Philippa Kaur

Having introduced our novel explore-and-fuse method and the Physarum Steiner Algorithm we shall dedicate this section to discussing how the algorithm’s parameters influence the model, and how the method can be used towards diverse applications.In what follows we shall consider how different parameters such as the different shapes of cells, as well as their number, influence the results obtained by the Physarum Steiner Algorithm. We shall then conclude the section by studying different applications that our methods have.Cell shapeAlthough13 and6 considered diamond shaped CELLs, we shall consider here CELLs with other shapes. The primary benefit of square cells is that their shape allows for more cytoplasm to be placed on the grid. As a result, the foraging phase is very fast so using square cells tends to result in shorter run times than using diamond-shaped cells. In addition, large square cells are able to more completely cover the standard square grid than diamond-shaped cells. On the other hand, diamond-shaped cells result in less cytoplasm and more time spent in the foraging stage. This gives the cytoplasm time to move towards a centralized location which results in better solutions.Example A In order to illustrate the above point, in Fig. 3a.i., we begin with squares that are tightly packed. Since the squares are so tightly packed (1 apart), if any piece of cytoplasm in a square is moved, it will lead to a connection with a neighboring cell. As a result, all the points are found very quickly. In fact, many of the squares are connected and part of the network even if they are not close to any of the points, as shown in Fig. 3 (a.ii.). Shrinking these extra squares takes a long time and can also result in long paths which are far out of the way as seen in Fig. 3a.iii.Example B In contrast to Example A, in Fig. 3b, we consider diamond-shaped cells. The cells start off diamond-shaped and with less overall cytoplasm than the square cells. The cells then spend quite a few iterations in the foraging phase. Although this does take time, it allows the cytoplasm to move towards a centralized location around the active zones as seen in Fig. 3 (b.ii.). When the cell finally proceeds to the shrinking phase, there is less cytoplasm to remove and no out of the way paths, resulting in shorter solutions. The downside to this is the increased time which in some cases can be very long (over 100 million iterations) and in some cases the algorithm may not even complete.The effect of multiple cellsIn what follows we shall examine the effects of the number of cells used. We run 10 trials on 10 grids for a total of 100 trials on each cell size and number of cells. For each trial, we measure the total amount or area of cytoplasm that is initially spawned. This is used to normalize the search area which is the number of squares in the grid (for example a (100 times 100) grid has search area 10,000).Success rate: The algorithm may sometimes be unsuccessful at connecting all the points. For example, the cells may miss a point early on and move far away from that point, making it almost impossible to ever find that point. There may also simply not be enough cytoplasm for two far away cells to fuse into one. For each number of cells (1, 9, 25, 100), we try various sizes/amounts of cytoplasm and compute the proportion of trials (out of 100) that successfully terminate within 10 million iterations.Figure 4(a) Proportion of trials that are successful versus the search area as a percentage of cytoplasm for trials with 1, 9, 25, and 100 cells. (b) Length of solutions versus the search area as a percentage of cytoplasm. (c) Number of iterations versus the search area as a percentage of cytoplasm. Failed trails excluded from graphs.Full size imageIn Fig. 4a, we see that the black line (100 cells) extends much further to the right than the cyan line (one cell). Thus, the more cells there are, the larger of a search area we can explore. This is mainly because with more cells, we can spread out our cytoplasm instead of having it be concentrated in certain areas.Solution length Another important metric to consider is the solution length. We measure how good the solution is by counting the amount of cytoplasm when the algorithm terminates. We ignore any cytoplasm that is part of a disjoint cell that does not contain an active zone, or in other words is separate from the cell that actually forms the tree. In Fig. 4b, we see that as the search area as a percentage of cytoplasm increases, the quality of the solution improves. This is because there is comparatively less cytoplasm to begin with. In addition, we see that as the number of cells increases, it is possible to find a better solution. This correlates with the earlier result shown in Fig. 4a that using more cells allows solutions to be found with less cytoplasm. Trials with 100 cells found the shortest solutions (rightmost data point).Run time The last metric we consider is the run time. We consider the true number of iterations the algorithm runs for. By true iterations, we account for the fact that in a parallel algorithm or set of real-world Physarum organisms, multiple cells will be introducing and moving bubbles at the same time. As a result, the iteration count is scaled by the number of disjoint cells. In Fig. 4c, we see that the more cells there are, the lower the number of iterations. This may be because with more cells, the cytoplasm is more spread out and therefore there are less out of the way points which may take a very long time to find. From the above analysis, we see that using more cells allows us to explore bigger search areas, find shorter solutions, and solve problems faster.ApplicationsThe behavior of Physarum and the models it has inspired have found many different uses among which are drug repositioning, developing bio-computing chips, approximating highways layouts, and designing subway systems2,8,9,10. In order to illustrate the operation of the Physarum Steiner Algorithm and demonstrate its applicability to real world problems, we consider the following:

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Network design We use the algorithm to develop a road network in the United States.

