1. Water Exchange between Individuals: The Resistance Function

1.1. The common stomach: Symmetric buffering and general form of the resistance function.

Let us define the water level w of a wasp to be the fraction of its crop volume that is filled with water. We assume that all wasps have the same maximum capacity for storing water. Consider a set of N CS wasps. We assume that all CS wasps have the same water level w so that the water level of the common stomach as a whole is also w.(Table 1)

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Table 1. Definition of variables.

https://doi.org/10.1371/journal.pone.0145560.t001

A key function of the common stomach is to act as a buffer or regulator for the flux of water into and out of the colony via a negative feedback mechanism. If we introduce a WF wasp into the system, then the WF wasp selects and attempts to interact with a CS wasp in order to deposit water into the common stomach. Let r(w) be the probability that the CS wasp (with water level w) will resist or refuse to interact with the WF wasp. That is, r(w) is the probability that the WF wasp will "fail" when the common stomach is at a water level w. If the WF wasp fails to interact, then we call the attempt a "failure." Otherwise, the interaction is a "success." We call r the resistance function, and we assume that r(w) strictly increases as w increases; that is, a CS wasp with a greater water level is more likely to resist the in-flux of water. We also designate r(1) = 1, meaning that a full CS wasp is guaranteed not to accept water.

Suppose now that we introduce a PF wasp into the system. The PF wasp selects and attempts to interact with a CS wasp in order to withdraw water from the common stomach for its pulp-collecting trip. Since the common stomach is a buffering system, we assume that the ideal water level for this function is w = 0.5, which may or may not be the actual water level of the common stomach. If w = 0.5 is the ideal water level for buffering, then the CS wasp regards the WF wasp and PF wasp symmetrically; there is no bias towards one or the other. Thus, we assume that the situation of the PF wasp is identical to that of the WF wasp, except for the following: While the situation of the WF wasp depends on the water level w, the situation of the PF wasp, since it attempts to withdraw water, depends on the emptiness level 1 − w. Hence, by symmetry, resistance towards a WF wasp by a CS wasp with water level w is equivalent to the resistance to a PF wasp by a CS wasp with water level 1 − w; if r_P is the resistance towards a PF wasp, then r_P(w) = r(1 − w).

A second key function of the CS wasp is to act as a mediator of water between the WF and PF wasps; the CS wasp accepts water from a WF wasp and then gives it to a PF wasp. We therefore assume in our model that a CS wasp with water level w is willing to interact with either a single WF wasp or a single PF wasp, and hence the CS wasp must resist either a WF wasp or a PF wasp but not both. That is, we exclude from our model the subset of CS wasps who refuse to interact. Furthermore, we assume that at any one time, a CS wasp is either willing to take more water or unload some water; that is, the CS wasp is willing to interact with either a WF wasp or a PF wasp, but not both. So, we have r(w) + r_P(w) = 1. But since r_p(w) = r(1 − w), then we have the following identity for the resistance function (Eq 1): (1)

This identity involves the expected resistance towards a WF wasp as a function of the water level of the common stomach. Although we can replace r with r_P and water level with emptiness level for a PF wasp, we choose to focus only on the WF wasp. By symmetry, whatever results we obtain for WF wasps at a water level w = x can be transferred to PF wasps at the water level w = 1 − x.

By Eq 1, r(1) = 1 implies r(1 − 1) = r(0) = 0, meaning that an empty CS wasp is guaranteed to accept water and a full CS wasp is guaranteed to give water. Also, for x = 0.5, we have r(0.5) + r(1 − 0.5) = 1 so that 2r(0.5) = 1, implying r(0.5) = 0.5. This means that a CS wasp with w = 0.5 is equally likely to accept or give away water. More generally, Eq 1 implies the symmetry that a 180° rotation of the graph y = r(x) about the point (0.5, 0.5) leaves the graph unchanged. Therefore, r has the general form (Eq 2): (2) for some strictly increasing and differentiable function g. Note that r is differentiable at w = 0.5: Let . So, we have , and notice that g′(x) = g′(1 − x) for x = 0.5.

We assume that the shape of the graph y = r(x) is sigmoid since it captures both of the key functions of the CS wasp; the sigmoid shape is a strong regulator and tends to drive the system to a symmetric state (assuming no other asymmetric factors at this point) but still allows for enough fluctuation for the mediation of water. Thus, we choose the function g to be exponential, which has the general form g(w) = a⋅b^w + c. Now, since g(0) = 0, we have 0 = a⋅b⁰ + c = a + c so that c = −a. This gives g(w) = a⋅b^w − a. Now, since g(0.5) = 0.5, we have 0.5 = a⋅b^0.5 − a, which results in . Hence, we have: where a is a free parameter. We find that as a increases, the steepness or sigmoidity of y = r(x) decreases. So, let σ = a⁻¹ (parameter 1) be the curvature parameter, so that a greater value of σ implies greater steepness of the curve (Table 1). This gives (Eq 3): (3)

Substituting this equation into Eq 2 then gives us the resistance function r. A greater water level w implies greater resistance and hence a greater probability that a WF wasp will fail to transfer water (Fig 1). As σ increases, the plot obtains a more "sigmoid" shape, which provides stronger water level regulation. The resistance function r for the PF wasp is y = r(1 − w); by symmetry, it is the reflection of the resistance function for a WF wasp about the line w = 0.5.

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Fig 1. Resistance r as a function of the water level of the common stomach w and the curvature (steepness) parameter σ for the WF wasp (see Eqs 2–4).

These functions describe the probability that a WF wasp fails in an interaction when the common stomach water level is w (under various values of σ).

https://doi.org/10.1371/journal.pone.0145560.g001

We also consider the limiting cases of the linear resistance function given by r₀(w) = w (σ = 0) and the step resistance function given by (Eq 4): (4)

Thus, for σ = ∞, a WF wasp is guaranteed to succeed for w < 0.5 and guaranteed to fail for w > 0.5.

1.2. The quantity of water transferred between individuals depending on the water level and resistance.

In this section, we derive how much water is transferred in one successful interaction between a forager and CS wasp as a function of common stomach water level. This will help to determine the number of successful interactions that a WF wasp must make to rid its water load. Suppose a CS wasp with water level w accepts water from a WF wasp. Let Δw be the amount of water (expressed as a fraction of crop volume) transferred from the WF wasp to the CS wasp in their interaction, and let Δw be a function of w. We assume that the CS wasp (and not the WF wasp) determines Δw; we assume that a WF wasp will always try to empty its load in one interaction (Δw = 1) (maximizing its task performance) but that the CS wasp is the one that ultimately decides how much water to accept from the WF wasp in accordance with its own internal state. As a second negative feedback mechanism to control water in-flux, we propose that Δw strictly decreases as w increases. Again, since a full CS wasp accepts no water, define Δw(1) = 0.

