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In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels. [ABSTRACT FROM AUTHOR]

By considering that the predators have some additional foods apart from the focal prey and the prey take refuge, a predator–prey model with the effect of seasonality in a fluctuating environment is investigated in this study. Gaussian white noises, created by random fluctuation of water level, are introduced on the prey’s growth rate and predator’s mortality rate. Both the deterministic and stochastic systems are analyzed mathematically as well as numerically. Our analytical findings show that the intensity of environmental noise, additional foods and prey refuge play significant roles in survival as well as extinction of both prey and predator populations in the aquatic system. Our numerical results show that the effectual level of additional foods, depending on water level, has positive as well as negative effects on the predator and prey populations. We find that both species strongly persist at certain water level and for the low intensity of noise. However, for an increased value of noise intensity, they persist weakly and then go to extinction. Overall, our findings show that the variations of water level together with additional food and white noise plays crucial roles in persistence and extinction of species in the ecosystem. [ABSTRACT FROM AUTHOR]

One of the fundamental questions in developmental biology is how a cell is specified to differentiate as a specialized cell type. Traditionally, plant cell types were defined based on their function, location, morphology, and lineage. Currently, in the age of single-cell biology, researchers typically attempt to assign plant cells to cell types by clustering them based on their transcriptomes. However, because cells are dynamic entities that progress through the cell cycle and respond to signals, the transcriptome also reflects the state of the cell at a particular moment in time, raising questions about how to define a cell type. We suggest that these complexities and dynamics of cell states are of interest and further consider the roles signaling, stochasticity, cell cycle, and mechanical forces play in plant cell fate specification. Once established, cell identity must also be maintained. With the wealth of single-cell data coming out, the field is poised to elucidate both the complexity and dynamics of cell states. [ABSTRACT FROM AUTHOR]

In this paper, I argue for an often-neglected solution to the conceivability argument: the reconciliatory response. Its advocates state that, even if zombies are metaphysically possible, it does not follow that all versions of physicalism are false. To make the reconciliatory response, we must construct a theory that counts as a version of physicalism (because it makes higher-level facts count as physical) but also allows for the metaphysical possibility of zombies. Call any physicalist theory that can make the reconciliatory response type-R physicalism. In this paper, I discuss one version of type-R physicalism: stochastic ground physicalism (SGP). First, I argue that type-R physicalism, construed as SGP, offers physicalists an attractive rationalist package that no other version of physicalism can provide. Second, I address two concerns that have been underexplored in the literature. First, the charge that SGP is incoherent because it fails to provide metaphysical explanations. Second, the charge that type-R physicalism is not a genuine form of physicalism because the supervenience of the phenomenal on the physical is a necessary condition for any formulation of physicalism. I argue that both concerns are ill-founded. [ABSTRACT FROM AUTHOR]

Cell-to-cell heterogeneity in products secretion is a bottleneck for diverse applications in bioengineering, including controlled drug release, biomaterial assembly, and production of recombinant proteins in bioprocesses. Experimental approaches aiming at characterizing cell-to-cell differences in gene expression and protein accumulation can be partially adapted to address heterogeneity in protein secretion. Significant technological bottlenecks need to be addressed to establish an efficient single cell analysis pipeline. Techniques designed for directing cell collective behavior (e.g., microfluidics or flow cytometry with feedback control) could be adapted for directing product secretion by cells. Cell-to-cell heterogeneity presents challenges across various fields, from biomedicine to bioproduction, where precise cellular responses are vital. While single cell technologies have significantly enhanced our understanding of population heterogeneity, the predominant focus has been on monitoring intracellular compounds. Recognizing the added complexity introduced by the secretion system, in this review, we first provide a systematic overview of the distinct steps necessary for driving protein secretion. We discuss the various sources of noise acting from the synthesized preprotein to the secretory protein released based on a Gram-positive cellular system as a model. We next explore the applicability of single cell technologies for monitoring protein secretion throughout these functional stages. We also emphasize the importance of applying these single cell technologies for monitoring protein secretion during bioproduction. [ABSTRACT FROM AUTHOR]

The Game of Life (GoL) is a paradigmatic computer simulation that exhibits the emergence of complex properties of the whole from relatively simple sets of heuristic rules operating at lower organizational levels. Therefore, it is widely understood as a valuable tool for investigating global properties of evolutionary processes. Cognizant of refined concepts that emerged in recent debates on the central role of historical contingency (historicity) in evolution, we modify the original GoL rules by introducing an updating feedback loop and a probability factor that reflect the degree of historicity in pattern evolution as both stochastic path dependence and sensitivity to initial conditions. We examine this trait in simulations of the emergence and breaking of bilateral and radial symmetries commonly observed in the evolution of life, most prominently as evolving body plans. We show that the implementation of historicity parameters leads to a more realistic sequential and gradual alternating emergence and the breaking of new symmetries than the original set of rules. Apart from its more realistic representations of evolutionary processes, the new approach allows for easier exploration of the parameter space, demonstrates the sequential and gradual emergence and breaking of symmetries, and provides a publicly available and modular simulation tool. We discuss the exploratory potential of the modified GoL platform we designed as an extended thought experiment to study the parameter space defining the historicity of biological processes. [ABSTRACT FROM AUTHOR]

The hypothesis of chaotic population dynamics was proposed in ecology by Robert May in the mid-1970s. At that time the idea was controversial, and it remains a fascinating and unsettled issue today. We report the results of a 20-year laboratory research programme that continued in the tradition of the pioneering ecologist Thomas Park using the Tribolium flour beetle model. We present biological evidence of complex population dynamics – including bifurcations, chaos, saddle nodes, phase switching, resonance effects, and multiple attractors – by using a low-dimensional difference equation model for Tribolium together with carefully designed, conducted, and statistically analysed experiments. The model, parameterized with data, also explains the results of historical Tribolium experiments, such as the classical competition studies of Thomas Park and his colleagues. Our research programme has inspired other studies using the Tribolium mathematical and laboratory model. This work was conducted by a multidisciplinary team, which included Jim Cushing. [ABSTRACT FROM AUTHOR]

