Most environments favor defection over cooperation due to natural selection. Nonetheless, the emergence of cooperation is omnipresent in many biological, social, and economic systems, quite contrary to the well-celebrated Darwinian theory of evolution. Much research has been devoted to better understanding how and why cooperation persists among self-interested individuals despite their competition for limited resources. Here we go beyond a single social dilemma since individuals usually encounter various social challenges. In particular, we propose and study a mathematical model incorporating both the prisoner’s dilemma and the snowdrift game. We further extend this model by considering ecological signatures like mutation and selfless one-sided contribution of altruist free space. The nonlinear evolutionary dynamics that results from these upgrades offer a broader range of equilibrium outcomes, and it also often favors cooperation over defection. With the help of analytical and numerical calculations, our theoretical model sheds light on the mechanisms that maintain biodiversity, and it helps to explain the evolution of social order in human societies. Numerous interdisciplinary researchers have long sought a way to understand how the tremendous biodiversity among species persists in nature despite the significant differences between them in terms of competitive capability. Darwinian evolution  always cWe construct a mathematical model on evolutionary multigames –27 by adWhile defection is the ultimate profitable strategy in the classical PD game, irrespective of the co-player’s decision, a player’s best strategy in the SD game depends on the co-player’s decision. Thus, we select two different games with two contrasting outcomes. The stable coexistence of both cooperators and defectors is the expected consequence in the SD game, resulting in the persistence of cooperative behavior. In comparison, cheaters are always encouraged to exploit the cooperative individuals in the set-up of the PD game. In our constructed model, unrelated individuals have their probabilistic selection to choose which game they want to play. We also consider the mutation –42 as anApart from that, to reveal the interplay between ecological and evolutionary dynamics, we inspect the influence of free space in our mathematical model. In the context of complex systems, free space –49 proveThe concurrence of ecological changes and rational players’ evolution, depicted in the present article, prevails through the same time scale, rather than the typical assumption that ecological processes are much faster than evolutionary processes . Our theC) and (ii) defection (D). Even they can play any of the two possible games (a) PD game and (b) SD game. They can adopt the PD game with probability p, and alternatively, they interact with other individuals by playing the SD game with the complimentary probability (1 − p). Both of these two-person games can be given by the following two payoff matrices A and B respectively, whereTo start with, we consider a simplistic assumption that each individual has two distinct choices, viz. (i) cooperation , and SPD = 0. Similarly, we choose TSD = β > 1, RSD = 1, SSD = 0, and PSD = −η ∈ , maintaining the relative ordering TSD > RSD > SSD > PSD for the standard SD game  is the probability of playing the PD game. Moreover, we consider free space F as an ecological variable, which contributes altruistically by helping others. Nevertheless, free space does not get any benefits by giving them reproductive opportunities. We incorporate this charitable role of free space by extending the 2 × 2 payoff matrix E to the 3 × 3 payoff matrix G as followsHere, G clearly reveals that the free space never earns any payoff for their selfless charitable act; however, it contributes a positive payoff σ1 and σ2 to the cooperators and the defectors, respectively. Since most of the game parameters  lies within the closed interval , we assume, for the sake of feasible comparison, σ1 and σ2 both lies within the interval . When σ1 and σ2 are equal to zero, free space will not contribute anything to anyone. However, whenever σ1 and σ2 attain positive values, individuals gain an additional payoff from free space.The matrix tInspired by the Malthusianism, we consider the following set of differential equations governing the changes in frequencies of cooperators and defectors as a function of time x and y are the normalized densities of cooperators and defectors, respectively. Let z be the available free space. Thus, we haveHere, x and y alone, one can easily capture the dynamics of the two strategies. We assume the birth rates of each individual depends crucially on the available free space as well as on their respective average fitness. Thus, we considerfC and fD are average fitness of the cooperators and the defectors, respectively. The average payoff of cooperators and defectors can be determined using the payoff matrix G of the multigames, and the relation Relation Note that the average fitness of free space isξ ∈  must lie within the domain  ×  maintaining the inequality 0 ≤ x0 + y0 ≤ 1, as they represent the frequencies of the two strategies. To determine the positivity of the proposed model c1 and c2 are the integrating constants depending on the initial densities x0 and y0. This proves that both x and y are non-negative. Now, to calculate the upper bound of x + y, we proceed as follows. The dynamical Before investigating the model x + y ≥ 0 and the mortality rate ξ is always positive, we have ξ(x + y)≥0. Thus, Since as per our previous analysis, c3 is the initial density dependent constant. Hence, we find thatx + y eventually remains bounded within the region . This boundedness within the closed interval  allows us to relate the possible emerging dynamics of the system x + y) is precisely one, the available free space is zero, as z = 1 − (x + y). i.e., there is no reproductive opportunity accessible to any individual in that situation with z = 0. When (x + y) = 0, then individually x = 0 and y = 0. Hence, all individuals die, and z is equal to one. Thus despite the presence of ample free space, all the individuals are extinct under that circumstances.Integrating both sides, we getξ = 0.38, β = 1.1, η = 0.85, p = 0.30, σ1 = 0.36, σ2 = 0.25 and μ = 0 in x0, y0) within the interval  ×  maintaining the inequality (x0 + y0)∈. We find that the dynamics switching between two stationary points, viz. E0 =  and E1 = . We analytically calculate the stationary points of the system μ = 0. We trace out four different stationary points,Next, we point out the multistable dynamics of our model E0 =  reveals all individuals die. This point is locally stable whenE1 =  exhibits a society free from any defectors. Here,x* ∈  represents that we have only left with defectors. Here, y* ∈  indicates the coexistence equilibrium offering the survival of cooperation and defection simultaneously. Here, x** satisfies the equation, x** and y** both should lie within the interval . The local stability yields the conditions for the stability of  areE0 and E1. Note that E1 leads to two different values for the selected values of the parameters, out of which  always remains locally stable, and  is unstable. Thus, we see the appearance of two different stationary points E0 (shown by red points) and E1 (displayed by violet points) in the basin of attraction portrayed in x0, y0) =  always helps the system E0. The system is never able to give rise to the survival of any cooperators and defectors without the presence of any individual at the beginning. Since the chosen initial point  itself is the stationary state, the system always stabilizes in the  stationary point irrespective of the parameter values. ξ = 0.15, β = 1.06, η = 0.85, p = 0.69, σ1 = 0.75, σ2 = 0.25, and μ = 0. Similarly, we find the system μ = 0) converges to E0 for the single initial condition  = . It is anticipated that the choice of x0 = 0 always leads to the cooperator-free steady state. The initial absence of cooperators in the mutation-free model will not entertain any exposure for the cooperators in the long run. The line of initial conditions x0 = 0 and y0 ≠ 0 produces the stationary state E2 (sea blue points in E1 (violet points) and E3 (yellow points) depending on the suitable choice of initial conditions  to   as follows,c4 is the integrating constant.Clearly, the chosen parameters satisfy the local stability criteria of both stationary points ions See . Thus, fσ1 = σ2 = μ = 0 in the absence of cooperators  as followsc5 is the integrating constant.Similarly, the system δx0 = δy0 = 0.01 and maintaining 0 ≤ x0 + y0 ≤ 1. To solve our proposed system 5 iterations with fixed integration step length δt = 0.01. The final point  is stored to decide the asymptotic dynamics of the system. Other initial conditions with y0 = 0 stabilize the dynamics in the defector-free state E1 (violet points). We also track a fair portion of the basin in μ = 0 converges to either E2 (sea blue points) or E3 (yellow points) depending on the choice of initial conditions.μ in ξ = 0.30, p = 0.30, β = 1.1, η = 0.85, μ = 0.02, σ1 = 0.30, and σ2 = 0.20. This non-zero μ leads to the disappearance of two stationary points, (i) the defector-free steady state E1, and (ii) the cooperator-free steady state E2. The symmetric mutation from one species to another species always gives them two feasible opportunities. Either both strategies survive or all the individuals perish for μ ≠ 0. We also mathematically derive two possible stationary points as follows,Although we choose the mutation-free model for the E0 = , which is stable under the conditions,The extinction equilibrium E3 =  becomes stable, ifThe interior equilibrium The compelling evidence of bistability under the same choice of parameters’ values is recognized in p, β, η, and ξ on the system μ = 0 in σ1 = σ2 = 0 in subfigures (a), (c) and (e). Also, we set σ1 ≠ 0 and σ2 ≠ 0 for the subfigures (b), (d) and (f). In the absence of mutation (μ = 0) and free space induced benefits (σ1 = σ2 = 0), the defector-free E1 exists for We examine the impact of the parameters β > 1 and ξ > 0, λ2 is always positive. Consequently, E1 is always unstable. This result is physically meaningful, as both the chosen games (PD and SD) do not encourage a defection-free society without any supporting mechanism. However, we find three different steady states in Since E0 =  (red points in subfigures (a), (c) and (e) of The extinction equilibrium E2 =  is the cooperator-free stationary point (sea blue points in subfigures (a), (c) and (e) of ξη(2p − 1)≤η2(2p − 1)2, and becomes stable if ξ > η(2p − 1)(2y** − 3y**2).E3 =  allows the coexistence of both cooperators and defectors (yellow points in subfigures (a), (c) and (e) of p, players are prone to play the PD game. Hence, we can not anticipate the survival of cooperators for large p. Thus, for large p, either the extinction equilibrium E0 or the cooperator-free stationary point E2 stabilizes in E3. Even, p − ξ parameter space, where the defector-free stationary point E1 stabilizes. The comparative study between the left and right column of With increasing β, the defectors are getting extra aid, and thus, we get the stabilization of the extinction equilibrium E0 (red zone) in p in both subfigures (a-b) permits only the existence of defectors, as people are playing mostly the PD game in such circumstances. Defectors are getting a favorable atmosphere in the PD game for our chosen PD game parameters’ values. A similar sort of stabilization of cooperator-free steady state (sea blue area) is observed in subfigures (c-d) for larger values of p. However, lower values of p provide the opportunity for playing the SD game. Hence, coexistence equilibrium (yellow region) stabilizes in p. Nevertheless, the inclusion of free space induced benefits helps to broaden the region of coexistence in the p − η parameter plane, as shown in σ1 and σ2 entertain the stabilization of a defector-free steady state (violet region). Hence, we can trace a fair portion of violet points in the p − ξ parameter space of the subfigure (f). However, too large a mortality rate reduces the opportunity of survivability of any individual, resulting in the stabilization of the extinction equilibrium E0. We focus on the effect of mortality rate ξ more elaborately in With increasing ξ constantly enlarge the chances of extinction. Thus, we observe a fair portion of the red region in σ1 and σ2 in the right column of this figure. These free space induced benefits encourage maintaining a defector-free society, as shown in σ1 = 0.75 > 0.25 = σ2 for p = 0.3 in ξ provide an opportunity to coexist for all strategies, which is observed in x0, y0) = . We plot the stability curve of all stationary points in Figs As expected, higher values of E2,(i) The brown dotted line for the cooperator-free steady state E1,(ii) the blue dashed line for the defector-free steady state E3, and(iii) the solid blue line for the interior equilibrium E0.(iv) the solid white line for the extinction equilibrium σ2 at 0.25 and examine the role of σ1. Since σ1 represents the free space induced benefits towards the cooperators, thus enhancement of its (σ1) value will help in promoting cooperation in the society. β’s value challenges the prevalence of cooperation. Thus, for a small σ1, we find a defector dominated society  stabilizes. For the intermediate choice of σ1, the stationary point E3 (yellow) yields the stable coexistence of both strategies. However, cooperators strive to keep in existence for the lower values of σ1, and we find the sea blue region of cooperator-free steady state. The parameter p indicates the probability of playing the PD game. Thus, p → 1− always gives defectors a more favorable environment to survive. That’s why we spot a sea blue portion in p and σ1. When p is small, people are prone to play the SD game; and thus, we get the coexistence of both strategies  in ξ always results in the extinction of both cooperators and defectors, and we locate a huge red region in the σ1 − ξ parameter plane of σ1 always promotes the cooperation strategy, and we obtain a violet zone of defector-free stationary point and a yellow region of interior equilibrium E3 in In σ2 − β parameter space produces the coexistence of both cooperators and defectors depending on other parameters’ values. Despite the increment of σ2 and β, the cooperators are able to survive along with the defectors due to our choice of other parameters’ values. The σ2 − η parameter space portrays that smaller values of σ2 can not provide any benefit to the defectors, and stabilize the defector-free stationary point E1 (violet region) irrespective of η’s value. However, if σ2 increases, it will yield a window of opportunity for the defectors to thrive. We observe a wide coexistence region (yellow region) and a small area of cooperator-free stationary point (sea blue region)in σ2 indicates the free-space induced benefits towards the defectors. So, it is expected that larger values of σ2 always enhance the chances of defectors’ survivability. Thus, we notice a cooperator-free region (sea blue zone) in both p enhance the chances of playing the PD game, where defectors get a favorable environment to survive. Thus, the cooperator-free sea blue region is found in σ2 − p parameter space. However, we trace a healthy portion of coexistence (yellow) too in μ ∈  in σ1 increases, the cooperators are getting a better environment for survival. We find a portion of stable coexistence equilibrium in each two-dimensional parameter space for large σ1 in x0, y0) = . There are a few places where the stability analysis fails to predict the stabilization of the extinction stationary point . This is mainly due to the multistable behavior of the proposed model Now we investigate the influence of bidirectional mutation on the long-term behavior of the nonlinear differential β, η, and p always provide the defectors a favorable environment to dominate the cooperators. However, beyond a critical value of σ1, both the cooperators and defectors can coexist, as shown in ξ always hinder the evolution of cooperators as well as of defectors. However, an intermediate choice of σ1 − ξ favors the successful evolution of both strategies, as portrayed through σ2, the defectors are not getting enough advantages to survive, and thus the extinction equilibrium  stabilizes in the red region of all subfigures of σ2 benefits the defectors; hence the defectors can survive beyond a certain threshold of σ2. The employed 50% mutation rate helps to flow a certain fraction of defectors into cooperators, and we have a moderate portion of coexistence state (yellow region) in 5 times with fixed integrating step length δt = 0.01. The last point is gathered to finalize the asymptotic state. All the codes to generate these figures are publicly available at Ref. . In fact, the cooperators’ density x is slightly better than that of y for the higher values of σ1 in σ2 in our proposed model σ2, the defectors will be benefited from free space, and hence, y (blue line) dominates x (red line) for larger values of σ2. But since σ1 is taken sufficiently large in this σ2 where x > y. Since the mutation rate Our mutation-induced model provides widespread coexistence of both strategies under suitable choices of parameters’ values. Such stable persistence of ecological communities strengthens the theory of concurrency. Our thorough analysis with several numerical simulations enhances our understanding of the mechanisms that drive the survival of cooperative behavior in the multigames consisting of both the Prisoner’s Dilemma and the Snowdrift games. The inclusion of altruistic behavior of free space along with the mutation in our proposed evolutionary model seems to be a natural course of action as observed in many realistic settings. Our findings clearly demonstrate that there exists an optimal probability of playing each game so that people are more likely to cooperate in such circumstances. Throughout the study, we have pointed out the positive impact of diverse factors (parameters) on significant improvement of cooperation. We have shown that the selfless contribution of free space promotes the coexistence of all strategies efficiently, even in the absence of mutation.In summary, our research indicates the proposed model Lastly, we discuss challenges related to our work that merits further investigation. Even though it is almost impossible to incorporate all complex, realistic relationships among social creatures using minimal modeling, it is nevertheless absolutely essential to investigate such models using elementary mathematical principles. In fact, the literature already provides a number of excellent models that offer a stimulating starting point for exploring many practical situations. Chen et al.  proposedOne should observe that our approach works absolutely fine with other social dilemmas. For any two two-person games of the form ony game , 76 . 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Is the manuscript presented in an intelligible fashion and written in standard English? </font> PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.Reviewer #1: YesReviewer #2: Yes********** <font color="black"> 5. Review Comments to the Author </font> Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. Reviewer #1: In their paper titled "Eco-evolutionary dynamics of multigames with mutations" the authors study an interesting problem, namely what are the options for multigames to yield favorable conditions for cooperation subject to mutations and selfless one-sided contribution of altruist free space. Research reveals a broader range of equilibrium outcomes, which often favor cooperation over defection. With the help of analytical and numerical calculations, the theoretical model the authors are proposing and studying sheds light on the mechanisms that maintain biodiversity, and also explain the evolution of social order in human societies.Understanding cooperation in social dilemmas is a long standing problem of universal interest across social and natural sciences, and new results in this research field are always well received. In this way, the current manuscript is well-suited for PLOS ONE, and I am sure it will attract citations and readers when published.I find the manuscript is clear and written with love to detail and care for presentation. The figures are also high quality and should be accessible to the wider audience, as are the results and the research in general. For these reasons I recommend publication in PLOS ONE subject only to minor revision.The following comments can be considered:1. Would the approach also work on other social dilemmas games, such that for example the snowdrift and the harmony game would be considered as multigames? And perhaps even more interestingly, would it work in group interactions. The authors can discuss this in the final section if they find of interest.2. Can the caption to Table 1 be made more comprehensive to say what is the meaning of physical significance and domains.3. More relevant works, such as  and [Physica A 591 (2022) 126804], can be cited to make the introduction more comprehensive.4. The writing is occasional burdened with somewhat awkward grammar and could be polished further before resubmission.5. Finally, perhaps the authors can make source available to allow others to reproduce the results.Reviewer #2: The manuscript is generally well written and easy to understand. The authors have proposed a math model that incorporates both prisoner’s dilemma and the snowdrift game. In my opinion, this is quite interesting to see the combination of these two theories. THe authors further extend this model by considering ecological signatures like mutation and selfless one-sided contribution of altruist free space. Their conclusion is that all of these offer a broader range of equilibrium outcomes, and it also often favors cooperation over defection. Their conclusion is well supported by their analytical derviations and numerical calculations. I have some minor queries:1) How is the steady state in fig 1 being determined?2) the manuscript is well written, and it will benefit the readers more if there is a section on the limitations and how these limitations can be overcome.3) I am happy to see that the code has been archived and made available. Can the readers include a section on future work to make the study more comprehensive?The manuscript can be readily accepted after the above minor queries are addressed.