We construct a relationship between w and Δw based on the following hypothetical scenario: Consider a hypothetical CS wasp that initially holds no water in its crop. Suppose the CS wasp successfully interacts with a series of WF wasps (one at a time) and accepts an average of Δw units of water from each WF wasp. By the resistance function, the CS wasp will eventually resist further interaction with another WF wasp. By this point, the CS wasp has some final or "terminal" water level (the terminal water level is an average because resistance is probabilistic). The terminal water level is therefore a function of the increment Δw. In particular, if we refer to the above scenario as "Scenario A", then we have that Scenario A "maps" the increment Δw to the terminal water level . We will say that the increment Δw "generates" the terminal water level .

Now consider a CS wasp with water level w. How much water will the CS wasp accept (assuming it does not resist)? That is, what is the value of Δw(w)? The central hypothesis of our model is that the value of Δw(w) is such that it "generates" the terminal water level . In other words, Scenario A "maps" the increment Δw(w) to the terminal water level . More explicitly, we propose that a CS wasp with water level w accepts Δw(w) units of water such that if a (different) hypothetical CS wasp is initially empty and accepts water in increments of Δw(w) units, then it will ultimately have a terminal water level of .

Our central hypothesis is based on the fundamental symmetry between WF wasps and PF wasps. Since a CS wasp with water level w accepts Δw(w) units of water from a WF wasp, then by symmetry a CS wasp with water level 1 − w transfers Δw(w) units of water to a PF wasp. But by our central hypothesis, Δw(w) "generates" the water level . Therefore, a CS wasp with water level 1 − w transfers Δw(w) units of water to a PF wasp, where Δw(w) is precisely the increment by which an empty hypothetical CS wasp must accept water from WF wasps in order to reach a terminal water level of .

The above scheme can be used to construct a mathematical relationship between w and Δw as follows: Let . Consider an initially empty CS wasp that has already accepted i − 1 water transfers of size Δw from WF wasps. Thus, the CS wasp has water level x_i−1 = (i − 1)Δw. By definition of the resistance function, the probability that the CS wasp will accept an i^th transfer is 1 − r(x_i−1) = r(1 − x_i−1) = r(1−(i − 1)Δw) (using Eq 1). Suppose the CS wasp continues to accept water until it has water level x. So, the CS wasp accepted transfers and refused the next transfer, the probability of which is r(x). To find the probability P_w(Δw, x) that an empty CS wasp accepts water in increments of size Δw until it has water level x, we multiply the probabilities of the CS wasp accepting the first transfers and resisting the next transfer:

This is a probability density function over x and therefore sums to 1 over x for a given Δw. It is similar to the geometric distribution but differs in that 1) there is an upper limit to the number of successes (once the CS wasp is full) and 2) the probability of success changes in accordance with the continuous resistance function. Every possible water level can be written as jΔw, . Notice that j = 0 is not possible since an empty CS wasp must accept the first transfer due to 1 − r(0) = 1. So, the average water level at which the in-flux of water into the CS wasp terminates is:

According to our central hypothesis, a CS wasp with water level w accepts an amount Δw of water such that the terminal water level of an initially empty CS wasp accepting water in increments of Δw is , and hence . Thus, we have the following relationship between Δw and w (Eq 5): (5)

Although we have w as a function of Δw, it is difficult to derive a closed-form expression for the inverse function that gives Δw in terms of w. So, to evaluate Δw(x), we adjust x until w(Δw) = x.

The equation Eq 5 reduces and simplifies in the special case of r_∞. This is significant because this limiting case defines a lower bound on Δw(w), which can be used to place a lower bound on the average water level of the common stomach. Let . Suppose n is even. So, . This means so that kΔw = 0.5. Thus, 0.5 is an integral multiple of Δw.

Since r_∞(x) = 0 for x ∈ [0, 0.5), a CS wasp is guaranteed to reach a water level x = 0.5 if it accepts water in increments of Δw. Now, since r_∞(0.5) = 0.5, there is a 50% chance that the CS wasp will accept another Δw. If it does not accept, then its final water level is x = 0.5. If it accepts, then its final water level is x = 0.5 + Δw, after which it can accept no more since r_∞(x) = 1 for x ∈ (0.5, 1]. Thus, the average water level is . Since , we have w = 1 − 0.5(1 + Δw) = 0.5(1 − Δw).

Suppose n is odd. So, . This means so that 2kΔw − Δw = 1 or kΔw = 0.5(1 + Δw). Since r_∞(x) = 0 for x ∈ [0, 0.5), the CS wasp is guaranteed to reach a water level x = 0.5(1 + Δw). Since 0.5(1 + Δw) > 0.5, r_∞(0.5(1 + Δw)) = 1 so that this is the final water level . So, . Therefore, we have for all the following relationship (Eq 6): (6)

Notice that Δw(0.5) = 0 so that a CS wasp with water level w = 0.5 neither accepts nor gives water. Thus, if the common stomach is 50% full, then all interactions stop, assuming no asymmetric factors exist (of course, this is only in the limiting case).

The amount of water Δw(w) transferred in one successful interaction decreases with the water level w (Fig 2). As σ increases, Δw(w) decreases for all 0.25 < w < 1; stronger resistance is accompanied by less water acceptance. Note that the plots for σ = 0 and σ = ∞ are natural upper and lower bounds on Δw(w) respectively. A similar function for a PF wasp on receiving water from a CS wasp can be obtained by reflecting y = Δw(w) about w = 0.5.

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Fig 2. The amount of water Δw transferred in a successful interaction by a WF wasp as a function of the water level of the common stomach w and the curvature (steepness) parameter σ (see Eqs 5 and 6).

https://doi.org/10.1371/journal.pone.0145560.g002

Notice that Δw(0) = 1 for all σ, which means that an empty CS wasp accepts all the water from a WF wasp, and a full CS wasp gives all of its water to a PF wasp. Thus, it is possible for an empty CS wasp to simply alternate between w = 0 and w = 1, although other factors break this alternation. Also, note that Δw(0.25) = 0.5 for all σ; when the common stomach is one-quarter full, a WF wasp will be able to transfer half of one crop load of water to an accepting CS wasp regardless of resistance (Fig 2). This can be seen as follows: Consider an empty CS wasp. Suppose it accepts water in increments of Δw = 0.5. Since the CS wasp is initially empty, then it is guaranteed to accept the first transfer. Since r(0.5) = 0.5 for all σ, there is a 50% chance that the CS wasp will accept a second water transfer. Thus, there is a 50% chance that the CS wasp will remain at x = 0.5 and a 50% chance that it will rise to x = 1. Thus, on average, we have so that .

2. Multiple Interactions of a Foraging Wasp with Common Stomach Wasps: Successes and Failures

2.1. The number of successes needed for a water forager to deliver its full load.

We showed that if the common stomach has water level w, then a WF wasp is able to deposit Δw(w) of its water into the common stomach per successful interaction with a CS wasp (see Eq 5). Thus, for a WF wasp to completely empty itself in one cycle of interactions, the number of successful interactions that it must make is given by (Eq 7): (7) where "ceil" denotes the ceiling function. The ceiling function is used because a WF wasp can only make an integral number of interactions, even if a bit of water remains for the last interaction. Here, we define one foraging cycle to be the set of interactions attempted for one cropload of water; it is the period between foraging trips. Again, the CS wasps force a WF wasp to make S successful interactions via resistance; since the CS wasps determine Δw, they determine S.