Complex systems (CSs) have garnered increasing attention in science education due to their prevalence in the natural world and their importance in addressing pressing issues such as climate change, pandemics, and biodiversity loss. However, the instruments for assessing one's CS knowledge are limited, and the knowledge of CSs among in-service teachers remains underreported. Guided by the complexity framework, we launched a study to develop and validate a survey instrument for measuring the knowledge of CSs among high school teachers and undergraduate students, as well as delineating the contours of their knowledge. In this article, we present the development and validation of a Complex Systems Knowledge Survey (CSKS), and we use the survey to compare the CS knowledge among 252 high school teachers and 418 undergraduate students in the United States. Our key findings include that (1) both high school teachers and undergraduates exhibit relatively low knowledge of decentralization and stochasticity, (2) undergraduates, especially those in non-STEM majors, demonstrate moderate to low knowledge of emergence, (3) few teachers and undergraduates differentiate between complicated and complex systems, and (4) teachers and undergraduates recognize CS examples across natural and social systems. The implications of our findings are discussed. [ABSTRACT FROM AUTHOR]

Abstract A novel control mechanism is presented for rural microgrids, standing out in the current literature with its advanced approach to load prioritisation and energy allocation. The system's main goal is to maximise energy supply to essential loads while effectively managing available resources. Distinct from traditional methods, this mechanism dynamically classifies loads according to user‐defined priorities, adjustable based on the control system's computational power and complexity. A critical feature is the utilisation of the Particle Swarm Optimisation (PSO) algorithm to optimise demand side management (DSM). This innovative approach leverages day‐ahead load and generation forecasts to ensure optimal energy distribution across load levels, maintaining continuous power supply to high‐priority loads and reducing blackout risks due to generation and load fluctuations. Analyses under stochastic scenarios demonstrate the robustness of the control action, with percentile‐based day‐ahead forecasting allowing for adaptation to significant variations in renewable energy generation patterns. The implementation results are significant, maintaining 100% supply continuity to essential loads throughout the day, even with generation fluctuations up to ‐20%. This marks a considerable improvement in load satisfaction, increasing it from 83% to 96%. A significant advancement in microgrid control is contributed, providing an adaptive, user‐centric approach that enhances load management and energy distribution, and facilitates more resilient and efficient microgrid systems in the face of highly variable renewable energy sources (RESs).

Abstract In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented.

The effects of species on one another sometimes depend on the order in which they arrive at a local site. This historical contingency, termed priority effects, can have a major influence on the assembly and functioning of ecological communities. Although historically studied within community ecology, phenomena consistent with priority effects appear in a variety of biological scenarios and across a wide range of spatial and temporal scales, but are referred to by many different but related terms. There exists a fragmented communication landscape between studies of priority effects across these disparate biological subfields, which is stifling a conceptual unification of how, when, and why priority effects occur. Our review provides a launchpad for developing a unified framework across a broad range of ecological contexts. Although primarily studied through the lens of community ecology, phenomena consistent with priority effects appear to be widespread across many different scenarios spanning a broad range of spatial, temporal, and biological scales. However, communication between these research fields is inconsistent and has resulted in a fragmented co-citation landscape, likely due to the diversity of terms used to refer to priority effects across these fields. We review these related terms, and the biological contexts in which they are used, to facilitate greater cross-disciplinary cohesion in research on priority effects. In breaking down these semantic barriers, we aim to provide a framework to better understand the conditions and mechanisms of priority effects, and their consequences across spatial and temporal scales. [ABSTRACT FROM AUTHOR]

Featured Application: Featured Application: Bioequivalence studies of highly variable drugs require the utilization of large numbers of volunteers. The EMA and FDA propose the utilization of scaled limits. In this study, we introduce the use of artificial neural networks, along with the typical 80–125% limits, as a tool for virtually increasing sample size and thus reducing the actual human exposure. The bioequivalence (BE) of highly variable drugs is a complex issue in the pharmaceutical industry. The impact of this variability can significantly affect the required sample size and statistical power. In order to address this issue, the EMA and FDA propose the utilization of scaled limits. This study suggests the use of generative artificial intelligence (AI) algorithms, particularly variational autoencoders (VAEs), to virtually increase sample size and therefore reduce the need for actual human subjects in the BE studies of highly variable drugs. The primary aim of this study was to show the capability of using VAEs with constant acceptance limits (80–125%) and small sample sizes to achieve high statistical power. Monte Carlo simulations, incorporating two levels of stochasticity (between-subject and within-subject), were used to synthesize the virtual population. Various scenarios focusing on high variabilities were simulated. The performance of the VAE-generated datasets was compared to the official approaches imposed by the FDA and EMA, using either the constant 80–125% limits or scaled BE limits. To demonstrate the ability of AI generative algorithms to create virtual populations, no scaling was applied to the VAE-generated datasets, only to the actual data of the comparators. Across all scenarios, the VAE-generated datasets demonstrated superior performance compared to scaled or unscaled BE approaches, even with less than half of the typically required sample size. Overall, this study proposes the use of VAEs as a method to reduce the necessity of recruiting large numbers of subjects in BE studies. [ABSTRACT FROM AUTHOR]

In this study, we delve into the dynamics of a generalist predator–prey system with prevalence of a disease in the prey population and the presence of predator-induced fear. In this intricate ecological web, the fear response triggered by predators influences the birth rate of susceptible prey population and also escalates intra-specific competition among them. In an effort to encapsulate a more realistic scenario, we extend our deterministic model to its stochastic counterpart by introducing environmental white noise that impacts the mortality rates of both prey and predator species. Comprehensive mathematical analyses are performed on both the deterministic and stochastic systems to elucidate their qualitative behaviors. Specifically, we derive conditions under which the stochastic system exhibits a unique global positive solution and explore the potential extinction of species within the ecosystem. To unravel the intricate dynamics numerically, we perform sensitivity of parameters, and construct one- and two-parameter bifurcation diagrams. Our investigations reveal that when the costs of fear surpass specific thresholds and disease incidence is high, both the susceptible and infected prey populations face extinction. Notably, the introduction of supplementary foods can lead to system’s destabilization, potentially pushing it towards a state where only predators can persist. Moreover, we observe that lower intensities of white noise have minimal impact on the system’s dynamics, while higher noise intensities introduce substantial fluctuations and may precipitate the extinction of species within the ecosystem. Furthermore, we elucidate the abundances of prey and predator species within the ecosystem through histogram plots. [ABSTRACT FROM AUTHOR]

