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

Here we provide a unified analysis of predator-prey biomass scaling in complex food webs. Doing so reveals a consistent sub-linear scaling pattern across levels of organization – from populations within webs to whole ecosystems – for freshwater, marine and terrestrial systems. This regularity in sub-linear predator-prey scaling among complex food webs from diverse ecosystem types has important implications for understanding energy flows in natural systems across large spatial gradients.Within food webs, predator-prey biomass scaling was characterised by a near three-quarter power scaling relationship ((bar{k}) = 0.71 across ecosystem types), revealing an approximately three-fold increase in predator biomass for every five-fold increase in prey biomass. When summing all predator and prey biomasses within a food web (Fig. 4), predator-prey scaling across webs followed a similar sub-linear scaling regime, with k ranging from 0.65 to 0.67 between ecosystem types. That is, biomass pyramids became systematically more bottom-heavy as pyramid size increased along a biomass gradient (Fig. 1a). These ecosystem-level patterns are quantitatively consistent with previous analysis of predator-prey biomass scaling among distinct trophic groups, which also found sub-linear scaling with k values between 0.66 to about 0.768,17,18. The approach we introduce here permits expanding these analyses to more complex omnivorous feeding relations both among populations within webs and across webs in diverse ecosystems. The similarity in the scaling exponents (and overlap in confidence intervals) of within- and across-web scaling suggest the existence of a general sub-linear scaling pattern, possibly signifying that fundamental constraints apply across levels of biological organization.These results beg the question: where do these sub-linear scaling patterns originate? We are not aware of any ecological theory that is sufficiently general to encompass the diversity of community types in which sub-linear biomass scaling is observed (Appendix S2). Size spectrum theory, which aims to explain the observation that, for whole ecosystems, biomass is approximately evenly distributed across logarithmic body size classes19,20 would appear to be particularity relevant. However, static size spectrum models typically assume that the predator-prey body mass ratio (PPmR) and trophic transfer efficiency (ratio of predator to prey production), whilst inherently variable21,22, do not vary systematically with prey biomass19,23. These measures indicate from which size class energy is obtained relative to predator body mass, and how efficiently that energy is utilized by any given predator to maintain its biomass. While these variables are thought to drive size spectra scaling3, they do not appear to be consistent with predator-prey biomass scaling observed in natural communities. Assuming both an even distribution of biomass across size classes, and a constant PPmR or transfer efficiency across a prey biomass gradient suggests an unchanging trophic biomass pyramid (all else being equal; Appendix S2), Therefore it is not clear how current size-spectrum models might account for sub-linear predator-prey biomass scaling.Predator-prey theory, on the other hand, which models the dynamics of feeding interactions, has traditionally focused on two distinct trophic levels, rather than on networks of highly omnivorous food webs24. Equilibrium predictions from a range of simple predator-prey models are also not consistent with sub-linear predator-prey scaling without additional and likely questionable assumptions (Appendix S2). Although predator-prey theory can be made to accord with our observed patterns, it requires tuning the scaling of prey growth or other terms of the model. Furthermore, questions remain about how best to simulate a biomass gradient as well as how models should be generalized to multi-trophic food webs (Appendix S2).Despite the lack of any general mechanism, it is reasonable to assume that predator biomass, at steady state, is maintained in proportion to prey production8,10. This would suggest that as prey biomass increases, their total production should scale near ~¾ to match the predator biomass they support. Density-dependent processes, such as competition for resources and other negative interactions among prey species, could thus cause per capita growth to decline sub-exponentially. We observed that changes in prey biomass were primarily driven by changes in prey density, rather than average prey body size, consistent with density dependent effects driving the sub-linear nature of predator-prey biomass relations, rather than allometric body mass effects. Clearly, however, ecological theory has further work yet to knit together the various patterns and processes to explain the consistency and generality of predator-prey scaling patterns.Addressing predator-prey biomass scaling from a food web perspective allowed us to assess which node properties were associated with greater predator-prey biomass ratios. Our results go beyond prior theoretical studies6,7 to provide empirical