To design a novel blocking screws (BSs) geometry and insertion method to treat distal tibia fracture with nailing and comparison of mechanical properties of novel and traditional screws.Twenty‐one synthetic left tibiae were sectioned to obtain 21 distal segments measuring 55 mm. Intramedullary (IM) 9‐mm tibial nails were advanced to 6 mm from the ankle joint. Two transverse and one anterior–posterior (AP) locking screws were inserted. Both medial–lateral (ML) BSs were placed 10 mm from the topmost interlocking screw. A custom‐made jig assisted in placing the novel and traditional BSs. The time spent in placing each BS was recorded. All the samples were repaired with an IM nail and without BSs, with two traditional BSs, and with two novel BSs. An initial loading from −150 to +150 N was applied to specimens in the ML direction at 185 mm from the nail end, followed by cyclic loading of the same for 10,000 cycles with failure‐to‐test loading of 350 N in the ML direction. The maximum displacement was measured at 80 mm from the nail end and recorded under initial loading. The damage of two kinds of BSs to the nail was recorded.Compared with average 5.21 min of the time of placing a traditional BS, the time spent in positioning a novel BS on the fracture model was 2.53 min. In the distal bone–implant constructs (BICs), the addition of traditional BSs decreased the maximum displacement of the BICs by 26.2%. The addition of the novel BSs decreased the displacement by 28.9%. All constructs survived 10,000 cycles without hardware deformation. The failure rate of the control group was significantly greater than that of the traditional group; however, the novel group was similar to the traditional group. The damage of the traditional BS to the nail was greater than that of the novel one.The novel and traditional BSs are comparably effective for increasing the primary mechanical stability of distal metaphyseal fractures after nailin. However, compared to the placement of a traditional BS, implanting a novel BS took more less time and caused less damage to the nail. Additionally, the most obvious advantage of the novel BS design and insertion technology was that the pressure and distance between it and the IM nail could be controlled by rotating the screw. These advantages of the novel BS will be beneficial for clinical application. This study introduces a new blocking screws (BSs) structure and compares the mechanical stiffnesses of novel and traditional BSs using a distal tibial fracture segment model treated with intramedullary nails. the most obvious advantage of the novel BS design and insertion technology was that the pressure and distance between it and the IM nail could be controlled by rotating the screw, which helped to shorten the adjustment time and reduce the occurrence of new fractures. By mechanical test, this study concluded that the stiffnesses of the novel and traditional BSs are equal. Distal tibia fractures are treated by various methods such as intramedullary (IM) nailing, plating, and external fixation, but an optimal treatment technique has not yet been established for clinical applicationet al. were the first to propose the use of a BS to assist the fixation of metaphyseal fractures for increased bone‐nail stiffnesset al.via nailingKrettek Studies have been conducted to investigate the proper placement of BSs, such as at an acute angle to the flared segment between the long axis of the displaced fracture fragments and aligned with the plane of the fractureThe first objective of this study is to describe the geometric construct of a novel BS. The novel BS geometry is very simple and improves the traditional BS screw tip by cutting it into a flat end. The second objective of this study is to introduce the placement method of a novel BS to improve its clinical application. The placement is parallel to the deformity of the fracture in a concave plane unlike the traditional perpendicular method. The point connecting with the nail of the flat end is its lateral plane. BS positioning adjustments can then be obtained by turning the BS instead of replacing it. This thread‐controlling adjustment strategy for metaphyseal fractures makes BS adjusting quite easy and avoids the need to accurately determine the entrance of the BS while avoiding additional injuries to soft tissues. However, it is unclear whether the mechanical property of the novel BS satisfies the need to enhance the BICs stability. Hence, the third objective of this study is to compare the mechanical stiffnesses of the two methods to supplement distal tibial metaphyseal fractures using small‐diameter IM nail fixations. Our null hypothesis states that the mechanical properties of additional BSs will be better than no BS; however, there will be no differences between the novel and traditional groups.n = 21) fourth‐generation composite Sawbones left tibiae with solid cancellous foam  and an expert tibial nail  were used for the investigation. Previous studies have confirmed that, compared with human bone, surrogates produce remarkable results for axial, compression, torsional, and bending stiffnesses, as well as for failure mechanisms under different loading conditionsn = 7) tibiae per treatment group. The treatment groups were constructs without any BS except for the pre‐planned screw path (control group), those with two bicortical traditional BSs placed in the anteroposterior (AP) position   Fig. , and thoup) Fig. .et al.The 9‐mm IM nail was inserted in an unreamed fashion using a standard techniqueet al.Loads ranging between −150 and +150 N (one‐third of the body weight of a 45.9‐kg person) were applied in the ML direction at 185 mm from the nail end after BIC fixation in a material testing machineversus displacement curve. Under the same loading, the smaller was the maximum displacement, the greater was the stiffness.The maximum displacement was used to evaluate the stiffnesses of the BICs. The BIC stiffness is an interesting mechanical parameter and is defined as the slope of the force et al.As mentioned earlier, loading from −150 to +150 N was then applied at a frequency of 1 Hz for 10,000 cycles in the ML direction, which represents the approximate number of steps taken over a 4–6 weeks period; i.e., the estimated interval for postoperative non‐weight bearingWe used the consumed time to compare the difficulty of placing a respective traditional and novel BS. The consumed time is an indirect index to evaluate the possible subsequent surgical complications.The damage to a nail is defined as the mark left on a nail by the friction of the drilling head and cutting of the screw thread. This indicator was used to evaluate the destruction to a nail when placing a traditional or novel BS.t‐tests were used to compare the placing time of the novel and traditional BSs. The difference was considered to be significant when the P value was less than 0.05.All data were initially tested for normality distribution using the Shapiro–Wilk statistical test. One‐way ANOVA tests (two‐tailed) and a least‐significant‐difference post‐hoc test were used to compare the results within the group. The measurements included time spent in placing a BS and displacement of the nail. Two independent sample F = 5.004, P = 0.018). Compared with 4.88 ± 1.20 mm (mean ± standard deviation) in the control group, the maximum displacement of the novel group was 3.47 ± 0.77 mm, which decreased by 28.9% . A comparison of the traditional and novel groups showed that they were not statistically significant. The increased stability of BIC in the novel BS was more than 2.7% greater than that of the traditional group, although no significant difference was observed.In the distal BICs, the addition of traditional BSs decreased the maximum displacement of the BIC by 26.2%, from 4.88 ± 1.20 mm (mean ± standard deviation) in the control group to 3.60 ± 0.72 mm in the traditional BS group . Time spent in the placement of the traditional BS required nearly twice as much time as the novel group.The average time needed to place a single BS on the fracture model in the new BS group was 2.53 min (range: 1.8–3.2 min), which was less than 5.12 min (range: 4.2–6.1 min) in the traditional group  and a functional point (the part touching the nail).The orientation of the traditional BS placement was perpendicular to the plane of the fracture deformity to enable the reduction and fixation of the flared segment. Thus, the effect of BS reduction and fixing the fracture segment was determined by its entry. The current difficulty involved the estimation of an appropriate entry point for placing the BSs. This endeavor remains a challenge for surgeons under conditions of skin shielding and fracture deformity. Although some precise methods for BS insertion have been described in recent literaturevia the movement of the anchored end. The current technique used to adjust the traditional BS requires resetting it, which requires even more time and consequential injury. To improve the placement and function of the BS, we modified its placement from a perpendicular position to the fragment movement plane parallel to it, thereby improving the placement and adjustment strategy. Meanwhile, the tip of the screw was designed to be flat to increase the connecting opportunity with the nail (Fig. et al. in 2020 described the clinical use of a fully threaded 3.5‐mm cancellous screw as a reduction tool for correcting frontal deformities along the coronal planeThe distance between the nail and the functional point was determined by the anchored end, resulting from the placement of the traditional BS perpendicular to the displacement of the flared segment. The tuning of the BS functional point was achieved ail Fig. . Koller et al.Our idea of using the BS as a reduction tool is similar to that of Koller et al.et al.This study demonstrates that both the novel transversal and traditional sagittal BSs, when placed adjacent to an IM nail, can increase the primary stability of distal tibia fractures. Using this model, the fracture was stabilized distally by two ML locking screws and one anterior–posterior locking screw. The fracture stiffness of the three locking screws was 26.2% less than that of the traditional BSs. The novel BSs increased construct stiffness up to 28.9%. Nevertheless, compared with the traditional group, no