Rayleigh-Taylor instability (RTI) is of critical important in a broad range of natural and industrial processes and is an intellectual challenge for theoretical studies. In this work, we analyze the scale-dependent linear and nonlinear Rayleigh-Taylor (RT) dynamics within the group theory approach. We link the governing equations, through an associated dynamical system based on space groups, to a momentum model based on scaling transformations. In doing so, we precisely derive expressions for the buoyancy and drag parameters of the momentum model, exactly integrate the model equations and determine solutions for bubbles and for spikes in both early-time and late-time regimes. In particular, we focus on the general situation in which the instability is driven by an acceleration having power-law time dependence. Our analysis provides extensive benchmarks for future research.
{"title":"An analysis of the buoyancy and drag parameters in Rayleigh-Taylor dynamics","authors":"D. Hill, S. Abarzhi","doi":"10.1051/mmnp/2023027","DOIUrl":"https://doi.org/10.1051/mmnp/2023027","url":null,"abstract":"Rayleigh-Taylor instability (RTI) is of critical important in a broad range of natural and industrial processes and is an intellectual challenge for theoretical studies. In this work, we analyze the scale-dependent linear and nonlinear Rayleigh-Taylor (RT) dynamics within the group theory approach. We link the governing equations, through an associated dynamical system based on space groups, to a momentum model based on scaling transformations. In doing so, we precisely derive expressions for the buoyancy and drag parameters of the momentum model, exactly integrate the model equations and determine solutions for bubbles and for spikes in both early-time and late-time regimes. In particular, we focus on the general situation in which the instability is driven by an acceleration having power-law time dependence. Our analysis provides extensive benchmarks for future research.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49338742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Because of mutations and selection, pathogens can manage to resist to drugs. However, the evolution of an asexual population (e.g., viruses, bacteria and cancer cells) depends on some external factors (e.g., antibiotic concentrations), and so understanding the impact of the environmental changes is an important issue. This paper is devoted to model this problem with a nonlocal diffusion PDE, describing the dynamics of such a phenotypically structured population, in a changing environment. The large-time behaviour of this model, with particular forms of environmental changes (linear or periodically fluctuations), has been previously developed. A new mathematical approach (limited to isotropic mutations) has been developed recently for this problem, considering a very general form of environmental variations, and giving an analytic description of the full trajectories of adaptation. However, recent studies have shown that an anisotropic mutation kernel can change the evolutionary dynamics of the population: some evolutive plateaus can appear. Thus the aim of this paper is to mix the two previous studies, with an anisotropic mutation kernel, and a changing environment. The main idea is to study a multivariate distribution of (2n) "fitness components". Its generating function solves a transport equation, and describes the distribution of fitness at any time.
{"title":"Adaptation of an asexual population with environmental changes","authors":"Florian Lavigne","doi":"10.1051/mmnp/2023024","DOIUrl":"https://doi.org/10.1051/mmnp/2023024","url":null,"abstract":"Because of mutations and selection, pathogens can manage to resist to drugs. However, the evolution of an asexual population (e.g., viruses, bacteria and cancer cells) depends on some external factors (e.g., antibiotic concentrations), and so understanding the impact of the environmental changes is an important issue.\u0000This paper is devoted to model this problem with a nonlocal diffusion PDE, describing the dynamics of such a phenotypically structured population, in a changing environment. The large-time behaviour of this model, with particular forms of environmental changes (linear or periodically fluctuations), has been previously developed. A new mathematical approach (limited to isotropic mutations) has been developed recently for this problem, considering a very general form of environmental variations, and giving an analytic description of the full trajectories of adaptation.\u0000However, recent studies have shown that an anisotropic mutation kernel can change the evolutionary dynamics of the population: some evolutive plateaus can appear. Thus the aim of this paper is to mix the two previous studies, with an anisotropic mutation kernel, and a changing environment. The main idea is to study a multivariate distribution of (2n) \"fitness components\". Its generating function solves a transport equation, and describes the distribution of fitness at any time.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42945831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Muhammad Fahim, M. Sajid, N. Ali, Muhammad Noveel Sadiq
This article examines a mathematical framework that describes the versatile behavior of heat and mass exchange in blood flowing through a narrowed vessel having multiple stenoses. The geometry of a channel having multiple stenoses with an asymmetrical axial axis and a symmetrical radial axis can be visualized by applying a suitable mathematical expression. The geometry of the chosen model considers the height and shape of stenoses. The modification in shape parameter is used to capture variations in the shape of the stenoses in the artery. The blood is supposed to be isochoric (incompressible), while its rheological behavior is characterized by Williamson’s fluid model. The transfer of momentum is analyzed using the equation of motion in cooperation with the continuity equation. In addition, the equations of heat conduction and diffusion are utilized, respectively, to illustrate the distributions of heat and mass. Simplified forms of momentum, mass, and heat transport equations are achieved by incorporating dimensionless quantities and moderate stenosis conditions. A well-known explicit finite difference approach is utilized to solve the emergent non-linear system of governing equations numerically. The influence of different evolving parameters on the flow field along with mass and heat distributions is illustrated through various plots.