In the special case of σ = ∞, the number of successes S_∞ needed for a WF wasp to empty itself is given by (Eq 8): (8) where Δw is given by Eq 6.

As the water level of the common stomach w increases, the number of successes needed S increases quickly (Fig 3). For clarity, the effect of the ceiling function (included in the definition of S) is not shown in Fig 3. A greater water level implies a greater number of successes needed for complete unloading since CS wasps accept less water at greater water levels. The plot is undefined when the common stomach is full since no successes can be made. As σ increases, S(w) increases for all 0.25 < w < 1; under stronger resistance, more successes are needed for complete unloading. Note that S(0.25) = 2 for all σ; when the common stomach is one-quarter full, 2 successes are needed for complete unloading. The plots for σ = 0 and σ = ∞ are natural lower and upper bounds respectively on S(w). For σ = ∞, S(w) is undefined for w ≥ 0.5 because Δw(0.5) = 0 (under σ = ∞) and successes are impossible for w > 0.5 due to resistance under σ = ∞.

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Fig 3. The number of successes S needed for a WF wasp to completely unload its full crop content while interacting with CS wasps as a function of the water level of the common stomach w and the curvature (steepness) parameter σ (Eqs 7 and 8).

For clarity, the effect of the ceiling function (used in Eqs 7 and 8) is not shown.

https://doi.org/10.1371/journal.pone.0145560.g003

A similar function for the number of successes needed S_P for a PF wasp to completely fill its empty crop can be obtained by reflecting y = S(w) about w = 0.5 since S_P(w) = S(1 − w) (where S was defined in Eq 7).

Until a WF wasp makes S successes, it attempts a number of interactions per cycle, some of which succeed and others fail due to resistance from the CS wasp. Let s be the number of successes that a WF wasp makes in one foraging cycle, where s is a function of the water level w of the common stomach (Table 1). It is not necessary that s = S, as we will see. Let f be a function such that f(s) is the expected number of failures that a WF wasp will make if it makes s successes. So, the total number of interactions per cycle that a WF wasp makes is n(s) = s + f(s). Now, we can define the failure rate (percentage of interactions that fail) for a WF wasp as follows:

In all of the above, we can replace s with s_P in the case of a PF wasp.

It is possible for a WF wasp to interact with the same CS wasp more than once per foraging cycle (although not consecutively). Thus, in the theory so far, it is possible for a wasp to fail every interaction that it attempts, so that it never completes the S successes needed for emptying and is trapped in the same foraging cycle forever; n = ∞. To avoid such infinities, we assume that there exists a function, call it cont, such that cont(s, f) is the probability that a WF wasp will attempt another interaction, having already made s successes and f failures. So, if the total number of interactions is high, then it is possible for a WF wasp to abort its cycle of interactions before it completes S successes. Thus, s(w) ≤ S(w). Note that in nature, a WF wasp that carries unloaded water and aborts the water foraging task tends to switch tasks, e.g., to a CS wasp, but our model does not address this behavior. We have investigated this in detail as reported previously [26, 30, 31, 35]. An explicit form of the cont function is presented next.

3. Constructing a Probability Distribution overall Success-Failure Combinations

Given a particular water level w, we now define the function P such that P(s, f) is the probability that a WF wasp will make s successes and f failures in its foraging cycle. The same probability holds for a PF wasp at a water level 1 − w. The construction of P incorporates the resistance function (r), the number of successes needed for emptying (S), and the probability that a wasp attempts another interaction (cont(s, f)). The distribution function P is a complete theoretical description of foragers and their interactions with CS wasps. We will use P to calculate quantities such as the average number of successes and average number of failures that a wasp makes at a given water level. These calculated quantities will later be used to study the effects of our parameters on the model and to compare the model predictions with empirical data.

Let R = r(w), with w a fixed water level. The probability that a WF wasp makes s successes is (1 − R)^s, and the probability that it makes f failures is R^f. The number of ways that a WF wasp can make s successes and f failures is . Based on these probabilities alone, the probability that a WF wasp will make s successes and f failures is .

We must modify the above probability result to incorporate the number of successes S needed for emptying. The necessary modification is understood by examining the interaction that concludes the foraging cycle of the WF wasp: If s = S, then the concluding interaction was necessarily a success, the probability of which is . In addition, a wasp with s = S continued after s − 1 successes and f failures but not after s successes and f failures. The probability of this is cont(s − 1, f) − cont(s, f). In contrast, s < S means that the last interaction was a failure, the probability of which is . Also, a wasp with s < S continued after s successes and f − 1 failures but not after s successes and f failures. The probability of this is cont(s, f − 1) − cont(s, f). We can collect these different probabilities into a function called last that describes the concluding interaction as follows:

Notice that we also defined the probability to be 0 if s + f = 0 to eliminate wasps who attempt no interactions.

Combining the above probabilities, the probability that a WF wasp will make s successes and f failures is proportional to the function:

To obtain the desired probability distribution P, We must normalize this function, which requires us to sum p(s, f) over all (s, f). The limits on the summation are 0 ≤ s ≤ S and 0 ≤ f < ∞. Because of the cont function, we know that . Thus, to reduce computation time, we can define a finite upper limit max(f) on f such that p(s, max(f)) ≈ p(s, ∞). Thus, 0 ≤ f ≤ max(f). We will use max(f) = 200, since we can be almost certain that no wasp will continue after 200 failures [26]. The normalization constant (A) is therefore:

With this, the probability that a WF wasp at a water level w will make s successes and f failures in its foraging cycle is given by (Eq 10): (10)

4. Number of Successful Interactions as a Function of the Common Stomach Water Level

In Section 1.1, we proposed that the water level determines the probability that a CS wasp will resist interaction and cause the attempted interaction to fail. In Section 1.2, we proposed that the water level also determines the amount of water that a CS wasp will accept/give, thereby determining the number of successes needed for a foraging wasp to complete its cycle. Together, these two propositions imply that the water level determines the number of interactions (successes + failures) needed for a wasp to complete its foraging cycle. We then proposed a programmed function of behavior (the cont function) that determines what fraction of these interactions a wasp is likely to complete before it decides to abort its cycle. Thus, in short, the common stomach determines the number of interactions needed, and the foraging wasp decides what fraction of these it will complete before aborting. Hence, the value of s is an indicator of the water level of the common stomach.