statistical differences were observed. The increased BIC stiffness of the novel BS was 2.7% greater than that of the traditional group. This could possibly lead to complaints from patients regarding the increased pressure resulting from the connection between the BS and nail. The traditional BS can only be used as an occupation screw to centralize the nail. However, the novel BS functions as a compression screw to reduce the nail and increase the construct stiffness. Breakage or backout of the novel screws was not observed until the end of each sample test. This may be explained by the fact that the force between the BS and the nail is less than the anti‐pull‐out strength of the interface between the bone and the semi‐cortical screws under a 150‐N load. In 2019, Ketata The 350‐N loading‐failure test resulted in the only new fracture, whereas the control group resulted in considerable nail bending and loosening Fig. . One impThe time required for novel BS insertion required less than 50% of that of the traditional BS. Clinically, the insertion of a traditional BS may require more time on account of the existence of complex soft tissues, fracture displacements, and the lack of precise markers and assistive instruments. Compared with the traditional BS, inserting the new one was easier owing to the use of interlocking as an obvious marker, the tuning of the BS location achieved by turning the screw, and the lack of any need to precisely position it. Moreover, the continuous friction against the nail when using the traditional BS caused damage because of threading and drill cutting. However, the contact between the novel BS and the nail was transient, and only miniscule damage was found in the novel BS group.With most tibial nails, two transverse locking screws are implanted to provide stability in the sagittal plane after reduction. However, these locking screws provide less stability in the frontal plane. Therefore, the direction of fracture translation is often along the axis of the locking nail in the frontal plane. For this reason, our model simulated the direction of fracture re‐displacement as an ML movement on the distal tibia fracture, which was fixed by nailing it with two transverse and one AP locking screws. Additional BSs were placed on the ML side to avoid translation of the ML interlocking screws. Additionally, the direction of the loading application was in the ML orientation.This study had several limitations. Only ML testing was conducted on the current transverse fracture model. Regarding the novel method for placing BSs, further study is needed to quantify the effects of axial fatigue loading. However, a prior biomechanical study confirmed that a single additional medial BS has no effect on the distal tibia fracture model when nailing with the BS under combined cyclic axial and torsional loads. Additionally, compared with the two distal locking screws, three have significant advantagesBased on the results of this mechanical study, we conclude that both the traditional and novel BS techniques increased the primary stability of distal metaphyseal fractures. They exhibited similar results in mechanical tests. However, the novel screw helped alleviate the difficulty of tuning of the BS during operation. The time spent inserting the new BS was significantly shorter than that of the traditional one, and the damage to the nail in the novel group was more subtle than that of the traditional one. Additionally, the obvious advantage of the novel BS is that the distance and pressure between it and the IM nail can be adjusted by turning the screw, which decreases the operation time and avoids the occurrence of new fractures. These advantages provide more benefits for the clinical application of BSs.None of the authors of this paper have a financial or personal relationship with other people or organizations that could inappropriately influence or bias the content of the paper.


Generation and control of humidity in a testing environment is crucial when evaluating a chemical vapor sensor as water vapor in the air can not only interfere with the sensor itself, but also react with a chemical analyte changing its composition. Upon constructing a split-flow humidity generator for chemical vapor sensor development, numerous issues were observed due to instability of the generated relative humidity level and drift of the humidity over time. By first fixing the initial relative humidity output of the system at 50%, we studied the effects of flowrate on stabilization time along with long term stability for extended testing events. It was found that the stabilization time can be upwards of 7 h, but can be maintained for greater than 90 h allowing for extended experiments. Once the stabilization time was known for 50% relative humidity output, additional studies at differing humidity levels and flowrates were performed to better characterize the system. At a relative humidity of 20% there was no time required to stabilize, but when increased to 80% this time increased to over 4 h. With this information we were better able to understand the generation process and characterize the humidity generation system, output stabilization and possible modifications to limit future testing issues. Replicating realistic environmental conditions on a small scale is essential when testing and developing chemical vapor sensors. In particular, the presence of water vapor, or relative humidity (RH), in the air is a real world challenge in almost all scenarios and must be taken into consideration as it can have a dramatic impact on a sensor’s performance.4. Therefore, it is possible that under completely dry conditions, very low RH, a sensor would detect a chemical, but under humid conditions the sensor may not due to the presence of hydrolysis breakdown products or the influence of the ambient RH.Ambient humidity can cause an instrument to produce a false alarm or fail to alarm  depending on the sensor type or algorithm. Moreover, it is also possible for the water vapor to interact with a target chemical vapor, analyte, through hydrolysis causing the chemical structure to change thus further complicating the problem6. The split-flow system presented was designed and built in-house from commercially available components and its performance monitored using in-house programmed software and commercially available transducers. While the system was simple to operate and control, several issues were observed during initial use and testing. Constant adjustments to the humidity control valves were required to counteract the fluctuations and instability in the humidity output and levels within a test chamber. Due to these observed fluctuations, the ability to perform long term experiments would be questionable, as constant adjustments to the system would be required to compensate for the instability in RH levels. The time required for the system to reach a stable RH level, equilibration time, is also unknown. Because of the instability of the humidity generated using this type of system, further investigation of the problem was required to gain a better understanding of the problem in order to mitigate the issues outlined above.During testing and evaluation, humidity can be generated and controlled using numerous commercial or non-commercial systems, including the split-flow design utilized for the work detailed in this paperWhile a split-flow design is an established method for generating humidity, the issues observed regarding stabilization time of a system and the resultant decrease in humidity are not well documented. Experiments and results documented herein shed light on an often overlooked or undocumented area of research that would likely be encountered by a user of this system type. This study only investigates one method of humidity generation, but the resulting information could be applied to other methods as they may be impacted by similar environmental and physical limitations.7. Expressing humidity is possible using a number of different measurements with RH being common. RH is the ratio of partial vapor pressure to equilibrium vapor pressure of water in the air at a specific temperature. The pressure dependency of humidity is related through Dalton’s law where the total exerted pressure  is equal to the sum of the partial pressures of the gases, as shown through Eq.  must decrease, as related through Eq. , i.e., ad in Eq. , where TIt should be noted that Eq.  provides9. By fixing these variables, the relationship between air flowrate, RH generated and the equilibration time can be investigated.During test and evaluation of the humidity generator, a controlled humid air stream was generated through sub-surface aeration of a water reservoir using a sparger, as shown in Fig. 10. The system described in this paper is of a split flow design constructed from commercially available components with the schematic shown in Fig. Many types of humidity generators and controller methods exist, they range from split flow, multi flow, to salt solution baths, and pressure systems4. Construction of the system is simple, can be built in-house using commercially available components, with the exception of the sparger, which was custom sized, but there are commercial alternatives available (Mott micro sparger tips). The use of a single MFC maintains the total flow volume of the system rather than multiple MFCs used for a multi flow design, removing a source of error in the total flow volume and the need to adjust and control multiple MFCs. When mixing the humid air stream with a chemical analyte supplied by a separate MFC, calculations are needed to determine the mixing ratios due to the additional supply of analyte stream. This is again simplified by using a single MFC as opposed to multiple MFCs to control the dry and wet streams separately. Safety is also a consideration for the generation system, where use of a split flow design maintains the system at near ambient pressure unlike a two pressure system that generates a pressurized saturated stream that is mixed with the dry stream in an expansion chamber. For this effort we did not address temperature management and instead operated at ambient laboratory temperatures of 21 ± 1 °C. Note that a temperature change of 1 °C would result in a RH change of approximately 3% which were observed, but