Stochasticity plays a significant role in the low‐power operation of a biological neural network. In an artificial neural network, stochasticity also contributes to critical functions such as the uncertainty quantification (UQ) for estimating the probability for the correctness of prediction. This UQ is vital for cutting‐edge applications, including medical diagnostics, autopilots, and large language models. Thanks to nonlinear variation of analogous Hall resistance with high computing velocity and low dissipation, a stochastic spin‐orbit‐torque (SOT) device exhibits significant potential for implementing the UQ. However, up until now, the application of UQ for stochastic SOT devices remains unexplored. In this study, based on SOT‐induced stochastic magnetic domain wall (DW) motion with varying velocity, a SOT synapse is fabricated that can emulate stochastic weight update following the Spike‐Timing‐Dependent‐Plasticity (STDP) rule. Furthermore, a stochastic SNN is set up, which, when compared to its deterministic counterpart, demonstrates a clear advantage in quantifying uncertainty for diagnosing the type of breast tumor (benign or malignant). [ABSTRACT FROM AUTHOR]

The Grey‐Scott model is a prominent reaction–diffusion system that has sustain consequential recent research about patterning creation in the reaction–diffusion systems. The present research work, explore the stability analysis of a proposed stochastic Grey‐Scott model, which recommend the randomness or unpredictability into the complex system. To take into account the different variations in reactant concentrations profiles, the stochasticity is modeled using the additive white noise terms. To investigate the stability characteristics of the stochastic Grey‐Scott model, we use the spectral collocation technique. Such mathematical perspective sheds light on the different system's behavior in the addition of white noise, and also it may have significance for comprehending processes in the real world that are controlled by comparable dynamics. [ABSTRACT FROM AUTHOR]

In recent years, there has been a growing interest in incorporating fractional calculus into stochastic delay systems due to its ability to model complex phenomena with uncertainties and memory effects. The fractional stochastic delay differential equations are conventional in modeling such complex dynamical systems around various applied fields. The present study addresses a novel spectral approach to demonstrate the stability behavior and numerical solution of the systems characterized by stochasticity along with fractional derivatives and time delay. By bridging the gap between fractional calculus, stochastic processes, and spectral analysis, this work contributes to the field of fractional dynamics and enriches the toolbox of analytical tools available for investigating the stability of systems with delays and uncertainties. To illustrate the practical implications and validate the theoretical findings of our approach, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

In this investigation, we explore a predator-prey system characterized by generalist predators exhibiting cooperative behavior during hunting, subject to nonlinear harvesting rate. Employing a Beverton-Holt type functional response to capture the impact of additional food sources on predator growth, our numerical findings affirm the destabilizing influence of hunting cooperation, excessive predation, and supplementary food sources. Conversely, the system demonstrates stabilization in response to increased prey species growth. We unveil the occurrence of both forward and backward bifurcations in the system due to the predator growth attributed to additional food sources, contingent on the degree of hunting cooperation. To broaden the scope, we extend our proposed model to its stochastic counterpart, introducing environmental white noises. Our numerical results anticipate that higher intensities of white noises lead to fluctuations of greater amplitude, while smaller intensities exhibit a more modest impact. Additionally, we elucidate the dynamics of prey and predator populations through histogram plots. The theoretical and numerical insights derived from this study provide a deeper understanding of the intricate dynamics within predator-prey systems of ecological communities, emphasizing the significance of additional food sources for cooperative predators subject to harvesting in both constant and fluctuating environments. [ABSTRACT FROM AUTHOR]

In this investigation, we delve into the dynamics of an ecoepidemic model, considering the intertwined influences of fear, refuge-seeking behavior, and alternative food sources for predators with selective predation. We extend our model to incorporate the impact of fluctuating environmental noise on system dynamics. The deterministic model undergoes thorough scrutiny to ensure the positivity and boundedness of solutions, with equilibria derived and their stability properties meticulously examined. Furthermore, we explore the potential for Hopf bifurcation within the system dynamics. In the stochastic counterpart, we prioritize discussions on the existence of a globally positive solution. Through simulations, we unveil the stabilizing effect of the fear factor on susceptible prey reproduction, juxtaposed against the destabilizing roles of prey refuge behavior and disease prevalence intensity. Notably, when disease prevalence intensity is too low, the infection can be eradicated from the ecosystem. Our deterministic analysis reveals a complex interplay of factors: the system destabilizes initially but then stabilizes as the fear factor suppressing disease prevalence intensifies, or as predators exhibit a stronger preference for infected prey over susceptible ones, or as predators are provided with more alternative food sources. Moreover, for the stochastic system, the oscillations tend to cluster around the coexistence equilibrium of the corresponding deterministic model when white noise intensity is low. However, with increasing white noise intensity, oscillation amplitudes escalate. Critically, very high levels of white noise can lead to the eradication of infection from the ecosystem. [ABSTRACT FROM AUTHOR]

Epigenetic changes are known to accrue in normal cells as a result of ageing and cumulative exposure to cancer risk factors. Increasing evidence points towards age-related epigenetic changes being acquired in a quasi-stochastic manner, and that they may play a causal role in cancer development. Here, I describe the quasi-stochastic nature of DNA methylation (DNAm) changes in ageing cells as well as in normal cells at risk of neoplastic transformation, discussing the implications of this stochasticity for developing cancer risk prediction strategies, and in particular, how it may require a conceptual paradigm shift in how we select cancer risk markers. I also describe the mounting evidence that a significant proportion of DNAm changes in ageing and cancer development are related to cell proliferation, reflecting tissue-turnover and the opportunity this offers for predicting cancer risk via the development of epigenetic mitotic-like clocks. Finally, I describe how age-associated DNAm changes may be causally implicated in cancer development via an irreversible suppression of tissue-specific transcription factors that increases epigenetic and transcriptomic entropy, promoting a more plastic yet aberrant cancer stem-cell state. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'. [ABSTRACT FROM AUTHOR]

