Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model. Numerical examples are provided to support our obtained results.
{"title":"Effect of fear on two predator-one prey model in deterministic and fluctuating environment","authors":"D. Mukherjee","doi":"10.5206/MASE/13541","DOIUrl":"https://doi.org/10.5206/MASE/13541","url":null,"abstract":"Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model. Numerical examples are provided to support our obtained results.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43359311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-12DOI: 10.1101/2021.02.10.21251500
Q. Griette, J. Demongeot, P. Magal
We provide a new method to analyze the COVID-19 cumulative reported cases data based on a two-step process: first we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.
{"title":"A robust phenomenological approach to investigate COVID-19 data for France","authors":"Q. Griette, J. Demongeot, P. Magal","doi":"10.1101/2021.02.10.21251500","DOIUrl":"https://doi.org/10.1101/2021.02.10.21251500","url":null,"abstract":"We provide a new method to analyze the COVID-19 cumulative reported cases data based on a two-step process: first we regularize the data by using a phenomenological model which takes into account the endemic or epidemic nature of the time period, then we use a mathematical model which reproduces the epidemic exactly. This allows us to derive new information on the epidemic parameters and to compute the effective basic reproductive ratio on a daily basis. Our method has the advantage of identifying robust trends in the number of new infectious cases and produces an extremely smooth reconstruction of the epidemic. The number of parameters required by the method is parsimonious: for the French epidemic between February 2020 and January 2021 we use only 11 parameters in total.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44597590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.
{"title":"Traveling waves in cooperative predation: relaxation of sublinearity","authors":"Srijana Ghimire, Xiang-Sheng Wang","doi":"10.5206/MASE/13393","DOIUrl":"https://doi.org/10.5206/MASE/13393","url":null,"abstract":"In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42185749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Natural disasters occur across the globe, resulting in billions of dollars of damage each year. Effective preparation before a disaster can help to minimize damages, economic impact, and loss of human life. This paper uses a game theory framework to set up a leader-followers model for resource distribution to several geographic zones before an adverse event. The researchers model population members who may choose to prepare in advance of an event by acquiring supplies, whereas others may wait until the last minute. Failure to prepare in advance could result in a significant loss due to the chance that supplies may no longer be available. Numerical simulations are run to determine how the leader should distribute supplies to maximize the preparedness of the overall population. It was found that population size is a significant factor for supply distribution, but the behaviour of individuals within a zone is also important. Much of the current resource allocation research focuses on the logistics and economics of supply distribution, but this paper demonstrates that social aspects should also be considered.
{"title":"A leader-followers game of emergency preparedness for adverse events","authors":"M. Nahirniak, M. Cojocaru, T. Migot","doi":"10.5206/MASE/11093","DOIUrl":"https://doi.org/10.5206/MASE/11093","url":null,"abstract":"Natural disasters occur across the globe, resulting in billions of dollars of damage each year. Effective preparation before a disaster can help to minimize damages, economic impact, and loss of human life. This paper uses a game theory framework to set up a leader-followers model for resource distribution to several geographic zones before an adverse event. The researchers model population members who may choose to prepare in advance of an event by acquiring supplies, whereas others may wait until the last minute. Failure to prepare in advance could result in a significant loss due to the chance that supplies may no longer be available. Numerical simulations are run to determine how the leader should distribute supplies to maximize the preparedness of the overall population. It was found that population size is a significant factor for supply distribution, but the behaviour of individuals within a zone is also important. Much of the current resource allocation research focuses on the logistics and economics of supply distribution, but this paper demonstrates that social aspects should also be considered.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45815678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In ecological communities, the behaviour of individuals and the interaction between species may change between seasons, yet this seasonal variation is often not represented explicitly in mathematical models. As global change is predicted to alter season length and other climatic aspects, such seasonal variation needs to be included in models in order to make reasonable predictions for community dynamics. The resulting mathematical descriptions are nonautonomous models with a large number of parameters, and are therefore challenging to analyze. We present a model for two predators and one prey, whereby one predator switches hunting behaviour to seasonally include alternative prey when available. We use a combination of temporal averaging and invasion analysis to derive simplified models and determine the behaviour of the system, in particular to gain insight into conditions under which the two predators can coexist in a changing climate. We compare our results with numerical simulations of the temporally varying model.
{"title":"Seasonal dynamics of a generalist and a specialist predator on a single prey","authors":"N. Bolohan, V. LeBlanc, F. Lutscher","doi":"10.5206/mase/13569","DOIUrl":"https://doi.org/10.5206/mase/13569","url":null,"abstract":"In ecological communities, the behaviour of individuals and the interaction between species may change between seasons, yet this seasonal variation is often not represented explicitly in mathematical models. As global change is predicted to alter season length and other climatic aspects, such seasonal variation needs to be included in models in order to make reasonable predictions for community dynamics. The resulting mathematical descriptions are nonautonomous models with a large number of parameters, and are therefore challenging to analyze. We present a model for two predators and one prey, whereby one predator switches hunting behaviour to seasonally include alternative prey when available. We use a combination of temporal averaging and invasion analysis to derive simplified models and determine the behaviour of the system, in particular to gain insight into conditions under which the two predators can coexist in a changing climate. We compare our results with numerical simulations of the temporally varying model.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70664502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modifiednumerical response. This numerical response accounts for the reduction in the her-bivore's growth and reproduction due to chemical defenses from plants. It is shownthat the system exhibits very rich dynamics including saddle-node bifurcations, Hopfbifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical sim-ulations are presented to illustrate the occurrence of multitype bistability, limit cycles,homoclinic orbits and heteroclinic orbits. We also discuss the ecological implicationsof the resulting dynamics.