Obstacle-avoidance We use the algorithm to solve the obstacle-avoiding Euclidean Steiner tree problem.

VLSI routing We use the algorithm to route connections between pads in chip design.

Topological surfaces We discuss the algorithm’s adaptability to varying surfaces and boundaries by considering topological surfaces such as the sphere, torus, Klein bottle, and (mathbb{RP}mathbb{}^2).

Road networks The Physarum Steiner Algorithm can be used to build a road network between the largest one hundred cities in the lower 48 United States (excluding Alaska and Hawaii). We use data32 containing the longitude and latitude of the 100 cities with the highest population to generate a rectangular grid of active zones.We spawn diamond-shaped cells of size 7 with a spacing of 1 as shown in Fig. 3. After many iterations, the final road network is shown in Fig. 5a. The algorithm is particularly suited to the problem of designing transportation systems because it first connects all the points before optimizing the network into a tree. The algorithm can thus be terminated early depending on how much redundant connectivity is desired in the transportation network.For example, in Fig. 5b, we have a network that still contains loops in high-traffic routes between the Bay Area, Los Angeles, and Las Vegas. If we allow the algorithm to continue running, we will get networks with fewer loops and eventually a tree.Figure 5Road network generated by the algorithm. (a) shows the final solution with no loops while (b) displays a solution that has some redundancy resulting from terminating the algorithm early.Full size imageWe believe that this algorithm can be applied to many similar problems such as designing fiber optic or electric cable networks. Moreover, as discussed in the last section, it will be very interesting to compare this study to that of33, where in vitro slime mold is used to investigate the construction of transportation networks over a USA map.Obstacle avoidance Due to the cellular automaton nature of this algorithm, it is straightforward to define boundaries or other obstacles that need to be avoided. This is very useful in cases where certain areas need to be avoided such as a lake or the boundary of a county. And, unlike the current standard obstacle-avoiding Euclidean Steiner algorithm27 which takes multiple hours for graphs with only 150 points, the run time of the Physarum Steiner Algorithm is not affected by the need to avoid obstacles.As an example, consider the boundary given in Fig. 6a. Here, the grey area represents the search area and the 100 white squares outlined in dark grey are the points. There are many possible real world situations similar to this. For example, the grey area could be a county and all the points represent homes that subscribe to a certain Internet service provider (ISP). The big white area in the center could be a lake and the smaller white area could be a dog park. The ISP company could utilize the Physarum Steiner Algorithm to find networks to lay fiber optic cables.Figure 6(a) Sample boundary map. Grey area is search area and small white squares are points. (b) Initial deployment of Physarum. (c) Solution at the end of the foraging stage. (d) The final network.Full size imageWe begin by deploying square Physarum cells of size 7 in Fig. 6b. In Fig. 6c, the cells begin to fuse, share intelligence, and find all the points. We choose a solution that still has some loops to increase reliability and ease of future modification to the network. Our final solution is shown in Fig. 6d. This solution is generated in 300,000 iterations and less than 30 seconds.VLSI Routing for VLSI (very large-scale integration) chip design19 is one of the largest real-world manifestations of the Steiner tree problem, especially as modern chips may contain upwards of 10 billion transistors. Solving the VLSI problem would require additional modification to the Physarum Steiner Algorithm since VLSI design is typically presented as a group Steiner tree problem and has very large problem sizes, the Physarum Steiner Algorithm. Due to the usage of a square grid in the Physarum Steiner Algorithm, the algorithm is easily applied to find rectilinear networks such as those required for routing chips. In addition, our empirical results suggest that it should scale well to the large problem sizes common in chip design. Using data from34, we consider a set of pads that need to be connected. In Fig. 7, we represent the pads as active zones and generate a tree between them.Figure 7(a) Graphical representation of 131-point VLSI data set34. (b) Routing solution obtained by the Physarum Steiner Algorithm.Full size imageTopological surfaces Finally, the Physarum Steiner Algorithm is easily applicable to finding Steiner trees on other topological surfaces. Given the nature of the algorithm, we are able to map coordinates on one edge to another. In Fig. 8, we use square identification spaces to find Steiner trees on the torus, sphere, Klein bottle, and (mathbb{RP}mathbb{}^2). These solutions on identification spaces can be seen on a torus and a sphere in Fig. 8a,b.Figure 8Steiner trees on topological surfaces we defined by identification space and obtained through our