Organisms act stochastically. A not uncommon view in the ecological literature is that this is mainly due to the observer having insufficient information or a stochastic environment—and not partly because organisms themselves respond with inherent unpredictability. In this study, I compile the evidence that contradicts that view. Organisms generate uncertainty internally, which results in irreducible stochastic responses. I consider why: for instance, stochastic responses are associated with greater adaptability to changing environments and resource availability. Over longer timescales, biologically generated uncertainty influences behavior, evolution, and macroecological processes. Indeed, it could be stated that organisms are systems defined by the internal generation, magnification, and record-keeping of uncertainty as inputs to responses. Important practical implications arise if organisms can indeed be defined by an association with specific classes of inherent uncertainty: not least that isolating those signatures then provides a potential means for detecting life, for considering the forms that life could theoretically take, and for exploring the wider limits to how life might become distributed. These are all fundamental goals in astrobiology. [ABSTRACT FROM AUTHOR]

Understanding how community assembly processes drive biodiversity patterns is a central goal of community ecology. While it is generally accepted that ecological communities are assembled by both stochastic and deterministic processes, quantifying their relative importance remains challenging. Few studies have investigated how the relative importance of stochastic and deterministic community assembly processes vary among taxa and along gradients of habitat degradation. Using data on 1645 arthropod species across seven taxonomic groups in Malaysian Borneo, we quantified the importance of ecological stochasticity and of a suite of community assembly processes across a gradient of logging intensity. The relationship between logging and community assembly varied depending on the specific combination of taxa and stochasticity metric used, but, in general, the processes that govern invertebrate community assembly were remarkably robust to changes in land use intensity. [ABSTRACT FROM AUTHOR]

ABSTRACT Dinoflagellates and diatoms are highly prevalent and ecologically important phytoplankton in coastal waters, greatly contributing to primary productivity in marine ecosystems. Although their composition and diversity have been extensively elucidated in the open ocean, their interaction patterns and community assembly in long-term eutrophic coastal waters remain poorly understood. This investigation aimed to elucidate the seasonal successional patterns of dinoflagellates and diatoms by 18S rRNA gene amplicon sequencing in a semi-enclosed bay. The results revealed that dinoflagellate and diatom communities have pronounced seasonal succession patterns, which are primarily associated with temperature. Furthermore, the most prevalent species throughout the year were Heterocapsa rotundata and Skeletonema costatum. Moreover, the assembly of dinoflagellate and diatom communities was mainly dominated by stochastic processes, with drift being the major factor. The co-occurrence of dinoflagellates and diatoms showed seasonal patterns, with the highest interactions observed in autumn. In addition, interactions of Syndiniales with dinoflagellates and diatoms highlighted the roles of parasites in eutrophic conditions. Flavobacteriaceae and Rhodobacteraceae are the bacterial taxa that most frequently interacted with dinoflagellates and diatoms, with interactions between dinoflagellates and bacteria being more complex than those between diatoms and bacteria. Overall, this study provides results that deepen our understanding of the phytoplankton dynamics in coastal eutrophic waters.IMPORTANCEDinoflagellates and diatoms are major phytoplankton groups in coastal waters. The composition and diversity of dinoflagellates and diatoms in the open ocean have been well documented; however, it remains uncertain to what extent their adaptation to long-term eutrophic conditions influences their response to environmental disturbances. Here, we investigated the interactions and assembly processes of dinoflagellates and diatoms in a eutrophic bay throughout the whole year. Our findings revealed that interactions between dinoflagellates and diatoms are primarily shaped by seasonal transitions, while prolonged eutrophic conditions tend to amplify stochastic processes in community assembly. These findings provide novel perspectives on the influence of long-term eutrophication on phytoplankton dynamics within eutrophic waters.

Abstract Stochasticity plays a significant role in the low‐power operation of a biological neural network. In an artificial neural network, stochasticity also contributes to critical functions such as the uncertainty quantification (UQ) for estimating the probability for the correctness of prediction. This UQ is vital for cutting‐edge applications, including medical diagnostics, autopilots, and large language models. Thanks to nonlinear variation of analogous Hall resistance with high computing velocity and low dissipation, a stochastic spin‐orbit‐torque (SOT) device exhibits significant potential for implementing the UQ. However, up until now, the application of UQ for stochastic SOT devices remains unexplored. In this study, based on SOT‐induced stochastic magnetic domain wall (DW) motion with varying velocity, a SOT synapse is fabricated that can emulate stochastic weight update following the Spike‐Timing‐Dependent‐Plasticity (STDP) rule. Furthermore, a stochastic SNN is set up, which, when compared to its deterministic counterpart, demonstrates a clear advantage in quantifying uncertainty for diagnosing the type of breast tumor (benign or malignant).

This article highlights current issues related to the problem of variability of wind power generation, which is becoming increasingly important as wind power’s contribution to the overall energy balance of many countries grows. Various aspects of wind flow volatility, including its random and deterministic components, are considered. The random component is associated with unpredictable changes in wind speed and direction, while the deterministic component is due to known patterns, such as daily and seasonal variations. The article proposes different methods for levelling the stochasticity of wind power flows. This is important for the stability and reliability of the power system, as it reduces the likelihood blackouts and ensures a more even distribution of load. The article also gives examples of the economic and environmental impact of using the proposed methods.

The Game of Life (GoL) is a paradigmatic computer simulation that exhibits the emergence of complex properties of the whole from relatively simple sets of heuristic rules operating at lower organizational levels. Therefore, it is widely understood as a valuable tool for investigating global properties of evolutionary processes. Cognizant of refined concepts that emerged in recent debates on the central role of historical contingency (historicity) in evolution, we modify the original GoL rules by introducing an updating feedback loop and a probability factor that reflect the degree of historicity in pattern evolution as both stochastic path dependence and sensitivity to initial conditions. We examine this trait in simulations of the emergence and breaking of bilateral and radial symmetries commonly observed in the evolution of life, most prominently as evolving body plans. We show that the implementation of historicity parameters leads to a more realistic sequential and gradual alternating emergence and the breaking of new symmetries than the original set of rules. Apart from its more realistic representations of evolutionary processes, the new approach allows for easier exploration of the parameter space, demonstrates the sequential and gradual emergence and breaking of symmetries, and provides a publicly available and modular simulation tool. We discuss the exploratory potential of the modified GoL platform we designed as an extended thought experiment to study the parameter space defining the historicity of biological processes.