{"title":"Dynamics of a plant-herbivore model with a chemically-mediated numerical response","authors":"Lin Wang, James Watmough, Fang Yu","doi":"10.5206/mase/11067","DOIUrl":"https://doi.org/10.5206/mase/11067","url":null,"abstract":"A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modifiednumerical response. This numerical response accounts for the reduction in the her-bivore's growth and reproduction due to chemical defenses from plants. It is shownthat the system exhibits very rich dynamics including saddle-node bifurcations, Hopfbifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical sim-ulations are presented to illustrate the occurrence of multitype bistability, limit cycles,homoclinic orbits and heteroclinic orbits. We also discuss the ecological implicationsof the resulting dynamics.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70664469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In order to determine the effectiveness of non-pharmaceutical interventions on an epidemic, we develop an agent-based model that simulates the spread of an infectious disease in a small community and its emerging phenomena. We vary parameters such as initial population, initial infected, infection rate, recovery rate, death rate, and asymptomatic rates, as inputs. Our simulations show that (i) random mass testing decreases the number of deaths, infections and time duration; (ii) as well as quarantines; (iii) social distancing lengthen outbreak period to an extent and helps flatten the epidemic curve; and (iv) the most effective combination of NPIs to minimize death, infection and duration is no mass testing, no social distancing and a total lockdown. Results of this study can aid decision makers in their policies to be implemented to have an optimal output.
{"title":"Determining the effectiveness of practicing non-pharmaceutical interventions in improving virus control in a pandemic using agent-based modelling","authors":"C. A. Buhat, S. Villanueva","doi":"10.5206/mase/10876","DOIUrl":"https://doi.org/10.5206/mase/10876","url":null,"abstract":"In order to determine the effectiveness of non-pharmaceutical interventions on an epidemic, we develop an agent-based model that simulates the spread of an infectious disease in a small community and its emerging phenomena. We vary parameters such as initial population, initial infected, infection rate, recovery rate, death rate, and asymptomatic rates, as inputs. Our simulations show that (i) random mass testing decreases the number of deaths, infections and time duration; (ii) as well as quarantines; (iii) social distancing lengthen outbreak period to an extent and helps flatten the epidemic curve; and (iv) the most effective combination of NPIs to minimize death, infection and duration is no mass testing, no social distancing and a total lockdown. Results of this study can aid decision makers in their policies to be implemented to have an optimal output.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48943517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.
{"title":"A procedure for deriving new ODE models: Using the generalized linear chain trick to incorporate phase-type distributed delay and dwell time assumptions","authors":"P. Hurtado, Cameron Richards","doi":"10.5206/mase/10857","DOIUrl":"https://doi.org/10.5206/mase/10857","url":null,"abstract":"Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70664420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Preface: Thematic Issue in Mathematical Biology and Applied Evolutionary Equations
前言：数学生物学与应用进化方程专题
{"title":"Preface: Thematic Issue in Mathematical Biology and Applied Evolutionary Equations","authors":"N. Vaidya, Dhruba R. Adhikari","doi":"10.5206/mase/13502","DOIUrl":"https://doi.org/10.5206/mase/13502","url":null,"abstract":"Preface: Thematic Issue in Mathematical Biology and Applied Evolutionary Equations","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48065761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The vadose zone is the portion of the subsurface above the water table and its pore space usually contains air and water. Due to the presence of infiltration, erosion, plant growth, microbiota, contaminant transport, aquifer recharge, and discharge to surface water, it is crucial to predict the transport rate of water and other substances within this zone. However, flow in the vadose zone has many complications as the parameters that control it are extremely sensitive to the saturation of the media, leading to a nonlinear problem. This flow is referred as unsaturated flow and is governed by Richards equation. Analytical solutions for this equation exists only for simplified cases, so most practical situations require a numerical solution. Nevertheless, the nonlinear nature of Richards equation introduces challenges that causes numerical solutions for this problem to be computationally expensive and, in some cases, unreliable. High order mimetic finite difference operators are discrete analogs of the continuous differential operators and have been extensively used in the fields of fluid and solid mechanics. In this work, we present a numerical approach involving high order mimetic operators along with a Newton root-finding algorithm for the treatment of the nonlinear component. Fully-implicit time discretization scheme is used to deal with the problem’s stiffness.
{"title":"High order mimetic difference simulation of unsaturated flow using Richards equation","authors":"Angel Boada Velazco, Johnny Corbino, J. Castillo","doi":"10.5206/mase/10874","DOIUrl":"https://doi.org/10.5206/mase/10874","url":null,"abstract":"The vadose zone is the portion of the subsurface above the water table and its pore space usually contains air and water. Due to the presence of infiltration, erosion, plant growth, microbiota, contaminant transport, aquifer recharge, and discharge to surface water, it is crucial to predict the transport rate of water and other substances within this zone. However, flow in the vadose zone has many complications as the parameters that control it are extremely sensitive to the saturation of the media, leading to a nonlinear problem. This flow is referred as unsaturated flow and is governed by Richards equation. Analytical solutions for this equation exists only for simplified cases, so most practical situations require a numerical solution. Nevertheless, the nonlinear nature of Richards equation introduces challenges that causes numerical solutions for this problem to be computationally expensive and, in some cases, unreliable. High order mimetic finite difference operators are discrete analogs of the continuous differential operators and have been extensively used in the fields of fluid and solid mechanics. In this work, we present a numerical approach involving high order mimetic operators along with a Newton root-finding algorithm for the treatment of the nonlinear component. Fully-implicit time discretization scheme is used to deal with the problem’s stiffness.","PeriodicalId":93797,"journal":{"name":"Mathematics in applied sciences and engineering","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70664458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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