********** <font color="black"> 6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.If you choose “no”, your identity will remain anonymous but your review may still be made public.Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy. </font> Reviewer #1: NoReviewer #2: No**********https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step. </div> While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool,  22 Jul 2022------------------------------------------------------------------------------------Reply to First Reviewer’s commentsPONE-D-22-17613------------------------------------------------------------------------------------.Reviewer #1 (Remarks to the Author): In their paper titled "Eco-evolutionary dynamics of multigames with mutations" the authors study an interesting problem, namely what are the options for multigames to yield favorable conditions for cooperation subject to mutations and selfless one-sided contribution of altruist free space. Research reveals a broader range of equilibrium outcomes, which often favor cooperation over defection. With the help of analytical and numerical calculations, the theoretical model the authors are proposing and studying sheds light on the mechanisms that maintain biodiversity, and also explain the evolution of social order in human societies.Understanding cooperation in social dilemmas is a long standing problem of universal interest across social and natural sciences, and new results in this research field are always well received. In this way, the current manuscript is well-suited for PLOS ONE, and I am sure it will attract citations and readers when published.I find the manuscript is clear and written with love to detail and care for presentation. The figures are also high quality and should be accessible to the wider audience, as are the results and the research in general. For these reasons I recommend publication in PLOS ONE subject only to minor revision.Response: We greatly acknowledge the referees for appreciating our work and gratefully acknowledge their insightful suggestions that helped considerably improve the manuscript. As far as the clarifications part is concerned, we have gone through every effort to improve the same in the revised version of the manuscript.Q.1. Would the approach also work on other social dilemmas games, such that for example the snowdrift and the harmony game would be considered as multigames? And perhaps even more interestingly, would it work in group interactions. The authors can discuss this in the final section if they find of interest.Response: We acknowledge the referee not only for appreciating our work but also for careful reading and thoughtful comments, which help to improve the manuscript in a significant way. The model formed in the manuscript with the help of two different two-person games can be formulated using any two-person games, not necessarily with the prisoner's dilemma and the snowdrift game only. The solution's existence, uniqueness, and boundedness can be similarly guaranteed using the calculations shown in the manuscript for any two different two-person games. We have checked it with two other two-person games, and if the reviewer advises, we will include the generic calculations in the manuscript. However, one should note that the stationary points alter in those cases as different games create a mathematical model with other functions. We strongly believe the role of mutation and the generous free space remain almost similar even for multigame with other two-person games. Incorporating group interaction leads to a potentially exciting problem that is out of scope for the present manuscript. We discuss how one can include group interaction by considering interdependent networks and games like the public goods game in the final section of the revised manuscript, as suggested by the anonymous reviewer.Q.2. Can the caption to Table 1 be made more comprehensive to say what is the meaning of physical significance and domains.Response: We have enhanced the caption of table 1 by briefly describing the meaning of physical significances and domains.Q.3 More relevant works, such as  and [PhysicaA 591 (2022) 126804], can be cited to make the introduction more comprehensive.Response: The references suggested by the referee concerning the evolution of cooperation in the presence of eco-evolutionary dynamics are indeed noteworthy in the context of our study. We have cited these articles in the discussion section, where a series of relevant publications are introduced.Q.4. The writing is occasional burdened with somewhat awkward grammar and could be polished further before resubmission.Response: In the revised manuscript, we have tried our best to improve the grammatical errors throughout our work.Q.5. Finally, perhaps the authors can make source available to allow others to reproduce the results.https://github.com/SayantanNagChowdhury/The-evolutionary-dynamics-in-ecological-multi-games-involving-mutations.Response: The data that support the findings of this study are openly available online on GitHubat-----------------------------------------------------------------------------------Reply to Second Reviewer’s comments -PONE-D-22-17613------------------------------------------------------------------------------------Reviewer #2 (Remarks to the Author):The manuscript is generally well written and easy to understand. The authors have proposed a math model that incorporates both prisoner’s dilemma and the snowdrift game. In my opinion, this is quite interesting to see the combination of these two theories. THe authors further extend this model by considering ecological signatures like mutation and selfless one-sided contribution of altruist free space. Their conclusion is that all of these offer a broader range of equilibrium outcomes, and it also often favors cooperation over defection. Their conclusion is well supported by their analytical derviations and numerical calculations. I have some minor queries:Response: We are thankful to the reviewer for appreciating our work and also for valuable suggestions.Q.1 How is the steady state in fig 1 being determined?Response: To determine the steady states of Figure 1, we have iterated the constructed model numerically with 2×〖10〗^6 iterations and a fixed integrating step length of 0.01. We used the 4th order Runge–Kutta method (RK4) for the numerical simulation. If the values of x and y remain greater than 0.0001, then we store those values as non-zero values. Otherwise, whenever x and y remain below 0.0001, we consider those values zero. Although we prove the values of x and y remain analytically within the closed interval , we still maintain that constraint in the numerical code too and store the values only when they stay bounded within this interval .Q.2. the manuscript is well written, and it will benefit the readers more if there is a section on the limitations and how these limitations can be overcome.Response: We greatly acknowledge the referee for appreciating our work. As per your valuable suggestion, we have added a section discussing the limitations and future perspectives. We also try to emphasize how one can overcome those limitations.Q.3. I am happy to see that the code has been archived and made available. Can the readers include a section on future work to make the study more comprehensive?Response: We acknowledge the referee not only for appreciating our work but also for careful reading and thoughtful comments, which help to improve the manuscript in a significant way.We have added a small section in the revised manuscript emphasizing the future works worth studying and need to be addressed in the future.AttachmentResponse to Reviewers.pdfSubmitted filename: Click here for additional data file. 26 Jul 2022Eco-evolutionary dynamics of multigames with mutationsPONE-D-22-17613R1Dear Dr. Ghosh,We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.Within one week, you’ll receive an e-mail detailing the required amendments. 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Nature Communications 10.1038/s41467-022-30079-z, published online 09 May 2022Correction to: The PDF version of this article contained an error in equation 12. The M matrix in equation 12 was incorrectly formatted as a row vector where they should form a square matrix.This has to be corrected only in the PDF version of the article, the HTML version is Ok.How it appears now:How it should appear:


G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle X with compatible actions of a reductive complex group on E, X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka.We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group  This is a survey of recent work, published since 2015, on equivariant Oka theory, mainly from our papers . Oka theOka theory has its roots in the pioneering work of Kiyoshi Oka. The Oka principle first appeared in his 1939 result that a holomorphic line bundle on a Stein manifold is trivial if it is topologically trivial. Oka theory was developed much further in the late 1950s to early 1970s, starting with Grauert’s foundational papers –10. OtheG on an affine variety X and proved his famous slice theorem. A consequence is that the quotient variety X is smooth, the map of X to G-fibre bundle over each stratum.A very brief review of the geometric invariant theory relevant here starts with the foundational 1973 paper of Luna . He studThe holomorphic version of the theory, for a reductive complex Lie group acting on a Stein space, was developed by Snow  and HeinMost of the work surveyed here can be summarised in seven main results, Theorems A–G. The following sections provide further details, relevant definitions, brief sketches of proofs, and other related results. In the final section, we list some open problems.G act holomorphically on a Stein manifold X. The categorical quotient G-orbits in X. Let G-finite holomorphic functions .Let a complex reductive group entclass1pt{minimaX and Y be Stein G-manifolds. If X and Y are G-biholomorphic, then X and Y have a common stratified quotient Q and G-biholomorphisms X and Y are locally G-biholomorphic over Q. Now our equivariant Oka problem is to see what kind of continuous or smooth G-isomorphism of X and Y implies that there is a G-biholomorphism.In Sects. G-diffeomorphism Q is called strict if it induces a biholomorphism between strongG-homeomorphism is somewhat involved and will be given in Sect. G-homeomorphism restricts to a G-biholomorphism q. A strict G-diffeomorphism is not necessarily a strong G-homeomorphism 18]). It  Schwarz  for \docchebauch . They shG be a reductive complex Lie group. Let X be a Stein G-manifold and V a G-module with common quotient Q. Suppose that V  is large or that X and V are locally G-biholomorphic over Q. Then X and V are G-biholomorphic.Let Note that this result does more than give a solution to the linearisation problem: it provides a sufficient condition for a Stein manifold to be biholomorphic to affine space.X be a reduced Stein space, G a complex Lie group, and A a complex Lie subgroup of the automorphism group of G. Then given a cocycle on X with values in A, we can produce a holomorphic group bundle X whose fibres are  isomorphic to G. As usual, trivial if it is isomorphic to E be a homogeneous holomorphic X, so E over X such that the action of each fibre of E is transitive. Ramspott proved that  Assume that X, E, and X and action of E. Then we say that X and that E is a holomorphic X. The next main theorem is an equivariant version of Ramspott’s theorem, with the stronger conclusion that the inclusion is a weak homotopy equivalence.If E be a homogeneous holomorphic X,  where K is a compact real Lie group whose complexification X,  and X. Then the inclusion of the space of E over X into the space of K-equivariant continuous sections is a weak homotopy equivalence.Let K-equivariance and K-equivariance is an appropriate condition on continuous sections; Note that for holomorphic sections, E is a holomorphic principal E). The other special case is the “uncoupled” case. It is a parametric Oka principle for equivariant maps from a Stein G/H, where the G/H can be quite general . NamelyTheorem X be a reduced Stein space and Y be a compact complex homogeneous space. Let K be a compact real Lie group whose complexification X and Y. Then the inclusion of the space of X to Y into the space of K-equivariant continuous maps from X to Y is a weak homotopy equivalence.Let For the basic theory of compact complex homogeneous spaces, we refer the reader to , Chap. 3Theorem K be a compact Lie group. Suppose that X and on a holomorphic group bundle X.Let K-X is topologically K-isomorphic to a holomorphic principal X.(a) Every topological principal X. Every continuous K-isomorphism K-isomorphism. In fact, the inclusion of the space of holomorphic K-isomorphisms K-isomorphisms is a weak homotopy equivalence.(b) Let X to an Oka manifold Y can be deformed to a holomorphic map. To adapt this result to actions of a reductive group G on X and Y, we need to find the right notion of Y being G-Oka. We define Y to be G-Oka if, for every reductive closed subgroup H of G, the submanifold H is Oka in the usual sense. E is a topological principal G-bundle over Q, then it has a holomorphic structure.If (G3)E and G-bundles which are continuously isomorphic, then they are holomorphically isomorphic.If In fact, more is true. (G4)G-bundles over Q, then there is a homotopy G-bundles with If Returning to our more general case where X and Y are not necessarily principal G-bundles, we want to find some analogue of the sheaf of groups Now suppose that G-fibre bundles over the strata of Q and, moreover, the fibre of each stratum S is not usually a group or even a homogeneous space.Our problem here is more complicated than in Grauert’s case since U be open in Q. Then we have the G-finite functions f of G-module. On a fibre G-finite functions G-submodule G-biholomorphism Q, induces algebraic G-isomorphisms of the fibres G-bundles are G-diffeomorphisms or G-homeomorphisms which behave reasonably on G-finite functions or on fibres.Let For G-diffeomorphism inducing the identity on Q. We say that strict if for all Let The definition is from ; in 20]20] we reX and Y be Stein G-manifolds locally G-biholomorphic over Q. Let G-diffeomorphism. Then there is a continuous deformation of G-diffeomorphisms, to a G-biholomorphism.K-invariaK-invariant strictly plurisubharmonic exhaustion function on X,  whose Kempf–Ness set R is a K-equivariant continuous strong deformation retract of X,  such that the deformation preserves the closure of each , p. 341 The following equivariant version of the covering homotopy theorem is used to prove an important fact about the Kempf–Ness set  By (a), we may take P is isomorphic to a constant bundle. Hence, once we identify the bundles K-bundle X,  the topological principal K-Let [As a consequence we have the following result.K acts real analytically on a Stein space X by biholomorphisms. Let G be a complex Lie group and X with fibre G,  whose structure group A is a Lie subgroup of the Lie automorphism group of G. Let K act continuously on A-bundle maps.[E be a topological K-X (not necessarily homogeneous). The restriction map from the space of continuous K-sections of E over X to the space of continuous K-sections of E over R is a homotopy equivalence.Let R to be a Kempf–Ness set as defined above. Next comes the central notion of an NHC-section.In the following, we take C be a compact Hausdorff space and C, such that N is a strong deformation retract of C. We define a sheaf V of K-equivariant NHC-sections of NHC-section of W, we mean a continuous map for every for every for every The topology on Let théorème principal [Now we formulate the relevant results from , first trincipal , p. 105.The topological group U is Runge in If , p. 324 K-K-bundles in [Next we state Heinzner and Kutzschebauch’s main result on the classification of principal ndles in ).K-X is topologically K-isomorphic to a holomorphic principal X.Every topological principal X. Let c be a continuous K-equivariant section of R. Then there is a homotopy of continuous K-equivariant sections R,  such that K-equivariant isomorphism from Let [P be the holomorphic principal E. Take a continuous K-section s of E over X. The preimage in P of its image in E is a topological principal K-Q of P. We have a topological K-isomorphism Q is topologically K-isomorphic to a holomorphic principal K-isomorphism First we prove that the inclusion K-isomorphism K-isomorphism over X. Applying the deformation to Q through topological principal K-P to a holomorphic principal E yields a deformation of s through continuous K-sections of E to a holomorphic section.By Theorem B be the closed unit ball in Now let P. We seek a global K-equivariant NHC-section E, maps X if R if Consider the holomorphic group Claim. On a sufficiently small saturated neighbourhood of each point of X, that is, locally over NHC-section exists.The proof of the claim is quite involved. It requires a detailed analysis of the equivariant local structure of the bundles involved. We will not attempt a summary, but refer the reader to .NHC-sections differ by a K-equivariant NHC-section of the holomorphic group P that fix NHC-sections together to produce On the intersection of two such saturated neighbourhoods, two such K-equivariant NHC-sections E obtained by letting R. Thus, we have a deformation B to By Theorem X. The problem is that the extension of continuous sections using the strong deformation retraction of X onto R does not give back the holomorphic sections for parameters in At this stage the continuous sections are defined over the Kempf–Ness set only, whereas the holomorphic sections are defined over all of  In fact, it suffices to show that The proof would now be done if we could show that the following commuting square has a continuous lifting. G-Oka manifold. Here G is a reductive complex Lie group. Let K be a maximal compact subgroup of G. It is natural to say that a G-manifold Y has the basic G-Oka property (G-BOP) if every continuous K-map from a Stein G-manifold X to Y can be deformed through such maps to a holomorphic map.We begin by motivating the definition of a Y satisfying G-BOP. First, if the G-action on X is trivial, then a K-map X to the submanifold Consider the following consequences of L be a closed subgroup of K. The complexification of L is a reductive closed subgroup H of G. Let X be a Stein H-manifold and consider the adjunctionY viewed as an H-manifold, G-manifold H-action X and Y. We conclude that if Y satisfies G-BOP, then Y also satisfies H-BOP, so by the above, Second, let Approximation and interpolation can easily be included in the above and we are led to the following definition.G act holomorphically on a manifold Y. We say that Y is G-Oka if the fixed-point manifold H of G.Let a reductive complex Lie group H to be the trivial subgroup, we see that a G-Oka manifold is Oka. On the other hand, the following example shows that an Oka G-manifold need not be G-Oka, even for Taking W of the involution X is smooth, given by the formula f such that W is Oka, then our promised example is f for which we have determined whether W is Oka or not.)