In this section, we place certain constraints on the water level w to help us with interpreting and using empirical data presented in later sections. The general behavior described in Section 2.1 allows us to draw a schematic sketch (i.e., the general trend) of the expected shape of the curve for y = s(w), where w is the water level of the common stomach (Fig 4). At w = 0, a WF wasp is guaranteed to succeed (no failures) so that s(0) = S(0) = 1. As w increases, S increases. To complete the larger number of successes S needed, a WF wasp must make more successes so that s increases. However, since r and therefore the failure rate f_% increase as w increases, it becomes more probable for a WF wasp to abort its cycle before completing S successes. Thus, s(w) < S(w) for w > 0. Eventually, f_% is so large that the WF wasp begins to abort its cycle with fewer and fewer successes, so that there exists a water level at which s(w) has its maximum value max(s) and after which s(w) decreases. At w = 1, failure is guaranteed so that s = 0 (Fig 4). By symmetry, the number of successes s_P that a PF wasp is expected to make at a water level w is given by s_P(w) = s(1 − w).

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Fig 4. Schematic sketch of the number of successes s that a WF wasp is expected to complete when the common stomach water level is w (solid curve).

An upper bound to s is the number of successes S that a WF wasp needs to completely rid its load (dashed curve). There exists a water level at which a WF wasp is expected to make a maximum number of successes max(s). The plot for the number of successes s_p expected for a PF wasp at common stomach water level w is obtained by s_p(w) = s(1 − w). These curves are meant to convey only a general trend (described in Section 2.1) and are not plots of equations.

https://doi.org/10.1371/journal.pone.0145560.g004

The schematic sketches of y = s(w) and y = s_P(w) show that these functions are symmetric about w = 0.5 so that their absolute maxima occur at water levels equidistant from w = 0.5 (Fig 5). Let w(x) be the water level such that s(w(x)) = x. Since s_P(w) = s(1 − w), 1 − w(x) is the water level such that s_P(1 − w(x)) = s(w(x)) = x. Let be the average water level of the common stomach taken over time. We assume that the average water level lies between the points at which the maximum numbers of successes occur: (Fig 5). We assume that the water level fluctuates symmetrically about . We also assume that the feedback mechanisms that regulate the water level are robust and timely such that the water level never fluctuates beyond the points at which max(s_P) and max(s) occur. Given that the common stomach is large and water flux is small in the sense that one foraging wasp is observed to make multiple interactions to withdraw/deposit at most one crop load of water, we are confident that our statement on the fluctuation of the water level is a safe assumption. It has been observed that the average numbers of successes (taken under fluctuating water level of the common stomach) that WF wasps and PF wasps make are 5.2 and 6.4 respectively [26]. Since 6.4 > 5.2, must be such that it is more frequent to find s_p(w) > s(w) than s_p(w) < s(w). The symmetric fluctuation of water about then implies (Fig 5). For w < w(max(s)), we assume that the graph y = s(w) is primarily concave-up. This implies that the average number of successes for WF wasps occurs at a water level greater than the average water level: . By symmetry, we have . Thus, . Since the graph y = s(w) has a turning point at its maximum, its concavity must decrease to 0 at some point. Thus, the plot of s(w) is more linear at w(6.4) than at w(5.2). By symmetry, s_p is more linear at 1 − w(6.4) than is s at w(5.2). This implies that is closer to 1 − w(6.4) than to w(5.2): . Solving for from this inequality, we get (Eq 13): (13)

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Fig 5. Schematic sketches for the number of successes s and s_p expected for a WF wasp and PF wasp to complete in one foraging cycle respectively when the common stomach water level is w.

The established average water level of the common stomach is (arrow), and it is assumed to have a value greater than where max(s_p) occurs and has a value less than w = 0.5. The empirical average numbers of successes [26] for a WF wasp and PF wasp (5.2 and 6.4 respectively) occur at water levels greater and less than the average respectively (this is deduced from the hypothesized shapes of the curves). Note that s = 5.2 might occur below, at, or above w = 0.5. Also, s_p = 6.4 occurs at a water level nearer to the average than does s = 5.2. Note that is located under a steeper section of y = s_p(w) than of y = s(w), so that the deviation in s_p is greater than the deviation in s (4.3 > 2.8) in agreement with empirical findings of Karsai and Wenzel [26].

https://doi.org/10.1371/journal.pone.0145560.g005

Note that since w(6.4) > w(5.2), w(6.4) − w(5.2) > 0 so that .

We want to derive an upper bound on s(0.5), which will later allow us to derive bounds on model parameters. Suppose 0.5 ≤ w(5.2). Since s is increasing, s(0.5) ≤ s(w(5.2)) = 5.2 < 5.8. Thus, s(0.5) < 5.8. Suppose, on the other hand, w(5.2) < 0.5. Since , we have 1 < w(5.2) + w(6.4), or 0.5 < 0.5(w(5.2) + w(6.4)). Let L be the secant line joining the points (w(5.2), 5.2) and (w(6.4), 6.4). Since L is increasing, L(0.5) < L(0.5(w(5.2) + w(6.4))). Since L is linear, L(0.5) < 0.5(L(w(5.2)) + L(w(6.4))) = 0.5(5.2 + 6.4). Thus, L(0.5) < 5.8. Since we assumed s to be concave-up, s(0.5) < L(0.5) so that s(0.5) < 5.8.

5. Correlation between Number of Successes and Interaction Time

To obtain a correlation between the number of successes and the number of failures that a forager makes in one cycle, we used data from Karsai and Wenzel [26], where the number of successes s was measured (S1 File). In this section, we fit a function to this collected data set (s, t), where s is the number of successful interactions that a foraging wasp was observed to make in one cycle and t is the time (in seconds) that it took for the wasp to complete its cycle (Fig 6). We will use this function to construct a failure rate function, which will carry empirical information. This will be the function against which we can compare theoretical results. By the symmetry of WF wasps and PF wasps, the relationship between s and t ought to be the same for both WF wasps and PF wasps; the asymmetry due to the "water leakage" should only result in the PF wasps having a higher average value of s. This allows us to group the data for WF wasps and PF wasps together.

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Fig 6. Empirical relationship [26] between the handling time t —i.e., total time needed for downloading (WF: grey circles) or uploading (PF: black squares) water in a single foraging cycle—and number of successes s completed during this behavior.

https://doi.org/10.1371/journal.pone.0145560.g006

Overall, handling time increases with number of successes (Fig 6). If the data is taken to suggest that the rate of increase increases with number of successes, then the source of the increasing rate must be an increasing number of failures with successes (assuming average time between any two interactions is fixed). In particular, number of failures increases at an increasing rate with number of successes. Since resistance increases with water level, this suggests that number of successes increases with water level up to the water level where the maximum number of successes occurs. Thus, this data provides empirical support for some of our hypotheses (Section 1.2).

We now want to fit a curve to the data and obtain t as a function of s. In Section 4, we show that for 1 ≤ s < max(s), there are two water levels at which a given value of s occurs (Fig 5). Since there are two different resistances at these two water levels, we find that for one value of s, there are two distinct values of n(s) (total number of interactions). If we assume that the time required to interact with one CS wasp is a constant Δt, then t and n are linearly related. Thus, for 1 ≤ s < max(s), there are two distinct values of t. Therefore, the curve relating t to s loops back to form a "double" curve with a single maximum (Fig 7). In order to simplify the fitting, we maintain the assumption from Section 4 that the water level never fluctuates beyond the points at which max(s_P) and max(s) occur. With this restriction, one value of s now has exactly one value of t associated to it, so that we can fit the data with a single curve. This single curve is not related to the double curve described above but instead replaces it completely.

