Modelling the leaping cycle by modified Lotka-Volterra equations with applications to technology and safety

ABSTRACT A bounded version of the Lotka – Volterra model based on logistic differential equations is suggested to emulate cyclic behaviour in complex evolutionary and other systems. The model provides reasonable approximations for the so-called leaping cycles and other oscillational processes when the main state variables are limited by physical constraints. The model represents a basic model-building block that can be used in system dynamics and other frameworks and applications that require fundamental understanding and modelling of principal features of complex cyclic processes. Two application areas – technological progress and industrial safety – are specifically considered here. The model is shown to emulate techno-economic cycles reasonably well and give an appropriate qualitative description of risk and safety variations in the industrial environment. Simulations using the suggested model are quantitatively compared with the results of agent-based simulations. While a simple model cannot match all details of complex agent-based simulations of the risk and benefit dilemma, the bounded Lotka – Volterra model still gives a reasonable approximation of the process. The conceptual implications of the model are, nevertheless, significant, pointing to common cyclic mechanisms taking place in complex evolutionary systems that may belong to different branches of science.


Introduction
Understanding and modelling complex systems remain one of the principal challenges for modern science and engineering.Among these systems, our interest is often focused on complex evolutionary systems -these systems involve some competition between elements in conjunction with some mechanisms of selection.In fact, most of the complex adaptive systems (Gell-Mann, 1994;Holland, 2006) we know are either evolutionary systems on their own or are outcomes of some kind of evolutionary development (Klimenko, 2013).Complex evolutionary systems have great diversity and represent systems of different physical natures -biological, social, economic, technological, etc. -but, despite obvious differences, these systems can have some common properties.The conceptual similarity of all evolutionary systems has been advocated by many distinguished thinkers - Tarde (1903), Koestler (1964), Richta (1967).Popper (2015), Dawkins (1976), Ziman (2000) -and is commonly accepted now.(Klimenko, 2008(Klimenko, , 2013;;Klimenko & Klimenko, 2019).
One can expect that the conceptual similarity noted above can (and perhaps should) be reflected in a similarity of some of the specific features associated with the evolutionary systems of different physical natures.For example, Coccia and Watts (2020) suggest that parasitism and symbiosis are common not only for biological but also for technological systems.
Many (if not most) complex evolutionary systems in addition to gradual progress and adaptation, exhibit cyclic behaviours and some of these cycles, which we call "leaping" (Klimenko, 2008), have a number of common qualitative features, despite being observed in evolutionary systems of different physical natures (see Figure A1).It is not surprising that different domains of scientific inquiry, which commonly have different terminologies, conceptualisations and ways of thinking, often refer to these cycles by different names ("life cycles", "surges", etc.).The term "leaping cycle" is used here to denote a generic cycle and avoid confusion with its specific implementations in different domains of knowledge.Leaping cycles in complex systems have been studied over the last decade and recently reviewed at a generic level (Klimenko & Klimenko, 2019).If these cycles have common features, and these commonalities are inherent and not coincidental, there should be a relatively simple, generic model that can reproduce such features.It was demonstrated that, if the evolution of a complex system is emulated by competing agents, the cycles can appear (and, in fact, do appear) when the competition between these agents is intransitive.Special competitive thermodynamics emerges in transitive competitions and rules out any cyclic behaviour.Intransitivity, therefore, is a gateway to real-world complexity, but intransitive relations (preferences) are much more difficult to investigate than those represented by conventional orders and semiorders (Klimenko, 2015).While mentioning significant progress in analysing complex agent-based models with intransitive competitions, we note that it would be desirable to find a relatively simple model that can still reflect some basic features of the leaping cycle.One of the principal features of the leaping cycle, which is used here, is the boundedness of its state variables since evolutionary systems tend to occupy a large but constrained domain and it is these constraints that lead to competition.
The main goal of the present work is to find a relatively simple (and, perhaps, the most simple) conceptual model in lines of system dynamics (Forrester, 2007) that still would be suitable to describe the principal features of the leaping cycles.Here, we do not target an exact description of all details associated with cyclic evolutions of complex systems -this task would be practically impossible -but attempt to follow the philosophy of system dynamics and reflect major components and principal conceptual interactions between them (Sterman, 2000).The suggested model is a modification of the classical Lotka -Volterra model (Fort, 2020;Lotka, 1910;Volterra, 1931), which, as discussed below, cannot reproduce leaping cycles in its original, non-modified form.While noting a wide spectrum of complex systems that can exhibit the leaping cycle, in this work we specifically consider only two important areas of application of the suggested model: techno-economic cycles (Perez, 2003) and cycles associated with risk and safety (Leveson, 2017) and demonstrates the ability of the model to reflect the principal features of the cycles.
Section 2 introduces a basic model for leaping cycles, which is called the bounded Lotka -Volterra model based on the original Lotka -Volterra model and logistic equations.Section 3 presents technoeconomic cycles or surges, gives their simplified interpretation and demonstrates the similarity of this interpretation and predictions of the bounded Lotka -Volterra model.Section 4 presents the application of the suggested model to safety, which is viewed from the systems perspective and tends to exhibit oscillations.In Section 5, the bounded Lotka -Volterra model is examined quantitatively, in comparison with much more complex agent-based simulations.Section 6 gives a summary of the ideas presented in this work.The Appendices present evidence of common cyclic behaviour in different domains of science.

Modelling the leaping cycle
Most complex evolutionary systems -technological, biological, economic, social, organisational, etc -tend to display at least some degree of cyclic behaviour.In many cases, these cycles have uneven rates of increases and decreases (e.g., gradual increases and sudden collapses) and are called "leaping".These cycles are often characterised by physical limits imposed on the main state variables leading to generally non-harmonic, bounded oscillations.For example, competitive strength (i.e., relative ranking as defined in Klimenko (2015)) is 1 for the most attractive elements (options) and 0 for the least attractive -this value cannot exceed one or fall below zero.Another example is given by technological surges of Perez (2003): the magnitude of development is limited for a given technological paradigm by resource availability and by economic and physical constraints.Recognising these constraints lead us to modification of the original Lotka -Volterra model that deals with bounded variables and can reproduce principal features of the leaping cycle.
Conceptually, the modified Lotka -Volterra model corresponds to a system with negative feedback and strict boundaries imposed on the state variables y i .Without loss of generality, we follow the conventional normalisation of logistic variables 0 < y i < 1. Choosing this normalisation is only a matter of convenience -all normalisations and corresponding variations of the models are conceptually equivalent.Þ; which satisfies 0 < ỹ < 1).Another convenient normalisation is À 1 < y i < 1; but it is not used in the present work since, as remarked above, state variables in complex evolutionary systems are often deemed to be nonnegative.