did not greatly impact the experiments. In summary, a split-flow design provides a simple solution to humidity generation and control using commercially available components.A split flow design offers several advantages over other common types of humidity generators and controllersInitial experiments were conducted to determine if the system could produce a steady stream of humid air, as well as the time required to reach a steady state, or equilibrium. An initial target RH of 50% was chosen, a common setting based on past experimental needs, with a flowrates in the range of 1 to 10 L/min. Prior to the start of the air flow, the wet valve was closed and dry valve was fully opened. The sparger was filled with deionized water to a predetermined level, and the desired air flowrate was set in the MFC. The air flow was started and the system was allowed to run until the reported RH level was below 2% to remove any condensed water remaining in in the lines from previous experiments. After the system was cleared of any residual moisture, the level was visually monitored and needle valves adjusted until the 50% RH target was reached and remained unchanged for approximately 1 min. The system was then allowed to run for a minimum of 12 h with the temperature and RH values recorded every second by the LabVIEW software.11. This level of fluctuation is within tolerances of many commercially available environmental chambers.Experiments were conducted for flowrates of 1, 2, 3, 4, 4.5, 5, 7, 8.5 and 10 L/min all initiated at the starting RH of 50%. The flowrates and range were chosen as they are common measurement rates for commercially available handheld gas sensors. Equilibration was defined as a change in RH of ± 2% to determine the equilibration time. This value was selected as this is a commonly observed fluctuation outdoorsAfter establishing an equilibration time for the system over a range of air flowrates, the observed decrease in RH over time was investigated. Repeating the initial steps from the previous experiments, the starting RH was set to about 57%, above the target RH of 50%, at a flowrate of 5 L/min and allowed to run for 12 h. The temperature and RH were monitored and recorded continuously, with the goal for the system to equilibrate at the target RH of 50%.To determine the effect of different starting RH values on time to equilibration and stability, a series of experiments were conducted where the starting RH was set to 20, 50 or 80%, with flowrates of 2, 5, 7, 8.5 or 10 L/min. These experiments were conducted in triplicate, with the system to running for 12+ h per replicate. The goal was to establish the relationship between, RH, flowrate and equilibration time.Initial equilibration time experiment results with a target RH of 50% and flowrates ranging from 1 to 10 L/min are shown below in Table At a flowrate of 1 and 2 L/min there was no significant change in RH over the duration of the experiment and thus no time for the humidity generator to equilibrate is required. As the flowrate of air increased above 3 L/min, there was continuous decrease from initial RH and the equilibrium RH with the greatest decrease around 7% which was observed at flowrates of 5 and 10 /min. The time to equilibrate ranged from about 210 to 500 min (3.5 to 8 h) over the flowrate range of 3–10 L/min. No obvious trends in the equilibration time or equilibrium humidity were observed, such as a linear decrease in humidity with flowrate. The equilibration time varied by as much as 150 min within the same flowrate experiments. There were increasing and decrease equilibration times observed with increase flow rate, e.g. at 5 L/min the equilibration time was 210–300 min, but at both and 7 L/min the equilibration time was greater. The volume of the sparger may have played a role in this as it has volume of only 2 L, with approximately 1.5 L of water. At 10 L per minute, five times the volume of air is being pushed through the sparger than its volume and more than six times the volume of water it contains. Later experiments will show that at lower humidity levels and thus lower air flow rates through the sparger, no equilibration time is required.For higher flowrates the volume of water in the sparger decreased slightly over longer test periods of up to 96 h, however no noticeable change in RH was observed due to the water volume changing. Therefore, longer term, multi-day experiments could be performed with minimal concern of effects due to the sparger volume decrease.12. The overall temperature change was less than 1 °C, presumably from variations in laboratory air temperature. This resulted in a RH fluctuation of less than 1% after equilibrium, which is within the tolerance defined above. While there is no direct evidence that evaporative cooling within the sparger occurred, it is possible that the drop in RH until equilibrium is reached is due to this effect. Additional temperature probes could be placed within the sparger air space and along the air flow path to determine this was in fact the source of the observed variation.Fluctuations in RH were observed to be inversely proportional to the temperature, as expected due to the dew point temperature relation described Eq. , and is 3To counteract the observed decrease in RH during equilibration, compensation could be made by overshooting the starting RH level and allowing the system to equilibrate to the target RH. With a target RH of 50% and an observed decrease of 7% in 5 experiments, denoted by the bold text of Table 9. While equilibration time was observed to be up to 8 h, the ability to start the system the prior day and allow it to come to equilibrium over night with little effect on performance negates some concerns associated with required long equilibration periods.Several experiments were allowed to run for up to 96 h to assess the long term stability of the generation system. Such long experimental runs support multi day testing and demonstrate that the humidity generation can be started the day before testing in order to allow the system equilibrate. At a flowrate of 2, 5, and 10 L/min, generation of humidity was stable for 86, 91, and 96 h, respectively, though longer generation period is possible as the sparger water volume only slightly decreased. As previously discussed, decrease in the sparger water volume did not have a noticeable effect on humidity generation or stability, at < 1% RH change, though this may not hold true on a larger scale, such as a pondTo study equilibration time and stability of RH environmental conditions other than 50%, experiments were conducted at 20 and 80% RH. Flowrates of 2, 5, 7, 8.5 and 10 L/min were used and the results are shown in Table For all of the experiments ran with a starting RH of 20%, there was no change observed in the RH through the course of the experiments, and therefore there was no equilibration time required. Unlike the experiments ran with a starting RH of 50% where at the lowest flow rates of 2 L/min there was no change in the RH, when the starting RH was increased to 80%, an equilibration time of at least 210 min was required. As the flow rate increased, so did the equilibration time, but for both the 50% and 80% start RH experiments at the highest flow rate of 10 L/min the equilibration time did decrease below the maximum observed value. The stabilization humidity level decreased with a range of 5 to 20%, with a greater decrease at the 80% RH starting point and higher flow rates of 8.5 and 10 L/min.9. This is supported by the results, comparing the lower 20% RH setting with the higher 80% RH, where at the lower setting there is no equilibration time, but at a higher humidity the sparger has to equilibrate due to the high air flow to create the more highly saturated water vapor. Another option to increase the saturated vapor pressure would be to increase the water temperature, however, this would require the rest of the system to be conditioned to avoid condensation. Ultimately, as the flowrate or initial RH setting increases, the stabilized RH drop also increases.For higher flowrates and humidity levels the lack of aerator surface area could limit the interaction between the air and water preventing enough water content in the air thus resulting in greater decrease in RH level when stabilization is achieved. Aeration efficiency can be increased by increasing the aerator surface area, decreasing pore size to create smaller, more abundant bubbles, and increasing the water column length of the sparger. These changes increase travel path of air or number of bubbles allowing for more interaction between the water and air, creating more water saturated airFurther experiments are needed to determine if the decrease in RH levels at stabilization can be compensated for at high flowrates and RH levels. The highest observed RH after stabilization with the system during initial testing was 81% at 5 L/min with the wet valve fully open and dry valve closed. Overall, lower and higher humidity levels are achievable and sustainable using the split flow system with the knowledge that at RH levels of 80%, there is a considerable decrease once equilibrium is reached.A split flow humidity generator and controller provides a simple method to create a controlled humidity stream for chemical vapor sensor testing. Initial RH instability in the system prompted further investigation into equilibration and the factors that affect the stability. Starting with a target of 50% RH, the time to achieve stability across a range of 1–10 L/min flowrate was tested and it was determined that as the flow went beyond 3 L/min stabilization could take several hours, though once reached, remained stable for over 90 h. This opens up the possibility to perform longer term, multiday sensor evaluations. One complication to the equilibration process is a drop in the stabilized RH, however this decrease can be compensated for as shown experimentally. Similar trends were observed with a starting RH of 20 and 80%, where stabilization took upwards of 8 h with a decrease from the starting RH level, but remained stable for several days after. Future work would look into different types of aeration media that may more efficiently generate saturated water vapor at higher air flowrates as well as automating the needle valves to adjust for humidity drift.