code. (a) Torus. (b) Sphere. (c) Klein Bottle. (d) (mathbb{RP}mathbb{}^2). Images generated using manim35.Full size imageConcluding remarksWe have presented here a novel explore-and-fuse approach to solve problems that cannot be solved by traditional divide-and-conquer.Our approach is inspired by Physarum, a unicellular slime mold capable of solving the traveling salesman and Steiner tree problems. Besides exhibiting individual intelligence, Physarum can also share information with other Physarum organisms through fusion. These characteristics of Physarum inspire us to spawn many Physarum organisms to independently explore the problem space and collect information in parallel before sharing the information with other organisms through fusion. Eventually, all the organisms fuse into one large Physarum that can then globally optimize using the knowledge collected earlier. Explore-and-fuse can be seen as a less rigid form of divide-and-conquer that can better handle problems that cannot be decomposed into independent subproblems.We demonstrate the explore-and-fuse approach on the Steiner tree problem by creating the Physarum Steiner Algorithm. This algorithm has the ability to incrementally find Steiner trees. The first solution tends to contain many loops that are removed with additional iterations of the algorithm. This incremental improvement is particularly useful for applications such as road and cable networks where some degree of redundancy in the connectivity is desired. In particular, it will be very interesting to compare our work to the the one done in33 where a protoplasmic network created by in vivo Physarum is considered to study and asses show the slime mold imitates the United States Interstate System. We foresee several applications of our algorithm in this direction, leading to similar findings to those appearing in the studies done in33.The algorithm operates on a rectilinear grid and is particularly applicable to rectilinear Steiner tree problems such as those that often arise in VLSI design. In addition, the algorithm performs well on the obstacle-avoidance Euclidean Steiner tree problem.In comparison to the existing Physarum-inspired Steiner tree algorithms described in Section “The Steiner tree problem”, the Physarum Steiner Algorithm uses a completely different mechanism. While the existing algorithms use a system of equations modeling the thickening of tubes as protoplasm flows through them, the Physarum Steiner Algorithm is based on modeling Physarum spatially moving around a grid and finding a tree between squares of the grid. In addition, it should be noted that the approach taking in existing algorithms would not work on the Euclidean Steiner tree problem as in the Euclidean Steiner tree problem, there are an infinite number of possible points that could be part of the Steiner tree (essentially any point in the plane). It would not be possible to write a system of equations representing the infinite possible points and edges. In the future, we believe further work could be done to improve the Physarum Steiner Algorithm. Since the Physarum Steiner Algorithm is an approximate algorithm, future improvements could be made so its approximations are closer to the actual optimal solution. In addition, it would be interesting to see this approach applied to other problems Physarum has been able to solve such as the traveling salesmen problem. More

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Study subjectsWe conducted foraging experiments on strepsirrhines (Nindividuals = 18) at the Duke Lemur Center (DLC), North Carolina, from February to November 201513. Our sample includes six fat-tailed dwarf lemurs (3–16 years of age, 3 males, 3 females), six gray mouse lemurs (3–7 years of age, all female), and six aye-ayes (17–32 years of age, 2 males, 4 females). Because these species are solitary and nocturnal, most animals were housed singly and were kept on a reversed light cycle such that they were active and could be tested during the day. Housing conditions were similar for all individuals, and they were all fed daily in a similar manner with a diet that included fruits, vegetables, meal worms, and monkey chow (details in13).All vervet data were collected on wild animals (Nindividuals = 12) at Lake Nabugabo, Uganda (0°22′–12° S and 31°54′ E) during four separate field seasons (April-June 2013, Double Trapezoid array, M group15; June–September 2013, Pentagon array, M group24; August–September 2015, Z-array, M group12; July–August 2017, Pentagon array, KS group25). M group was composed of between 21–28 individuals, containing 2–3 adult males, 7–9 adult females, 2 subadult males, 1–3 subadult females, and 9–12 juveniles and infants. KS group was composed of 39–40 individuals including 5 adult males, 11 adult females, 3 sub-adult males, 5 sub-adult females, and 15–16 juveniles and infants. All individuals were reliably identified based on natural features (details in12,15,24,25). Outside of foraging experiments, wild vervets were not provision fed.All Japanese macaque data (Nindividuals = 10) were collected at the Awajishima Monkey Centre (AMC), Awaji Island, Japan (34°14′43.6″ N and 134°52′59.9″ E) between July and August 2019 (Z-array26). AMC is a privately-run tourist and conservation center visited by a large group of free-ranging Japanese macaques (~ 400 individuals) called the “Awajishima group”47. The group is composed of different-aged individuals of both sexes, with bachelor males and bachelor male groups living around the periphery48. The Awajishima group forages on wild foods for much of their dietary requirements but is also provision-fed a combination of wheat and soybeans, supplemented with peanuts, fruits, and vegetables twice daily for ~ 10 months of the year (details in47,49,50).Study designNavigation arraysThe strepsirrhines and vervets were tested on a “double-trapezoid” shaped multi-destination array with six feeding platforms13,15, modified from17 (Fig. 1a), where there were 720 possible routes (6!). Three different double-trapezoid arrays were built to account for differences in body size: one for the smaller dwarf and mouse lemurs, one for the mid-sized aye-ayes, and one for the larger, wild vervets. Arrays were scaled such that the distance from platform 1–2 (the shortest distance between targets) was approximately twice the body length of the subject species. Vervets were additionally tested on a Z-shaped array with six feeding platforms (720 possible routes, Fig. 1b12), and a pentagon-shaped array with five feeding platforms (120 possible routes, Fig. 1c24,25,46). Japanese macaques were tested on an identically sized Z-array26.Figure 1Design of the navigational arrays used, with (a) the Double Trapezoid array used for Cheirogaleus medius, Microcebus murinus, Daubentonia madagascariensis, and Chlorocebus pygerythrus. Three different arrays were built and scaled to the body size of animals (see “Methods”). (b) The Z-array used for C. pygerythrus and Macaca fuscata. The same size array was used for both species because they are similar in adult body lengths (vervet mean range from four sites: 34.5–42.6 cm51, Japanese macaque mean range from six sites: 48.9–59.7 cm52. (c) The Pentagon used for C. pygerythrus. Distances here are unitless but roughly proportional to the body size of each species tested. Created in R version 4.0.4 and ProCreate.Full size imageFor strepsirrhine trials, DLC staff captured individuals in their enclosures and transported them in padded crates to the testing room. The dwarf and mouse lemur array was set up in a specially designed box (0.91 × 1.83 m) with a small compartment to contain strepsirrhines for rebaiting between trials. The aye-aye array was set up on the ground in a room measuring 2.44 × 4.27 m, where subjects stayed during the duration of their daily trials13. Vervet and macaque trials occurred when individual monkeys voluntarily left their group to participate in foraging experiments alone. Vervet arrays were set up using wooden feeding platforms (0.75 m long, 0.75 m wide × 0.75 m high) placed in an outdoor clearing measuring roughly 10 × 14 m in the home range of the study group. Japanese macaque arrays were also set up using small wooden feeding tables (0.40 m long, 0.30 m wide, 0.21 m high), covered in green plastic labeled with the platform number. Two identical arrays were built in neighbouring provision-feeding fields at the AMC (Near Lower Field: ~ 10 × 35 m, and Far Lower Field: ~ 15 × 45 m).In these studies, all platforms were baited with a single food item. The reward used varied by species (strepsirrhines: grape piece, apple piece, honey, agave nectar, or nut butters, vervets: slice of banana, piece of popcorn; macaques: single peanut or piece of sweet potato). Strepsirrhines have sensory adaptations for using olfaction to locate food53, while the cercopithecoids are heavily reliant on vision to locate resources54, so we ensured that each platform was baited with identical food items within a trial that smelled and looked the same to avoid biasing where the animals chose to go. Platforms for the wild monkeys were not rebaited between trials until all animals were ≥ 20 m away and the entire sequence could be rebaited before their return15,24,25,26.For all species, we started a trial when the tested individual entered the array and took the reward at a platform. We then recorded each successive platform visit (including revisits to empty platforms) until all rewards had been collected ending the trial. In our analyses, we included a total of 852 trials collected over six navigational experiments, completed by 40 unique individuals (18 lemurs, 12 vervets, 10 macaques) (Table 2).Table 2 Individuals and trial sample size included in the analysis.Full size tableData simulationsIn addition to empirically collected data, we simulated agents learning to travel efficiently in the same set of arrays using a simple iterative-reinforcement learning model based on the one used by Reynolds et al.6 to test for traplining behavior in bumblebees. In this model, agents move randomly between locations in an array until they visit all locations, then reset for another trial. If the agent completed a trial by travelling less distance than on previous trials, the probability of the agent repeating location-to-location transitions that occurred in that trial increased for future trials by a reinforcement factor. Initial transition