Bounded Lotka -Volterra model
The simple oscillator, which is given by is suitable for describing harmonic waves but, obviously, does not fit more complex patterns of the leaping cycle.Here a i and b i (i ¼ 1; 2) are model parameters.The classical model that can describe oscillations in complex systems is the Lotka -Volterra model (Fort, 2020;Lotka, 1910;Volterra, 1931) that is specified by the following system of equations In addition to harmonic waves, this model specifies a wider class of oscillating processes.Traditional interpretation of the model is the competition between predator and prey: y 1 is the number of predator and y 2 is the number of prey.The mathematical properties of this model are well known and do not need to be discussed.We only mention that y 1 and y 2 oscillate while remain positive y 1 ,y 2 > 0; i.e., without crossing the lines y 1 ¼ 0 and y 2 ¼ 0. This model cannot match the evolution of a complex system exhibiting a leaping cycle since the oscillations are generally not bounded from above.In order to replicate the principal features of the leaping cycle, the model needs to be modified, leading to the bounded Lotka -Volterra model Unlike the original Lotka -Volterra model, the solutions of this model are bounded 0 < y i < 1, i ¼ 1; 2, as needed for our purposes.In the rest of this work, we investigate whether the bounded Lotka -Volterra model can provide a reasonable qualitative and quantitative approximation for the leaping cycle.
The logistic integrator, which is conventionally given by and, according to Sterman (2000), corresponds to the causal loop diagram shown in Figure 1, is present on the left-hand side of equations (3).Causal diagrams are commonly used in system dynamics to reveal principal physical relations between major state variables.The positive sign of the input signal ϕ corresponds to the growth of the parameter y; while negative ϕ decreases y.The variable y remains bounded by the infimum y ¼ 0 and the supremum y ¼ 1 due to the reinforcement (R) and balancing (B) loops shown in the figure.If ϕ is fixed, the logistic equation is conventionally interpreted as governing constrained propagation of basic epidemic or innovation.The relevance of the logistic integration and the original Lotka -Volterra model was pointed out by Blanco (1993), although in a different context.This solution of equation ( 4) for ϕ ¼ 1 is conventionally represented by the logistic function y ¼ ð1 þ e À t Þ À 1 .This function and, occasionally, its more complex variations (e.g., the sequential dual logistic growth) are used in applications Meyer (1994); Kucharavy and De Guio (2008).These variations are conceptually and functionally different from the model represented by (3) and considered here.In line with the generalised forms of the Lotka -Volterra model (Fort, 2020), we can introduce -the multi-variable, generalised form of the bounded Lotka -Volterra model, where i; j ¼ 1; 2; . . .; n.
The bounded Lotka -Volterra model ( 3) is determined by four parameters a 1 ; a 2 ; b 1 and b 2 : Rescaling time t can be used to remove one of the constants a 1 or a 2 .The system requires specification of the initial conditions, say The model can be used to emulate basic cyclic behaviours in complex systems.The following subsection outlines a variety of oscillations that can be emulated by the bounded Lotka -Volterra model, while a number of qualitative and quantitative physical examples of modelling leaping cycles are given in the rest of this article.

The phase space and oscillations
The system of equations (3) can be integrated by eliminating dt from (3) and separating the variables, producing the following integral of motion where for i ¼ 1; 2: The point y 1 ¼ b 1 and y 2 ¼ b 2 is the stationary point of the system where the function Fðy 1 ; y 2 Þ reaches its global minimum F min .Since the function Fðy 1 ; y 2 Þ is infinite at the boundaries y 1 ; y 2 ¼ 0; 1; these boundaries are not achievable but can be closely approached in simulations.
Figure 2 shows the lines of constant Fðy 1 ; y 2 Þ -i.e., the phase trajectories -for the same values of the parameters as in Figure 7.The stationary point is shown by the dot.The solutions around this dot and away from the boundaries are close to harmonic  (Sterman, 2000).oscillations (waves).If one of the logistic integrators, say the first equation in (3), is replaced by conventional integration dy 1 =dt ¼ y 2 À b 2 ð Þa 2 =4, then integration of the system requires that is just above F min ; the cycle is similar to the harmonic oscillations around the stationary point 3a, where F � 1:47 and F min � 1:39).If Fðc 1 ; c 2 Þ is large, the solution closely approaches one or more boundaries (Figure 3b, where F � 5:39).If at least one of the initial conditions c 1 or c 2 is close to its boundary (i.e., c i !þ0 or c i ! 1 À 0), then Fðc 1 ; c 2 Þ is large.Large values of a 2 =a 1 increase variations and the rate of change of y 1 and decrease variations and the rate of change of y 2 (see Figure 3c), and vice versa (see Figure 3d).Note that, since variations of y 1 in Figure 3d are small and are well within the logistic boundaries 0 and 1, logistic integration of y 1 is similar to conventional integration.Hence, a feedback loop involving one logistic and one conventional integrator is a special case of the model.If the variables are small y 1 ; y 2 � 1, then the model effectively reproduces the behaviour of the original Lotka -Volterra model.The values of b 2 above its median value of b 2 ¼ 1=2 introduce temporal asymmetry into oscillations: y 1 grows slowly but diminishes quickly (Figure 3e) and vice versa for b 2 < 1=2.The inversion of this logic (due to the opposite signs in equations ( 3)) describes the effect of b 1 on the oscillations of y 2 (Figure 3F).The parameters used in simulations are shown in Figure 3(a-f) are listed in Table 1.In the same way as the original Lotka -Volterra model can emulate harmonic oscillations by selecting suitable initial conditions, the bounded Lotka -Volterra model can emulate both the original Lotka -Volterra model and harmonic oscillations by selecting initial conditions ensuring that the solutions do not approach the logistic boundaries.