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Fig 7. Schematic sketch of a possible fit to the handling time t vs. number of successes s data shown in Fig 6 assuming that the common stomach water level w fluctuates on both sides of the water level at which the maximum number of successes is made (see Fig 4).

If the water level could fluctuate beyond the point where max(s) occurs, then two distinct water levels could have the same number of successes. These distinct water levels correspond to different handling times t since the number of failures varies with water level. Thus, for 1 ≤ s < S, one value of s corresponds to two values of t (as shown). The double curve depicted is ultimately not used for fitting, since we assume the water level never fluctuates to the extent described above (Section 4).

https://doi.org/10.1371/journal.pone.0145560.g007

We believe that the curve of best fit is strictly increasing and concave-up. Indeed, if this is the case, then we have the following: Since t is concave-up, we have t″(s) > 0. Since n is linearly related to t, we have n″(s) > 0. Since n(s) = s + f(s), we have f″(s) > 0. Thus, the failure rate f_% increases as s increases. If f_% increases, then it is probable that r increases as s increases. Furthermore, w increases as r increases. Thus, the function that relates s and w is increasing, which is in fact the premise of our model. We can therefore regard the time-success data as the basis for our model. The converse, however, is not true; our model does not require that t be concave-up, although other curves will not increase at the rate necessary to fit the data. Thus, we use the concave-up exponential function to fit the data (R² = 0.516, n = 69, p < 0.001), and we get the following (Eq 14): (14)

6. Deriving Number of Failures as a Function of Successes from Time-Successes Data

Given the relationship in Eq 14 between the number s of successful interactions made and the total interaction time t(s) required, it is possible to derive a relationship between s and the accompanying number of failures f(s) as follows: The time needed for a WF wasp to make one success is t(1). The value s = 1 indicates that the water level is w = 0 so that the number of failures accompanying s = 1 is f(1) = 0. Thus, the time required for a WF wasp to find and attempt its first interaction (success or failure) is t(1). Now, consider a WF wasp that makes s successes. This WF wasp makes f(s) failures and therefore attempts a total of n(s) = s + f(s) interactions. The total time required for a WF wasp to attempt n(s) interactions is t(s). We know from above that the first interaction takes time t(1). Thus, the time required for a WF wasp to leave the first CS wasp and to complete its interactions with the remaining n(s) − 1 CS wasps is t(s) − t(1). In this time, the WF wasp moves from one CS wasp to another n(s) − 1 times (the number of intervals between the n(s) CS wasps). Let Δt (parameter 3) be the average time needed for a WF wasp to move from one CS wasp to another CS wasp and to attempt interaction with the latter. Since each of the n(s) − 1 intervals between the n(s) CS wasps takes time Δt, we have (n(s) − 1)Δt = t(s) − t(1). Solving for Δt and noting that n(1) = 1 + f(1) = 1, we have the following (Eq 15): (15)

The value of Δt is an unknown constant that is independent of s and w. In Karsai and Wenzel [26], the average time between two successes was measured to be 9.6 seconds so that Δt < 9.6. It is apparent from this equation that Δt is the slope of the plot of the linear equation relating t and n.

Solving the above equation for n(s) yields:

Since f(s) = n(s) − s, we have the following relationship between the number of successes s and the number of failures f that a wasp makes (Eq 16): (16)

Let us note for a later section that for all s. As in the case of t, the domain of f is 1 ≤ s ≤ max(s). In all of the above, the variable s can be replaced with s_P in the case of a PF wasp. This means that the relationship between successes and failures is the same for both a WF wasp and PF wasp.

We cannot use the function f to achieve numerical outputs unless we know the value of Δt. If a point (x, y) such that f(x) = y is provided, then we can calculate Δt and use this value to make f usable for numerical evaluations.

7. Equating Theoretical-Based and Empirical-Based Functions within Reasonable Error

Our theoretical model (Eqs 9–11) depends on two free parameters σ and K. The function (Eq 16) derived from empirical data depends on one free parameter Δt. To find values for these parameters, we require that the theoretical and empirical functions agree with each other within reasonable error. We define the function (Eq 17): (17) where and are Eqs 11 and 12, and f is Eq 16. Note that S and r both depend on σ. The function E gives the percent error between the theoretical number of failures and empirically derived number of failures f for a given number of successes at a specified water level w and under specified values of the model parameters. For theory and empirical functions to agree, we require |E| ≤ 0.05. That is, the percent error between f and must be within 5%. This restriction will allow us to find the region of biologically feasible values for our model parameters.

To find the values of the model parameters, it is necessary to understand whether E increases or decreases with each of r, K, and Δt. We observe by testing Eqs 10 and 11 that the following identity holds independently of σ, K, and Δt (Eq 18): (18)

Thus, the resistance or probability of failure at a water level is equivalent to the colony-level failure rate as computed in terms of the average number of successes and average number of failures. In particular, if r(w) is increased, then we observe by testing Eqs 11 and 12 that decreases and increases. Since f′(s) > 0, decreasing implies decreasing . Thus, by Eq 17, . If K is increased, then we observe by testing Eqs 11 and 12 that and increase proportionately (since r is held constant). Suppose and increase by a factor a (a > 1) so that we have . Since f″(s) > 0 (noted after Eq 16), f(as) = bf(s), b > a. Thus, , . Hence, by Eq 17, . If Δt is increased, then and are unaffected (neither depends on Δt) and decreases (by Eq 16). Therefore, by Eq 17, . These findings will be used in the proceeding sections.

1. Bounds on and Approximate Values of the Model Parameters

In this section, we use our mathematical model to derive bounds on and approximate values of our model parameters. These values will then be used in proceeding sections to understand the relationship between the common stomach water level and the interactions that foragers attempt with CS wasps.

We first derive an upper bound on Δt. Consider the water level w = 0.5. By definition, r(0.5) = 0.5 so that (by Eq 18). Thus, . By Eq 15, we have:

Recall the restriction (Section 4). Because Δt strictly increases with , implies 0 < Δt < 3.74.

We now derive bounds on σ. Consider the water level w = 0.25. Recall that S(0.25) = 2 for all σ (Fig 3). Assume K > 0.1 (improving on K > 0) so that the probability that a WF wasp continues after a failure is at least 10%. Let K = 0.1 (its lower bound) and Δt = 3.74 (its upper bound). Consider the equation E(2, r(0.25), 0.1, 3.74) = 0.05. Solving for r(0.25), we get r(0.25) = 0.170…. Since , r(0.25) > 0.170… implies E < 0.05. To satisfy E > −0.05 (since we demand |E| < 0.05), we can have K > 0.1 or Δt < 3.74 (since and . On the other hand, r(0.25) < 0.170… implies E > 0.05. To satisfy E < 0.05 (since we demand |E| < 0.05), we must then have K < 0.1 or Δt > 3.74, neither of which is possible. Thus, r(0.25) > 0.170…. This implies σ < 5.52…. Since we assumed σ ≥ 0 (Section 1.1), we have approximately 0 ≤ σ < 5.53. Note that this range of values is approximately the range for which S(0.5) = 6. That is, under a resistance function with a σ value in the deduced range, WF wasps must make six successful interactions to completely unload their water when the common stomach water level is w = 0.5.