Savanna ecosystems are shaped by the frequency and intensity of regular fires. We model savannas via an ordinary differential equation (ODE) encoding a one-sided inhibitory Lotka–Volterra interaction between trees and grass. By applying fire as a discrete disturbance, we create an impulsive dynamical system that allows us to identify the impact of variation in fire frequency and intensity. The model exhibits three different bistability regimes: between savanna and grassland; two savanna states; and savanna and woodland. The impulsive model reveals rich bifurcation structures in response to changes in fire intensity and frequency—structures that are largely invisible to analogous ODE models with continuous fire. In addition, by using the amount of grass as an example of a socially valued function of the system state, we examine the resilience of the social value to different disturbance regimes. We find that large transitions (“tipping”) in the valued quantity can be triggered by small changes in disturbance regime. Given the ecological, economic and cultural value of savannas as well as their precarious ecological role, savanna ecosystems are a frequent target of modeling investigations  a continuous differential equation that governs the system except for at a sequence of discrete impulses; (2) the impulse function describing the jumps or kicks in the state of the system; and (3) impulse mechanisms that describe when the impulses happen, either at certain times, at certain states or both  bifurcations. These bifurcations curves separate different bistability regions. The curves are described in Sect. Using the methods described in Sect. mentclass2pt{minimFigures entclass1pt{minimaIn addition, we identify limit points curves. The locations of these curves from numerical continuation are shown in black (solid lines) in Figs. In Theorems n et al. .x through a transcritical bifurcation whenIn part of the entclass1pt{minimax and y are nonzero.The following two theorems give necessary conditions for tree–grass savanna coexistence fixed points where both A necessary condition for a transcritical bifurcation associated with a grass–tree coexistence fixed point crossing into the first quadrant is met wheny almost always nonzero, corresponding to the h(y) that reduces the multiplicity of the Let V and h(y). By integrating Eq. h(y) becomesLet entclass1pt{minimamentclasspt{minimaThe bifurcation condition in Theorem mentclass2pt{minimA necessary condition for a transcritical bifurcation associated with a grass–tree coexistence fixed point crossing into the first quadrant at  is met whenx almost always nonzero, corresponding to the g(x) that reduces the multiplicity of the For all values of the disturbance parameters,  is a forest-only fixed point of the system, i.e., there is a forest only fixed point whenU and g(x) Setting Let entclass1pt{minimaNote that in Theorem The bifurcation condition in Theorem Examining the impact of different disturbance regimes on the long-term behavior of the system involves varying entclass1pt{minimaentclass1pt{minimaIn Fig. The existence of the grass-only bifurcation curve and limit point curves at higher values of We derive a continuous system in the limit of small frequent disturbances. Many models of savanna ecosystems use this kind of simple disturbance regime, rather than using an impulsive disturbance model , we set the social value to x itself for this example. At disturbance parameter sets with bistability, we choose the highest grass state available as the social value. The socially valued state is shown in Fig. The amount of grass, x. Returning to Of note in this figure is the stark transition from low to high social value along the curve of saddle-node bifurcations to the left of the cusp point. In Fig. For this example of a socially valued function (the amount of grass), the bistability region defines a disturbance parameter space within which the socially valued function exhibits resilience to parameter variation. The system state is not resilient to a shift that leads to a disturbance regime outside of bistability region, in that restoring the old disturbance regime does not restore the system state. Under this model, woody encroachment due to a change in disturbance would require more than just restoring the original fire regime. It would also require actively removing trees and promoting grass to return the system to the basin of attraction of the high-grass state.Our previous analysis in this paper has relied upon identifying the long-term behavior of the system. However, we must take into account the possibility of transient timescales (Morozov et al. In Fig. In this section, we compare the results of our model to other models cited in this paper.In field observations, savanna systems exhibit bistability between attracting states and have an attracting tree–grass coexistence state (Staver et al. In addition finding mechanisms that generate a stable savanna state, savanna modeling efforts also focus on capturing bistabilities in the system. For example, Staver et al.  use obseSpatially homogenized models, either continuous or stochastic, capture a mean-field average of trees and grass over the whole area of a savanna. Some, like the model in this paper, focus on biomass, while others model the fraction of coverage for different plants. In contrast, spatially extended models have the potential to capture patterns of trees and grass within the savanna, as well as the boundaries between savanna and forest (Wuyts et al. This analysis of a simple one-sided Lotka–Volterra interaction with an impulsive disturbance focused on bifurcations in the disturbance space. Understanding sensitivity of the predictions of the model to the structure of the underlying Lotka–Volterra interaction, including to parameters in that system, would extend these results to a broader range of savanna systems. In addition, it would make it possible to separately assess the robustness of model predictions in the face of structural model errors, which are inevitable in ecological modeling.It would be possible to focus further work using this model on questions of woody encroachment, as transitions between grassy and woody states are observed in the model. Creating a spatially extended impulsive model could also be of interest, particularly in light of the ease of biome recovery in spatial models. In addition, the analysis of transient behaviors could be placed within a more general framework. Given that many analyses focus on long-term behavior, expanding tools for understanding transient would be worthwhile.


The description of spatial patterns in species distributions is central to research throughout ecology. In this manuscript, we demonstrate that five of the most widely used species‐level spatial patterns are not only related, but can in fact be quantitatively derived from each other under minimal assumptions: the occupancy area curve, Taylor's Law, the neighborhood density function, a two‐plot variant of Taylor's Law and two‐plot single‐species turnover. We present an overarching mathematical framework and derivations for several theoretical example cases, along with a simulation study and empirical analysis that applies the framework to data from the Barro Colorado Island tropical forest plot. We discuss how knowledge of this mathematical relationship can support the testing of ecological theory, suggest efficient field sampling schemes, highlight the relative importance of plot area and abundance in driving turnover patterns and lay the groundwork for future unified theories of community‐level spatial metrics and multi‐patch spatial patterns. We show that several widely used single species spatial metrics  are all derivable from each other under minimal assumptions. We provide example equations showing this relationship and demonstrate its application to simulated and empirical data. Describing the spatial patterns formed by individuals in a population is one of the most important and widespread research activities across ecology. Knowledge of species‐level spatial patterns is used to draw inferences about the possible underlying ecological mechanisms operating in populations and communities, design sampling schemes for characterising populations and identify general laws that are common across ecological systems Brown, . A diverAt the most general level, such spatial metrics can be divided into three broad categories. The first category includes metrics that describe plot‐based spatial scaling, which is the change in some property of a population with the area of a plot being surveyed. Perhaps, the most important of these scaling metrics is the occupancy area curve  orderly, such that no two points can occur at the same location, (2) homogeneous, such that the statistical properties of the pattern, particularly the first‐order and second‐order intensity (defined below), are the same across the entire landscape and (3) isotropic, such that there is no directionality in the pattern. Together, these three common assumptions define a “well‐behaved” point pattern. One or more plots or quadrats of area F(x), where F may be a Greek or upper case Roman letter, to denote a metric or intermediate variable whose shape is a function of one or more variables x. We reserve the notation P(X=Y) specifically for the probability that a random variable X takes the value Y.In all equations below, we use the notation λ, which describes the overall density of points in the landscape and can be calculated as the number of points in the entire pattern divided by the area of the landscape. The second intermediate statistic is the second‐order intensity, λ2(r), which represents the probability that two locations separated by distance r both contain a point. The third intermediate statistic is the quadrat count distribution , which describes the probability of observing n individuals in a plot of area A that is small relative to the landscape. A two‐plot joint quadrat count distribution for the simultaneous abundance in two plots, each of area A, can be defined as P , where D is the distance between the nearest edges of the two plots.There are many methods of describing point patterns quantitatively ,The focus of this manuscript is on the relationship between five single‐species spatial metrics: the occupancy area curve, Taylor's Law, the neighborhood density function, the two‐plot variant of Taylor's Law and two‐plot turnover. The mathematical relationships between these five metrics are shown in summary form in Figure Ω(r), can be calculated as ] = E[n2]–E[n]=σ2–μ + μ2 where n is a random variable representing the number of points in a plot and μ and 2σ are the mean and variance of the quadrat count distribution. Second, the quantity E[R] can also be calculated as a double integral of the second‐order intensity function over all possible locations of points in the plot (see Supporting Information 1). Following the notation of Diggle  to μ(A).The second‐order intensity can then be linked to the quadrat count distribution as follows. First, the expected number of pairs of points in a plot, f Diggle , the rel Diggle, . This eqC, in abundance between two plots as a function of both plot area A and the distance between the plots D. The relationship between the two‐plot Taylor's Law and the second‐order intensity can be derived using similar logic to the above as C, is given byP(n=0|A) is the probability of a plot of area A containing zero individuals, which is given by the quadrat count distribution. The