Technological change
Modelling cyclic features of complex systems is not trivial -even a qualitative representation of the patterns observed in complex systems is complicated and often allows for different interpretations.Among many complex evolutionary systems, economy and technology conventionally receive the most attention and are carefully monitored in modern society.To avoid uncertainties associated with different fields of knowledge (e.g., those presented in Figure A1), we explore the first principal features of leaping cycles in a particular field, specifically, for long techno-economic cycles.The history of mankind is marked by technological development, which can be called technological evolution since this complex process involves competition and selection between technological ideas, inventions and innovations (Koestler, 1964;Schuster, 2016;Ziman, 2000).While the initial pace of technological progress was fairly slow, it gained momentum during the industrial revolution and  7.
the pace of progress had an accelerating trend ever since.The seminal works of Kondratiev (1984), Schumpeter (1947), Freeman and Louca (2001) and Perez (2003) indicate that technological development is not only progressive but also cyclic.One needs to note a large and still rapidly growing number of publications dedicated to evolutionary aspects of technological development: Ziman (2000), Arthur (2009), Solé et al. (2011), Schuster (2016), Coccia (2020a)and many others.Kondratiev (1984) was the first to observe long waves in industrial activity.These waves were initially deemed to be mere oscillations with a characteristic period of around 50 years as shown in Figure 4. Schumpeter, Freeman and Perez subsequently introduced a different understanding of the Kondratiev wave as the destruction of the old and the emergence of the new technological generation.It seems that the theory of technological surges, which was developed by Perez (2003) and is followed here, gives the most accurate explanation for the phenomenon of long cycles in economic activity.This theory indicates that each cycle is associated with the invention, development and growth of a new generation of technologies (i.e., technological paradigm shift) that proceeds through several common phases: gestation, irruption, frenzy, synergy and maturity.These phases are illustrated in Figure 4.

Perez's theory of technological surges
A new cluster of technologies is conceived during the gestation period, which may last for a long time.During this period, new ideas and inventions are typically not known to the general public, and become visible only after a "big bang" event demonstrating the viability of the new technology.During the following irruption phase, the new technology begins to attract attention and investment but still exists within the old technological paradigm, which generates most of the revenue although becoming less and less attractive for investors.Once the dominance of the old technological generation is challenged, the new technology attracts substantial financial capital, which is keen not to miss emerging opportunities.This marks the transition from the irruption phase to the frenzy phase.This transition is characterised by the exponential growth of the new industries and the emergence of the first signs of the decline of the retiring industries.The rate of investment into new technology keeps growing and, towards the end of the frenzy phase, reaches unsustainable levels, Table 1.Parameters of the bounded Lotka -Volterra cycles shown in Figure 3.
1, 1 0.5, 0.5 0.5, 0.7 b) 1, 1 0.5, 0.5 0.9999, 0.7 c) 1, 2 0.5, 0.5 0.9999, 0.7 d) 5, 0.2 0.5, 0.5 0.5, 0.01 e) 1, 1 0.5, 0.75 0.9999, 0.7 f) 1, 1 0.75, 0.5 0.9999, 0.7  violating physical constraints and overshooting all notions of balanced equilibrium between financial capital and a productive economy.This results in a financial bubble that inevitably collapses at the turning point.The crisis associated with this collapse often delivers a final blow to old and nonviable companies, clearing the space for the ensuing rise of a new technological generation.The turning point is usually characterised by a market correction and, possibly, the introduction of new regulations aimed at abating the excesses of the frenzy phase.This underpins the subsequent slower but more balanced and more productive growth of the synergy phase, where technology and capital work together for the benefit of society.During the maturity phase, the current generation of technologies approaches its limits when any further growth becomes increasingly more problematic.As profit margins decrease, institutions begin to look for new opportunities that may involve cost reduction, expansion into new domains or new technological solutions.Finally, a new attractive technological paradigm eventually emerges and the cycle, which lasts around 50 years, is subsequently repeated.Similar ideas can be found in many other publications (Devezas, 2006).Devezas and Corredine (2001) summarised major existing theories of the long wave pointing to the relevance of the logistic integrator to the effects of the propagation of innovation driving these waves.In fact, the logistic equation was also mentioned in the original letters written by Kondratiev from the Soviet prison in the 1930s (Devezas, 2006, p. 133).Glazyev and Kharitonov (2009) mostly follow the theory developed by Perez but with some simplifications: only three phases -the embryonic phase, the phase of growth and the phase of maturity -are identified in the surges.The statistical analysis of Korotayev and Tsirel (2010) confirmed the probabilistic significance of the long (Kondratiev) waves but identified the Kuznets waves with frequency doubling of the Kondratiev waves.This is consistent with Perez's subdivision of the long wave into installation and deployment periods as shown in Figure 4. Coccia (2010) analysed the shape of the long waves and determined that, statistically, these waves have an asymmetric shape.Hilbert (2020) reviewed the dynamic of the long waves (surges) and pointed to shifts in sources of energy associated with the waves.
Since the beginning of the industrial revolution in the 18th century, there have been five industrial surges, while it seems that we are witnessing the beginning of the 6th surge.These surges, which are characterised in Figure B1, are conventionally grouped into 3 industrial revolutions.Surges 2 and 4 completed the technological transformation that began in surges 1 and 3.The emerging surge 6 is often referred to in popular literature as the 4th industrial revolution.Specifying the timing of these surges is not a simple task since technological surges are disturbed by financial and political events, wars and natural cataclysms.The disturbances induced by the two world wars were especially strong accelerating the development of some technologies and delaying the development of others.That is why realistic technological cycles never have perfect theoretical shapes and, if considered under some magnification, would be a superposition of other cycles of smaller magnitudes and durations.The timing of the technological surges is given in Figure B1.