Cognitive flexibility can enhance the ability to adjust to changing environments. Here, we use learning simulations to investigate the possible advantages of flexible learning in volatile (changing) environments. We compare two established learning mechanisms, one with constant learning rates and one with rates that adjust to volatility. We study an ecologically relevant case of volatility, based on observations of developing cleaner fish Labroides dimidiatus that experience a transition from a simpler to a more complex foraging environment. There are other similar transitions in nature, such as migrating to a new and different habitat. We also examine two traditional approaches to volatile environments in experimental psychology and behavioral ecology: reversal learning, and learning set formation (consisting of a sequence of different discrimination tasks). These provide experimental measures of cognitive flexibility. Concerning transitions to a complex world, we show that both constant and flexible learning rates perform well, losing only a small proportion of available rewards in the period after a transition, but flexible rates perform better than constant rates. For reversal learning, flexible rates improve the performance with each successive reversal because of increasing learning rates, but this does not happen for constant rates. For learning set formation, we find no improvement in performance with successive shifts to new stimuli to discriminate for either flexible or constant learning rates. Flexible learning rates might thus explain increasing performance in reversal learning but not in learning set formation, and this can shed light on the nature of cognitive flexibility in a given system. [ABSTRACT FROM AUTHOR]

Multivariate normal distribution is base for many statistical techniques, including ordinary least square inference. Here we show that in order to make research on Internationalization of Companies, more flexible approach is needed, namely partial least squares (PLS). It is a nonparametric technique, used in Structural Equation Modeling (SEM), which makes no distributional assumptions and also may be applied with small sample sizes. In this study we discuss on regularity conditions for PLS from the perspective of semi-continuous covariance which fills the gap in the current studies. The stochastic aspects, especially those related to usage of PLS-SEM, can be well integrated to the topologically grounded regression, where jumps in the covariances can occur. The purpose of the research is to analyze and understand the impact of the Incentive System (IS) for Innovation, within the scope of the National Strategic Reference Framework (QREN) 2014-2020, on the Internationalization of Portuguese Small and Medium Enterprises (SMEs). We study stochastic aspects of theoretical model which aggregates the variables Product Innovation, Marketing Innovation, Organizational Innovation and Working Conditions as determinants of Internationalization of Companies. Data were collected based on a quantitative methodology, through a self-completion questionnaire using the Likert psychometric scale, which registered 120 participants. Organizational Innovation (exogenous latent construct) and Product Innovation have shown a statistically significant indirect effect on the Internationalization of Companies (endogenous latent construct) through Marketing Innovation. The latter has a direct effect on the Internationalization (target construct). However, Working Conditions has the greatest impact on Internationalization, meaning that measures such as increasing wages, decreasing the use of temporary work and precarious work conversion into labor effective relations have a very relevant direct effect on the Internationalization of Portuguese SMEs. [ABSTRACT FROM AUTHOR]

The modern conditions of airport operation are considered from the point of view of reducing tariffs for services by reducing the costs of special equipment operation. The main attention is paid to the analysis of the influence of the costs of spare parts and materials used in its maintenance on the cost of special equipment use. The processes of special equipment utilization at the airport are considered as stochastic, and the logistics system at the airport is considered as a mass service system. It is taken into account that the stochasticity of the system is conditioned not only by the properties of the processes, but also by the randomness of the factors influencing them. In this regard, to manage the system under consideration, the list of factors that determine the intensity of resource consumption, as well as the regularities of their changes is analyzed. The main attention is paid to the necessity to find a compromise between uninterrupted supply, which requires an increase in the size of orders, and the reduction in the volume of stocks, which increases the cost of working capital of the aviation enterprise. The influence of climatic conditions of operation on the consumption of resources for the maintenance of special equipment at the airport is also analyzed. The analysis has shown that the determination of the need for resources for the maintenance of special equipment at airports is currently planned on the basis of the actual consumption for the previous year and it is not practically taken into account that the consumption varies seasonally. In this regard, it is noted that the relevance of research aimed at developing methods for planning the need of special transport services of modern airports in spare parts and consumables for maintenance, taking into account the variation in the intensity and operating conditions of special equipment. The analyzed approaches to the development of such methods show the need to consider the process of determining the need for spare parts and consumables for maintenance of special equipment as non-stationary, due to seasonal changes in the intensity and operating conditions. On this basis the basic principles of development of theoretical provisions and practical recommendations for determining the size of stocks of spare parts and consumables for maintenance of special equipment are proposed. [ABSTRACT FROM AUTHOR]

A recent car-following experiment (Experiment-I) showed that the speed standard deviation grows in a concave way along a platoon of vehicles following one another. This paper investigates the growth pattern of traffic oscillations in a platoon in another car-following experiment (Experiment-II) and compares it with that in Experiment-I. It is shown that the speed standard deviation also exhibits concave growth characteristics in Experiment-II. The hypothesis test show that there is no significant difference in growth pattern of speed standard deviation between the two experiments. However, since drivers are asked to follow closely in Experiment-II, the acceleration standard deviations, the driver sensitivities, and the strength of stochastic factors are larger than the ones in Experiment-I. Furthermore, a new comparison experiment is performed which indicates that traffic oscillation growth in the platoon in the set of experiments on a circular track is likely to be equivalent to that on a straight track. [ABSTRACT FROM AUTHOR]

Antiferromagnet (AFM) has currently participated in the spin‐orbit torque (SOT) technology due to its great potential to be applied to the field‐free SOT switching and to promote the thermal stability of MRAM. However, the effect of varying AFM ordering on the SOT switching and the associated properties is still not comprehensively understood. This work reports how an AFM ordering modifies the strength of Dzyaloshinskii–Moriya‐interaction (DMI) in a heavy metal (Pt)/FM (Co)/AFM (IrMn) trilayer and its effects on SOT switching. Increasing the AFM ordering reflects the enhanced exchange bias through increasing IrMn thickness appears to significantly reduce the DMI strength of the trilayer. Controlling the IrMn thickness appears to serve as a unique switch to activate memristivity/stochasticity in the devices via tailoring AFM ordering on exchange bias: The strong AFM ordering via increasing IrMn thickness enables to increase the stability of multi‐levels for SOT switching, which promotes the memristivity for neuromorphic application. On the contrary, the weak AFM ordering via reducing IrMn thickness will lead to significant stochasticity for the physically unclonable functionality. This work demonstrates an intrinsic tuning over the AFM ordering will serve as a switch to turn the SOT device into a stochastic/memristive cell to bridge probabilistic and neuromorphic computing. [ABSTRACT FROM AUTHOR]