relationship between the probability of plot occupancy and area is known as the occupancy area curve  is the joint probability that both plots are empty. This turnover metric can also be interpreted as the conditional probability that one plot is occupied, given that the other is occupied.Similar to the Sorensen index of community turnover or distance decay =A (see Supporting Information 1). As such, variation in plot area is equivalent to variation in expected abundance within a plot of fixed physical area.In the results below, units are defined such that As described above, this unified framework can be used to derive any four of the five spatial metrics given knowledge of the fifth. We demonstrate this mathematical relationship with several concrete theoretical examples, in which the exact form of one metric is assumed and the forms of the remaining four metrics are derived.a is a constant often in the range of 0.1 to 100 and b is a constant commonly ranging from 1 to 2 . CSR can be understood as a limiting case of the examples above in which Expressions for Taylor's Law, the neighborhood density function and the two‐plot Taylor's Law variant for four theoretical examples are shown in Table a=0.88 and b=1.51 for the Taylor's power law examples, reflecting the median values of these parameters from a recent review  and α=2.30 and β=0.71 for the Gaussian second‐order intensity example, which are equal to the median values of these parameters across 171 species in the Barro Colorado Island dataset (see below). For the two‐plot Taylor's Law and turnover, the metrics are shown as a function of distance D for a fixed plot area A=1.Figure ρ, the number of individuals per cluster is drawn from a Poisson distribution with mean μ and the location of individuals within a cluster is defined by a bivariate normal distribution with variance 2σ around a cluster center.We next test our mathematical framework by using a Thomas process model  in the scaled units used in our analysis (see Results), is given by the Gaussian function , medium  and high  idealized species. These parameters were used to simulate 200 replicate point patterns for each idealized species in a 500 m by 1000 m landscape, matching the dimensions of the BCI plot.We simulate point patterns for each of three idealized ‘species’. We chose parameters for each idealized species based on empirical data from the Barro Colorado Island (BCI) tropical forest plot (see below). We began by dividing the species at BCI into three abundance tertiles. We fitted Thomas process models to each BCI species using the D holding plot area constant. For each idealized species, we then calculate the median value of the metric at each A or D across all 200 simulations and use this median metric for comparison to theoretical predictions.For each simulation, we directly evaluate the shape of all five spatial metrics. An intermediate‐sized small plot of side length 25 m is used for calculations of two‐plot Taylor's Law and turnover, which are evaluated as functions of interplot distance α and β from the simulations. In reviewing the simulated data, however, we found that the median fitted α and β parameters or each idealized species deviated slightly from the target values for the low  and medium  species. We used these recovered parameter values to generate predictions for these idealized species.We generated theoretical predictions for each metric by using the equations derived for a species exhibiting a Gaussian second‐order intensity function, along with known values of Boswell and Patil  note thaFigure α and β for each combination of species and metric (see Supporting Information 3 and 4).Finally, we test the applicability of our framework to an empirical data set using data from the Barro Colorado Island tropical forest plot . As units for each species were scaled such that λ=1, these correlations do not reflect simply a variation in abundance between species. Overall, the correlations involving Taylor's Law and the two‐plot Taylor's Law are the weakest, suggesting that in practice, the shape of the remaining spatial metrics will not be as well‐predicted from knowledge of these particular metrics. Figures SI1.3–6 provide additional information on how variation in α and β parameters relate to the shapes of each of the five metrics.Figure The framework above demonstrates the mathematical relationships between five important spatial metrics that are frequently considered separately. Expressing the quantitative relationships between these metrics requires two general assumptions. Deriving the relationship between Taylor's Law, the neighborhood density function and two‐plot Taylor's Law requires the assumption that the point pattern is orderly, homogeneous and isotropic. Relating these three metrics to single‐plot occupancy and two‐plot joint occupancy additionally requires the assumption of a parametric form for the bivariate quadrat count distribution, which we here assume to be negative binomial. As shown above, the ability to predict one metric from the known shape of another in practice will depend on several factors, including the extent to which a species’ point pattern meets these two assumptions, the sensitivity of the shapes of empirical metrics to changes in parameter values, which empirical metric is used to estimate parameters, and the ability to statistically recover accurate empirical parameter values from a point pattern.a and b parameters will exhibit identical parametric forms of all five spatial metrics, regardless of their density or abundance, with the exact numerical form of the metric scaled by a constant (see Supporting Information 1).The equivalence of area units and species density in the framework above highlights the additional observation that there is only a single, universal set of spatial metrics for any set of species that share one of the metrics. For example all species exhibiting a Taylor's power law with the same This mathematical relationship leads to at least four important implications for both basic and applied ecology. First, from the perspective of theory testing, this framework highlights that a theory that predicts one of the five metrics above also inherently makes predictions about the other four metrics. This provides an additional means of testing such theories with empirical data. As one concrete example, the Maximum Entropy Theory of Ecology predicts that the quadrat count distribution will take the form of a geometric distribution across spatial scales, a prediction that appears broadly consistent with empirical data Harte, . An applSecond, from the perspective of experimental design, this framework highlights that, at least in principle, data collected using one sampling scheme can be used to determine the shape of metrics that have traditionally been thought to require a different sampling approach. One useful translation would be to use nested measurements of plot abundance for a species, a type of data used to construct nested species–area relationships, to predict the second‐order intensity function or single‐species turnover, two metrics that otherwise require spatially explicit data to measure directly. The simulation study above suggests that this particular translation may be error‐prone given the overlap in Taylor's Law across parameter sets. In contrast, however, the empirical study suggests that among a large group of species, the shape of Taylor's Law does have significant predictive power in inferring the shape of other distance‐based spatial metrics.Third, from the perspective of spatial ecology, the shape of the predicted single‐species turnover metrics highlights the potentially pervasive dominance of area over distance in determining spatial turnover. Although turnover is often analyzed in terms of the distance between plots, the equations above demonstrate that plot area may have a stronger influence on the value of thesemetrics than interplot distance. Figure SI1.2 shows that for the two‐dimensional Taylor's power law and Gaussian second‐order intensity examples, the turnover metric is more strongly influenced by plot area than by distance, particularly when the plot area is relatively large. Since the community‐level Sorensen index is the sum of the individual species turnover metrics, this result suggests that the shape of the species abundance distribution in a community may be the dominant influence on community turnover, rather than the distance between two plots. We note that this finding rests on the assumption that a species’ point pattern is homogenous, stationary and isotropic, an assumption that may break down at larger spatial scales where habitat heterogeneity and gradients become more prominent.Finally, from the perspective of theory development, this framework lays the groundwork for future theories that can relate spatial metrics at the community level. For example, the single‐species framework presented here could be extended to the community level through the addition of a metacommunity species abundance distribution. Using the assumption of spatial independence between species that is common to many macroecological theories (see McGill, Another community‐level expansion of this framework lies in the construction of a multi‐plot species–area relationship. To begin, note that the probability that a species is simultaneously absent from both of a pair of plots is part of the calculation of the two‐plot turnover metric above. This probability depends both on plot area and the distance between plots. As above, a metacommunity abundance distribution can be used with this probability to construct a species–area relationship for a pair of plots separated by a fixed distance as the area of either or both plots change. This logic can be further generalized to a landscape with more than two plots, where any given plot may change in area, where the network‐wide probability of presence is calculated using a vector of means and covariance matrix for the abundance of each species in the network. In this more general case of the species–area relationship, the change in species richness with the area will depend on the particular location within the network where the area is added or subtracted.To support the expansion of this framework, several of the assumptions made above could be relaxed. The negative binomial quadrat count distribution could be replaced by other distributions that can be parameterized from plot mean, variance and covariance, such as the lognormal distribution (Limpert et al., In summary, this work presents a mathematical framework demonstrating the relationships between five species‐level metrics of spatial pattern: the occupancy area curve, Taylor's Law, the neighborhood density function, a two‐plot variant of Taylor's Law and two‐plot turnover. This framework presents new avenues for theory testing, expands investigators’ options for spatial sampling and demonstrates the importance of plot area in driving patterns of turnover. The results also lay the groundwork for expanded, unified spatial theories that explicitly incorporate habitat heterogeneity and fragmentation at the community level.All authors designed and participated in developing the theoretical framework described here. J.K. and M.B. wrote the manuscript with contributions from K.W.Supplementary MaterialClick here for additional data file.Supplementary MaterialClick here for additional data file.Supplementary MaterialClick here for additional data file.Supplementary MaterialClick here for additional data file.