Technological surges and their simulations
The technological cycle that is shown in Figure 5(b) represents a sequence of Perez surges and is, essentially, similar to a product life cycle, but applied not to a single product but to a large cluster of technologies.These surges are different from the original interpretation of the Kondratiev waves shown in Figure 5(a).The small kink in the middle of each surge indicates possible overheating and correction, which is related to overshooting trends common for share markets.This overheating is present in some cycles (not necessarily technological) and absent in others.Although conventional tools of system dynamics can model overshooting, we, as shown in Figure 5(c) neglect this feature for the sake of transparency.The technological cluster leading a particular surge usually replaces an old technology, which either disappears completely or loses dominance but survives in a specialised small niche.Most technological clusters do not collapse and disappear at every surge but continue over two (or more) great surges until they are replaced by new technology.In this case, the collapse of obsolete technology is usually quick -as an example, one can recall the rapid demise of the industry producing steam locomotives (Klimenko, 2008).The duration of the rising and declining phases of the long waves has been analysed by Coccia (2010), who concluded that the former tends to last longer than the latter.This effect is even more pronounced for the waves of disruptive innovation associated with specific technologies (Coccia, 2020b).Overall, we must recognise that technologies are constrained by physical laws and the availability of resources, and, for a given technological generation, have only a limited capacity to grow.According to the concept of creative destruction by Schumpeter (1947), old technologies have to be removed to give way to a new generation of technologies and further technological progress.We arrive at important constraints imposed on the production amplitude of the surge (denoted here by y 1 ): it must be positive but for a given level of technology cannot exceed a certain limit.In real life, technological development becomes problematic for mature industries but still continues, albeit at a slow pace.In our simplified conceptual consideration, we assume the existence of a strict upper limit (which, as previously discussed, can be set to unity without loss of generality).
The simplified version of a technological cycle, which is shown in Figure 5(c), follows only the leading technology (i.e., the curve corresponds to different technological clusters in the different surges).While a single surge followed by a decline (as in Figures 4 and  5(b)) may be referred to as the "life cycle", we use the term "leaping cycle" for the connected sequence of technological leaps (Figure 5(c)) to avoid confusions with other common use of the term "life cycle".The second curve shown in Figure 5(c) is the relative investive attractiveness of the current leading technology, which is characterised by a variable y 2 oscillating between 0 (unattractive technology) and 1 (the most attractive technology).Once the advantages of a new technology become clear, its appeal rapidly increases, reaching the highest possible levels and attracting enthusiasts and investors.This stage is characterised by initially exponential and then increasingly saturated production growth.As the current technology reaches maturity, its further substantial growth becomes rather difficult or impossible and its attractiveness for investors declines.These investors begin to look for alternatives and, once such a promising alternative is found and demonstrated, its attractiveness rapidly increases.The old technology shrinks (in absolute or relative terms) or, possibly, collapses, while the new technology forms the basis for another technological surge.Note the existence of negative feedback in this system: high attractiveness brings investments and increases production while reaching the maximum production amplitude decreases attractiveness and, consequently, investments.Availability of production capacity and investments is a precondition for the emergence of new and highly attractive technology.While harmonic Kondratev waves shown in Figure 5(a) can be modelled by conventional oscillators the modelling of the leaping cycle evidently needs alternative treatment.
According to Perez (2003Perez ( , 2010)), the growth of production within a given technological paradigm is determined by the investive attractiveness of the current technology; and vice versa: technological unattractiveness can lead to the decline (or even collapse) of the production.This dependence is shown by the causal loop diagram in Figure 6.The change in technological attractiveness is determined by two factors saturation, which refers to abundant production leading to technological maturity and eventually to effective retirement, and innovation, which implies the emergence of a new technological generation.While the decline is understood as growth taken with the negative sign, increase and decrease in technological attractiveness correspond to two conceptually different processes: saturation and innovation.This reflects the physics of the process -the old technological generation is not reinvigorated but replaced by a new generation causing a technological paradigm shift.As explained above, our present consideration neglects this difference assuming that innovation is effectively equivalent to negative saturation, as illustrated in Figure 6.Numerical implementation of this diagram gives the bounded Lotka -Volterra model, which can be used to emulate basic cyclic behaviours in such complex systems.Figure 7 presents the solution of equations (3) for a 1 ¼ 2; a 2 ¼ 1; b 1 ¼ 0:75; b 2 ¼ 0:9; c 1 ¼ 0:5; c 2 ¼ 0:3.One can see that y 1 and y 2 correspondingly resemble the amplitude and the attractiveness shown in Figure 5.

Applications of the leaping cycle model to risk and safety
This section considers a spectrum of issues associated with risk and safety, focusing on systemic properties, which commonly exhibit cyclic oscillations, and examining the possibility of approximating these oscillations by the model.As in the previous section, we consider a qualitative agreement between the model and systemic interpretation of safety variations.