Dysbiosis in the gut microbiota has been intimately implicated in shrimp (Litopenaeus vannamei, Penaeidae) diseases. However, considering the variety of shrimp diseases and the variability in experimental conditions, studies addressing common features of the gut microbiota−shrimp disease relationship are limited. Through an unbiased subject‐level meta‐analysis framework, 463 shrimp gut bacterial communities from 27 studies were re‐analysed, including six lifestages and eight diseases of shrimp, with the causal agents of viral, bacterial, eukaryotic, and unknown pathogens. Shrimp lifestages and diseases were the predominant factors governing the gut microbiota. After ruling out the top lifestage‐ and disease‐specific discriminatory amplicon sequence variants (ASVs) from the gut microbiota, the top 27 disease common‐discriminatory ASVs were identified, contributing to an overall accuracy of 95.9% in diagnosing shrimp health status. By using these optimisation procedures, the accuracy of our diagnosis model was unbiased by shrimp lifestage, specific disease, sampling size, hypervariable region and sequencing platform. The shrimp eight diseases consistently and significantly increased the relative importance of stochasticity, the relative abundance of pathogenic potentials and diversified core ASVs, whereas decreased the diversity and stability of gut microbiota. Collectively, these findings illustrate the effectiveness of meta‐analysis in determining the robust and common features of the shrimp gut microbiota in response to diverse diseases. In particular, disease common‐discriminatory ASVs could accurately diagnose shrimp health status, although the data are divergent in biotic and technical variances. [ABSTRACT FROM AUTHOR]

Despite the well known scale‐dependency of ecological interactions, relatively little attention has been paid to understanding the dynamic interplay between various spatial scales. This is especially notable in metacommunity theory, where births and deaths dominate dynamics within patches (the local scale), and dispersal and environmental stochasticity dominate dynamics between patches (the regional scale). By considering the interplay of local and regional scales in metacommunities, the fundamental processes of community ecology—selection, drift, and dispersal—can be unified into a single theoretical framework. Here, we analyze three related spatial models that build on the classic two‐species Lotka–Volterra competition model. Two open‐system models focus on a single patch coupled to a larger fixed landscape by dispersal. The first is deterministic, while the second adds demographic stochasticity to allow ecological drift. Finally, the third model is a true metacommunity model with dispersal between a large number of local patches, which allows feedback between local and regional scales and captures the well studied metacommunity paradigms as special cases. Unlike previous simulation models, our metacommunity model allows the numerical calculation of equilibria and invasion criteria to precisely determine the outcome of competition at the regional scale. We show that both dispersal and stochasticity can lead to regional outcomes that are different than predicted by the classic Lotka–Volterra competition model. Regional exclusion can occur when the nonspatial model predicts coexistence or founder control, due to ecological drift or asymmetric stochastic switching between basins of attraction, respectively. Regional coexistence can result from local coexistence mechanisms or through competition‐colonization or successional‐niche trade‐offs. Larger dispersal rates are typically competitively advantageous, except in the case of local founder control, which can favor intermediate dispersal rates. Broadly, our models demonstrate the importance of feedback between local and regional scales in competitive metacommunities and provide a unifying framework for understanding how selection, drift, and dispersal jointly shape ecological communities. [ABSTRACT FROM AUTHOR]

Relatively little is known about whole-plant fungal communities (mycobiome) and associated soil nematodes, especially with respect to woody plant seedlings and disturbance caused by forest harvesting. In a growth chamber experiment, we tested simulated clear-cut soil conditions on shoot biomass, total soil nematode density, and the shoot and root mycobiome of Douglas-fir, Pseudotsuga menziesii (Mirb.) Franco, seedlings. Soil treatments included unamended bare soil and soil amended with root segments of kinnikinnick, Arctostaphylos uva-ursi (L.) Spreng., pinegrass, Calamagrostis rubescens Buckley, or P. menziesii seedlings. We used next-generation Illumina sequencing and the PIPITS pipeline to obtain fungal taxa used for mycobiome community richness and Jaccard-based taxonomic normalized stochasticity ratio to assess mycobiome community assembly stochasticity. Total nematode density, measured from Baermann funnel extractions, increased in soils supplemented with A. uva-ursi or C. rubescens root segments. Root mycobiomes were more stochastic in the A. uva-ursi than P. menziesii or the bare conditions, whereas the shoot mycobiome was more stochastic in the C. rubescens treatment than in the P. menziesii treatment. Our results suggest that refugia plants impact the phyto-biome, in this case plant-associated nematodes and the stochasticity of root and shoot mycobiome community assembly, while not showing noticeable impacts on above-ground plant growth. [ABSTRACT FROM AUTHOR]

BACKGROUND: Individual lifespans differ. Some of those differences are due to heterogeneity, some to stochasticity. Some of the heterogeneity is due to socioeconomic, physiological, or environmental differences; some to unobserved latent factors. All of these are, from time to time, called inequality. OBJECTIVE: This paper aims to clarify the relations between heterogeneity, stochasticity, inequality of opportunity, and inequality of outcome in a wider context than has yet been attempted. METHODS: A population is divided into groups differing in their demographic rates. Markov chain or life table methods provide the moments of longevity for each group. A mixing distribution describes the relative abundance of groups. The variance in longevity is partitioned into within-group and between-group components. The approach applies to longevity, healthy longevity, lifetime reproductive output, and other outcomes. RESULTS: Important socioeconomic factors make only a small contribution to the variance in longevity, most of which is due to individual stochasticity. Some exceptions, in laboratory studies of insect populations and interspecies comparisons in biodemography, are explored. CONCLUSIONS: Important socioeconomic factors make only a small contribution to the variance in longevity, most of which is due to individual stochasticity. Some exceptions, in laboratory studies of insect populations and interspecies comparisons in biodemography, are explored. CONTRIBUTION: Recognizing the role of stochasticity clarifies the source and the implications of this important source of variance.

