In some recent work, we have introduced some efficiency functionals to account for optimal dispersal strategies of predators in search of food. The optimization parameter in this framework is given by the L'evy exponent of the dispersal of the predators. In this paper, we apply our model to the case of foragers with finite lifetime (i.e., foragers which need to eat a certain amount of food in a given time, otherwise they die). Specifically, we consider the case in which the initial distribution of the forager coincides with a stationary distribution of the targets and we determine the optimal L'evy exponent for the associated efficiency functional. Namely, we show that if the Fourier transform of the prey distribution is supported in a sufficiently small ball, then the optimizer is given by a Gaussian dispersal, and if instead the Fourier transform of the prey distribution is supported in the complement of a suitable ball, then the ballistic diffusion provides an optimizer (precise conditions for the uniqueness of these optimizers are also given).
{"title":"Lévy flights, optimal foraging strategies, and foragers with a finite lifespan","authors":"S. Dipierro, Giovanni Giacomin, Enrico Valdinoci","doi":"10.1051/mmnp/2024015","DOIUrl":"https://doi.org/10.1051/mmnp/2024015","url":null,"abstract":"In some recent work, we have introduced some efficiency functionals to account for optimal dispersal strategies of predators in search of food.\u0000The optimization parameter in this framework is given by the L'evy exponent of the dispersal of the predators.\u0000In this paper,\u0000we apply our model to the case of foragers with finite lifetime (i.e., foragers which need to eat a certain amount of food in a given time, otherwise they die).\u0000Specifically, we consider the case in which the initial distribution of the forager coincides with a stationary distribution of the targets and we determine the optimal L'evy exponent for the associated efficiency functional.\u0000Namely, we show that if the Fourier transform of the prey distribution is supported in a sufficiently small ball, then the optimizer is given by a Gaussian dispersal, and if instead the Fourier transform of the prey distribution is supported in the complement of a suitable ball, then\u0000the ballistic diffusion provides an optimizer\u0000(precise conditions for the uniqueness of these optimizers are also given).","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141683803","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}
T. Dobroserdova, Lyudmila Yurpolskaya, Yuri Vassilevski, Andrey Svobodov
Personalized blood flow models are used for optimization of the Fontan procedure. In this paper we discuss clinical data for model initialization. Before the Fontan procedure patients undergo CT or MRI examination. Computational domain of interest is reconstructed from this data. CT images are shown to have a better spatial resolution and quality and are more suitable for segmentation. MRI data gives information about blood flow rates and it is utilized for setting boundary conditions in local 3D hemodynamic models. We discovered that the MRI data is contradictory and too inaccurate for setting boundary conditions: the error of measured velocities is comparable with blood velocities in veins. We discuss a multiscale 1D3D circulation model as potentially suitable for prediction of the Fontan procedure results. Such model may be initialized with more reliable data (MR measurements of blood flow in aorta and ultrasound examination of easily accessible vessels) and take into account collateral and fenestration blood flows which are typical for Fontan patients. We have calculated these flow rates for several patients and demonstrated that such flows occur systematically.
{"title":"Patient-specific input data for predictive modeling of the Fontan operation","authors":"T. Dobroserdova, Lyudmila Yurpolskaya, Yuri Vassilevski, Andrey Svobodov","doi":"10.1051/mmnp/2024013","DOIUrl":"https://doi.org/10.1051/mmnp/2024013","url":null,"abstract":"Personalized blood flow models are used for optimization of the Fontan procedure. In this paper we discuss clinical data for model initialization. Before the Fontan procedure patients undergo CT or MRI examination. Computational domain of interest is reconstructed from this data. CT images are shown to have a better spatial resolution and quality and are more suitable for segmentation. MRI data gives information about blood flow rates and it is utilized for setting boundary conditions in local 3D hemodynamic models.\u0000 \u0000We discovered that the MRI data is contradictory and too inaccurate for setting boundary conditions: the error of measured velocities is comparable with blood velocities in veins. We discuss a multiscale 1D3D circulation model as potentially suitable for prediction of the Fontan procedure results. Such model may be initialized with more reliable data (MR measurements of blood flow in aorta and ultrasound examination of easily accessible vessels) and take into account collateral and fenestration blood flows which are typical for Fontan patients. We have calculated these flow rates for several patients and demonstrated that such flows occur systematically.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141355312","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}
Frederico José Ribeiro Pelogia, Henrique Mohallem Paiva, Roberson Saraiva Polli
Contribution: This study provides insights into COVID-19 dynamics by employing a phenomenological model representing multiple epidemiological waves. It aims to support decision-making for health authorities and hospital administrators, particularly in optimizing Intensive Care Unit (ICU) bed management and implementing effective containment measures. Background: Given the intricate complexity of ICU environments, utilizing a mathematical model to anticipate occupancy is highly beneficial and might mitigate mortality rates associated with COVID-19. The study focuses on the evolution of intensive care patient numbers across multiple epidemiological waves in Italian regions. Methodology: Our methodology involves the application of a low-complexity phenomenological model with an efficient optimization procedure. ICU occupancy data from five populous Italian regions are utilized to demonstrate the model’s efficacy on describing historical data and providing forecasts for two-week intervals. Findings: Drawing from the analyzed ICU occupancy data, the study establishes the effectiveness of the proposed model. It successfully fits historical data and offers accurate forecasts, achieving an average relative RMSE of 0.51% for the whole fit and 0.93% for the predictions, across all regions. Beyond the immediate context, the model low complexity and efficient optimization make it suitable to diverse regions and diseases, supporting the tracking and containment of future epidemics.