[ore, Oka . The fixThe expected basic properties of the equivariant Oka property are easily established straight from the definition or from basic properties of Oka manifolds.G act holomorphically on a complex manifold Y. G acts trivially on Y,  then Y is G-Oka if and only if Y is Oka.If Y is G-Oka and H is a reductive closed subgroup of G,  then Y is H-Oka with respect to the restriction of the action to H.If If G-Oka, then G-Oka with respect to the diagonal action.If G-retract of a G-Oka manifold is G-Oka.A holomorphic Y is the increasing union of G-Oka G-invariant domains, then Y is G-Oka.If [Here are three ways to construct new equivariantly Oka manifolds from old. The first two are from . The locG be a complex reductive group and let H be a reductive closed subgroup of G. Y is an H-manifold, then G-action is G-Oka if and only if Y is H-Oka.If F. Assume that Y,  Z,  and F are G-manifolds and G-equivariant. Further assume that Z is Stein and F is G-Oka. Then Y is G-Oka if and only if Z is G-Oka.Let Y is covered by G-Oka G-invariant Zariski-open subsets, then Y is G-Oka. If Let The first main theorem of  is TheorG is finite, the Luna strata in Q, where the subvariety k. Each difference G-maps from a given map Since Step 1. Suppose that we have a homotopy of G-maps, to a holomorphic G-map G-invariant neighbourhood of X.G-module. The holomorphic map extends by Cartan’s extension theorem followed by averaging. Finally, the homotopy extends since Y is an absolute neighbourhood retract in the category of metrisable G-spaces.For Step 2. There is a homotopy of G-maps, to a holomorphic G-map This step may be reduced to an application of Forstnerič’s Oka principle for sections of branched holomorphic maps ([G-Oka property. It is the property ascribed to the G-Oka manifold Y in part (a) of Theorem Y in part (b) is called the basic G-Oka property with interpolation (G-BOPI), and the property ascribed to Y in part (c) is called the basic G-Oka property with approximation and jet interpolation (G-BOPAJI). The definition of the basic G-Oka property with approximation (G-BOPA) should be obvious. The following result combines [G-The remarkable fact that Oka manifolds can be defined in many nontrivially equivalent ways points to the concept being natural and important. The same has been proved to some extent in the equivariant setting. Above we defined the basic G,  the following properties are equivalent: G-BOPA, G-BOPI, G-BOPAJI, G-G-Oka property.For a complex manifold with an action of a finite group We believe that generalising Theorem G be a complex reductive group and K a maximal compact subgroup of G. Let X be a Stein G-manifold and Y be a G-Oka manifold. Assume that X has a single slice type, that is, the quotient map G-fibre bundle. Then every K-equivariant continuous map G-equivariant holomorphic map.Let Y is said to be elliptic—Gromov’s definition [E on Y, such that G acts on Y, then we say that s is a G-spray if the action on Y lifts to an action on E by vector bundle isomorphisms such that both s and the projection Y is G-elliptic if it carries a dominating G-spray.We now turn to the equivariant versions of two fundamental properties in Oka theory. A manifold finition  marked tG-ellipticity, also known as G-Y satisfies G-G-map f from a Stein G-manifold X to Y, there is a holomorphic G-vector bundle E over X and a dominating G-spray f. This means that E over x, and The weaker notion of relative Y is G-Runge if for every Stein G-manifold X and every G-invariant Runge domain X, the closure of the image of the restriction map G-maps G-maps G is the trivial group, the G-Runge property of Y is one of the equivalent formulations of the Oka property. When G is reductive, the property of a domain X being G-invariant and Runge can be described in several equivalent ways [We say that ent ways , Sect. 6G-Oka property hold for both G-ellipticity and the G-Runge property [G-homogeneous space is G-Runge. This is what we know about the relationships between the three properties.Analogues of the basic results about the G be a reductive complex group and Y a G-manifold. Y is G-elliptic, then Y is G-Runge.If Y is G-Runge, then Y is G-Oka.If G is finite and Y is Stein and G-Oka, then Y is G-elliptic.If Let The proof of (a) is somewhat involved: it is an equivariant version of Gromov’s linearisation method, sketched in . The proY be a G-homogeneous space and take the trivial G-vector bundle G acts on its Lie algebra G-spray, so Y is G-elliptic. By Theorem Y is G-Runge.Now let G-biholomorphisms G-diffeomorphisms a weak homotopy equivalence with respect to the appropriate topology? The same question for strong G-homeomorphisms.Does the parametric version of Theorem X does not have the infinitesimal lifting property?Is there a counterexample to Theorem Does Theorem Build approximation and interpolation into Theorem Show that the weak homotopy equivalence in Theorem G? Does it hold for actions for which all the G-orbits are closed?Does Theorem Does the parametric version of Theorem G be a reductive group and Y be a G-Oka manifold. Is YG-Runge?Let


Communications Medicine 10.1038/s43856-022-00209-1, published online 06 December 2022.Correction to: There was a mistake in Table 5 of this article. The specificity (%) value in the final row of Table 5 (the ‘Spheroid’ row) read ‘100’, but should have read ‘67’. There was also a typographical error in the Background section of the Abstract. ‘Organ-on-a-Chip technology has the potential improve success in drug development pipelines, as it can recapitulate organ-level pathophysiology and clinical responses’ should have read ‘Organ-on-a-Chip technology has the potential to improve success in drug development pipelines, as it can recapitulate organ-level pathophysiology and clinical responses’. Both mistakes have been corrected in the HTML and PDF versions of the article.


In wireless networks, MAC scheduling methods can be divided into two types according to the implementation model: centralized and distributed scheduling. By reasonably designing MAC scheduling policies, both centralized and distributed schedulers can ensure a reliable throughput capacity region, i.e., realizing throughput-guaranteed. However, it can be found that some existing throughput-guaranteed scheduling schemes cannot further ensure bounded end-to-end average delay, and the reason for this phenomenon has not been deeply analyzed. In practical communication networks, throughput and delay are equally important. Based on this idea, the existing MAC scheduling strategies are investigated systematically in this paper from two aspects of throughput and delay, and their performances are evaluated and compared through both theoretical analysis and simulation experiments. The work of this paper provides a theoretical basis for the improvement of MAC scheduling technology in the next-generation wireless networks. Depending on the way of obtaining the permission to use wireless channels, MAC layer protocols for wireless networks can be divided into two categories: competitive protocol and scheduling policy. The characteristic of the competitive protocol is that the node participates in the channel competition to win the permission to use the wireless channel. CSMA/CA protocol in IEEE 802.11 standard is a typical competitive protocol. Differently, the access permission of the wireless channel under the scheduling policy is pre-assigned in order to avoid the competition and conflict in transmission. The essence of MAC scheduling technology is to use STDMA (Space Time Division Multiple Access) to determine the scheduled time-slot for each link and allocate conflicting links into different time-slots . Let lq(n) denote the queue length of link l in time-slot n. Assume that the capacity of link l is fixed and denoted by lc. Indicator variable 𝜑l(n) is used to indicate whether link l is scheduled in time-slot n. If l is scheduled and lq(n) > 0, 𝜑l(n) = 1; otherwise, 𝜑l(n) = 0. Thus, the queue update process of any link l is shown in Packets sent from the source node must be queued and forwarded to the destination node. The MAC scheduling method divides the time axis into time-slots and completes the input and output of the queue in each time-slot. Nodes are required to maintain data queues in both single-channel and multi-channel networks. The queue update action includes the arriving and sending of packets. According to the scheduling algorithm, when a queue is scheduled in this time-slot, data will leave the queue and be sent to the destination or to the next-hop node. Let + is the projection on  < ∞, for any l ∈ {1, ..., L}.In network queue analysis, it is usually assumed that the queue length at the initial time is finite, that is, After the Lyapunov function is constructed, Foster’s Criterion is needed to determine the weak stability of the system . The criCriterion 1.For positive integer H and constantif there exists a nonnegative Lyapunov function V(n) > 0 and a set0 with finite states such thatΓholds for any time-slot n, then the aperiodic countable Markov chain (n) is positive recurrent and ergodic where E[V(n + H)q V(n) │(n)] is defined as Lyapunov drift.q(n) is positive recurrent  < ∞; otherwise, if q(n) ∉ Γ0, E[V(n + H)V(n)│q(n)] δ. Based on Criterion 1, weak stability can be deduced. Therefore, strong stability can lead to weak stability, not vice versa.For the proof of Criterion 2, one can refer to . Accordiλ ∈ γ Ω (0 < γ < 1), the scheduling policy is then considered to be throughput-guaranteed and its efficiency ratio is γ. In fact, the efficiency ratio γ determines the throughput performance of a MAC scheduling policy.In this paper, we use Ω to represent the maximum throughput capacity region that can be ensured by any scheduling strategy. Under a particular MAC scheduling mechanism, if the network system remains weakly stable when the input load This section uses Criterion 1 and Criterion 2 to evaluate the throughput and delay performances of several typical scheduling policies.For centralized scheduling in single-channel networks, the performance of the MWS  algorithAlgorithm implementation: In each time-slot n, the central control node in the network determines the schedule S represents the set of all possible schedules under a particular interference model and the schedule vector φ = . Obviously, the realization of MWS requires a central node to execute the scheduling process on the premise of knowing all the queue length information in the network and to determine the schedule in each time-slot through Equation (10). Hence, the implementation complexity of MWS is high.Performance evaluation of MWS: Assume that the input load vector λ locates strictly inside the maximum capacity region. That is, there exists an L-dimensional vector ε with all elements equal to ε such that λ + ε ∈ Ω holds. For time-slot n, the following Lyapunov function is constructed aseference  and obtaε.It can be seen from the analysis above that MWS can achieve the maximum capacity region and can ensure the strong stability of the network when the input load locates inside the capacity region. The upper bound of average end-to-end delay is given by (15), which depends on the arrival rates and the parameter Algorithm implementation: MS is a typical distributed scheduling strategy. Under MS, any queue that meets lq(n) > lc is scheduled in time-slot n unless another queue in its interference set I(l) has already been scheduled. Hence, employing MS can ensurel. MS does not require the centralized control node and each link decides whether to participate in the data transmission at the current time-slot. The implementation complexity of MS is much lower than that of the MWS policy.Performance evaluation of MS: Assume that l, we use the following Lyapunov function:ω and