probabilities were inversely proportional to the distance between two locations. Unlike Reynolds et al.6 our simulated agents started at a random location and were not required to return to that location to complete the trial. This matches the trial structure used in our experiments (open-TSP), and reflects multiple central place foraging patterns in primates55. Finally, agents could not return to the location they had just come from, using an “avoid the last location” behavioral heuristic observed in nectivores56,57, which prevented agents from getting stuck in “loops” between two locations (S1 Simulation Validation).Within each of the arrays used to collect empirical data, we ran simulations with reinforcement factors of 1 (no reinforcement), 1.2 (mild reinforcement), and 2 (strong reinforcement). For each array and reinforcement factor combination, we ran 100 agents that each completed 120 trials, where there was an equal probability of starting each trial at any location. Then, for each array and reinforcement factor combination, we ran 100 additional simulations per species tested in the given array, where the probability of starting a trial at any location was equal to the empirically observed location-starting probabilities of the respective species.These simulations were designed to help us test predictions of our two hypotheses regarding primate learning and decision making within the arrays. If primates learn to solve navigational arrays efficiently by reinforcing movements between platform pairs, they should exhibit overall greater receptiveness in their sequences of location visits than reinforcement factor 1 simulations, and a greater decrease over time in total distance travelled to complete the arrays. If primates are pre-disposed to navigate arrays using heuristics, they should exhibit shorter distances travelled on initial trials than in simulations.Data analysisFrom the raw sequences of locations visited in each trial, we calculated two metrics: minimum distance traveled, and the proportion of platform revisits that occurred within identical 3-platform visit sequences (determinism-DET)18. All calculations were done using R version 4.0.458 and packages rstan59 and tidyverse60. A fully reproducible data notebook containing this work, as well as all analyzed data, is available at https://github.com/aqvining/Do-Primates-Trapline. All figures were created by AQV in R version 4.0.4 and ProCreate.Distance traveledTo calculate minimum distance traveled, we created a distance matrix for each resource array containing the relative linear distance between any two resource locations. These minimum linear distances approximate the distances traveled by the animals, which may not necessarily be linear. We then summed the linear distances for all transitions made in a trial. Because resource arrays were scaled to the subject species’ body size, these relative distances were standardized.DeterminismGiven a sequence of observations, Ayers et al.63 defines determinism (DET) as the proportion of all matching observation-pairs (recurrences) that occur within matching sub-sequences of observations (repeats) of a given length (minL). This metric has been previously used to distinguish sequences of resource visitation generated by traplining behaviour from sequences generated by known processes of random movement within a given resource array18,61,62. It has several advantages in the analysis of foraging patterns, including the ability to detect repeated sequences between non-consecutive foraging bouts, imperfect repeats in sequences (i.e., omission or addition of a particular site), and distinguishing between forward- and reverse-order sequence repeats63.We adapted the methods of63 to calculate the number of recurrences and repeats generated by the sequence of location visits in each trial of our experiments and simulations. Based on an analysis of the sensitivity of DET scores to the parameterization of minL, we set minL to three for our calculations (S2 Sensitivity Analysis).Statistical analysesLearning ratesWe modelled distance travelled as a function of trial number, species, and individual. Metrics of animal performance on learned tasks are known to follow power functions over time and experience64, so we a priori applied log transformations to distance travelled and trial number, then fit a linear model. Thus, in the resulting model, the intercept can be interpreted as an estimated distance travelled on the first trial and the slope can be interpreted as the exponent of a learning curve. We modelled species and individual effects on the intercept by summing an estimated grand mean (µ0), species level deviation (µsp,j), and individual level deviation (µid,i). We treated species and individual level effects on the learning rate parameter (slope) the same way, summing a grand mean (b0), species level deviation (bsp,j), and individual level deviation (bid,i). We estimated additional parameters for the variance of individual level deviations in intercept and slope (σµID and σbID, respectively). Finally, after finding residuals in an initial analysis to have variances predicted by trial number and species, we estimated a separate error variance for