This article is an editorial discussing the topic of ecological modeling in population, community, and ecosystem dynamics. The authors explain that the purpose of the special topic was to gather a sample of modeling works to understand the current trends and research directions in the field. The articles covered a wide range of topics, including evolutionary dynamics, biogeochemical cycles, species interactions, and uncertainty. The authors highlight the underlying concern of uncertainty in the majority of the articles and emphasize the importance of understanding and including uncertainty in ecological theories. They conclude by encouraging readers to find inspiration for their own research from the published articles and to contribute to tackling uncertainty in ecological systems. [Extracted from the article]

The bioequivalence (BE) of highly variable drugs is a complex issue in the pharmaceutical industry. The impact of this variability can significantly affect the required sample size and statistical power. In order to address this issue, the EMA and FDA propose the utilization of scaled limits. This study suggests the use of generative artificial intelligence (AI) algorithms, particularly variational autoencoders (VAEs), to virtually increase sample size and therefore reduce the need for actual human subjects in the BE studies of highly variable drugs. The primary aim of this study was to show the capability of using VAEs with constant acceptance limits (80–125%) and small sample sizes to achieve high statistical power. Monte Carlo simulations, incorporating two levels of stochasticity (between-subject and within-subject), were used to synthesize the virtual population. Various scenarios focusing on high variabilities were simulated. The performance of the VAE-generated datasets was compared to the official approaches imposed by the FDA and EMA, using either the constant 80–125% limits or scaled BE limits. To demonstrate the ability of AI generative algorithms to create virtual populations, no scaling was applied to the VAE-generated datasets, only to the actual data of the comparators. Across all scenarios, the VAE-generated datasets demonstrated superior performance compared to scaled or unscaled BE approaches, even with less than half of the typically required sample size. Overall, this study proposes the use of VAEs as a method to reduce the necessity of recruiting large numbers of subjects in BE studies.

Abstract Phenotypic heterogeneity is very common in genetic systems and in human diseases and has important consequences for disease diagnosis and treatment. In addition to the many genetic and non-genetic (e.g., epigenetic, environmental) factors reported to account for part of the heterogeneity, we stress the importance of stochastic fluctuation and regulatory network topology in contributing to phenotypic heterogeneity. We argue that a threshold effect is a unifying principle to explain the phenomenon; that ultrasensitivity is the molecular mechanism for this threshold effect; and discuss the three conditions for phenotypic heterogeneity to occur. We suggest that threshold effects occur not only at the cellular level, but also at the organ level. We stress the importance of context-dependence and its relationship to pleiotropy and edgetic mutations. Based on this model, we provide practical strategies to study human genetic diseases. By understanding the network mechanism for ultrasensitivity and identifying the critical factor, we may manipulate the weak spot to gently nudge the system from an ultrasensitive state to a stable non-disease state. Our analysis provides a new insight into the prevention and treatment of genetic diseases.

With the development of the social economy, the importance of energy and environmental issues is becoming more and more prominent. Distributed power sources (DG) have the advantages of low carbon and environmental protection, flexible control, and low investment cost, but their nodes in the grid are random, which will increase the complexity and uncertainty of grid operation after grid connection. In this paper, we study the planning path of DGs in distribution networks and the protection technologies involved. Firstly, we focus on the impact of DG access on the protection of distribution networks and the impact of different access capacities and access locations of DG on the protection of distribution networks. Second, after studying the current status of the application of traditional genetic algorithms in distribution network fault location, a two-layer planning model based on the modified extreme learning machine algorithm (MPSO-ELM) is proposed, with the load demand guaranteed by the first layer of power allocation and the fault location incorporated into the distribution network protection scheme in the second layer. Finally, simulations are performed to compare and demonstrate the advantages of this MPSO-ELM algorithm. The simulation results show that the MPSOELM algorithm converges faster than the traditional genetic algorithm by 7s, the average operating speed of power nodes is 3.61ms faster than the traditional genetic algorithm by 5.59ms, and the accuracy in fault location is improved by 23.14%. The simulation results verify the comprehensiveness, accuracy, and high efficiency of the ELM algorithm for double-layer planning of distribution networks containing distributed power supplies, which is important for improving the level of power grid construction and avoiding the risk of power grid construction.

Abstract Antiferromagnet (AFM) has currently participated in the spin‐orbit torque (SOT) technology due to its great potential to be applied to the field‐free SOT switching and to promote the thermal stability of MRAM. However, the effect of varying AFM ordering on the SOT switching and the associated properties is still not comprehensively understood. This work reports how an AFM ordering modifies the strength of Dzyaloshinskii–Moriya‐interaction (DMI) in a heavy metal (Pt)/FM (Co)/AFM (IrMn) trilayer and its effects on SOT switching. Increasing the AFM ordering reflects the enhanced exchange bias through increasing IrMn thickness appears to significantly reduce the DMI strength of the trilayer. Controlling the IrMn thickness appears to serve as a unique switch to activate memristivity/stochasticity in the devices via tailoring AFM ordering on exchange bias: The strong AFM ordering via increasing IrMn thickness enables to increase the stability of multi‐levels for SOT switching, which promotes the memristivity for neuromorphic application. On the contrary, the weak AFM ordering via reducing IrMn thickness will lead to significant stochasticity for the physically unclonable functionality. This work demonstrates an intrinsic tuning over the AFM ordering will serve as a switch to turn the SOT device into a stochastic/memristive cell to bridge probabilistic and neuromorphic computing.