Safety as a systemic property
While safety and reliability are related conceptually these terms are not synonymous and denote different properties.As explained by Leveson (2017), reliability is the property of a particular element to perform its function without (or with a very small probability of) failures, while safety is an emergent property of the whole system that allows us to avoid serious consequences even when elements of this systems fail.Reliability does not always imply safety and vice versa.Safety reflects interactions of many factors: conceptual, technical, human and environmental.A reliable principal element of a complex industrial system may create conditions when operating staff is unfamiliar with the possibility and indications of its failure since such failures are rare events.Staff that is not familiar with an unusual critical situation can make incorrect decisions under stress.Subsequent investigations of operator actions often create a hindsight bias: correct decisions that can eventually be found in the safety of post-accident analysis are not necessarily identifiable in extreme conditions requiring immediate action.An example of a systemic failure is given by the 1979 accident at the Three Mile Island nuclear power plantthe third most disastrous nuclear accident in the world (Leveson, 2017).A series of events including minor mistakes and violations lead to the tripping of the turbine and the reactor, followed by excessive pressure build-up in the reactor cooling system.The pressure was reduced by venting the coolant but the relief valve was not closed leading to a critical loss of coolant and partial reactor meltdown.The operator closed the valve and believed that it is closed since the "valve open" indicator went out.In fact, the valve was stuck open and the indicator reflected the state of the control devices and not the actual position of the valve.The operator did not have experience with such conditions and believed that the indicator reflects the valve position.The Three Mile Island accident was clearly contributed by a seemingly minor but rather important design flaw: using control indicators that can mislead the operators.If the reflected the actual valve position, this would be a much safer design.Note that there is no such thing as 100% safety in the real world -it is still possible that the valve actuator and the valve indicator fail at the same time, but the probability of such an event is very small: P event ¼ P act P ind ; where P act and P ind are the corresponding failure probabilities for the actuator and the indicator.
Since safety is an emergent property of complex systems, it inevitably allows for different perspectives, which may lead to different conclusions.This is a common property of all complex systems that often require using different perspectives for analysing different features of the systems or for pursuing different goals -complex systems are complex because we do not have any fixed standard methodology for describing and predicting their behaviours.This means that the problem of safety is multifaceted, and many different approaches can be reasonable, depending on the case under consideration and the goals of the analysis.For example, safety is usually associated with substantial legal or regulatory requirements.Satisfying these requirements guarantees a strong legal position in case of an accident but does not necessarily bring the probability of such an accident to its minimum.Introducing extensive paperwork is often a good measure for legal protection but can divert the attention of the staff from actual risks and potential dangers.While many alternative approaches to safety investigations can be reasonable and useful, investigation biases are often more associated with the interests of pressure groups than with the investigation methodology.Technical competence, experience, impartiality and independence of the investigators is the best way to ensure objective outcomes.One also needs to remember that requirements of safety and efficiency are often contradictory.A practical decision-maker needs to strike a reasonable balance between these requirements, which inevitably becomes a complex problem when dealing with complex systems.

The safety cycle
Since safety is an emergent systemic property according to Leveson (2017), it is subject to common behavioural features observed in complex systems, which often involve cyclic behaviour.This problem was considered by Dulac et al. (2005), who analysed a safety cycle for NASA missions considering three major parameters success, risk and safety priority.The suggested model exhibits oscillations with success leading to a reduction in safety priority leading to increased risk.In this work, we follow similar ideas and consider only two parameters safety effort and safety -this model is more simple and seems to fit generic industrial conditions.An increase in safety effort leads to an eventual increase in safety, while a good safety record and the absence of accidents result in a gradual decline in the safety effort.When the safety effort becomes critically low, this eventually results in an accident -this corresponds to a sharp fall in safety, creating a strong impetus for increasing safety effort, which eventually results in increased safety.The cycle is then repeated.This mechanism is illustrated by the causal loop diagram in Figure 8.The two bounded variables.
-safety and safety effort -form a balancing loop generating cycles.
Note that the actual safety level often remains unknown: we do not know the exact probability of an accident unless a long record of accidents is observed under the same conditions.This however is impractical as an accident is typically followed by measures that reduce the probability of further accidents.Generally, it is difficult to investigate and assess the probability of rare events, and accidents are by definition rare events.Therefore in most cases, we deal with perceived or estimated safety which serves as a proxy for actual safety, which remains unknown.Perceived safety falls dramatically after an accident while the actual safety can be deemed to remain the same until new safety measures are implemented.It is clear that the safety y1 is physically bound by 0 < y1 < 1 where 0 corresponds to minimal safety and inevitable accidents and 1 corresponds to maximally safe conditions where the probability of accidents is negligibly small.The second variable, safety effort y 2 , is also bounded.First, unless one wishes to consider sabotage, the safety effort is always positive, y 2 > 0: Second, the safety effort cannot be increased indefinitely: practically, the excessive effort would not have any additional positive effect.Therefore we interpret y 2 as the effective safety effort, which is subject to the condition y 2 < y ðsupÞ 2 . Without loss of generality, we can always renormalise y 2 and set y ðsupÞ 2 ¼ 1.Since we have oscillations of the bounded variables, 0 < y 1 < 1 and 0 < y 2 < 1, we are able to use the suggested model to describe these oscillations.Figure 9 presents the model simulations for a 1 ¼ 1; a 2 ¼ 2; b 1 ¼ 0:9; b 2 ¼ 0:8; c 1 ¼ 0:1; c 2 ¼ 0:995.The environment is perceived safe when y 1 ! 1 and this state lasts for a long time leading to complacency, resulting in a slow decline in the safety effort, which usually remains unnoticed.When the safety effort becomes critically low, accidents are bound to happen, leading to a sudden drop in perceived safety after an accident.This promptly brings the safety effort to its maximum levels, which prevents further accidents and eventually improves the perceived safety.The cycle is subsequently repeated.

Reproducing the leaping cycle for the agent-simulated risk-benefit dilemma
Unlike in the previous sections, where the bounded Lotka -Volterra model is compared with our understanding of qualitative features of the leaping cycle, this section examines the quantitative agreement between the model and a complex simulation of a cyclic evolutionary process involving 10,000 agents.The model and the agent simulation have radically different numbers of degrees of freedom -2 and 20,000 correspondingly -yet, we expect that the model may be able to describe some principal features of the simulated cycle.It is important to stress that the simulation is selected exactly as in previous publications (Klimenko, 2014b,a), only the model (and not the simulation) parameters are adjusted as needed.
In the risk-benefit dilemma, different strategies represented by agents compete to achieve higher benefits while, as much as possible, avoiding risk.The agents competing to achieve higher benefits are bound, however, to undertake higher risks.Initially, small increases in risk can be ignored, but as the overall risk becomes higher and higher until it reaches a critical value, the system loses its stability and collapses into a low-risk, low-benefit state.The cycle is then repeated: benefits and risks tend to grow until the system collapses again.The cyclic mechanism discussed above is representative of leaping cycles in complex evolutionary systems with intransitive competition (Klimenko & Klimenko, 2019;Klimenko, 2012Klimenko, , 2015) ) and, as in other leaping cycles, has a gradual increase, a brief stabilisation and a sudden collapse.In this section, we compare this cycle with the cycle produced by the bounded Lotka -Volterra model.