each species (σε,sp) and weighted the standard deviations of the resulting error distributions by dividing them by the square root of one plus the trial number.We set regularizing priors on the model parameters, assuming distances travelled would remain within one order of magnitude of the most efficient route, but not setting any strict boundaries. For the grand mean of the intercept, we used a normal distribution centered around twice the minimum possible distance required to visit all platforms in the array, with a variance of one. For the grand mean of the slope and all species and individual level deviations to the slope and intercept, we used normal distributions centered at zero with variance of one. For all error terms, we used half-cauchy priors with a location parameter of zero and a scale parameter of one. The full, hierarchical definition of the model is given in Eq. (1).$$Distance sim {mu }_{0}+ {mu }_{sp,j}+ {mu }_{id, i}+left({b}_{0}+ {b}_{sp, j}+ {b}_{id,i}right)Trial+ epsilon$$$${mu }_{0} sim mathrm{N}(4.78, 1)$$$${mu }_{sp}, {b}_{0}, {b}_{sp} sim mathrm{N}(mathrm{0,1})$$$${mu }_{id} sim mathrm{N}(0, {sigma }_{mu ID})$$$${b}_{id} sim mathrm{N}(0, {sigma }_{bID})$$$$epsilon sim mathrm{N}(0, {sigma }_{epsilon ,sp}/sqrt[2]{1+Trial})$$$${sigma }_{mu ID}, {sigma }_{bID}, {sigma }_{epsilon } sim mathrm{Half Cauchy}(mathrm{0,1})$$DeterminismTo compare DET between species, and between empirical and simulated data, we created a binomial model of expected repeats generated in a trial given the number of recurrences (Eq. 2).$$Repeats sim binom(Recursions, DET)$$$$DET= {logit}^{-1}(alpha)$$$$alpha={a}_{0}+Sp+Src+ Int+ID$$$${a}_{0}, Sp, Src, Int sim mathrm{N}(0, 1)$$$$ID sim mathrm{N}(0, {sigma }_{ID})$$$${sigma }_{ID}sim mathrm{Half Cauchy}(mathrm{0,1})$$where a0 is the mean intercept, Sp is one of four coefficients determined by the species (simulations are of the “species” which was used to assign its starting-location probabilities), Src is one of four coefficients determined by the source (empirical data and each level of reinforcement factor), Int is one of 16 interaction coefficients (each possible combination of Sp and Src), and ID is a varying effect of the individual. Because the length of a sequence affects DET, we limit our analysis of DET to the sequences generated by a subject’s or an agent’s first ten trials. Subjects that completed fewer than ten trials were excluded from this portion of the analysis. More

To derive the relationship among macro-level ecological variables, which would constitute an ecological analog of the thermodynamic equation of state, we introduce a fourth state variable, B, the total biomass in the community. The ecological analog of the thermodynamic equation of state, an expression for biomass, B, in terms of S, N, and E, arises if we combine METE with a scaling result from the metabolic theory of ecology (MTE)18,21. In particular, we assume the MTE scaling relationship between the metabolic rate, (varepsilon ,) of an individual organism and its mass, m: (varepsilon sim {m}^{3/4}). Without loss of generality22, units are normalized such that the smallest mass and the smallest metabolic rate within a censused plot are each assigned a value of 1. With this units convention, the proportionality constant in this scaling relationship can be assigned a value of 1. From the definition of the structure-function, it follows23 that averaging the biomass of individuals times the abundance of species, nε4/3, over the distribution R and multiplying by the number of species gives the total ecosystem biomass as a function of S, N, and E. Explicitly:$$B=Smathop{sum}limits_{n}nint dvarepsilon ,{varepsilon }^{4/3}R(n,varepsilon {{{{{rm{|}}}}}}S,N,E)$$
(1)
Both the sum and integral in the above equation can be calculated numerically, and Python code to do so for a given set of state variables S, N, and E, is available at github.com/micbru/equation of_ state/.We can also approximate the solution to Eq. 1 analytically (Supplementary Note 2) to reveal the predicted functional relationship among the four state variables. If E > > N > > S > > 1:$$B=cfrac{{E}^{4/3}}{{S}^{1/3}{{{{{rm{ln}}}}}}(1/beta )}$$
(2)
where (capprox (7/2)Gamma (7/3)) ≈ 4.17 and (beta) = ({lambda }_{1}+{lambda }_{2}) is estimated13,22 from the relationship (beta {{{{{rm{ln}}}}}}(1/beta )approx S/N). Equation 2 approximates the numerical result to within 10% for 5 of the 42 datasets analyzed here, corresponding to N/S greater than ~100 and E/N greater than ~25. Multiplying the right-hand side of Eq. 2 by (1-1.16{beta }^{1/3}) approximates the numerical result to within 10% for 33 of the 42 datasets analyzed here, corresponding to N/S greater than ~3 and E/N greater than ~5. The inequality requirements are not necessary for the numerical solution of Eq. 1, which is what is used below to test the prediction.Empirical values of E and B can be estimated from the same data. In particular, if measured metabolic rates of the individuals are denoted by ({varepsilon }_{i},) where i runs from 1 to N, then E is given by the sum over the ({varepsilon }_{i}) and B is given by the sum over the ({{varepsilon }_{i}}^{4/3}.) Similarly, if the mass, mi, of each individual is