge about the state of the environment or other decision variables. Competitors will decide to choose a foraging patch using the information on the competitors' distribution, on the resource supply rate of each patch, and on the competitors' intake rate (payoff). The uncertainty of such information affects the prediction of the average distribution of competitors. Furthermore, according to the type of the resource variance, not only undermatching but also overmatching may occur. "Undermatching" means that competitors underuse the good patch relative to matching of means and vice versa. As a specific case, Recer et al. (1987) have shown that if the resource supply rate follows a Markov process in one patch and it is a constant in a second patch, competitors with perfect information underuse the variable patch relative to matching of means. Earn and Johnstone (1997) have also shown that if the coefficient of variation of the resource supply rate is constant for all patches, competitors with perfect information underuse the good patch relative to matching of means.

Previous continuous input studies have tested matching of means, though the resource supply rate contains some degree of variance (Earn and Johnstone, 1997). Therefore, the tests might derive a conclusion in error. It is necessary to clarify the conditions of matching of means to derive the adequate prediction to test an ideal free theory.

This article gives an analysis of the extended model of Recer et al. (1987) and that of Earn and Johnstone (1997). I show that the prediction based on the IFD of the average number of competitors is dependent on (1) the uncertainty of information used for decision making and (2) the type of the variance in the resource supply rate.

The model
Suppose that two patches are available to a group of competitors. Let R_i be the resource supply rate in the patch i (i = 1, 2). The food supply rates are random variables:

where q_i are the average food supply rates and _i are independent random variables with E[_i] = 0 and var[_i] = _i². E and var denote the expectation and variance, respectively. The food supply rate of each patch fluctuates in dependently around q_i with a variance of _i². I have ignored here the cases in which _i are dependent random variables (except for a brief consideration for the perfectly synchronized fluctuation in Appendix C).

There are N_i competitors in patch i and a total of n competitors overall (n = N₁ + N₂). All the competitors are equal in food acquisition ability and can move between patches without cost. Competitors distribute themselves so each obtains the highest resource intake. Assuming continuous input conditions, a competitor's intake rate in a patch is equal to the supply rate divided by the number of competitors present. Note that there is no depletion of resources in continuous input conditions, because this assumes that food items arrive continuously and are consumed immediately (Parker and Sutherland, 1986; Tregenza, 1994). If there is a depletion of resources, the input-matching rule no longer holds, even if the resource supply rate of each patch is a constant. In this case, we have to apply an extended model such as the interference model (Sutherland, 1983). In this study, I ignore risk-sensitive foraging such as the minimization of the death rate (see Caraco et al., 1980; Houston and McNamara, 1997), though it also causes deviation from the input-matching rule.

The mean proportion of competitors in patch i, E[N_i]/n, is one of the IFD predictions of most interest. Here, I define

Equation 2 indicates that competitors underuse the good patch relative to matching of means in the condition of undermatching of means, and competitors overuse the good patch relative to matching of means in the condition of overmatching of means.

Uncertain information model
The uncertain information model assumes that competitors cannot trace the variation of the resource supply rate and thatcompetitors have to choose a foraging patch with an expectation of the resource availability for each patch. Here I examine two submodels with different levels of the uncertainty in the information for patch choice: (1) when there is uncertainty in the information on the supply rate and (2) when there is uncertainty in the information on the supply rate and the competitors' distribution.

Uncertainty in the information on the supply rate
Suppose that competitors know the distribution of the other compe titors, but know only the exact expectation of the resource supply rate for each patch. For example, trout in mountain torrent streams will know the distribution of the other individuals using visual cues but have to expect that the availability of food items drifting to each site will be unpredictable (e.g., Fausch, 1984). Recer et al. (1987) assumed this for the case that resource input fluctuates faster than competitors respond. This assumption is also similar to that of Abrahams' (1986) perception limit model in which competitors know the distribution of the other compe titorsbut do not know the exact resource supply rate and have a limited ability to resolve differences between patch qualities.