{"title":"Multi-wave modelling and short-term prediction of ICU bed occupancy by patients with COVID-19 in regions of Italy","authors":"Frederico José Ribeiro Pelogia, Henrique Mohallem Paiva, Roberson Saraiva Polli","doi":"10.1051/mmnp/2024012","DOIUrl":"https://doi.org/10.1051/mmnp/2024012","url":null,"abstract":"Contribution: This study provides insights into COVID-19 dynamics by employing a phenomenological model representing multiple epidemiological waves. It aims to support decision-making for health authorities and hospital administrators, particularly in optimizing Intensive Care Unit (ICU) bed management and implementing effective containment measures. Background: Given the intricate complexity of ICU environments, utilizing a mathematical model to anticipate occupancy is highly beneficial and might mitigate mortality rates associated with COVID-19. The study focuses on the evolution of intensive care patient numbers across multiple epidemiological waves in Italian regions. Methodology: Our methodology involves the application of a low-complexity phenomenological model with an efficient optimization procedure. ICU occupancy data from five populous Italian regions are utilized to demonstrate the model’s efficacy on describing historical data and providing forecasts for two-week intervals. Findings: Drawing from the analyzed ICU occupancy data, the study establishes the effectiveness of the proposed model. It successfully fits historical data and offers accurate forecasts, achieving an average relative RMSE of 0.51% for the whole fit and 0.93% for the predictions, across all regions. Beyond the immediate context, the model low complexity and efficient optimization make it suitable to diverse regions and diseases, supporting the tracking and containment of future epidemics.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141104312","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}
We propose a hierarchy of mathematical models for the numerical simulation of active thin structures in a viscous fluid and its application to mucociliary transport. Our aim is to simulate large forests of cilia and analyze the collective dynamics arising in the flow, as well as their impact on the efficiency of the mucus transport. In a 3d model we describe the cilia individually and study their joint actions on the fluid. The model is built upon a 3d Stokes problem with singular source terms that represent the action of the 1d cilia on the fluid, including the background flow (making the problem nonlocal). Surface tension between the periciliary layer and the mucus is taken into account. From the 3d model we also derive a 1d space averaged model, describing the dynamics of the mean velocity of the mucus that is propelled by the cilia, hence allowing lower computational costs and still providing useful characterization of the efficiency of the transport. Mathematical properties of the models (existence and uniqueness of solutions in suitable functional spaces) are analyzed. Numerical simulations highlight the influence of critical parameters on the efficiency of the mucociliary transport in the case of dense forests of cilia.