Balancing risks and benefits
The system under consideration involves competition of agents j ¼ 1; 2; 3; . . .; n, each of them wishes to attain the higher benefit f j but keep minimal risk r j .Yet, as shown in Figure 10, high benefit without associated risk is not accessible: the area f j > r 1=3 j is prohibited.To boost the benefits one needs to take a higher risk in real-world conditions.These conditions may reflect sharemarket agents taking higher risks to achieve or retain high profits or represent industrial companies, which are pushed by competition to reduce the safety margins of their products in order to achieve higher performance (Klimenko & Klimenko, 2019;Klimenko, 2014a).
In the simulations, each of agents j ¼ 1; . . .; n represents a strategy ðr j ; f j Þ that competes with strategies of the other agents so that the winning strategy is assigned to the losing agent with some variations (we can call them mutations -see Klimenko (2012) for details).The system evolves towards better and better strategies, although the exact evolution of the system depends on preferences for higher f and lower r.The relation ðr j ; f j Þ � ðr i ; f i Þ indicates that strategy j is preferable over strategy i, -this preference can be equivalently expressed by a corresponding coranking function ρð. ..Þ; so that (Klimenko, 2015) If the strategies ðr j ; f j Þ and ðr i ; f i Þ are equally preferable then The notation ðr j ; Preferences may have various features and properties, but we must stress the importance of transitivity in this context.Preferences can be transitive, provided for any choice of strategies j, i and k, or intransitive when there exist strategies j, i and k so that Intransitivity is common in mathematics, economics and other areas (Poddiakov & Valsiner, 2013) although conventional theories tend to presume transitivity.As discussed below, the evolutions of complex competitive systems principally depends on transitivity or intransitivity of the competition rules (Klimenko, 2012;2013, Klimenko, 2015)).
The most conventional approach for specifying preferences is introducing a utility, u j ¼ uðr j ; f j Þ; while defining ρ ji ¼ u j À u i , that is It is clear that the preference generated by defining any utility is necessarily transitive and this imposes principle restrictions on the simulations.The fundamental Debreu theorem (Debreu, 1954) demonstrates the inverse statement is also true: equivalent utility can always be introduced for a preference, provided the preference is transitive and topologically continuous.A formal analysis of competitive evolutions under transitive conditions indicates that these evolutions can be characterised by monotonically increasing competitive potential, which is most important for us, cannot be cyclic (Klimenko, 2012(Klimenko, , 2014b)).The widespread use of utilities (which in different fields of knowledge might be called "fitness", "target function", etc.) in evolutionary simulations does not permit emulation of cycles despite the fact that, as demonstrated in Appendix 7, the leaping cycles are common in complex evolutionary systems in the real world (Klimenko, 2013).Therefore, we must ensure the intransitivity of the preferences and select an appropriate coranking function ρð. ..Þ: Let us assume that the benefit is selected to be proportional to its utility that is ρððr; f 2 Þ; ðr; f 1 ÞÞ ¼ f 2 À f 1 for a fixed r -this condition simply defines scaling of the variable f .Hence, assuming some homogeneity with respect to the variable r, we can write where φðrÞ is an arbitrary monotonically increasing function satisfying φð0Þ ¼ 0: The monotonicity is required due to our preference for lower risks.It is easy to see that the linear function φðrÞ ¼ αr corresponds to the utility defined and, therefore, relations ( 9) and ( 15) define a transitive preference.If the function φðrÞ is non-linear, then the corresponding preference is intransitive.Indeed according to the Debreu theorem, the transitive preference must have a utility that, according to our scaling assumption for variable f ; takes the form u j ¼ f j À ψðr j Þ, where ψðrÞ is an arbitrary monotonically increasing function.Since ρ ji ¼ 0 demands that u j ¼ u i and vice versa, we easily obtain for any arbitrary r j and r i and the corresponding f j and f i that ensure ðr j ; f j Þ,ðr i ; f i Þ.If φðrÞ is non-linear, no choice of ψðrÞ is possible to ensure that φðr 2 À r 1 Þ ¼ ψðr 2 Þ À ψðr 1 Þ for any r 2 and r 1 .Indeed, differentiation of this equality with respect to r 2 and putting r 1 ¼ 0 We have just proven the following proposition: Proposition 5.1.
The coranking function ρððr j ; f j Þ; ðr i ; f i ÞÞ ¼ f j À f i À φðr j À r i Þ defines a transitive preference if and only if the function φðrÞ is linear.

Agent-based simulations versus the bounded Lotka -Volterra model
In this subsection, we consider evolutionary simulations using 10,000 agents for two corankings defined by As it follows from the proposition proven in the previous subsection, the first expression corresponds to a transitive preference while the preference associated with the second expression is intransitive.The coefficient α ¼ 1=3 is used in (17) to match the maximal values achieved in transitive and intransitive simulations.Utility is then defined u j ¼ f j À r j =3 for all cases and is equivalent to coranking (17) but is different from ( 18): intransitive preferences do not allow for the introduction of an equivalent utility.
Figure 11(a) illustrates evolution of utility u ¼ f j À r j =3 and risk r for transitive competition.Once the highest utility is reached, the system remains in this state indefinitely.The evolution of the system for the case of intransitive competition, which is shown in Figure 11(b) appears to be cyclic: as the system is pushed into the areas of high risk, at some stage, the defensive strategy of low risk and low benefit becomes more attractive.The system then collapses into a defensive state (Klimenko, 2014b,a).Note that during the growing phase of the simulations, the evolutions are very similar, almost indistinguishable for the transitive and intransitive cases but the overall behaviours are fundamentally different as illustrated by Figures 11  (a) and 11(b).The intransitive system suddenly collapses and the cycle is then repeated but, as discussed previously, the transitive system cannot collapse.
The transitive behaviour in Figure 11(a) can be approximated by logistic integration (4) and logistic function and does not need oscillating models.Our main question is about the intransitive case: whether complex cyclic behaviour can be represented by the bounded Lotka -Volterra model.First, since the model variables are bounded by 0 and 1, the physical variables displayed in the figure need to be linearly rescaled to match this range -as this is done in Figure 11, which shows 2r=3 as risk and 1:6u as a utility.The causal loop diagram shown in Figure 12 reflects that chasing high benefits boosts utility but also increases the associated risks, while high risks force the competing agents to switch from aggressive to defensive strategies, leading to low benefits and low utility.Note that this effect is possible only in intransitive competitions since utility is compelled to increase in transitive competitions.
The bounded Lotka -Volterra cycle that corresponds to the causal loop diagram in Figure 12 is shown for a 1 ¼ 1; a 2 ¼ 1; b 1 ¼ 0:2; b 2 ¼ 0:8; c 1 ¼ 0:9; c 2 ¼ 0:7 in Figure 11(c) and provides a reasonable approximate description of the process.Growing benefit increases utility but, due to physical constraints, also results in a rapidly increasing risk.When the risk is high, the system collapses and utility promptly drops to its low level associated with low benefit and low risk.The utility then grows again repeating the cycle.While the bounded Lotka -Volterra model of the leaping cycle provides a reasonable approximation for this evolutionary system, it cannot possibly match all details of complex evolutions.For example, the physical collapse of the system involves a simultaneous decline of both benefit and risk, while the fall of utility (and benefit) slightly pre-empts the reduction of risk according to the bounded Lotka -Volterra model.