We now derive bounds on K and a lower bound on Δt. Assume to the contrary that K ≤ 0.957… (K = 0.957… is the value that results in s(0.5) = 5.2 under the deduced S(0.5) = 6). The restriction K ≤ 0.957 implies max(s) < 8 for all 0 < σ < 5.53; in the cases of both σ = 0 and σ = 5.53, we observe that K = 0.957 is not large enough to have max(s) > 8. We know, however, that the average number of successes that a WF wasp makes is 5.2 ± 2.8, which implies that max(s) > 5.2 + 2.8 = 8, which is a contradiction. Thus, K > 0.957…. Since increases with K, we have . Also, S(0.5) = 6 and together imply K < 0.990…. Therefore, 0.957… < K < 0.990. Just as how implies Δt < 3.74, implies 3.52 < Δt. Hence, 3.52 < Δt < 3.74; the average time between two attempted interactions (success or failure) is between 3.52 and 3.74 seconds.

We can use the bounds 3.52 < Δt < 3.74 and 0.957… < K < 0.990… to strengthen the bounds on σ. Consider the inequality E(2, r(0.25), 0.957…, 3.74) < 0.05. By the same reasoning used in the second paragraph of this section, we get r(0.25) > 0.232…, which implies σ < 0.644…. We can improve this bound further. Recall w(x) is the water level at which Δw = x (Eq 5), where Δw = 1 / S. For every S, let K_min(S) and K_max(S) be the values of K solved from E(S, r(w(1 / S)), K, 3.52) = −0.05 and E(S, r(w(1 / S)), K, 3.74) = 0.05 respectively. For every S, we have the bounds K_min(S) < K < K_max(S). To satisfy these bounds for all S, we need max(K_min(S)) < K < min(K_max(S)). We observe that max(K_min(S)) = K_min(7) and min(K_max(S)) = K_max(2). Requiring the inequality E(2, r(0.25), K_min(7), 3.74) < 0.05, we get σ < 0.516. This also implies K > K_min(7) > 0.985… > 0.957…, where the exact value of K_min(7) depends on σ. Since we still have K < 0.990…, we require 0.990…< K_max(2) to guarantee that K can be arbitrarily close to 0.990…. The definition of K_max(2) then implies E(2, r(0.25), 0.990…, 3.74) < 0.05, which yields σ < 0.494…≈ 0.494. Thus, 0 ≤ σ < 0.494.

Note that 0 ≈ 0.494 (e. g. r(0.25) ≈ 0.236 for σ = 0.494…). In particular, the average percent error between r(w) (σ = 0.494…) and w (σ = 0 is:

Thus, we approximate σ as σ = 0; that is, we conclude that the resistance function is approximately linear. The biological implications of this are mentioned in Results section 2. Now σ = 0 implies K_min(7) = 0.986… so that approximately 0.986 < K < 0.990. Thus, the probability that a forager aborts its foraging cycle after any given failure is very low; out of every 100 failures observed (over any set of foragers), about 98 or 99 of them will be followed by another attempt at interacting, and 1 or 2 of them will be followed by an abortion of the foraging cycle. For every value of K between 0.986 and 0.990, Δt seems to vary with the water level. However, Δt varies strictly within its bounds 3.52 and 3.74. That is, although Δt varies, it is always within 3% of 3.63, so that we can still regard Δt as approximately constant.

2. Natural Bounds on the Number of Successes Needed for Complete Unloading of Water, and Interpreting the Linear Resistance Function

As described in Section 1.2, the amount of water Δw transferred from a WF wasp to a CS wasp and the number of successful transfers S needed for complete emptying are bounded both above and below. The bounding property results from the way in which Δw depends on the resistance function (Section 1.2). The resistance function with σ = 0 results in an upper bound on Δw and therefore a lower bound on S, while the resistance function with σ = ∞ results in a lower bound on Δw and hence an upper bound on S for w < 0.5 (Fig 8). Thus, for nonzero water levels below 50%, a WF wasp must make multiple successful interactions (finite number) to completely rid its load. Although a half-full CS wasp has room to accept half of a crop load, it accepts less than this amount (Δw(0.5) < 0.5). In particular, for σ = 0, a WF wasp must make 6 interactions to completely rid its load when the common stomach is half-full. Thus, a linear resistance function might explain why foragers make on average about 5–6 successful interactions. Although Δw under σ = ∞ is undefined for w > 0.5, r_∞(w) = 1 for w > 0.5; since a WF wasp is guaranteed to be rejected by a CS wasp with w > 0.5 (under σ = ∞), no water is transferred. A resistance function with a high value of σ is sensitive to changes in water level about w = 0.5 and encourages water in-flux at small w. Thus, even under r_∞, a non-zero value of Δw is expected for w < 0.5. This result is significant because there are natural restrictions on the number of successful interactions needed regardless of σ; these bounds are independent of empirical data and is built into the system. Consider, for example, the water level w such that s(w) = 5.2. This implies S(w) > 5.2. Since S_∞(x) > S(x) for 0 < x < 0.5, S_∞(w) > 5.2. By Eq 8, this is , or . Solving for w, we get w > 0.4. Thus, a WF wasp can make 5.2 successes only at water levels greater than 40%. As another example, in our model, an intrinsic feature of the system is that at w = 0.5, at least six successes are needed for a WF wasp to completely empty itself, guaranteeing multiple interactions.

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Fig 8. Lower and Upper bounds on the amount of water Δw that a WF wasp is expected to transfer to an accepting CS wasp (8A) and on the number of successes S needed for a WF wasp to completely rid its load (8B) when the common stomach water level is w.

These bounds follow from the bounds on the assumed resistance function: Resistance with σ = 0 (black solid line) defines an upper bound on Δw and hence a lower bound on S. Resistance with σ = ∞ (grey dashed line) defines a lower bound on Δw and hence an upper bound (for w < 0.5) on S.

https://doi.org/10.1371/journal.pone.0145560.g008

In this context, the result that σ ≈ 0 (Results Section 1) is significant because it implies that S(w) is minimized for every w; σ = 0 minimizes the number of successes needed for complete unloading, and therefore maximizes the efficiency of water transfer. Note that while water transfer efficiency is maximized, the lower bound on S is such that multiple interactions are still necessary for complete unloading. Thus, as our model suggests, the system may have evolved such that the necessity for multiple interactions is built-in but also minimized within that constraint.