In this case, when N_i competitors stay in patch i, a competitor expects the resource intake rate (payoff) in the patch to be E[R_i]/N = q_i/N_i. All competitors are expected to go to the patch where the expected resource intake rate is the highest. As a result, the average intake rate is equal in both patches at equilibrium:

where P_i is the intake rate and n_i is the number of competitors in patch i at equilibrium. P_i is a random variable (var[P_i] = _i²/n_i²). No individual can improve its average intake rate by moving once an equilibrium state has been reached (n_i are constants). Because N_i = n_i at equilibrium, then E[N_i]/n = q_i/(q₁ + q₂), which indicates matching of means.

Uncertainty in the information on the supply rate and the competitors' distribution
Suppose that all competitors know only their own payoff (food gain) and cannot observe both the distribution of the other competitors and the resource supply rate of each patch. Furthermore, suppose that competitor j chooses patch i at trial t with a probability _i,j(t) anand each competitor learns the probability from its own previous payoffs. In this case, the average proportion of competitors at patch i for _i,j(t) is E[N_i]/n = _j=1ⁿ _i,j(t)/n. Note that the stochasticity of N_i is not caused by the stochasticity of resources, but by the stochasticity of decision making [_i,j(t)]. I here assume the learning rule to decide _i,j(t) is the evolutionarily stable (ES) learning rule (Harley, 1981). Harley (1981) showed that if the learning rule is an ES learning rule, it is a rule for learning evolutionarily stable strategies (ESSs). Harley (1981) also proved the following for _i,j(t) at the ESS:

where P_i,j(t) is the payoff of competitor j in patch i at trial t, and the right arrow means "asymptotically approaches." The payoff P_i,j(t) which the competitor chooses is R_i(t)/N_i(t). For another patch which is not used, P_i,j(t) = 0. In this case, the proportion of competitors at patch i for the ES learning rule becomes (appendix A):

Matching of means is, therefore, always predicted for competitors with the ES learning rule.

I also examined a case in which competitors adopt the relative payoff sum (RPS) learning rule (Harley, 1981), using Monte Carlo simulation (Appendix A). The RPS learning rule is a concrete rule that approximates the ES learning rule (Harley, 1981). After the initial phase of the learning period, the mean proportion of competitors in patch i, E[N_i]/n, become almost equal to q_i/(q₁ + q₂) (Appendix A). This result indicates matching of means for competitors with the RPS learning rule.

Perfect information model
Suppose that all the competitors have perfect information on the environment. This assumption is the same as that of Earn and Johnstone (1997). In this model, competitors know the stochastic supply rate of resources. Therefore all competitors trace the environment and obtain the equal resource intake rate in all patches constantly (equal intake prediction):

where P_i and N_i are also random variables that have a stochasticity caused by the stochasticity of resources. The mean and variance of the resource intake rate are E[P_i] = E[(R₁ + R₂)/n] = (q₁ + q₂)/n and var[P_i] = (₁² + ₂²) /n².

From Equation 6, the IFD theory also predicts that the proportion of competitors should be equal to the proportion of resources constantly (input-matching rule). This can be expressed as

By taking the average of both sides of Equation 7,

Equation 8 indicates that the perfect information model does not always predict matching of means. That is, the input-matching rule does not always yield matching of means. The prediction of the average distribution of competitors depends on the relative size of the resource variance between patches. When the resource variance is small, Equation 8 becomes

where the first term of the right-hand side is the prediction of matching of means, and the second term of the right-hand side is the bias from matching of means (Appendix B). The bias term of Equation 9 indicates that if q₁ > q₂ and if the resource variance in patch 1 (good patch) is sufficiently larger than that in patch 2 (poor patch), the model predicts undermatching of means. In contrast, if the resource variance in patch 2 (poor patch) is larger than (or equal to) that in patch 1 (good patch), the model predicts overmatching of means. If the resource variance for each patch, _i², is proportional to the average of the supply rate in the patch, q_i, the bias term of Equation 9 is 0 (i.e., matching of means). Icall this type of the resource variance the type II variance. Equation 9 also indicates that if the average of the supply rate is equal between patches (q₁ = q₂), competitors underuse the patch with larger variance. If patch 2 does not vary temporally ( _2^2 = 0), the variable patch (patch 1) is always underused by competitors. This result is consistent with that of Recer et al. (1987), which is explained briefly in the introduction.