measured, then B is the sum over the mi and E is the sum over the mi3/4. In practice, for animal data, metabolic rate is often estimated by measuring mass and then using metabolic scaling, while for tree data, metabolic rate is estimated from measurements of individual tree basal areas, which are estimators5 of the ({varepsilon }_{i}).With E and B estimated from the same measurements, the question naturally arises as to whether a simple mathematical relationship holds between them, such as E = B3/4. If all the measured m’s, are identical, then all the calculated individual (varepsilon {{hbox{‘}}}s) are identical, and with our units convention we would have E = B. More generally, with variation in masses and metabolic rates, the only purely mathematical relationship we can write is inequality between E and B3/4: (E=sum {varepsilon }_{i}ge (sum {{{varepsilon }_{i}}^{4/3}})^{3/4}={B}^{3/4}). Our derived equation of state (Eq. 2) can be interpreted as expressing the theoretical prediction for the quantitative degree of inequality between E and B3/4 as a function of S and N.A test of Eq. 1 that compares observed and predicted values of biomass with data from 42 censused plots across a variety of habitats, spatial scales, and taxa is shown in Fig. 1. The 42 plots are listed and described in Table S2 and Supplementary Note 3. The communities censused include arthropods and plants, the habitats include both temperate and tropical, and the census plots range in area from 0.0064 to 50 ha. As seen in the figure, 99.4% of the variance in the observed values of B is explained by the predicted values of B.Fig. 1: A test of the ecological equation of state.Observed biomass is determined by either summing empirical masses of individuals or summing empirical metabolic rates raised to the ¾ power of each individual. Predicted biomass is determined from Eq. 1 using observed values of S, N, and E. The quantity ln(predicted biomass) explains 99.4% of the variance in observed biomass. Units of mass and metabolism are chosen such that the masses of the smallest individuals in each dataset are set to 1 and those individuals are also assigned a metabolic rate of 1. The shape of the marker indicates the type of data, and the lighter color corresponds to higher species richness. Data for all analyses come from tropical trees39,40,41,42,43,44,45, temperate trees30,31,32,33,46,47,48, temperate forest communities27,49, subalpine meadow flora28, and tropical island arthropods50.Full size imageFigure 2 addresses the possible concern that the success of Eq. 1 shown in Fig. 1 might simply reflect an approximate constancy, across all the datasets, of the ratio of E to B3/4. If that ratio were constant, then S and N would play no effective role in the equation of state. Equation 1 predicts that variation in the ratio depends on S and N in the approximate combination S1/4ln3/4(1/(beta (N/S))). In Fig. 2, the observed and predicted values of E/B3/4 calculated from Eq. 1, are compared, showing a nearly fourfold variation in that ratio across the datasets. The equation of state predicts 60% of the variance in the ratio.Fig. 2: The explanatory power of diversity and abundance.The observed ratio E/B3/4 is plotted against the ratio predicted by Eq. 1. Of the fourfold variability across ecosystems in that ratio, 60% is explained by the variability in the predicted combination of diversity and abundance. The shape of the marker indicates the type of data, and the lighter color corresponds to higher species richness. Data for all analyses come from tropical trees39,40,41,42,43,44,45, temperate trees30,31,32,33,46,47,48, temperate forest communities27,49, subalpine meadow flora28, and tropical island arthropods50.Full size imageFigure 3 shows the dependence on S and N of the predicted ratio E/B3/4 over empirically observed values of S, N, and E. We examined the case in which S is varied for two different fixed values of each of N and E (Fig. 3a) and N is varied for two different fixed values of S and E (Fig. 3b). The value of E does not have a large impact on the predicted ratio, particularly when E > > N. On the other hand, the predicted ratio depends more strongly on N and S.Fig. 3: The theoretical prediction for the ratio E/B3/4 as a function of S and N.The biomass B is predicted by holding E fixed along with one other state variable. In a N is fixed and S is varied, and in b S is fixed and N is varied. The fixed values are chosen to be roughly consistent within a range of the data considered. The color of the lines represents the corresponding fixed value of N or S, while the solid and dashed lines represent different fixed values of E.Full size imageThe total productivity of an ecological community is a focus of interest in ecology1, as a possible predictor of species diversity24 and more generally as a measure of ecosystem functioning25. By combining the METE and MTE frameworks, we can now generate explicit predictions for certain debated ecological relationships, including one between productivity and diversity. Interpreting total metabolic rate E in our theory as gross productivity, then in the limit 1 More