Discussion and conclusion
Complex evolutionary systems tend to display cyclic behaviours that are not harmonic and are often characterised by gradual growth and sudden collapses.While these cycles may have different physical mechanisms, the observed similarities are reflected by the common concept of the leaping cycle.The existence of such similarity is confirmed by system dynamics that offers a physical mechanism and a relatively simple model -the bounded Lotka -Volterra model -that can emulate the basic behaviour observed in such cycles.Complex evolutionary systems deal with large numbers of elements or agents implying that relative characteristics of the elements (e.g., relative rankings that are conventionally bounded by 0, which corresponds to the weakest element, and 1, which corresponds to the strongest element) tend to determine the evolution of the systems.The original Lotka -Volterra model cannot reproduce the leaping cycles but the suggested modification of the model (i.e., the bounded Lotka -Volterra model) is shown to emulate the basic features of the leaping cycles reasonably well.
The bounded Lotka -Volterra model enforces that each of the two state variables y 1 and y 2 oscillate within their limits y further characterised by another four parameters and two initial conditions.The bounded Lotka -Volterra cycle is shown to approximate, at least at a conceptual level, a wide range of bounded oscillatory motions, including techno-economic surges and oscillations of safety.We also demonstrate that the bounded Lotka -Volterra model is a reasonable approximation for agent-based evolutionary simulations of a leaping cycle.These simulations are performed for a system of competing agents that has 20,000 degrees of freedom, while the bounded Lotka -Volterra model has only two degrees of freedom.
One cannot expect that the model based on relatively simple logistic equations can recover all intricate details of the evolution of complex systems -this generally needs more complex agent-based models.In this work, however, we follow the philosophy of system dynamics and note that the emulation of some principal properties is sometimes sufficient for general understanding and practical analysis.Besides simplicity, practicality and ease of use, the models of system dynamics often bring something that is even more important: fundamental conceptualisation and understanding.The suggested model reflects the most basic mechanisms enacting cycles in complex evolutionary systems.The existence of a relatively simple model that can characterise, at least qualitatively, complicated cyclic processes in very complex and diverse evolutionary systems has profound implications pointing to conceptual similarity between cyclic processes taking place in different, seemingly unrelated branches of scientific knowledge.This finding supports our hypothesis that leaping cycles appear to be common for complex evolutionary systems.
Complex systems can and, perhaps, should be considered and examined from different perspectives (Klimenko, 2022).Seemingly contradictory statements with respect to complex systems can correspond to different perspectives and be valid at the same time.In most cases, complex systems are too complicated to be modelled with all minor details.This requires simplifications reaching compromise between refinement and performance.Different modelling approachese.g., system dynamics and agent-based models -serve different purposes and are very useful in their own ways.A relatively simple model based on a modification of the Lotka -Volterra equations allowed us to demonstrate the fundamental similarity of cyclic oscillations that belong to different fields of knowledge.
(2) The actual process of gestation of each technological revolution goes far back before its big-bang, though for the present purposes the visible crystallisation is the most important (Perez, 2003, note 83, Part I, chpt.6(B),p.63).(3) A new ethnos (often having an old name) transforms the landscape with a new way of adapting to natural conditions.This happens, as a rule, during the incubation period of the upsurge phase and is not reflected in historical sources (besides myths).[. ..]This is a "trigger mechanism", which does not always lead to the emergence of a new ethnos since termination of the process by extraneous force is always a possibility (Gumilev, 1978, Part 4, chpt. XVI(6), p.162).

Creative minority as a driving force
(4) Of these, one of the most important is the willingness to try out truly radical innovations as improvements (Perez, 2003, Part I, chpt. 3(C), p.31).(5) . . .passionarity is the ability and desire to change the environment, or, using the language of physics, to overcome inertia of the aggregate state of the environment (Gumilev, 1978, Part 6, chpt. XXIII(2), p.205).The main feature of passionarity is that it is contagious (Gumilev, 1978, Part 6, chpt XXIV (3), p.213).[. ..] passionarity can manifest itself in a variety of traits of character, with equal ease generating feats and crimes, creation, good and evil, but without leaving a place to inaction and calm indifference (Gumilev, 1978 (Kuhn, 1962, chpt VIII, p.84). ( 8) At the beginning of that period, the revolution is a small fact and a big promise; at the end, the new paradigm is a significant force, having overcome the resistance of the old paradigm and being ready to serve as propeller of widespread growth (Perez, 2003, Part I, chpt. 4, p.36). ( 9) The opening up of new markets, foreign or domestic, and the organisational development from the craft shop and factory to such concerns as U. S. Steel illustrate the same process of industrial mutation -if I may use that biological term -that incessantly revolutionises the economic structure from within, incessantly destroying the old one, incessantly creating a new one.This process of Creative Destruction is the essential fact about capitalism (Schumpeter, 1947, Part II, chpt. 7, p.73). ( 10) The largest mass extinctions produce major restructuring of the biosphere wherein some successful groups are eliminated, allowing previously minor groups to expand and diversify (Raup, 1994, abst (ii)).