Even when the resource variance is large, the nature of the model is also retained qualitatively. To show this, I examine three typical types of the variance for the resource supply rate, R_i: type I (_i² = q_i²v_I²), type II (_i² = q_iv_II²), and type III (_i² = v_III²), where v_I², v_II² and v_III² are constants. The relative size of the resource variance in the poor patch increases in ascending order of the types (i.e., type I < type II < type III). With type I variance, the coefficient of variation of the supply rate is a constant (= v_I) for all patches. Earn and Johnstone (1997) regarded the type I variance as a standard. If R_i is theconvolution of independent and identical unit random variables, R_i has the type II variance (e.g., Hoel, 1971). Therefore, in continuous input experiments which introduce food items at regular intervals, if each independent food input event has an identical average and variance in the re source profitability (e.g., the size variation in the food items), the total variance in the resource supply rate becomes the type II variance. The Poisson distribution also provides the type II variance. In the type III variance, the resource variance of thepoor patch is equal to that of the good patch. Note that the standard statistical models such as Student's t test assume the type III variance.

For the type I variance, the perfect information model predicts undermatching of means (Earn and Johnstone, 1997; Appendix C). In contrast, for the type III variance, the model predicts overmatching of means (Appendix C). When the resource supply rate follows a gamma distribution, the type II variance always causes matching of means (Appendix D). Because the gamma distribution which is decided by two parameters has flexibility to approximate the actual distribution (see Hoel, 1971), the type II variance will yield the prediction close to matching of means in many other distributions. Numerical analysis using a computer revealed that a log-normal distribution with the type II variance also predicted matching ofmeans approximately (results not shown).

Figure 1 shows the relationship between the proportion of resources, q_i/(q₁ + q₂), and the average proportion of competitors, E[N_i]/n, for the type I, II, and III variances. Asignificant degree of undermatching or overmatching of means occurs only when there is a large variance of the resource supply rate. For the type I variance, significant bias occurs due to extremely large coefficients of variation (v_I = 1); however, there is hardly any bias when the coefficient of variation is about 0.5 (Figure 1). When the variance of the resource supply rate is small, the bias for the type I variance is not more than one-tenth of the square of the coefficient of variation of the resource supply rate (Appendix B).

View larger version (23K): [in this window] [in a new window]   Figure 1 The numerical prediction of the perfect information model. Relationship between the proportion of resources and the average proportion of competitors for the variance type I ([– - – -] vI = 0.5; [–––] vI = 1) and types II and III ([– – – –] vIII = 0.5 and q = 2;[- - - -] vIII = 1 and q = 2), assuming gamma distribution for the resource supply rate (appendix C). Matching of means (denoted by a solid line) is always predicted from the type II variance. Note thattheprediction is symmetrical for the point (0.5, 0.5)

 

	DISCUSSION

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
Implications for future tests of the ideal free distribution
The results here provide a perspective for experimenters who test the continuous input model under conditions where the resource supply rate is stochastic. If the experiment is set so thata competitor learns the expectation of the patch quality using its previous experiences and chooses a patch using the expectation (e.g., Hakoyama and Iguchi, 2001; Harley, 1981; Milinski, 1994), the uncertain information model is appropriate. Competitors will adopt some learning rule for decision making that approximates the ES learning rule (Harley, 1981). Milinski (1994) showed that sticklebacks use long-term memory for decision making concerning patch choice in population foraging. In the uncertain information model, matching of means is always predicted regardless of the resource variability. In a study by Hakoyama and Iguchi (2001), the patches in the experimental tanks were separated by opaque blocks that prevented fish on one side of the compartment from directly viewing the arrival of food items on the other side. Therefore, the fish in a compartment could not know the food input event in the other compartment. The fish also could not anticipate a feed event in their own patch because the timing of feedings ina patch was set at random. In this case, fish cannot trace thevariation of the resource supply rate and have to assess the patch quality using only previous experience.