{"title":"Heat and mass diffusion to Williamson fluid streaming through a tube with multiple stenoses while subjected to periodic body acceleration","authors":"Muhammad Fahim, M. Sajid, N. Ali, Muhammad Noveel Sadiq","doi":"10.1051/mmnp/2023021","DOIUrl":"https://doi.org/10.1051/mmnp/2023021","url":null,"abstract":"This article examines a mathematical framework that describes the versatile behavior of heat and mass exchange in blood flowing through a narrowed vessel having multiple stenoses. The geometry of a channel having multiple stenoses with an asymmetrical axial axis and a symmetrical radial axis can be visualized by applying a suitable mathematical expression. The geometry of the chosen model considers the height and shape of stenoses. The modification in shape parameter is used to capture variations in the shape of the stenoses in the artery. The blood is supposed to be isochoric (incompressible), while its rheological behavior is characterized by Williamson’s fluid model. The transfer of momentum is analyzed using the equation of motion in cooperation with the continuity equation. In addition, the equations of heat conduction and diffusion are utilized, respectively, to illustrate the distributions of heat and mass. Simplified forms of momentum, mass, and heat transport equations are achieved by incorporating dimensionless quantities and moderate stenosis conditions. A well-known explicit finite difference approach is utilized to solve the emergent non-linear system of governing equations numerically. The influence of different evolving parameters on the flow field along with mass and heat distributions is illustrated through various plots.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46266865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"MMNP - next generation","authors":"V. Volpert","doi":"10.1051/mmnp/2023017","DOIUrl":"https://doi.org/10.1051/mmnp/2023017","url":null,"abstract":"","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42592085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population of Aedes mosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya... in a wide area. We use a system of reaction-diffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-called hair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speed c to push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space. Moreover, we succeed in constructing a 'forced' traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results.
{"title":"A control strategy for the Sterile Insect Technique using exponentially decreasing releases to avoid the hair-trigger effect","authors":"N. Nguyen, Alexis L'eculier","doi":"10.1051/mmnp/2023018","DOIUrl":"https://doi.org/10.1051/mmnp/2023018","url":null,"abstract":"In this paper, we introduce a control strategy for applying the Sterile Insect Technique (SIT) to eliminate the population of Aedes mosquitoes which are vectors of various deadly diseases like dengue, zika, chikungunya... in a wide area. We use a system of reaction-diffusion equations to model the mosquito population and study the effect of releasing sterile males. Without any human intervention, and due to the so-called hair-trigger effect, the introduction of only a few individuals (eggs or fertilized females) can lead to the invasion of mosquitoes in the whole region after some time. To avoid this phenomenon, our strategy is to keep releasing a small number of sterile males in the treated zone and move this release forward with a negative forcing speed c to push back the invasive front of wild mosquitoes. By using traveling wave analysis, we show in the present paper that the strategy succeeds in repulsing the population while consuming a finite amount of mosquitoes in any finite time interval even though we treat a moving half-space. Moreover, we succeed in constructing a 'forced' traveling wave for our system moving at the same speed as the releases. We also provide some numerical illustrations for our results.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47032317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain. Asymptotic analysis is applied to determine the permeability of a tissue with a roll of blood vessels.