Because we assumed that all CS wasps have the same water level (Section 1.1), we interpret the resistance function as the average resistance supplied by the common stomach as a whole. Thus, since σ = 0, the average resistance function is approximately linear. This does not imply that the resistance function of every individual is necessarily linear; it is possible that resistance values from different water levels with σ > 0 average to a linear function (Fig 9). Suppose there is variance in the distribution of water among CS wasps. Suppose, for example, that 50% of the CS wasps have w = 0, and the other 50% of the CS wasps have w = 0.5. The average water level of the common stomach is then 0.25, and the average resistance is . Thus, on average, r(0.25) = 0.25, regardless of σ. Therefore, the system may be minimizing S(w) for every w by setting the water distribution among CS wasps such that the average resistance function is linear. This allows for the possibility that (1) the individual resistance function has σ > 0 and that (2) individual resistance functions may vary among CS wasps (σ may vary from CS wasp to CS wasp). By (1), the system can be both a strong buffer (by individuals) and could still allow for the easy flux of water (on average). By (2), an exact value of σ for individual resistance functions might not be critical so long as these individual functions average to a linear function.

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Fig 9. Schematic plot of how nonlinear individual resistance functions r averaged over different water levels of the common stomach w might yield a linear average resistance function.

Suppose individual CS wasps have sigmoid resistance functions (black solid line). Suppose half of the CS wasps are empty (r(0) = 0) and the remaining half are half-full ((r(0.5) = 0.5). The average resistance of the whole common stomach lies on the secant line (grey dashed line) joining (0, 0) and (0.5, 0.5); the average water level is w = 0.25, and the average resistance is r = 0.25. This means that a linear average resistance function does not imply linear individual resistance functions. Thus, the result that σ = 0 holds for the common stomach as a whole but does not necessarily hold for individual CS wasps.

https://doi.org/10.1371/journal.pone.0145560.g009

3. Performance of Foragers Predicted by the Model

In this section, we first examine the approximate number of successes s and s_P that a WF wasp and PF wasp are expected to make respectively when the common stomach water level is w (Fig 10).

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Fig 10. Plot of the numbers of successes s and s_p that a WF wasp (black) and PF wasp (grey) are expected to complete in one foraging cycle when the common stomach water level is w.

The plots are based on σ = 0 and K = 0.988 (Results Section 1). The plot for the WF wasp is an approximation obtained by joining the right endpoints of a step function. The plot for the PF wasp is symmetric to the plot for the WF wasp about the line w = 0.5.

https://doi.org/10.1371/journal.pone.0145560.g010

Since S (number of successes needed) is an integer while resistance r and the parameter K are real numbers, the plot of s(w) should be step-like. It is, however, sufficient for our analysis to consider only the plot of the right end-points of the steps of s(w). Note that s = 5.2 occurs at water level w > 0.5 (Fig 10), which is a consequence of our simplification of the plot of s(w). Since s_p(w) = s(1 − w), the plot for s_p(w) is obtained as the reflection of the plot for s(w) over w = 0.5. In general, the shapes of s(w) and s_p(w) agree with their hypothesized shapes (Fig 5).

Now let w(x) be the water level such that s(w(x)) = s; that is, let w(x) be the water level at which a WF wasp is expected to make x successes. We see that w(5.2) = 0.514 and w(6.4) = 0.568. So, 1 − w(6.4) = 0.438, which is the water level at which s_p = 6.4 (Fig 10). We can substitute these values into Eq 13 to achieve the following bounds on the average water level :

Hence, the average water level has a value slightly less than 50%. Although this was qualitatively derived in part from the empirical observation that WF wasps and PF wasps make on average 5.2 and 6.4 successes respectively [26], we used our model to determine at what water levels these average values occur. We can now conclude the following: The average water level is slightly less than 50%, and because the resistance function is linear and WF wasps are very unlikely to abort their foraging cycles (large K), a particular relationship between w and s emerges. This relationship is also founded on the precise way that resistance and amount of water accepted by a CS wasp are related (Section 1.2). Under this precise relationship, the water level fluctuates about its average value such that we observe the emergent averages 5.2 and 6.4. Thus, the values 5.2 and 6.4 as average numbers of successes are indicators of more fundamental parameters that underlie the system.

We now observe that y = s(w) has a maximum value of 17.158 at w = 0.800 (Fig 10). This implies that y = s_p(w) has the same maximum value at w = 0.200. It is not necessary that the common stomach water level ever reaches the value w = 0.200 or w = 0.800. So, at most, 17.158 is simply an upper bound on the maximum number of successes that a forager is expected to make. For the same reason that the average number of successes is greater for PF wasps than for WF wasps (since ), the maximum number of successes should be greater for PF wasps than for WF wasps. In data collected from real colonies, the largest value of s is 12, and the top two largest values of s_p are 15 and 19 (Fig 6) [26]. Thus, our result that s_p ≤ 17 is approximately accurate for PF wasps. Note that in the case of K = 0.990 (the upper bound on K), the maximum number of successes predicted is 19. So, let max(s_p) = 17, and for WF wasps, let max(s) = 12. We see that w(12) = 0.694 (Fig 10), and we already showed that 1 − w(17) = 0.200. If the common stomach water level fluctuates symmetrically about the average water level , then is the midpoint of 1 − w(17) and w(12): . Note that this is consistent with the bounds we obtained on in the previous paragraph.

The number of failures f that a WF wasp is expected to make increases with the common stomach water level w (Fig 11).

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Fig 11. The number of failures f that a WF wasp is expected to make in one foraging cycle when the common stomach water level is w.

https://doi.org/10.1371/journal.pone.0145560.g011

Note that the plot of failures for a PF wasp can be obtained by reflecting y = f(w) about w = 0.5. Because we showed that the common stomach water level fluctuates between the water levels at which a PF wasp and WF wasp are expected to make their maximum numbers of successes (w = 0.200 and w = 0.694 respectively), it is sufficient to plot f(w) only for w ≤ 0.800; we include water levels between 0.694 and 0.800 because the reflection of this region is included in the plot for a PF wasp. We see that as the common stomach water level increases, a WF wasp is expected to make more and more failures. The maximum number of failures that can be expected is about 71 at about w = 0.8 compared to 30 at w = 0.694. Although 71 is large, such extreme water levels are probably infrequent. Such large values, however, help to explain the relationship between number of successes and handling time; handling time appears to increase exponentially with success number, and this is because increasing success number implies increasing water level (Fig 10) which in turn implies increasing failure number (Fig 11). Since , , and since r(w) = w (σ = 0), . Thus, f(w) increases with w at a faster rate than does s(w). Given that f is concave-up, we expect the average value of f to be greater than the value of f at and at w = 0.476. Thus, the average number of failures is greater than f(w = 0.476) = 4.1. Further, depending on the concavity of f, it is possible that its average value could occur at a water level w > 0.514, where w = 0.514 is the water level at which the average number of successes (5.2) is made. Hence, more failures on average than successes does not necessarily imply that , since it is also consistent with . The possibility that more failures are made on average than successes is left open.