This paper presents a method for optimal pressure sensor placement in water distribution networks using information theory. The criterion for selecting the network nodes where to place the pressure sensors was that they provide the most useful information for locating leaks in the network. Considering that the node pressures measured by the sensors can be correlated , a subset of sensor nodes in the network was chosen. The relevance of information was maximized, and information redundancy was minimized simultaneously. The selection of the nodes where to place the sensors was performed on datasets of pressure changes caused by multiple leak scenarios, which were synthetically generated by simulation using the EPANET software application. In order to select the optimal subset of nodes, the candidate nodes were ranked using a heuristic algorithm with quadratic computational cost, which made it time-efficient compared to other sensor placement algorithms. The sensor placement algorithm was implemented in MATLAB and tested on the Hanoi network. It was verified by exhaustive analysis that the selected nodes were the best combination to place the sensors and detect leaks. Finding a suitable sensor placement is a fundamental problem for monitoring water distribution networks (WDNs) because it is impossible to install sensors at each point of the geographic area covered by the distribution system. A WDN comprises hundreds of nodes; however, only a few sensors can be installed in certain carefully selected nodes. Then, the main question is how to select the optimal sensor placement. Finding an answer to this problem is not trivial because the selected nodes must capture the most relevant information to estimate hydraulic variables at non-measured points and provide essential information for different supervision algorithms, e.g., for leak localization ,2. OftenN nodes, a sensor placement consists of a selection i-th node, whereas In a mathematical/computational context, the placement of pressure sensors is a mixed-integer programming problem. In this problem, for a network with S, then the number of possible sensor placements is reduced to Combinatorial analysis shows that there are Usually, sensor placement focused on leak localization is addressed with an optimization approach from synthetic pressure data obtained by simulation. Some authors have focused on minimizing the number of undetectable leaks ,20, wherThe huge computing time in networks with hundreds and thousands of nodes using optimization-based methods and the high dependence on the selected leak localization method has motivated the present work. In this new proposal, it is not considered how specific leak localization methods will use the information provided by the sensors, but rather that the sensor placement method only focuses on the sensors capturing as much information related to the leaks as possible. The proposed method consists of a heuristic algorithm to select the subset of nodes where to place the sensors, seeking to maximize the relevance of the information captured by the sensors while minimizing the redundancy between the pressures in the selected nodes. Both metrics, relevance and redundancy, are defined in terms of information theory.An important contribution of this work is the reduction in computing time for sensor placement, compared to methods based on metaheuristic optimization. Another relevant contribution is the nondependence of the sensor placement on the leak localization method used, which allows the use of the same sensor placement with different localization methods. Some aspects not yet covered in this work are the possible heterogeneity of the sensors  and the influence of the measurement noise in the optimal placement, but they are considered as future work.The rest of the document is organized as follows: in E has probability P, its information content is defined by:I is defined by the base of the logarithm, b, which is called “bit” if X with probability function x as a result when measuring X is given by:In Shannon’s information theory (IT), the self-information of a random variable is defined according to the unexpectedness of its values . Thus, tentropy is used:X, that is, the sum of the self-information of each of its possible values weighted by its probability of occurrence.To quantify the average information that a random variable contains, considering all its possible values, the X and Y.The mutual information of two random variables, sometimes called “information gain”, measures the amount of information obtained from one of the random variables by observing the other one. For example, in a practical application of WDN monitoring, the mutual information between two node pressures would indicate how much information about the pressure at one node is gained by knowing the pressure at the other one. In probabilistic terms, the mutual information determines how different the joint distribution of X and Y, defined over the space X and Y, whereas X and Y, respectively. The mutual information =∑yX and Y are independent.Furthermore, For continuous random variables, the summations in  are repln bits, so that the domain of each variable is reduced to Due to the difficulty in modeling the probability densities and subsequently evaluating the double integrals in , a simpland then  is appliThe proposed sensor placement method is based on a dataset of node pressures that collects typical variations due to leaks of different sizes in all network nodes. The pressure dataset is obtained from simulations with the hydraulic model of the network in . Each prfeature selection stage. To select the features (the subset of nodes where the sensors will be placed), an algorithm is proposed that seeks to maximize the relevance of the selected features (node pressures) for the response variable (leaky node), while each of them avoids capturing information already contributed by the others, that is, minimizing redundancy.In the context of machine learning, the placement of pressure sensors is a The following definitions of relevance and redundancy, proposed in , are useDefinition 1.A metric of the relevance of the subset of node pressures where x is any feature in (Relevance)Definition 2.A metric for information redundancy in a feature subset where x and (Redundancy)M different leakage scenarios are simulated in a network containing N nodes, the result of the simulation is a collection of NM-dimensional vectors, x and N candidate nodes . In addition, an output vector, y in ,,M differor, y in , is geneThe exhaustive search for the optimal subset of sensors, posed in  was consNext, a heuristic algorithm is proposed, which orders the node pressures according to their importance to explain the different leak classes (leaky nodes). The first node pressures in the output list correspond to the nodes with the highest importance for explaining the leak positions according to the information contained in the dataset. The sequential selection of nodes starts from an empty subset and, at each iteration, adds the best-ranked node among those that are still available to be selected. At each iteration, the relevance of each available feature (node pressure) with respect to the output (leaky node) and its redundancy with respect to the variables that have been previously selected is evaluated using the following equations, adapted from  and 8)::8):(9)ReSince maximizing relevance and simultaneously minimizing redundancy represents a multiobjective problem, a combined relevance/redundancy index (RRI) is defined that increases with increasing relevance and also with decreasing redundancy, so the problem is expressed as a single objective to be maximized:The complete node ranking process is formally expressed in Algorithm 1. When the process finishes, the nodes where to place the sensors are taken from the first positions in the list Algorithm 1: Node ranking based on information theory.Data: Set with all node pressures, Result: Set with ordered node pressures,  Initialization:  The number of sensors to place for leak localization purposes is determined by the equipment available in most cases. The minimum number of sensors for a successful leak localization method will depend on how that method uses the available information, the measurement noise, as well as the quality, resolution and calibration of the sensors. If there are enough resources to intensively instrument the network, it must be taken into account that increasing the number of sensors does not always lead to better performance in locating leaks. To determine how many sensors should be placed, it is suggested to start from the ranking obtained by Algorithm 1, and run a marginal analysis with the leak localization method to be used. Starting from one sensor (the best ranked), the number of sensors is progressively increased and the leak localization performance is evaluated for each new set of sensors until adding a new sensor no longer represents a significant benefit for locating leaks.It should be noted that Algorithm 1 does not take into account the geographical distribution of the nodes, since relevance and redundancy depend only on the mutual information between node pressures. This means that the network topology is what determines the amount of mutual information rather than the distance between sensors .Algorithm 1 was implemented in MATLAB and tested on the Hanoi network . The modIn order to build the pressure dataset, leaks of different magnitude were simulated at each junction node using the EPANET 2 simulation program  through k-nearest neighbors (k-NN) and quadratic discriminant analysis (QDA). These leak localization methods are based on classifiers that recognize directional patterns in pressure residuals using supervised learning techniques, as described in [To assess the optimality of the sensor placement obtained from Algorithm 1, leak localization tests were carried out using two machine learning methods that used the pressures in the selected nodes as features (input variables). The methods used were the ribed in .Through the marginal analysis, suggested at the end of fined in . The AccThe results in Further tests were made on larger networks, e.g., in some midsize sectors of the Madrid network. Although, as noted above, there may be different sensor placements that lead to a good performance in locating leaks, the one obtained by Algorithm 1 has the advantage of being calculated in less time than the methods based on metaheuristics and that it is not linked to a specific leak location method, so changing the leak location method does not imply changing the location of the sensors, which would be impractical.This paper has presented a technique for finding optimal sensor placements from information theory using a sequential forward selection, maximizing the relevance and minimizing the redundancy of the selected node subset. The proposed technique is computationally less expensive than other methods reported in the literature because the proposed technique operates directly on the values of node pressures without performing calculations for leak localization in the implementation of the algorithm. The optimality of the sensor placement obtained with the proposed method was extensively tested by simulation with the Hanoi network. It was found that the selection of nodes where to place sensors using information theory produced the best combination of pressure variables to locate leaks using different machine learning methods.An implicit assumption in the proposed algorithms is that all network nodes have the same availability to place the sensors. However, in practice, some specific nodes may have placement priority over others; for example, critical nodes (points of minimum pressure) and nodes that supply essential services  could be monitored as a priority. It may also occur that some nodes already have a sensor installed and that previous partial placement must be held, or that the conditions in a node are physically adverse and instrumentation is avoided. These circumstances warrant adjustments to the proposed sensor placement algorithm that may lead to future work. Another possible working line is the combination of heterogeneous sensors where different sensing specifications are included  or where the sensors measure different physical magnitudes .