B are positive numbers and K, where K is the interference degree [function:V(n)=∑l=1Algorithm implementation: Different from MS, the distributed Q-SCHED strategy divides each time-slot into scheduling-slot and transmission-slot, and further divides the scheduling-slot into M mini-slots, as shown in l picks the backoff time from the set {1, 2, …, M + 1} according to the following probability:I represents the backoff time selected by l and the parameter lp is calculated byM + 1 means that the link is not scheduled in this time-slot. Once the backoff time expires, the link starts transmitting unless there have been other links in its interference set that are sending data.In each time-slot, link Performance evaluation of Q-SCHED: We introduce the following Lyapunov function:μ > 0, if the input load of the network satisfiesH such that [V(n) ≥ Φ. Since {q(n): V(n) < Φ} is a set with finite states, we can obtain that the Markov chain {q(n)} is positive recurrent using Criterion 1. Hence, the network is weakly stable. It should be noted that ensuring weak stability under the condition (23) implies that the Q-SCHED policy is throughput-guaranteed. However, strong stability is not achieved, and Q-SCHED cannot provide a bounded end-to-end average delay. On the other hand, based on the random access technology as shown in O(logL) number of iterations [uch that EV(n+H)−erations .For scheduling in multi-channel networks, performances of the straightforward extension of MWS , distribAlgorithm implementation: MWS in single-channel networks can be directly extended to multi-channel networks. Similar to MWS, the scheduling indicator vector n according tol operating on channel c and l is scheduled on channel c, and 0 otherwise.Performance evaluation: Literature [q(n)} can be achieved. That is, the network system remains weakly stable when λ ∈ Ω. In fact, if we construct the following Lyapunov functionterature  has provrring to  that MWSAlgorithm implementation: Authors of [thors of  proposedI(ζ), ζq(n) and ζc represent the interference set, queue length and capacity of Tuple link ζ respectively. According to MS, we haveζ.After the Tuple-based model is established, the MS algorithm is implemented based on Tuple links. Suppose Performance evaluation: To evaluate the performance of Tuple-based MS, the following Lyapunov function is constructed:ζ, one can achieve the one-step Lyapunov drift κ and B1 are positive numbers and Y, where Y denotes the interference degree under the Tuple-based model.H such thatV(n) ≥ Φ. Hence, the network should be weakly stable by using Criterion 1 under DA-LC. The efficiency ratio of DA-LC can be arbitrarily close to that of Tuple-based MS. Therefore, compared with Tuple-based MS strategy, DA-LC cannot guarantee a bounded end-to-end average delay. However, the implementation complexity of DA-LC is much lower and does not increase with the number of Tuple links since it adopts the random access technology.DA-LC algorithm is a combination of the Tuple-based model and the Q-SCHED strategy in a single-channel network. Specifically, the multi-channel environment is first virtualized into a single-channel network using the transformation process shown in λ. In a single-channel environment, all links share one available channel to transmit data. In a multi-channel environment, the interface (network card) of a node can switch between four different channels with the channel capacity of 1, 1, 2, 2 respectively. The simulation parameters adopted in our experiments are shown in In this section, we use NS-2.31 software to simulate actual network scenarios and compare the throughput and delay performances of several existing wireless scheduling strategies. We use the topology shown in For a multi-channel network environment, In the next-generation wireless networks, MAC scheduling is becoming the preferred technology of MAC layer. In order to better design the scheduling algorithms and verify their performances in different scenarios, this paper summarizes a complete set of methods, including the concepts of strong/weak stability and their judgment methods. By using these methods, this paper analyzes the performances of typical MAC scheduling policies, including throughput performance and delay performance. Through theoretical analysis and simulation experiments, we verify that strong stability can provide lower end-to-end delay than weak stability in both single-channel and multi-channel networks. The research work of this paper provides a theoretical basis for the design and evaluation of MAC scheduling policies.


In hypothesis testing, we first set up a null hypothesis and an alternative hypothesis. For example, if we are interested in comparing proportions between two groups, then the null hypothesis would be stated as “the proportions are the same” and the alternative hypothesis as “the proportions differ” between the two groups. We then gather data to test the veracity of the null hypothesis. In particular, the data allow us to calculate the p-value, which is defined as “the probability under a specified statistical model that a statistical summary of the data would be equal to or more extreme than its observed value” [p-value is lower than the 0.05 significance level, we reject the null hypothesis and accept the alternative hypothesis. At the beginning of our research training, we learned about hypothesis testing, d value” , and prap-value has become the most frequently used statistic in the biomedical research field—a sine qua non for determining if a study finding is real or due to chance, if a new drug is effective or not, or if a health technology should be approved or denied by regulatory authorities. However, in recent years, there is a growing discussion that p-values and “statistical significance” , albeit useful, have often been misused and misinterpreted in the biomedical field [p-values in clinical research. This methodology, often referred to as “null hypothesis significance testing”, originated in the early 20th century with the ideas of Fisher . Since tal field ,7,8,9,10al field  will furTo begin with, let us assume that a single-center, randomized, double-blind, placebo-controlled trial is performed to determine if a new drug reduces mortality in 400 patients hospitalized with severe COVID-19 pneumonia, who are otherwise receiving standard-of-care. The occurrence of death at 28 days following randomization is the trial’s primary endpoint. The numbers of patients who died or survived at 28 days, in each of the two study groups, are given in p-value that is higher than the cut-off threshold of 0.05 . Equivalently, the Risk Ratio is 0.70 , and the 95% CI includes the null value of one , or because the null value falls within the confidence interval. A statistically non-significant result does not “prove” the null hypothesis [Assuming that the particular study suffers no systematic error (bias) or confounding, we can employ significance testing or confidence interval (CI) methods to examine whether there is statistical evidence against the null hypothesis . Using the data from pothesis . This ispothesis . Let us now assume that a subsequent multi-center, randomized, double-blind, placebo-controlled trial assessed the efficacy of the same experimental drug in 1700 patients hospitalized with severe COVID-19 pneumonia . The Risk Ratio is 0.70 , but the 95% CI now excludes the null value of one  while the first one did not? Again, the answer is “no”. We cannot make such a claim! This is another misunderstanding about p-values. Especially when the study is underpowered and/or of a small sample size , the p-value has a large variance across repeated samples  [p-value obtained in a study to be indicative of the likelihood of getting the same results in future studies, conducted on the same topic, is not correct.Using the data from ulation) . This isp-value is probably “the most misunderstood, misinterpreted, and occasionally miscalculated index in all of biomedical research” [It is wrong to claim that a statistically non-significant result shows “no association”, when the confidence interval includes clinically important decreases (or increases) in risk. Similarly, it is wrong to claim that two such studies  are in contrast. Yet, such errors are very common in the biomedical literature. Despite its seeming simplicity, the esearch” . p-values, and improve the conduct of quantitative science: (1)p-values can indicate how incompatible the data are with a specified statistical model.” Based on the observed data, the researcher either rejects or “fails to reject” the null hypothesis.“(2)p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.” They can only tell us how consistent the data are with the null hypothesis.“(3)Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.” Decision-making must not rest solely on a p-value. Researchers should also consider a broad range of other factors, e.g., study design, data quality, related prior evidence, plausibility of mechanism, real-world costs and benefits, etc.“(4)Proper inference requires full reporting and transparency.” Researchers should not conduct multiple, unplanned analyses of the data and (selectively) report only those with certain p-values . This is usually called cherry-picking, data-dredging, significance-chasing, significance-questing, selective-inference, or p-hacking, and leads to a spurious excess of statistically significant results in the biomedical literature.“(5)A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.” For example, smaller p-values do not necessarily imply the presence of larger or more important effects, and larger p-values do not imply a lack of effect or lack of importance.“(6)By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.” Analysis of the data should not end with the calculation of a p-value. Depending on the context, p-values can be supplemented by other measures of evidence that can more directly address the effect size and its associated uncertainty .“Driven by increasing concern, the American Statistical Association (ASA) has issued a statement  with sixp-values in applied research. Importantly, it also provides great references and supplementary material for anyone who would like to delve further into this topic. The ASA’s statement  outlinedp-values [A comprehensive understanding of statistical concepts improves the quality of clinical research. We strongly encourage clinical researchers to involve biostatisticians and methodologists early on in their studies; to ensure a sound study design and a cohesive data collection strategy to minimize biases; to prespecify a data analysis plan that can address potential misuse and misinterpretation of p-values ; and to p-values .