evidence that populations of highly omnivorous predators, as well as predator populations that feed down the food web on smaller, more productive, prey (i.e. a high predator-to-prey body mass ratio), tend to attain higher biomass stocks than predicted by their prey biomass alone. Interestingly, the role of these variables in driving predator biomass deviations appear to vary between ecosystem types: predator biomass increases more strongly with PPmR in rock pool webs, whereas predator omnivory only proved to correlate with predator biomass residuals in soil webs (Fig. 3). Further research would be instructive to understand if these are general patterns across different types of terrestrial and aquatic ecosystems. For instance, whilst rock pool webs display very similar network topology and PPmR scaling as open marine webs25,26, we might expect different scaling patterns in pelagic marine webs where trophic interactions take place in three dimensions21, where ontogenetic diet shifts are common27, and where food chains are long13. Adapting our food-web approach to quantify biomass scaling among size classes would likely be informative for tackling these additional complexities. Whilst predator biomass was associated with PPmR and omnivory (in soil webs), the consistent sub-linear predator-prey scaling regime within ecosystem types and across levels of organization, suggests that density dependent population growth might be the overriding driver of predator-prey biomass scaling.The regularity in predator-prey scaling we observed could provide insight into baselines for the biomass structure of natural communities, which could be informative for assessing the effects of environmental impacts within ecological communities and ecological status. For instance within webs, deviations away from these baselines in the form of smaller power-law exponents (shallower slopes) could reflect local perturbations (e.g. acidification, warming, over-exploitation) which have a disproportionate impact among larger organisms at higher trophic levels28. Predator-prey biomass scaling could therefore offer a complementary approach to body size distributions and size spectra for evaluating ecosystem health29. A similar approach could be applied for scaling relations within species, where the same species occur in multiple webs. Doing so could reveal how the biomass of a given predator species responds to variation in prey availability. For instance, among the stream food webs studied here, two common fish species displayed the characteristic near ¾-power scaling pattern, whilst the biomass of salmonids, and particularly brown trout (Salmo trutta), was invariant with prey biomass across webs (Fig. S4). These results are consistent with previous work in these sites which has highlighted the importance of terrestrial prey for subsidizing the biomass production of these apex predators30,31. Deviations from the expected general scaling pattern could therefore be valuable for identifying the importance of environmental factors that permit some species an ‘escape’ from the predator-prey power law (see also32), and offers promising avenues for future research.Our study, which takes a first step towards investigating predator-prey biomass scaling in complex food webs, has some notable limitations. First, information on the weighting of feeding links was not available for the food webs studied here, and a more comprehensive investigation should require specific interactions strengths and vulnerabilities of each species, data that is, as yet, unavailable. Although our results are robust to alternative assumptions in how these factors are treated (Table S5), any systematic variation in feeding interactions could play an important role. Second, information on the biomass of all basal resources was also not generally available, so our analysis focused on higher trophic predators feeding on (animal) prey. While our approach may equally apply more generally to consumers and resources (e.g. aquatic snails feeding on biofilm), further work is required to test the generality of the empirical patterns we observed using more detailed datasets where this information, and data on interaction strengths, is widely available.Overall, our study reveals a consistent sub-linear predator-prey scaling regime in complex food webs and makes a strong case for the existence of a systematic form of density-dependent population growth that governs the biomass structure of freshwater, marine and terrestrial ecosystems. The highly conserved predator-prey scaling we observed within and across food webs implies a relatively simple scaling-up of predator and prey population biomasses across levels of biological organization. These general patterns in energy flow between predator and prey could facilitate improvements in modelling trophic structure and community dynamics, as well as the corresponding ecosystem functions4,5. Our findings suggest sub-linear predator-prey biomass scaling holds within complex omnivorous food webs, urging ecologists to understand the origin of this large scale, cross-system pattern. More

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