The Nuclear Non-Proliferation Treaty (NPT) is a central element of the global nuclear order, the primary goal of which is to prevent nuclear war. But this understanding is being threatened by a number of developments. Frustration about the lack of nuclear disarmament and concerns about humanitarian consequences of nuclear weapons led to the negotiation of the Treaty on the Prohibition of Nuclear Weapons (TPNW). The new treaty has further exposed existing fault lines within the NPT and exacerbated unresolved conflicts over the proper approach to disarmament and the weighting of the NPT pillars. Currently, disagreements over the compatibility of the two treaties and the approach to the TPNW in particular divide the membership of the NPT. At the same time, real proliferation cases test the regime’s ability to act, as norm enforcement is regularly hampered by interference from the great powers. These developments—the absence of genuine disarmament, disputes among NPT members, competition between the TPNW and the NPT, and actual nuclear proliferation—are part of a comprehensive crisis and destabilization of the non-proliferation regime. We argue that three developments are necessary to restabilize the regime: a cooperative resolution of the power struggles among the major powers; a depolarization of the inter-group divisions in the NPT; and a return to the principle of nuclear war prevention as a common maxim for action. In 1995, the Review and Extension Conference of the Treaty on the Non-Proliferation of Nuclear Weapons (NPT) decided, without a vote, to extend the treaty indefinitely, after the initial duration of 25 years ended. Optimism about the non-proliferation regime abounded. The 2000 Review Conference (RevCon) resulted in a substantial consensus on all three pillars of the NPT—non-proliferation, peaceful uses of nuclear energy, and, most significantly, nuclear disarmament, including 13 practical steps. Modest, but steady progress of the regime seemed foreordained.Twenty years later, the situation has deteriorated dramatically. The members of the treaty are more divided than ever Müller . NuclearThe absence of meaningful disarmament, disputes within the NPT membership, competition between the NPT and the TPNW, and real nuclear proliferation mark the four aspects of a comprehensive regime crisis.The article shows that these crisis dimensions are closely intertwined, even though they are often considered separately: if the crisis intensifies in one area, this has a direct impact on the other dimensions. It follows that only a comprehensive response to the regime crisis, one which considers all dimensions, can stabilize and sustain the global nuclear order. In light of this, the article proceeds as follows: first, we analyze the place of the NPT in the global nuclear order. We then address the various crisis dimensions: the changing geostrategic context and the breakdown of nuclear arms control; intra-regime conflicts; the relationship between the TPNW and the NPT; and current proliferation crises. Based on this analysis, we develop a series of suggestions for positive development of the regime.order; but what they envisage may be no more than their own rule over the rest; or something that gives them security with little regard to the security of everybody else; or a system shaped by an unifying ideology which others dislike, etc.. There may be actors who try to set up a cooperative order in the common interest, or others who at least want to build an order of compromise among many differing interests. Today, we can find evidence of all these notions of order, and the existing order is the result of this panoply of actions pulling and pushing in many different directions.The term “order” requires a caveat. Order invokes the idea of something purposeful and human-made. This association is only half true. What we call global order is the vector of the interplay of actions of quite a large of number actors—including both states and non-state entities. Some actors may indeed aim at some form of Whether we look at today’s world from a moralistic or a rationalistic perspective, global order is subjected to the imperative of preventing nuclear war. All the objectives of the many different actors will become irrelevant if this ultimate human catastrophe ever becomes reality Lodgaard . It tookWe can distinguish between four practical operative elements of this order:Thus, the NPT has been a key part of the efforts to avoid a nuclear Armageddon. Now though, it seems as though the actors have lost sight of this central objective of the treaty, it is something that has disappeared from their collective memory. This amnesia is strange, as the NPT’s preamble starts with a paragraph characterizing nuclear war as an immense catastrophe for humankind.The NPT is the only existing international legal instrument which obliges its five nuclear weapon states parties to pursue complete nuclear disarmament. This undertaking has two aspects: First, it represents a quid pro quo for the vast majority’s voluntary renunciation of the most powerful weapon of the time. Second, the commitment of the NWS to disarm is another contribution to the endeavor to prevent nuclear war: when there are no more nuclear weapons, they can no longer be used. The majority of the non-nuclear weapon states (NNWS) regard nuclear disarmament as the ultimate guarantee that nuclear war will not occur, the core objective of the NPT.When the NPT was initiated, negotiated, and concluded, the achievement of the above objective was being jeopardized by two processes: First, an increasing number of states possessed basic technical capabilities which could result, sooner or later, in the development and production of nuclear weapons. Second, the nuclear superpowers, the US and the Soviet Union, were involved in a nuclear arms race. The NPT addressed the first problem, bilateral arms control dealt with the second one. The emerging nuclear order was thus characterized by a threefold self-constraint: (a) The NNWS renounced nuclear weapons; (b) the NWS imposed limits on their nuclear competition and tried to manage their geopolitical rivalry so as to avoid deadly crises prone to escalation; (c) the NWS agreed to a legal obligation to accept further limitations on their nuclear arsenal and eventually their complete elimination.William Walker has called this order an “enlightenment project” Walker : understenlightenment project seemed to prevail. Geopolitical conflicts of the Cold War were solved. A sequence of nuclear and non-nuclear arms control and disarmament measures of unprecedented density ended the arms race. The Soviet nuclear legacy did not lead to proliferation: Ukraine, Belarus, and Kazakhstan agreed to transfer the nuclear weapons on their territories to Russia and to accede to the NPT and the NNWS, as did all the other fission products of the former Soviet Union and Yugoslavia. Important holdouts such as Spain, Argentina, Brazil, Algeria, Saudi Arabia, and most other Arab states decided to join the NPT. South Africa dismantled its nuclear weapons and related facilities and became party to the NPT, and other African states, which had remained outside the treaty because South Africa had, soon followed suit. China and France, who had originally refused to sign the treaty, ratified the NPT as well. It was in the context of this wave of real disarmament—and the expectation of more—and the apparent irresistibility of the NPT that a conference of parties decided to make the duration of the treaty indefinite in 1995.In the decade from 1986 to 1996, the The euphoria did not last, however. Rivalry between the US and Russia reemerged in Europe and extended to the Middle East, Northern Africa, the Caucasus, and Central Asia. China’s rise to power added another contender, and exacerbated US-Chinese rivalry in East and Southeast Asia, as well as in South Asia where India had become more open to an entente with the US Walker . China’sInevitably, this plurilateral power rivalry had military aspects. As the competitors were NWS, nuclear weapons regained prominence in national doctrines and strategies. The “cessation of the nuclear arms race” as called for in NPT Art. VI came to an end. Nuclear arms control and disarmament grinded to a halt and was even reversed by the termination of the ABM-; INF- and START II-Treaties. The specter of nuclear war hung over the horizon once again , the horizontal structure of the NPT—the three pillars of non-proliferation, disarmament, and peaceful uses of nuclear energy—has long given rise to serious disputes. Historically, the three pillars must be understood as equal and symmetrical; they cannot be separated from one other and should be understood as mutually reinforcing Joyner . Althougfavorable conditions  ons Ford . While mons Ford . In bothDiscussions about the interpretation of the NPT’s most important norms are as old as the treaty itself. In addition to the relative weighting of the NPT’s pillars, opinions also differ on the readings of Article VI and the preamble, the interpretation and scope of the “inalienable right” to the peaceful uses of nuclear energy (Article IV), the handling of cases of nuclear proliferation and non-compliance (see below), and the approach towards a WMD-free zone in the Middle East. The most controversial questions are whether the nuclear disarmament clause is indeed an obligation and whether the ultimate goal of the treaty is to foster progress in nuclear disarmament or to prevent nuclear proliferation Miller . CloselyThe only controversy as heated as the issue of disarmament concerns the universalization of the treaty, a dispute which is particularly evident in the contestation regarding the establishment of a WMD-free zone in the Middle East. The discussions on the WMD-free zone are a peculiar legacy of the indefinite extension of the NPT in 1995 . During the negotiations on the indefinite extension of the NPT, Egypt used the NWS’ desire for the Arab Union to endorse a final agreement as leverage to achieve a quid pro quo on the nuclear asymmetry in the Middle East Miller . As a reThese issues, which have caused a great deal of controversy in the past, must now be seen against the backdrop of two opposing trends: on the one hand, the formation of a broad civil society and state front which aims to delegitimize nuclear weapons altogether; on the other, the burgeoning great power conflict between the US, Russia, and China, coupled with ongoing modernization of their nuclear arsenals (Erästö and Cronberg grand bargain: The self-restraint of the NNWS (perpetuated by the indefinite extension of the NPT in 1995) is no longer balanced by the compliance of the NWS with their disarmament obligations. This reinforces the justice issues that are at the heart of the treaty and that have existed since its inception . The NWS’ narrative ignores that the growing rivalry between the major powers makes success in nuclear disarmament or arms control highly unlikely. The nuclear powers’ constant denial of their disarmament obligations seriously damages the NPT’s n Müller . The uneAs a result of the current focus on the TPNW, thematic areas within the NPT that would usually be more conducive to compromise, such as peaceful use of nuclear energy, are either thrust aside or run the risk of being politicized as well. It is against this backdrop that Gustavo Zlauvinen, the President-Designate of the 2022 RevCon, has announced that the conference will be tailored to the issue of peaceful use. While this strategy could help to find a common basis for discussion, it is unlikely that the conflicts over nuclear disarmament and the TPNW can be suppressed for long. The conflict over the nuclear-weapon-free zone in the Middle East is also likely to retain its significance and role as a possible