As an interesting expansion of the uncertain information model, we can consider a situation in which individuals are uncertain of what happens in the patch they do not occupy but are aware of what is happening in their own patch. In this case, individuals are likely to assume a long-term average (so, effectively zero variance) in the other patch, but observe their own, which would effectively mean that they would overswitch.

If the experiment is set up so that competitors can choose a patch after sufficient assessment of the environment, the perfect information model is appropriate. Experimenters can test the effect of the resource variability on the IFD prediction by controlling the variability of resources in continuous input conditions. For example, I propose an experiment consisting of a series of trials in which competitors choose between two feeding patches. Food supply rate of patches is an independent random variable, and it changes between trials. In each trial, competitors are informed of the supply rate of each patch by a cue of the supply rate (e.g., a cue light; see Hakoyama and Iguchi, 1997) and are faced with a choice of two patches before the feeding. When the distribution of competitors between patches is fixed after a sufficient time to assess the cue of the supply rate, the researcher records the competitors' distribution and feeds food items to competitors. After the initial learning period, competitors will trace the resource variability, and then some biases from matching of means will be observed. Milinski (1979) showed that sticklebacks switch their distribution according to the change of the resource availability, and animals seem to be able to track clear (or long-term) changes of the resource environment. A large variance of the resource supply rate is necessary for the experiment because only a large variance of the resource supply rate causes a significant undermatching (or overmatching). In this case, the approximate formula assuming a small resource variance (Appendix B) is inadequate for the IFD prediction, and investigators should use the formula assuming a gamma distribution for the resource variability (Appendix D). If the resource variance is type I variance, the regression formulas derived by Earn and Johnstone (1997) are available.

Implications for the undermatching of previous tests
Many continuous input tests have shown the underuse of thegood patch relative to the input-matching prediction (Abrahams, 1986). Several reasons for this deviation from theIFD prediction have been proposed (reviewed by Kacelnik et al., 1992): (1) perception limit (Abrahams, 1986;Hakoyama and Iguchi, 1997), (2) an ideal despotic distribution (IDD; Fretwell, 1972), (3) unequal competitive abilities (Houston and McNamara, 1988; Parker and Sutherland, 1986), and (4) perpetual switching of feeding site at equilibrium (Houston and McNamara, 1987; Houston et al., 1995; Regelmann, 1984). The perfect information conditions with a specific type of the resource variance (e.g., type I variance) might also lead to undermatching in the previous tests. However, because the variability of resources causes not only undermatching but also input-matching and overmatching, it is not possible to discuss the effect of the variability of resources on bias without knowing the type of the variability of resources. As an example of experiments with knowing the type of the variability of resources, Recer et al. (1987) showed that mallard ducks, which probably tracked the environment, underused the variable patch relative to matching of means. Humphries et al. (1999) also conducted an experiment in which competitors chose between the constant rate patch and the variable patch and found overall that fish spend more time in the constant patch. This result might be ascribed to the perfect information conditions. In contrast, if an experiment has satisfied the assumption of the uncertain information model or the type II variance (e.g., Hakoyama and Iguchi, 2001), undermatching is not due to the effect of the variability of the resource supply rate. Because, as mentioned in "The Model" section, a convolution of independent and identical unit of random variables yields a type II variance, the type II variance might be natural in previous continuous input experiments. Hakoyama and Iguchi (2001) provided food as a convolution of independent and identical unit of random variables (food input events); therefore the resource variance was type II. In general, however, reanalysis of published tests is difficult due to the lack of direct information on the uncertainty in the decision making and the type of the resource variance.

Implications of this study for general IFD models
Does the condition of matching of means for the IFD in continuous input conditions hold true for the IFD in general conditions or for the extended IFD models (see Kacelnik et al., 1992)? The generality differs between the two continuous input models (the uncertain information and the perfect information model). If assumptions of the uncertain information model are satisfied, the prediction for the IFD in continuous input conditions holds true even for the other general IFD models such as the interference model (Sutherland, 1983); numerical predictions from a stochastic model are also equal to that from the original deterministic model.In contrast, if assumptions of the perfect information model are satisfied, the quantitative prediction for the IFD in continuous input conditions (e.g., matching of means for the type II variance) will not hold true for the extended IFD models because the other parameters in the general models will also affect the bias in numerical predictions. For example, because the interference model has an additional parameter, interference constant (Sutherland, 1983), the condition of matching of means for the stochastic version of the interference model is determined by the interaction of the typeof the resource variance and the value of the interference constant. A detailed analysis of each extended model isbeyond the scope of this study. In any case, this elemen talmodeling in continuous input conditions will help clarifyaframework for considering the distribution of competitors under conditions where the resource supply rate is stochastic.