Nature 10.1038/s41586-022-05197-9 Published online 19 October 2022Correction to: ħk values shown in Fig. 1a,b and Fig. 2c–e, the modified constant ħ was omitted in error. In addition, the Peer review information mistakenly linked to an earlier version of the article manuscript and has since been removed. The changes are reflected in the HTML and PDF versions of the article.In the version of this article initially published, for the


Euterpe oleracea Mart.), a species belonging to the Arecaceae family, has been cultivated for thousands of years in tropical Central and South America as a multipurpose dietary plant. The recent introduction of açaí fruit and its nutritional and healing qualities to regions outside its origin has rapidly expanded global demand for açaí berry. The health-promoting and disease-preventing properties of this plant are attributed to numerous bioactive phenolic compounds present in the leaf, pulp, fruit, skin, and seeds. The purpose of this review is to present an up-to-date, comprehensive, and critical evaluation of the health benefits of açaí and its phytochemicals with a special focus on cellular and molecular mechanisms of action. In vitro and in vivo studies showed that açaí possesses antioxidant and anti-inflammatory properties and exerts cardioprotective, gastroprotective, hepatoprotective, neuroprotective, renoprotective, antilipidemic, antidiabetic, and antineoplastic activities. Moreover, clinical trials have suggested that açaí can protect against metabolic stress induced by oxidation, inflammation, vascular abnormalities, and physical exertion. Due to its medicinal properties and the absence of undesirable effects, açaí shows a promising future in health promotion and disease prevention, in addition to a vast economic potential in the food and cosmetic industries.The açaí palm ( Euterpe oleracea Mart.), a species belonging to the palm tree (Arecaceae) family, is native to several countries in the Amazon region of tropical Central and South America, including Brazil, Ecuador, and Venezuela [The açaí palm . Furthermore, açaí has been shown to exhibit anticancer, antiatherogenic, antimicrobial, antinociceptive, anticonvulsant, antileishmanial, and antiaging activities. Moreover, research has indicated the tissue-protective effects of açaí on several organs, such as the heart, liver, lungs, kidneys, and brain ,23,24,25Açaí, popularly known as açaí-do-Pará, açaizeiro, or açaí-de-toceira, is a palm tree a native Notably, the disease-preventing effects of açaí are related to its composition of bioactive phytochemicals. The major constitutive phytochemicals of açaí are present throughout various parts of the plant.In the açaí fruit, the polyphenols are the most significant constituent of the chemical profile. Major secondary polyphenol metabolites include anthocyanins (ACNs) and proanthocyanidins (PACs), in addition to other flavonoids. Several phenolic acids , flavonoids , lignans, and procyanidin oligomers have been reported in the phytochemical profile of açaí fruit ,31,32. Tp-hydroxybenzoic acid, vanillic acid, syringic acid, ferulic acid) and a flavonoid, (+)-catechin, in açaí oil , such as 3,4-dihydroxybenzoic acid; çaí pulp . It has çaí pulp . Assays çaí pulp . SimilarO-caffeoylquinic acid, 4-O-caffeoylquinic acid, and 5-O-caffeoylquinic acid [O-caffeoylshikimic acid, 4-O-caffeoylshikimic acid, and 5-O-caffeoylshikimic acid (C-glycosides (ACGs), a group of flavonoids, including: 6,8-di-C-hexosyl apigenin; 6,8-di-C-hexosyl apigenin sulfate; 6-C-hexosyl-8-C-pentosyl apigenin isomers; 6-C-glucosyl luteolin, or homoorientin; 6-C-pentosyl-8-C-hexosyl apigenin isomers; 8-C-glucosyl luteolin; and 6-C-glucosyl apigenin  guidelines was maintained throughout the process of data collection [This review was designed on the basis of the following focal question: “What are the described health effects of açaí?” A literature search through PubMed, Cochrane, Embase, and Google Scholar databases was conducted to identify studies performed with açaí in relation to its health benefits, implications in disease, or both. Keywords directing the investigation included llection ,44. The Eligible studies published from 2004 to 2022 were included in this review. Exclusion criteria were non-English language studies, unpublished data, poster presentations, and conference abstracts. For each clinical trial study, the detection, selection, and reporting of bias was utilized to carefully consider and evaluate risk of bias. Patient inclusion, intervention category, outcomes analyses, missing events, and data were also examined. Assessments of both bias and quality were performed in alignment with directives of the Cochrane Handbook for Systematic Reviews of Interventions . The desNumerous preclinical studies have revealed the effects of açaí in vitro  and in vAmong the health implications of açaí included in this discussion, antioxidant and anti-inflammatory faculties have been documented most frequently in the current literature. A large quantity of in vitro evidence exists in support of the antioxidant capacity of several compounds  present in açaí ,142,143.In a similar manner, Earing et al.  evaluateCCD-18Co cells to investigate the inflammatory protein expression, ROS suppression, and anti-inflammatory activity of açaí phenolic compounds. Açaí extract was noted to decrease COX-2, TNF-α, and NF-κβ expression induced by lipopolysaccharide (LPS) in human colon myofibroblasts. Because the downregulation of biomarkers, such as COX-2, TNF-α, and NF-κβ, is vital to the reduction of inflammation, these results provide further evidence of anti-inflammatory activities of phenolic compounds in açaí. Another in vitro study performed by Machado et al. [Injury, toxins, infection, genetic defects, and trauma can induce resident immune cell activation . Subsequo et al.  demonstro et al. ,57,148. Pain can have a negative impact on quality of life, as well as the performance of daily activities. Currently, nonsteroidal anti-inflammatory drugs (NSAIDs) and opioids are common pharmacological options of the treatment of pain. The anti-inflammatory, antihypertensive, antioxidant, and vasodilatory activities of açaí has resulted in the exploration of its value as an antinociceptive and analgesic agent ,118,149.Escherichia coli, Pseudomonas aeruginosa, Streptomyces aureus, and Enterococcus faecalis [E. coli [Due to growing antimicrobial resistance, the pharmaceutical field is constantly seeking new alternatives to oppose relevant pathogens. The high polyphenol content of açaí has been associated with antimicrobial activity . One stufaecalis . EOO com[E. coli . OverallS. aureus and found it to be effective against planktonic cells and biofilms of this microorganism. Furthermore, hydroalcoholic extracts produced from dried pulp, leaves, and seeds of açaí demonstrated significant antimicrobial activity against Clostridium perfringens, S. aureus, and P. aeruginosa [Leishmania amazonensis and Leishmania infantum and found a reduction in the number of promastigotes, augmented synthesis of ROS, and the induction of cell death phenotypes in both species. The use of açaí juice for 72 h not only engendered protective effects against L. amastigotes and strongly diminished IL-17 levels, but also reduced the number of intracellular amastigotes in macrophages infected with L. amastigotes and L. infantum [Dias-Souza et al.  studied ruginosa . Hence, ruginosa  evaluateinfantum . Of noteGastric ulcers are one of the most common conditions afflicting humans. Current treatments for gastric ulcers include H2-receptor antagonists, M1-receptor blockers, and proton pump inhibitors. However, these drugs can be costly, have health-associated side effects, and result in relapse. For this reason, more efficacious and inexpensive therapeutics for gastric ulcers are in high demand. Cury et al.  investigThere are limited investigations examining the impact of açaí berry on cognitive function or brain health. The experiments available have established that açaí largely confers its neuroprotection through antioxidant and anti-inflammatory mechanisms, restoration of mitochondrial function, and inhibition of toxic protein aggregation ,154,155.2O2). Analysis of ABTS and deoxyribose assays revealed no differences in antioxidant activity amongst the various açaí genotypes. All hydroethanolic extracts were found to reduce ROS produced by H2O2 in SH-SY5Y cells, elucidating their neuroprotective effects [The anti-inflammatory potential of açaí extract was evaluated via an in vitro microglial model that substantiated its modulation of nucleotide oligomerization domain (NOD)-like receptor pyrin domain-containing 3 (NLRP3) inflammasome proteins and antioxidant pathways, augmentation of anti-inflammatory cytokines, and reduction of pro-inflammatory cytokines . Microgl effects .The vulnerability of the brain to oxidative stress, in addition to its energy requirements for neurotransmission, greatly increases the risk of mitochondrial dysfunction in neural tissue . Data suNeurodegeneration in the setting of NDDs can cause significantly altered neuron viability. This can lead to the overproduction, inadequate clearance, and ultimately, aggregation of toxic proteins in or around brain tissue . The deaBrain injury from seizures can result in neuronal cell death . FactorsDrosophila melanogaster fed a high-fat diet. This study found that açaí supplementation increased the transcription of genes related to a small heat-shock-related protein and two detoxification genes. Açaí also reduced transcription of phosphoenolpyruvate carboxykinase (PEPCK), a critical gene in the appropriate regulation of gluconeogenesis. Moreover, açaí increased the lifespan of females subjected to oxidative stress, suggesting that açaí improves survival [Many neurodegenerative diseases eventually lead to symptoms of depression. Because depression has been linked to oxidative stress, the antidepressant effects of açaí berry have also been explored . Açaí jusurvival . Açaí masurvival ,158,159.Cocos nucifera L. saturated fat in rats. Although no alterations in triglycerides were noted, there were reductions in total cholesterol and low-density lipoprotein cholesterol (LDL). A treatment regimen of both açaí oil and simvastatin, a common anticholesterolemic drug, also prevented the formation of atheromatous plaques in the vascular endothelium of rats [Dyslipidemia is precipitated by disruptions in lipid metabolism that result in chronically elevated serum lipids and, therefore, increased predisposition to CVDs, obesity, atherogenic processes, diabetes, and metabolic syndrome (MetS) ,160. Die of rats ,109. Com of rats  evaluate of rats . For thi of rats ,120.Souza et al.  proposedGlobally, the most prevalent liver condition is nonalcoholic fatty liver disease (NAFLD) . NAFLD iAnother study revealed that the use of açaí seed flour could prevent obesity-induced hepatic steatosis in mice . In thisThe effects of filtered açaí pulp on the expression of paraoxonase (PON) activity in rats with NAFLD have also been investigated ,167. RatAlcoholic liver disease (ALD) is the most prevalent cause of advanced liver disease and has been linked to high levels of global mortality and morphological consequences, including alcoholic hepatitis, steatosis, fibrosis, cirrhosis, and hepatocellular carcinoma . The resWorldwide, T2DM is a serious public health crisis reaching epidemic proportions . DiabeteCardiovascular risk factors such as hypertension, dyslipidemia, obesity, diabetes mellitus, and MetS promote endothelial injury due to oxidative stress . This enROS play an intricate role in oxidative stress and accompanying myocardial injury. Increased quantities of ROS during heart ischemia can elicit cell membrane destruction, lipid peroxidation, and dysfunction in the antioxidative defense system. For this reason, açaí has been studied as a prospective vasodilatory, anti-inflammatory, and antifibrotic agent . It has CVDs are currently the leading cause of morbidity and mortality among adults. Although scientific progress has identified a spectrum of different risk factors for cardiovascular pathologies, the current state of prevention, and even treatment, of CVDs is suboptimal ,168,169.Figueiredo et al.  treated Pontes et al.  demonstrChronic kidney disease (CKD) is a general term for a group of heterogenous disorders that affect kidney structure and function. Over time, considerable research has been conducted to properly define, classify, and treat CKD. Ultimately, data has indicated that prevention is the most effective way to avoid the development and progression of renal disorders ,170. Becda Costa et al.  found thOver time, the global frequency of cancer, a group of diseases characterized by abnormal cell growth, has continued to increase. Therefore, cancer prevention has become a topic of paramount importance within the scientific field. The chemopreventive and anticarcinogenic potency of açaí has been linked to its ability to decrease the viability of cancer cells, as well as reduce the incidence of tumors and tumor cell proliferation ,172,173.Sibuyi et al.  investigFurthermore, Monge-Fuentes et al.  studied O-glucoside, cyanidin 3-rutinoside) and carotenoids, especially β-carotene and lutein. Hence, lyophilized açaí pulp was found to have the potential to exert antitumor activity. Additionally, Fragoso et al. [Helicobacter felis in mice.Fragoso et al.  evaluateo et al.  demonstro et al.  studied o et al.  studied o et al.  found tho et al.  concludeStoner et al.  investigIn addition to preclinical studies, numerous clinical trials have evaluated the effect of açaí on different aspects of human health. These studies are summarized in Euterpe edulis Mart.) juices on biochemical and antioxidant parameters in humans. The participants were indicated to drink 200 mL/day of açaí juice for four weeks. The authors observed posttreatment improvement in HDL levels and antioxidant enzyme activities  in their test subjects. However, there was a noteworthy limitation of this trial. Although this report indicated that the patients were asked about adverse events throughout the experimental process, no information was provided with regards to the results for this question.De Liz et al.  performeAnother trial showed that the consumption of 200 g of açaí pulp/day over a four-week timeframe afforded no modifications in total cholesterol, LDL and HDL, triglycerides, or apolipoprotein B (ApoB) in healthy female patients, besides the demonstration of significant antioxidant actions. Nonetheless, increased ApoA-I was observed, suggesting an improvement in the metabolism of this lipoprotein. The favorable actions on plasma HDL metabolism and antioxidant defenses indicate a potential use of açaí as an antilipidemic agent .Additionally, Alqurashi et al.  investigCarvalho-Peixoto et al.  considerGale et al.  performeA study performed by Oppitz et al.  revealedAranha et al.  conducteIn addition, Kim et al.  evaluateIn overweight subjects, Udani et al.  revealedFurther, Kesller et al.  investigOn the whole, current scientific literature is deficient in information regarding the toxicity of açaí. Marques et al.  publisheAdditionally, Ribeiro et al.  tested aHowever, Marques et al.  used ratMoreover, Caiado et al.  exploredLactobacillus rhamnosus, the fruit can be easily used as a suitable matrix for products that use this microorganism as a probiotic agent [Açaí has enormous economic potential in many fields of industry. In the food industry, açaí may be used in the production many goods, ranging from probiotic beverages to desserts to ice cream. Due to its potent effects in preventing morphological changes in principally ic agent ,189,190.ic agent . Of noteic agent .Moreover, in the cosmetic industry, açaí has been used to create multicomplex emulsions that enhance the photoprotective effects of sunscreens formulas. Due to its antioxidant capacity, açaí also has potential as an ingredient in various other skincare products, such as skin lotions ,193,194.In the biotechnological industry, açaí-derived ultrafine fibers have been used to compose innovative pH sensors for the visual monitoring of food quality . AdditioOur results show that açaí has medicinal properties and the economic potential for widespread use throughout the food and cosmetic industry. The fruit presents a rich phytochemical profile composed of phenolic compounds, quinones, terpenes, and norisoprenoids, all of which are related to its health-promoting and disease-preventing potential. In vitro and in vitro studies demonstrated that açaí possesses antioxidant and anti-inflammatory effects; exerts cardioprotective, gastroprotective, hepatoprotective, neuroprotective, and renoprotective activities; improves hyperinsulinemia and dyslipidemia; and shows antineoplastic actions. Additionally, açaí exerts antimicrobial and antiparasitic effects. Clinical trials have demonstrated that açaí protects against prostate cancer, MetS risk factors, and auditory dysfunctions. Moreover, its derivatives, such as berry extracts, whole fruit extracts, seed extracts, and phytochemically enriched extracts, have no hepatotoxicity, cardiotoxicity, or nephrotoxicity, strengthening its safety and health potential.However, our work had some limitations. Firstly, numerous included studies did not specifically define the composition of the utilized açaí. Secondly, a limited number of clinical studies have been conducted on this fruit, especially concerning direct human consumption. In addition, the studies included in this paper are heterogeneous due to the variety of forms of açaí administration used  and in different doses. In short, the accumulated scientific evidence on the properties of açaí, in the absence of undesirable effects, awards this plant a promising future in health promotion and disease prevention. Moreover, there is a vast economic potential of açaí in the pharmaceutical, food, and cosmetic industries.


Correction to: Doc Ophthalmolhttps://doi.org/10.1007/s10633-022-09904-9xIn the original article, the formatting of “<<” and “>>” was indicated with different symbols.For example,and,Unified symbols are used to indicate greater than and less than.The original article has been corrected.


Asher L, Birhane R, Weiss HA, et al. Community-based rehabilitation intervention for people with schizophrenia in Ethiopia (RISE): results of a 12-month cluster-randomised controlled trial. Lancet Glob Health 10: e530–422022; —In this Article the acknowledgments have been corrected to add “MDS was not employed by the Wellcome Trust at the time of the study.” This correction has been made as of May 10, 2022.