{"title":"Pressure boundary conditions for viscous flows in thin tube structures: Stokes equations with locally distributed Brinkman term","authors":"G. Panasenko, K. Pileckas","doi":"10.1051/mmnp/2023016","DOIUrl":"https://doi.org/10.1051/mmnp/2023016","url":null,"abstract":"The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain. Asymptotic analysis is applied to determine the permeability of a tissue with a roll of blood vessels.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47574423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. Developing new approaches that help control the spread of infectious diseases is a critical issue for public health. Such approaches must consider the available resources and capacity of the healthcare system. In this paper, we present a new mathematical approach to controlling an epidemic model by investigating the optimal control that aims to bring the output of the epidemic to target a desired disease output yd = (yid)i∈{0,...,N}. First, we use the state-space technique to transfer the trajectory controllability to optimal control with constraints on the final state. Then, we use the fixed point theorems to determine the set of admissible controls and solve the output trajectory controllability problem. Finally, we apply our method to the model of a measles epidemic, and we give a numerical simulation to illustrate the findings of our approach. Mathematics Subject Classification. — Please, give AMS classification codes —.
{"title":"OUTPUT TRAJECTORY CONTROLLABILITY OF A DISCRETE-TIME SIR EPIDEMIC MODEL","authors":"Lahbib Benahmadi, M. Lhous, A. Tridane, M. Rachik","doi":"10.1051/mmnp/2023015","DOIUrl":"https://doi.org/10.1051/mmnp/2023015","url":null,"abstract":"Abstract. Developing new approaches that help control the spread of infectious diseases is a critical issue for public health. Such approaches must consider the available resources and capacity of the healthcare system. In this paper, we present a new mathematical approach to controlling an epidemic model by investigating the optimal control that aims to bring the output of the epidemic to target a desired disease output yd = (yid)i∈{0,...,N}. First, we use the state-space technique to transfer the trajectory controllability to optimal control with constraints on the final state. Then, we use the fixed point theorems to determine the set of admissible controls and solve the output trajectory controllability problem. Finally, we apply our method to the model of a measles epidemic, and we give a numerical simulation to illustrate the findings of our approach.\u0000Mathematics Subject Classification. — Please, give AMS classification codes —.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43927263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nader El Khatib, O. Kafi, D. Oliveira, A. Sequeira, J. Tiago
Atherosclerosis, as a result of an inflammatory process, is the thickening and loss of elasticity of the walls of arteries that is associated with the formation of atherosclerotic plaques within the arterial intima, which present a double threat. A piece of vulnerable plaque can break off and be carried by the bloodstream until it gets stuck; and plaque that narrows an artery may lead to a thrombus that sticks to the blood vessel's inner wall. The purpose of the present article is to compare effects across different atheromatous plaque material assumptions on hemodynamics and biomechanics within a partly patient-specific computational domain representing an atherosclerotic artery. A full scale 3D FSI numerical model is implemented and different material hyperelastic assumptions are considered for comparison purposes. The 3D realistic geometry is reconstructed from a medical image. This technique may be useful, specially with the recent advances in computer-aided design (CAD), medical imaging, and 3D printing technologies that have provided a rapid and cost efficient method to generate arterial stenotic biomodels, making in vitro studies a valuable and powerful tool. To understand our results, hemodynamic parameters and structural stress analysis were performed. The results are consistent with previous findings.