	APPENDIX A

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
Proof of matching of means for the evolutionarily stable learning rule
Suppose that competitors adopt the ES learning rule to choose a foraging patch. At the ESS, the cumulative payoff is equal for all competitors:

and the cumulative payoff at patch i at the ESS is

Therefore, from Equation 4, the proportion of competitors at patch i at the ESS becomes:

Then, matching of means is always predicted for competitors with the ES learning rule.

Simulations for the relative payoff sum learning rule
Harley (1981) developed the RPS learning rule and showed the rule realizes an IFD when the resource supply rate of each patch is a constant. I here explain the RPS learning rule briefly and show the results of the Monte Carlo simulation when the resource supply rate is variable.

The RPS learning rule is expressed as:

where _i,j(t) is the probability that competitor j chooses patch i (i = 1,2) at trial t (t = 1,2,...); y_i is residual value of behavior to use patch i; x is memory factor; P_i,j(t) is payoff (food gain) of competitor j on trial t at patch i. Equation A3 indicates the posterior probability _i,j(t) is gradually modified by the relative payoff sum.

For simulations, I assumed the resource supply rate R_i(t) follows the gamma distribution (see Appendix D). For the equilibrium state after the initial phase of learning period (at t = 2000), I calculated the mean proportion of competitors in patch i, E(N_i)/n = _j=1ⁿ_i,j(2000)/n. Independent simulations are repeated 30 times for each of the type I and III variance. I set parameters as q₁ = 45, q₂ = 15, n = 30, x = 0.995, r_i = 2, v_I² = 0.2 and v_III² = 135. I compared the E[N₂]/n with both the prediction from matching of means [= q₂/(q₁ + q₂) = 0.25] and the predictions from the perfect-information model. The perfect information model with the type I variance or type III variance predicts a bias from matching of means (E[N₂]/n is 0.268 for the type I variance and 0.234 for type III variance; Appendix D).

The mean (±SE) E[N₂]/n of 30 simulations was almost equal to the prediction from matching of means (type I: 0.250 ± 0.001: type III 0.250 ± 0.001), and did not significantly differ from the prediction of matching of means (one-sample sign tests; type I: s⁺ = 13, s^– = 17, p =.5847; type III: s⁺ = 17, s^- = 13, p =.5847). In contrast, the mean E[N₂]/n for the type I variance was significantly lower than the prediction from the perfect information model (one-sample sign test; s⁺= 0, s^– = 30, p <.0001). The mean E[N₂]/n for the type III variance was significantly higher than the prediction from the perfect information model (one-sample sign test; s⁺ = 30, s^-=0, p <.0001).

	APPENDIX B

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
Predictions from the perfect information model when the resource variance is small
When _i is small (_i << q_i), we can derive an approximate formula of the expected proportion of individuals in patch 1. The proportion of resources in patch 1 can be expanded as

By neglecting the higher-order terms with respect to _i and by taking the average of both sides of Equation A4,

in which I used E[_i] = 0 and the covariance of independent variables ₁ and ₂ is equal to zero (cov[₁,₂] = 0). So, from Equations 2, 8, and A5, when q₁>q₂,

Because the type II variance satisfies q_{1 2}² = q_{2 1}², it provides matching of means.

For the type I variance, Equation A5 is

where ß = q₁/(q₁ + q₂) (0 < ß < 1). Because |ß(1 - &beta)(1 - 2ß)v_I²| < 0.096v_I² for 0 < ß < 1, the bias is not more than one-tenth of the square of the coefficient of variation of the resource supply rate.