Animals from embryos to adults experiencing stress from climate change have numerous mechanisms available for enhancing their long-term survival. In this review we consider these options, and how viable they are in a world increasingly experiencing extreme weather associated with climate change. A deeply understood mechanism involves natural selection, leading to evolution of new adaptations that help cope with extreme and stochastic weather events associated with climate change. While potentially effective at staving off environmental challenges, such adaptations typically occur very slowly and incrementally over evolutionary time. Consequently, adaptation through natural selection is in most instances regarded as too slow to aid survival in rapidly changing environments, especially when considering the stochastic nature of extreme weather events associated with climate change. Alternative mechanisms operating in a much shorter time frame than adaptation involve the rapid creation of alternate phenotypeswithin a life cycle or a few generations. Stochastic gene expression creates multiple phenotypes from the same genotype even in the absence of environmental cues. In contrast, other mechanisms for phenotype change that are externally driven by environmental clues include well-understood developmental phenotypic plasticity (variation, flexibility), which can enable rapid, within-generation changes. Increasingly appreciated are epigenetic influences during development leading to rapid phenotypic changes that can also immediately be very widespread throughout a population, rather than confined to a few individuals as in the case of favorable gene mutations. Such epigenetically-induced phenotypic plasticity can arise rapidly in response to stressorswithin a generation or across a few generations and just as rapidly be "sunsetted" when the stressor dissipates, providing some capability to withstand environmental stressors emerging from climate change. Importantly, survival mechanisms resulting from adaptations and developmental phenotypic plasticity are not necessarily mutually exclusive, allowing for classic "bet hedging". Thus, the appearance of multiple phenotypes within a single population provides for a phenotype potentially optimal for some future environment. This enhances survival during stochastic extreme weather events associated with climate change. Finally, we end with recommendations for future physiological experiments, recommending in particular that experiments investigating phenotypic flexibility adopt more realistic protocols that reflect the stochastic nature of weather. [ABSTRACT FROM AUTHOR]

Neural phenotypes are the result of probabilistic developmental processes. This means that stochasticity is an intrinsic aspect of the brain as it self-organizes over a protracted period. In other words, while both genomic and environmental factors shape the developing nervous system, another significant--though often neglected--contributor is the randomness introduced by probability distributions. Using generative modeling of brain networks, we provide a framework for probing the contribution of stochasticity to neurodevelopmental diversity. To mimic the prenatal scaffold of brain structure set by activity-independent mechanisms, we start our simulations from the medio-posterior neonatal rich club (Developing Human Connectome Project, n = 630). From this initial starting point, models implementing Hebbian-like wiring processes generate variable yet consistently plausible brain network topologies. By analyzing repeated runs of the generative process (>107 simulations), we identify critical determinants and effects of stochasticity. Namely, we find that stochastic variation has a greater impact on brain organization when networks develop under weaker constraints. This heightened stochasticity makes brain networks more robust to random and targeted attacks, but more often results in non-normative phenotypic outcomes. To test our framework empirically, we evaluated whether stochasticity varies according to the experience of early-life deprivation using a cohort of neurodiverse children (Centre for Attention, Learning and Memory; n = 357). We show that low-socioeconomic status predicts more stochastic brain wiring. We conclude that stochasticity may be an unappreciated contributor to relevant developmental outcomes and make specific predictions for future research. [ABSTRACT FROM AUTHOR]

This experiment was undertaken to determine the adaptation of Lilium L. cultivars to North-Western part of Türkiye and to define morphologic plant growth characteristics at aboveground part, including flower life and flower abnormalities. The bulbs of the three Lilium L. cultivars, 'Profundo', 'Valverde', and 'Vonq', were sown to perlite in pots in June, 2021, and evaluated for emergence and flowering rates, stem and plant height, stem and flower diameter, leaf and flower number, flower abnormality rates, flower fragrance degree and flower life in 2022, after one year adaptation without removing the previous sown bulbs. As the results of the cultivars differences these were determined that, at the adaptive year; the emergence rates varied between 60.42-89.58%, flowering rates changed between 43.75-87.50%, stem heights changed between 40.63-48.25 cm, plant heights changed between 52.50-59.75 cm and these data showed statistically importance. In addition to the data in the adaptive year; stem diameters, leaf numbers, flower numbers flower fragrance degree and flower longevities also showed statistical importance between cultivars. The flower abnormality rates were found between 8.3-25% in cultivars although no statistical results. Interestingly a striking, stochastic case also was observed from a flower with 8 tepals, 8 stamen and 4 lobed stigma at 'Valverde' contrary to well-known flower formula. It has been concluded that 'Profundo' has more remarkable result in terms of longevity of flower compared to the other two varieties while 'Valverde' and 'Vonq' shared the same group with some precedence in most of the agro-morphologic characteristics e.g. plant height, stem height and stem diameter. Lilium L. 'Vonq' showed that the highest adaptive capacity among the cultivars with the highest emergence rate (89.58) in all sowed bulbs and flowering rate (87.50) in all emerged bulbs under the study conditions in the open-air. [ABSTRACT FROM AUTHOR]

The influence of mangrove systems in advancing the process of sedimentation is very well studied over the years. However, understanding the actual mechanism of accretion at a specific point within a mangrove forest is difficult, due to the stochasticity of sediment distribution and deposition. Factors that influence the rate of sediment accretion within the mangrove forest such as plant litter accumulation and bioturbation were investigated in Pichavaram mangrove forest, India. The annual rate of vertical accretion was recorded as 1.5 ± 0.9 cm year-1 for the site at elevation 0.4 m, and 0.9 ± 0.04 and 0.9 ± 0.05 cm year-1 for the other two sites which are at -0.8 and -1.2 m respectively. Results show that the sites were accreting sediments but at varying rates, inter-site variability was low but the factors influencing the processes were different. Plant litter contributed significantly to the accretion, and organic matter estimates also supported this observation. A higher rate of organic matter accumulation was observed in one of the sites, which is not regularly inundated by tides. On the contrary, in sites which were regularly washed by tides, accretion was seen in both allochthonous and autochthonous inputs and the annual rate of vertical accretion was almost the same. [ABSTRACT FROM AUTHOR]

Mathematical modeling in epidemiology, biology, and life sciences requires the use of stochastic models. In this paper, we derive a competitive two-strain stochastic SIR epidemic model by considering the change in state of the epidemic process due to an event. Based on the density-dependent process theory, we construct a six-dimensional deterministic model that can be used to describe the diffusion limit of the stochastic epidemic on a heterogeneous network. Furthermore, we show the explicit expressions for the variances of infectious individuals with strain 1 and strain 2 when the level of infection is increasing exponentially. In particular, we find that the expressions of the variances are symmetric. Finally, simulations for epidemics spreading on networks are performed to confirm our analytical results. We find a close agreement between the simulations and theoretical predictions. [ABSTRACT FROM AUTHOR]