{"title":"A NUMERICAL 3D FLUID-STRUCTURE INTERACTION MODEL FOR BLOOD\u0000\u0000FLOW IN A MRI-BASED ATHEROSCLEROTIC ARTERY","authors":"Nader El Khatib, O. Kafi, D. Oliveira, A. Sequeira, J. Tiago","doi":"10.1051/mmnp/2023014","DOIUrl":"https://doi.org/10.1051/mmnp/2023014","url":null,"abstract":"Atherosclerosis, as a result of an inflammatory process, is the thickening and loss of elasticity of the walls of arteries that is associated with the formation of atherosclerotic plaques within the arterial intima, which present a double threat. A piece of vulnerable plaque can break off and be carried by the bloodstream until it gets stuck; and plaque that narrows an artery may lead to a thrombus that sticks to the blood vessel's inner wall. The purpose of the present article is to compare effects across different atheromatous plaque material assumptions on hemodynamics and biomechanics within a partly patient-specific computational domain representing an atherosclerotic artery. A full scale 3D FSI numerical model is implemented and different material hyperelastic assumptions are considered for comparison purposes. The 3D realistic geometry is reconstructed from a medical image. This technique may be useful, specially with the recent advances in computer-aided design (CAD), medical imaging, and 3D printing technologies that have provided a rapid and cost efficient method to generate arterial stenotic biomodels, making in vitro studies a valuable and powerful tool. To understand our results, hemodynamic parameters and structural stress analysis were performed. The results are consistent with previous findings.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44575919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leadingn cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, heterogeneous populations) allows the communicable disease to become endemic even if the basic reproduction number is less than $1$. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information.
{"title":"A minimal model coupling communicable and non-communicable diseases","authors":"M. Marvá, E. Venturino, Carmen Vera","doi":"10.1051/mmnp/2023026","DOIUrl":"https://doi.org/10.1051/mmnp/2023026","url":null,"abstract":"This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leadingn cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity.\u0000Our results show that considering the non-communicable disease (in the end, heterogeneous populations) allows the communicable disease to become endemic even if the basic reproduction number is less than $1$. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42609988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates a three-strategy (cooperators, toxin producers, and cheaters) $N$-player division of labor game in bacterial populations. We construct the replicator equation to discuss the evolution of the frequency of the three strategies. Firstly, we prove that the interior equilibrium is always unstable, the three strategies cannot coexist. Secondly, according to Sotomayor's theorem, the system undergoes transcritical bifurcation. Furthermore, the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter (toxin rate, absorption rate, toxin quantity, etc) is analyzed. In summary, high toxicity rates, high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low, and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist. All players perform the task. Finally, the triangle diagrams and three-dimensional diagrams are presented, which show the initial conditions of the three strategies also affect the dynamic results. As the amount of toxin increases, the range of players who choose to perform tasks widens.
{"title":"Analysis of dynamic evolution process of the $N$-player division of labor game model","authors":"Hairui Yuan, Xinzhu Meng","doi":"10.1051/mmnp/2023013","DOIUrl":"https://doi.org/10.1051/mmnp/2023013","url":null,"abstract":"This paper investigates a three-strategy (cooperators, toxin producers, and cheaters) $N$-player division of labor game in bacterial populations. We construct the replicator equation to discuss the evolution of the frequency of the three strategies. Firstly, we prove that the interior equilibrium is always unstable, the three strategies cannot coexist. Secondly, according to Sotomayor's theorem, the system undergoes transcritical bifurcation. Furthermore, the sensitivity of the two-dimensional evolutionary state diagrams to the third parameter (toxin rate, absorption rate, toxin quantity, etc) is analyzed. In summary, high toxicity rates, high levels of toxins, and low levels of competition tend to promote cooperation. All players choose to perform the task, and the cheater disappears. When the absorption rate of cooperators is high enough, only cooperators exist in the population over time. When the absorption rate of the cooperator is low, and the absorption rate of the toxin producer is greater than the threshold, the cooperator and the toxin producer coexist. All players perform the task. Finally, the triangle diagrams and three-dimensional diagrams are presented, which show the initial conditions of the three strategies also affect the dynamic results. As the amount of toxin increases, the range of players who choose to perform tasks widens.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43619389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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