If σ is increased from 0, then S(w) increases for all w > 0 (Fig 3 and Results Section 2) and r(w) decreases for 0 < w < 0.5. From , it is unclear whether f(w) increases or decreases with σ for w < 0.5. We expect, however, that f(w) decreases for w < 0.5 since r(w) → 0 for w < 0.5 as σ → ∞. This decrease in failures, however, will not significantly increase the efficiency of water transfer since we will see that WF wasps are able to unload most of their water at low water levels even under σ = 0. On the other hand, for w > 0.5, as σ is increased, we expect f(w) to increase since both the number of successes needed S and the resistance r increase for w > 0.5. This means that σ = 0 minimizes the number of failures expected for a WF wasp at water levels greater than 50%. Although this seems to contradict the idea that negative feedback in the form of failures should be larger for w > 0.5, we see that the expected number of failures is large even for σ = 0. Thus, it seems that larger failure expectancy would not benefit the system significantly and that the minimum number of failures, which is itself large for high water levels, is sufficient. Further, this reinforces the fact that the common stomach is not only a buffer but also a mediator of water between WF wasps and PF wasps; if water flux was over-restricted by a very large σ value, then the system would not be an efficient mediator of water. The value σ = 0 might be able to provide both strong negative feedback and still allow for efficient mediation of water. Also, as mentioned in Results Section 2, it is possible that the individual resistance function is stronger than the average resistance function, so that the plot of failures might be steeper for individual CS wasps.

We now examine the fraction of one crop load of water that a WF wasp is expected to unload in one foraging cycle when the common stomach water level is w (Fig 12). Recall that one foraging cycle begins when the WF wasp arrives with a full load of foraged water and ends when the WF wasp either empties its crop or aborts the cycle. In one foraging cycle, when the water level is w, a WF wasp transfers Δw(w) of its crop per successful interaction and makes s(w) such successes. Thus, the total amount of water (fraction of one crop) that the WF wasp successfully unloads is s(w)Δ(w). By Eq 7, this is approximately (completed fraction of successes needed for complete emptying). Note that for PF wasps, the amount of water that a PF wasp is able to receive in one foraging cycle at water level w is . We see that for most water levels, a WF wasp is expected to unload almost all of its water into the common stomach (Fig 12). Around w = 0.7, the fraction of unloaded water begins to decrease rapidly. This is because although S(w) continues to grow at w = 0.7 (Fig 8), s(w) begins to plateau to its maximum at w = 0.7 (Fig 10). For w > 0.8, the WF wasp is not able to unload much of its water; it aborts its job due to the high resistance from the CS wasps.

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Fig 12. The fraction of one crop load of water that a WF wasp is expected to successfully transfer in one foraging cycle when the common stomach water level is w.

https://doi.org/10.1371/journal.pone.0145560.g012

We want to focus on the result that a WF wasp unloads approximately 50% of its load when w = 0.8, which is approximately where the plot of s(w) has its maximum value. Although we said that the water level remains strictly below w = 0.694, it is still relevant to study the plots at w = 0.800 because we are equating the behavior of a WF wasp at w = 0.800 to the behavior of a PF wasp at w = 0.200, and the water level can reach w = 0.200.

The result that WF wasps are expected to unload about half of their load at the water level where s(w) has its maximum value is significant because it might provide a biological explanation for the evolution of the value of the parameter K. Suppose we decrease K below 0.988. Since this increases the probability of abortion, it decreases the number of successes expected for a forager; s(w) decreases for all w. The function S(w), however, is independent of K. Thus, decreasing K decreases . Since there exist water levels for which under K = 0.988, values of K below 0.988 will imply some range of water levels over which WF wasps unload less than half of their crop loads. It could be possible, however, that the system evolved such that WF wasps and PF wasps can be expected to unload and fill respectively at least half of their crops at every common stomach water level within its fluctuating range. Thus, it is possible that K = 0.988 evolved to ensure a certain minimum on the average amount of unloading (for WF wasps) and loading (for PF wasps). If K is increased above 0.988, then WF wasps will be expected to unload more than 50% at all water levels below w = 0.8. This does not, however, provide any advantage; at low water levels, WF wasps are already efficient (Fig 12), and at higher water levels, increasing the in-flux of water by increasing K is counter to the negative feedback of the resistance function. Thus, K = 0.988 ensures that foragers are "at least" 50% successful with their task at all feasible water levels. This idea is explained in greater detail in the next section.

Finally, we examine the fraction of one crop of water that a WF wasp is expected to transfer on average per interaction (success or failure) when the common stomach water level is w (Fig 13). Since is the fraction of one crop that a WF wasp is expected to unload in one foraging cycle and since one foraging cycle contains s + f interactions, the average amount of water transferred per interaction is . This is = = . Thus, when the common stomach water level is w, a WF wasp is expected to unload water at a rate of crop loads per interaction (this value must be a fraction of one crop load). Since we are assuming that each interaction, including the time needed to move from one CS wasp to another, takes an average time of Δt, the rate of transfer per interaction is a measure of the time rate of water unloading. For PF wasps, the rate at which water is expected to be transferred to a PF wasp per interaction is . We see that decreases with w (Fig 13). Since σ = 0, we have , and we notice that 1 − w decreases with w while S(w) increases with w. Thus, the decreasing trend is mathematically expected. Biologically, the rate at which a WF wasp unloads water per interaction decreases as the common stomach water level increases. Note that this rate does not depend on the parameter K since neither r nor S depends on K. Since both r and S depend on σ, and since the value of σ is probably "encoded" into the behavior of CS wasps and not foragers (even if the σ value resulted from system-level interactions), the rate of loading and unloading per interaction is determined by the common stomach. For w = 0.25, if σ is increased, then S remains the same, S(0.25) = 2 (Fig 3), while 1 − r(0.25) increases. Thus, if σ is increased, then the rate at which water flows into the common stomach is increased at w = 0.25. We expect a similar trend for other water levels near w = 0.25. Thus, it seems that the system would be more efficient if σ was larger, but we found that σ is set at its minimum value of 0. Part of the resolution to this question might be as follows: At w = 0.5, if σ is increased, then r(w) remains the same since r(0.5) = 0.5 for all σ, while S(0.5) increases with σ. Thus, if σ is increased, then will decrease for both WF wasps and PF wasps by symmetry. Thus, since the common stomach water level is probably w = 0.5 more frequently than it is w = 0.25, the system may have evolved so that the flow rate of water in and out of the common stomach per interaction is maximized at or near w = 0.5. In particular, since S(0.5) = 6 for σ = 0, this maximum flow rate is crop loads per interaction. This maximum flow rate should maximize the efficiency with which the common stomach mediates water from WF wasp to PF wasp, while it seems that the resistance function is still "strong" enough to regulate the common stomach water level from fluctuating too much.

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Fig 13. The amount of water that a WF wasp is expected to unload per interaction (normalized) when the common stomach water level is w (the amount of water is given as a fraction of one crop load).

This is the rate at which water is transferred into the common stomach per interaction: .

https://doi.org/10.1371/journal.pone.0145560.g013