The generalist accumulates extracellular nitriteWe first tested whether the generalist accumulates substantial extracellular nitrite under our experimental conditions, and thus creates a niche for the specialist. To accomplish this, we grew the generalist alone in bioreactors with anoxic ACS medium amended with 12 mM nitrate as the growth-limiting substrate and measured the extracellular concentrations of nitrate and nitrite over time. We performed these experiments at pH 6.5 (strong nitrite toxicity) and 7.5 (weak nitrite toxicity).We observed a substantial accumulation of extracellular nitrite regardless of the pH (Fig. 3A, B). When grown at pH 6.5 (strong nitrite toxicity), extracellular nitrite accumulated to a concentration comparable to the initial nitrate concentration (measured maximum extracellular nitrite concentration, 11.8 mM; measured initial nitrate concentration, 12.0 mM) and was subsequently consumed to below the detection limit (Fig. 3A). When grown at pH 7.5 (weak nitrite toxicity), extracellular nitrite again accumulated to a concentration comparable to the initial nitrate concentration (measured maximum extracellular nitrite concentration, 11.7 mM; measured initial nitrate concentration, 12.9 mM) and was subsequently consumed to below the detection limit (Fig. 3B). During growth at pH 6.5, substantial nitrite consumption did not begin until a prolonged period of time after nitrate consumption was complete, resulting in a relatively long duration of nitrite availability (Fig. 3A). During growth at pH 7.5, in contrast, substantial nitrite consumption began immediately after nitrate consumption was complete, resulting in a relatively short duration of nitrite availability (Fig. 3B). The longer duration of nitrite availability at pH 6.5 indicates that the duration of the niche created by the generalist for the specialist depends on pH.Fig. 3: Growth and nitrogen oxide dynamics of the generalist in batch culture.We grew the generalist alone in a bioreactor at A pH 7.5 (weak nitrite toxicity) or B pH 6.5 (strong nitrite toxicity) under anoxic conditions with nitrate as the growth-limiting substrate. Blue squares are measured extracellular nitrate concentrations, yellow triangles are measured extracellular nitrite concentrations, and black circles are measured cell densities. We measured extracellular nitrate and nitrite concentrations with IC and cell densities with FC. C Measured durations of nitrite availability for the generalist growing in batch culture. We grew the generalist alone in 96-well microtiter plates under anoxic conditions with nitrate as the growth-limiting substrate. Open symbols are durations of nitrite availability at pH 6.5 and closed symbols are durations of nitrite availability at pH 7.5. Each symbol is an independent biological replicate.Full size imageTo routinely quantify the duration of nitrite availability, we grew the generalist alone with varying amounts of nitrate as the growth-limiting substrate. We then quantified the length of time from when the growth rate with nitrate was maximum to when the growth rate with nitrite was maximum. This cell density-based proxy measure is valid because the growth of the generalist is directly linked to the consumption of nitrate and nitrite (Fig. 3A, B). The cell density of the generalist was initially linearly correlated with nitrate consumption at both pH 6.5 (strong nitrite toxicity) (two-sided Pearson correlation test; r = −0.96, p = 1.5 × 10–8, n = 15) (Fig. 3A) and 7.5 (weak nitrite toxicity) (two-sided Pearson correlation test; r = −1.00, p = 2.2 × 10–16, n = 30) (Fig. 3B). After nitrate was depleted, the cell density of the generalist became linearly correlated with nitrite consumption at both pH 6.5 (strong nitrite toxicity) (two-sided Pearson correlation test; r = −0.97, p = 3 × 10–4, n = 7) (Fig. 3A) and 7.5 (weak nitrite toxicity) (two-sided Pearson correlation test; r = −0.97, p = 6.8 × 10–10, n = 16) (Fig. 3B). We further validated our cell density-based approach by testing for concordance with our IC-based direct measures of the duration of nitrite availability. We observed a significant positive and linear relationship between the cell density- and IC-based measures (two-sided Pearson correlation test; r = 0.999, p = 0.023, n = 3) (linear regression model; slope = 1.19, intercept = −2.31, r2 = 0.99) (Supplementary Fig. S2), which further validates our cell density-based approach to routinely estimate the duration of nitrite availability.Using our cell density-based approach, we found that the duration of nitrite availability was significantly longer at pH 6.5 (strong nitrite toxicity) than at 7.5 (weak nitrite toxicity) regardless of the initial nitrate concentration (two-sample two-sided t-tests; Holm-adjusted p  0.92, Holm-adjusted p  0.6), and thus followed model predictions (Fig. 4A). However, when the specialist was initially rare (measured initial log rS/Gs of –3.19, –2.65, and –0.88), the relative abundances of the specialist continuously decreased between the third and twelfth transfers (Mann–Kendall trend tests; tau = –0.61 to –0.89, p  0 were dominated by phenotype C (dominant ancestral phenotype with a long time delay between nitrate and nitrite consumption), while generalist isolates from co-cultures with initial rS/Gs  More