showstopper at NPT review conferences, depending on how Egypt, the “political director” of the Arab League, plays its cards. In the current situation, it is questionable whether the NPT community can use the additional time, gained due to the postponement of the 2020 RevCon because of the Covid-19 pandemic, to resolve their conflicts. What could bring them together is perhaps a renewed commitment to the NPT’s founding mission—the prevention of nuclear war—, an unequivocal commitment to the mutually supportive relationship of the treaty’s three pillars, an agreement on some tangible practical disarmament steps,weapons of mass destruction category, chemical and biological weapons  established an Open-Ended Working Group (OEWG) to discuss new paths to nuclear disarmament, and, on the basis of the OEWG’s report, created a negotiation body to develop a legal instrument for prohibiting nuclear weapons. The negotiations were completed by July 7, 2017, after just two rounds of talks, and the UNGA supported the opening of the TPNW for signature on September 20, 2017  and have almost categorically rejected the TPNW and any discussions on it  or about the political conditionality (or lack thereof) of disarmament undertakings. The TPNW, in particular, is thus a symbol of the deadlock between the demands of many NNWS for “tangible” disarmament, which goes beyond singular disarmament events, and the step-by-step approach to disarmament as propagated by the NWS and the NNWS allied to them. Disputes among the members of the NPT are now centered almost exclusively on the role, legitimacy, and desirability of the TPNW. Cleavages within the NPT community have, to some extent, become part of the debate on the ban treaty. This thematically narrow approach has only increased the polarization within the NPT. The question of whether the TPNW should be supported or condemned has turned into a question of faith . Under pressure from the Soviet Union, Pyongyang joined the NPT in 1985. That it took the country more than six years to conclude a comprehensive safeguards agreement (CSA) with the IAEA already raised suspicions as to the country’s compliance, which could, however, not be addressed because the NPT’s compliance/enforcement mechanism Findlay , p. 39 dThe Syrian proliferation case is also grounded in the dependence of the NPT’s compliance and enforcement system on P5 agreement and support. Suspicions were raised when the Israeli Air Force destroyed an alleged plutonium production reactor complex in 2007, which Syria had allegedly built with the help of North Korea but not reported to the IAEA. Ensuing inspections found evidence of undeclared activities, leading to a report by the IAEA Board of Governors affirming Syrian non-compliance with its safeguards obligations. Even while Syria indicated its willingness to accept inspections at the suspected site, after it had been cleaned, Damascus refused to allow investigations at other locations. Since then there has been no progress on the matter. Because Syria has not concluded an AP with the IAEA, the Agency is only able to verify the accuracy of inventories of declared nuclear materials, but not the absence of undeclared nuclear activities Heinonen , p. 23.The third ongoing case of proliferation concern is also characterized by the fact that, for decades, Iran violated its safeguards obligations in order to acquire nuclear weapons capability at least up until 2003  when it comes to effectively dealing with non-proliferation cases under the NPT. The lack of favorable political environment—i.e. great power cooperation—makes it difficult to enforce norms Müller . While tGreat power interference has also hindered norm enforcement in the Syrian and North Korean proliferation cases. Russia has prevented a thorough investigation of Syria’s nuclear activities after the Israeli Air Force bombed a reactor that was under construction. Not only did China cast a negative vote when the IAEA found North Korea to be in non-compliance with its safeguards obligations in April 2003 and referred the case to the UNSC. It was also due to Chinese influence that the language of the resolution that was finally adopted by the Council, as well as all the subsequent resolutions, remained rather weak. Neither UNSC sanctions, nor the Six-Party-talks  initiated in 2003 could prevent North Korea from steadily developing its nuclear capabilities. Its first nuclear test in 2006 was followed by further tests in 2009, 2013, 2016, and 2017. China agreed to some sanctions against North Korea after each of the tests, but prevented sanctions that would have been harsh enough to make Pyongyang reconsider its course.These unresolved proliferation cases could contribute to drive further proliferation. Nowhere is this more visible than in the Middle East. In addition to Iran, whose attempt to at least significantly delay the achievement of whatever military nuclear ambitions exist was destroyed by the Trump administration’s irreconcilable hostility, Syria, Turkey, and Saudi Arabia have given practical or verbal notice of their conditional readiness to abandon the virtuous path of non-proliferation. It remains to be seen in which direction the nuclear energy program of the VAE will turn. A flicker of nuclear ambition is also visible among America’s East Asian allies.In this section, we seek to outline the conditions for a restabilization of the non-proliferation regime and to draw the boundaries of a possible area of agreement between both sides of what is currently a deep divide. This does not necessarily mean that these conditions will (or can) be met, as this depends on the contingencies of the negotiation process and the further development of the antagonists’ positions. This chapter is therefore not intended to be a strategic prescription (which is a matter for campaigners and diplomats) or a blueprint for an agreement (the specifics of which entirely depend on the negotiating parties and the dynamics of their interaction). Rather, the focus is on determining where windows of opportunity may exist. Whether or not these opportunities will be seized depends on the actors on the ground.Three changes are needed to restabilize the non-proliferation regime: First and most importantly, the ongoing power struggle between the great powers needs to be settled in a cooperative manner, including the nuclear dimension and the initiation of a process of plurilateral arms control and disarmament. This process has to emanate from the two states that possess the largest arsenals, the US and Russia. The decision to extend the New START Treaty for five years is an important first step as it reinstates much needed provisions for transparency and fostering confidence-building norms. Such a process should also aim to gradually include the other nuclear weapon states as recognized by the NPT and eventually, the countries possessing nuclear weapons outside the regime. While specific regional solutions are likely to be needed for the cases of Israel and North Korea, the P5 process appears to be an appropriate forum for sustained engagement with India and Pakistan, particularly with regard to working toward a mutual understanding of non-proliferation and disarmament priorities and discussing risk reduction measures. A unified approach is also needed to deal with regional challenges to non-proliferation. Another development that is indispensable for a cooperative settlement of current power struggles is that the great powers overcome their quest for narrow geopolitical interests and acknowledge their responsibilities when it comes to upholding international security (Leveringhaus and Hurrell Second, the differences between the various groupings in the NPT need to be overcome Miller . This isThird, it is of utmost importance that the principle of the prevention of nuclear war once again becomes the guiding principle behind the actions of NWS and NNWS. This step may pave the way for further changes in the direction of a peaceful regulation of the nuclear order. There are many possible ways of acknowledging this principle. Suggestions for responsibility-based approaches Lodgaard , pursuinNow, 50 years since it entered into force, the future of the NPT is more uncertain than ever. Temporary optimism about progress in disarmament has given way to disillusionment. Yet the destabilization of the NPT cannot simply be blamed on ban treaty enthusiasts nor on the geostrategic situation and looming great power conflicts. The growing rivalry between the US, Russia, and, more recently, China makes progress on disarmament increasingly unrealistic. Instances of nuclear proliferation put both the interplay between the great powers and the efficacy of the NPT’s compliance and enforcement mechanisms to the test. All this further exacerbates the deep divisions within the NPT community. The dichotomies within the NPT regime are evident not only when it comes to the question of how to handle the TPNW. However, finding the right approach to dealing with the TPNW could prove crucial for the future of the NPT. The TPNW is a direct product of the NPT’s shortcomings: the hierarchical structure of the NPT, the unequal treatment of its parties, and the disputes on the correct interpretation of the NPT’s norms have fueled the conflicts in the NPT community. Above all, it was the deep dissatisfaction of a large number of NNWS with the disarmament record of the NWS that gave rise to the TPNW. The response of the NNWS—by means of the TPNW—is a powerful normative comment on the state of nuclear disarmament as well as the NPT. Yet, the TPNW alone cannot guarantee sufficient security in a world without nuclear weapons. Nor will it lead to nuclear disarmament without the willingness and cooperation of the NWS.The competition between the TPNW and the NPT, brought about by the misguided policies of the proponents of either side, has real-world consequences for the relationship between the NWS and the NNWS and the stability of the whole NPT regime. A further destabilization of multilateral nuclear arms control could exacerbate smoldering proliferation crises and reinforce the nuclear component of regional conflicts in the Middle East and East Asia alike. A crack in the already fragile foundation of the NPT would also affect ongoing attempts to rescue or revive bilateral nuclear arms control. Multilateral and bilateral nuclear arms control and disarmament are closely interwoven; the end of one can all too easily entail the end of the other. The entwinement of bilateral and multilateral arms control and disarmament is, in turn, the best chance we have of reducing the risk of nuclear war. A stabilization of the NPT and a channeling of the great power conflict into more cooperative forms could pave the way for nuclear disarmament more generally.de-emotionalized while reminding all groups that no one wins if the regime fails. After all, “from its inception, the NPT was more than a body of international law. It was a political settlement among disparate states, and a statement of the values, norms and aspirations that should henceforth guide their behaviour” (Walker Given the current geostrategic situation, a cooperative process of plurilateral reconciliation hardly seems possible. Such a process would have to emanate from the United States and Russia; yet neither state seems willing to take the first step. As a consequence, other NWS, with China leading the way in pursuit of its own geopolitical agenda, can all too easily point to the nuclear superpowers’ failures in order to mask their own shortcomings. A reconciliation process jointly directed by the two nuclear superpowers would also help the NPT to overcome its extreme polarization. Yet, to mitigate the differences between the various groups within the NPT, the debate would have to be ” Walker . Reinvig