When _i is small (_i << q_i), we have also an approximate formula of the variance of the proportion of individuals in patch 1 using Taylor's expansion:

	APPENDIX C

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
The condition of the matching of means for the perfect information model
Earn and Johnstone (1997) proved that for the perfect information model with type I variance there is always undermatching of means. Here, I give a slightly different proof of this result, and I also prove that with type III variance there is always overmatching of means.

The proportion of resources in patch 1 can be expressed as

By taking the average of both sides of Equation A9,

Thus, from Equations 2, 8, and A10, when q₁ > q₂, if

where q = q₁ + q₂.

For the type I variance, I write _i = q_ix_i, where x_i are independent and identical random variables following E[x_i] = 0, var[x_i] = v_{\rm I}^2 , and x_i > -1. Here random variables are transformed to u₁ = x₁ + x₂ + 2 and v₁ = x₁ – x₂. For all x₁ and x₂, u₁ > 0, u₁ + v₁ > 0, and u₁ – v₁ > 0. Note that the joint probability density function f₁(u₁,v₁) is an even function of v₁ for all u₁ and v₁, because v₁ = x₁ - x₂ and -v₁ = x₂ - x₁ are equally probable. If q₁ > q₂,

in which I used E[qu₁v₁/(q²u₁² - (q₁ - q₂)²v₁²)] = 0, because the mean of an odd function of v₁ is zero.

For the type III variance (_i are independent and identical random variables), random variables are transformed to u₂ = ₁ + ₂ + q and v_2 = \varepsilon _1 - \varepsilon _2. For all ₁ and ₂, u₂ > 0, u₂ + v₂ > 0, and u₂ - v₂ > 0. The joint probability density function, f₂(u₂,v₂), is also even in v₂ for all u₂ and v₂. If q₁ > q₂,

where = ₁ + ₂, and in which I used the mean of an odd function of v₂ is 0 (E[qv₂/2u₂] = 0),

and

Equations A12 and A13 indicate that undermatching of means for the type I variance and overmatching of means for the type III variance occur.

I here assumed _i are independent and identical random variables, but if ₁ perfectly correlates with ₂, matching of means is always predicted from Equation A11.

	APPENDIX D

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
Predictions from the perfect information model when the resource supply rate follows the gamma distribution
Let the resource supply rate R₁ and R₂ be independent random variables with a gamma distribution. Since the gamma distribution is decided by two parameters, it has flexibility as a model of actual distribution (Hoel, 1971). The exponential distribution and chi-square distribution are specific cases of the gamma distribution. The probability density function of R_i (i = 1, 2) is

where _i is the shape parameter, _i is the scale parameter, and is the gamma function. E[R_i] = q_i = M_ii and var[R_i] = _i² = _ii².

To begin with, the random variables R₁ and R₂ are transformed to new variables U and V:

then the Jacobian is

so

where g(u,v) is the joint probability density function of the random variables U and V. The probability density function of V = R₁/(R₁ + R₂) is

where Be is the Euler beta function. The mean proportion of competitors (= the mean proportion of resources) is

and the variance is

where F is the hypergeometric function (see Abramowitz and Stegun, 1972). To calculate these statistics with hypergeometric functions, mathematical software such as Mathematica (Wolfram Research, Inc.) and MATLAB (MathWorks, Inc.) are convenient.

Note that if the resource variance is the type II (_i² = q_iv_II²), ₁ = ₂ = v_II² is satisfied. If ₁ = ₂, V follows a beta distribution with parameters ₁ and ₂ [h(v) = (1 - v)^₂-1 v^₁-1/Be(₁,₂)]. In this case,

Then, if the resource supply follows a gamma distribution, the perfect information model with the type II variance always predicts matching of means.

	ACKNOWLEDGEMENTS

 
I thank T. Azumaya, H. Tanaka, and B. Wood for useful comments on the manuscript. This work has been supported by CREST (Core Research for Evolutional Science and Technology) of the Japan Science and Technology Corporation (JST) (J. Nakanishi is principal investigator). Partial financial support was provided by the Japan Ministry of Agriculture, Forestry and Fisheries and the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B), 14760128, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

 
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Abramowitz M, Stegun I, 1972. Handbook of mathematical functions. New York: Dover.

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