Explosive growth (irruption)
(11) Each technological revolution is an explosion of new products, industries and infrastructures that gradually gives rise to a new techno-economic paradigm (Perez, 2003, Part I, chpt. 2, p.9). ( 12) The upsurge phase is always linked to expansion in the same way as gas increases its volume when heated (Gumilev, 1978, Part 8, chpt.XXXI (2), p.285).As the upsurge phase transits into the acme phase, the objective of expanding territories appears as inevitably as boiling of water at 100 C and normal pressure.(Gumilev, 1989, chpt.V (4), p.151).( 13) Every social thing, that is to say, every invention or discovery, tends to expand in its social environment (Tarde, 1903, chpt.I(III), p.17). ( 14) . . .this explosive, punctuated model conforms to the typical pattern of vertebrate evolution characterised by rapid diversification following a major extinction event (Feduccia, 2003, abstract).

Breakdown (turning point)
(17) When the economic structural tensions that make the bubble unsustainable come to a head, the outcome is written on the wall: some form of breakdown followed by a serious recession (Perez, 2003, Part II, chpt. 11(A), p.118).( 18) While passionarees are most numerous in the acmatic phase, they are replaced by opportunists in the breakdown phase, which leads to a sharp decrease in the numbers of passionarees (Gumilev, 1978, Part 6, chpt. XXVI(1), p.223).The essence of the breakdown phase is in divisions between people, leading to a loss of unity within the superethnos (Gumilev, 1989, chpt. VIII(11), p.262).( 19) The nature of a breakdown can be summed up in three points: a failure of creative power in the creative minority, which henceforth becomes a merely "dominant" minority; an answering withdrawal of allegiance and mimesis on the part of the majority; a consequent loss of social unity in the society as a whole (Toynbee, 1946, Argument, IV (XIII), p.924).( 20) Numerous examples of so-called "explosive" evolution show such a period of rapid divergence followed by a sharp drop in evolutionary rate (Simpson, 1944, chpt. IV(2), p.139).The radiation as such has definitely ended, both as regards its expanding phase, radiation strictly speaking, and its contracting or weeding out phase (Simpson, 1953, chpt. VII (4), p.230).

Golden age (synergy)
(21) If, at this turning point, the institutional adjustment is successfully achieved, what follows may be a golden age.It can be a period of full employment and widespread productive investment, a period when production is at centre stage, when at last the benefits of the system begin to spread down and an era of "good feeling" sets in.The best face of capitalism can then be seen.(Perez, 2003, p.5, Part I, chpt. 1) During synergy, investment concentrates in the core countries, where the whole economy is flourishing and opportunities across the complete industrial spectrum now abound (Perez, 2003, Part I, chpt. 6(B), p.64) (22) At this stage a new civilisation, with its institutions, its beliefs, and its arts, may be born.In pursuit of its ideal, the race will acquire in succession the qualities necessary to give it splendour, vigour, and grandeur (Le Bon, 1896, Book III, chpt.V, p.119).
(23) We will call this phase "golden autumn" to distinguish it from the rainy days that follow.During the golden autumn, fruits are collected and riches are accumulated.Stability and prosperity can be interrupted only by external wars.Great thinkers, painters and writers are tolerated and fed (Gumilev, 1989, chpt.IX(1), p.270).

Maturity, decline and extinction
(24) This is the twilight of the golden age, though it shines with false splendour.It is the drive to maturity of the paradigm and to the gradual saturation of markets (Perez, 2003, p.54, Part I, chpt, 5(E)).Technologies gradually lose dynamism and markets begin to stagnate, the surge of growth moves to the near periphery and later even to the farther peripheries (Perez, 2003, Part I, chpt, 6(B), p.64). ( 25) The obscuration phase is dominated by opportunists, who gradually push away people of integrity and balance (Gumilev, 1989, p.298, chpt. X(1)).The society corresponding to this phase, which is called "obscuration", is not democratic.Ostensibly, the social organisation remains the same as in the previous phase, but is now characterised by negative selection.[. ..] mediocrity is valued over talent, ignorance is preferred to education, dishonesty is desired more than integrity [. ..] (Gumilev, 1978, Part 8, chpt. XXXV(1), p.329).
Transitions between phases are gradual and, as a rule, not visible to contemporaries (Gumilev, 1978, Part 8, chpt. XXXV(3), p.332).( 26) The tenacity with which universal states, when once established, cling to life is one of their most conspicuous features, but it should not be mistaken for true vitality.
It is rather the obstinate longevity of the old who refuse to die (Toynbee, 1946, part VI, chpt. XXIII, p.560)

Figure 3 .
Figure 3. Modelling cycles with the different values of the simulation parameters shown inTable 1. Solid black line -y 1 ; dashed blue line -y 2 .

Figure 4 .
Figure 4.The structure of technological surges, as introduced by Perez (2003) further developing ideas of Kondratiev, Schumpeter and Freeman.Kondratiev's original wave is shown for comparison.

Figure 5 .
Figure 5. Technological surges: a) Kondratiev wave; b) sequence of innovation surges followingPerez (2003), a single life cycle is shown by a thick line, red stars --the "big bang" events; c) continuous treatment of the surges (i.e., the leaping cycle) neglecting overheating of the turning points and focusing only on the leading technological cluster, solid black line -the amplitude of the surge, dashed blue line -relative attractiveness of the current technology for investors.

Figure 6 .
Figure 6.Casual loop diagram for the technological surges of Perez.

Figure 8 .
Figure 8. Casual loop diagram for the safety cycle.

Figure 10 .
Figure 10.Domain for agent-based simulations of the risk-benefit dilemma.The red cloud of particles is the location of agents at 1000 time steps and the blue lines show the evolution of the mean agent position in time.

Figure A1 .
Figure A1.Correspondence between phases of the leaping cycles denoted in different fields of knowledge.
The phase space of the cycle showing the lines of constant Fðy 1 ; y 2 Þ and the location of F min (red dot).The choice of the simulation parameters is the same as in Figure

Product cycle Perez's theory of technological surges Kuhn's paradigm shiŌ theory EvoluƟonary biology Toynbee's civilisaƟon theory Gumilev's theory of ethnogenesis Generic leaping cycle
. (27) After having exerted its creative action, time begins that work of destruction from which neither gods nor men