{"title":"Mathematical Modelling of Natural Phenomena","authors":"A. Decoene, Sebastien Martin, Chabane Meziane","doi":"10.1051/mmnp/2024010","DOIUrl":"https://doi.org/10.1051/mmnp/2024010","url":null,"abstract":"We propose a hierarchy of mathematical models for the numerical simulation of active thin structures in a viscous fluid and its application to mucociliary transport. Our aim is to simulate large forests of cilia and analyze the collective dynamics arising in the flow, as well as their impact on the efficiency of the mucus transport. In a 3d model we describe the cilia individually and study their joint actions on the fluid. The model is built upon a 3d Stokes problem with singular source terms that represent the action of the 1d cilia on the fluid, including the background flow (making the problem nonlocal). Surface tension between the periciliary layer and the mucus is taken into account. From the 3d model we also derive a 1d space averaged model, describing the dynamics of the mean velocity of the mucus that is propelled by the cilia, hence allowing lower computational costs and still providing useful characterization of the efficiency of the transport. Mathematical properties of the models (existence and uniqueness of solutions in suitable functional spaces) are analyzed. Numerical simulations highlight the influence of critical parameters on the efficiency of the mucociliary transport in the case of dense forests of cilia.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141110272","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}
We propose a general model to investigate the effect of the distinct dispersal coefficient {for the} infected and susceptible hosts on the pathogen dynamics. The mathematical challenge lies in the fact that the investigated model is partially degenerate and the solution map is not compact. The spatial heterogeneity of the model parameters and the distinct diffusion coefficients induce infection in the low-risk regions. In fact, as infection dispersal increases, the reproduction of the pathogen particles decreases. The dynamics of the investigated model is governed by the value of the basic reproduction number $R_0$. {If $R_0leq1$, then the} pathogen particles extinct, and for $R_0>1$ the pathogen particles persist, and we guarantees of the existence of at least one positive steady state. The asymptotic profile of the positive steady state is shown in the case when one or both diffusion coefficients for the host tends to zero or infinity.
{"title":"Generalities on a delayed spatiotemporal host-pathogen infection model with distinct dispersal rates","authors":"Djilali salih","doi":"10.1051/mmnp/2024008","DOIUrl":"https://doi.org/10.1051/mmnp/2024008","url":null,"abstract":"We propose a general model to investigate the effect of the distinct dispersal coefficient {for the} infected and susceptible hosts on the pathogen dynamics. The mathematical challenge lies in the fact that the investigated model is partially degenerate and the solution map is not compact. The spatial heterogeneity of the model parameters and the distinct diffusion coefficients induce infection in the low-risk regions. In fact, as infection dispersal increases, the reproduction of the pathogen particles decreases. The dynamics of the investigated model is governed by the value of the basic reproduction number $R_0$. {If $R_0leq1$, then the} pathogen particles extinct, and for $R_0>1$ the pathogen particles persist, and we guarantees of the existence of at least one positive steady state. The asymptotic profile of the positive steady state is shown in the case when one or both diffusion coefficients for the host tends to zero or infinity.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140971096","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}
Atlantic cod collapsed in the late 20th century after being harvested heavily for 50 years. This paper aims to design conservation guidelines for the cod population, which is diminishing due to predation by grey seals and cannibalism. For this purpose, we first designed a continuous time ecological model (with and without the Allee effect) using a system of differential equations consisting of juvenile Atlantic cod, adult Atlantic cod, and grey seals. The developed model has set forth global existence, non-negativity, and long-term behavior. Subsequently, to handle the extinction problem cost-effectively, Pontryagin's principle is employed to construct the optimal control, which is then numerically solved using an iterative forward–backward method. We numerically explored the impact of the Allee effect on cod survival within the original model and its two extended versions (i) stochastic and (ii) reaction-diffusion, to thoroughly understand the possible consequences wherein a population has cannibalistic tendencies. The numerical comparison between the non-Allee and Allee models (Ordinary, Stochastic, Reaction-Diffusion) reveals that the Allee effect may significantly promote recovery and benefit the cannibalistic population. We adopted a partial rank correlation coefficient (PRCC) to conduct a global sensitivity analysis to estimate the most sensitive parameters responsible for cod prevalence.
{"title":"The role of Allee effect in cannibalistic species: An action plan to sustain the declining cod population","authors":"Parimita Roy, Sanjoli Jain, Mohamed Maama","doi":"10.1051/mmnp/2024007","DOIUrl":"https://doi.org/10.1051/mmnp/2024007","url":null,"abstract":"Atlantic cod collapsed in the late 20th century after being harvested heavily for 50 years. This paper aims to design conservation guidelines for the cod population, which is diminishing due to predation by grey seals and cannibalism. For this purpose, we first designed a continuous time ecological model (with and without the Allee effect) using a system of differential equations consisting of juvenile Atlantic cod, adult Atlantic cod, and grey seals. The developed model has set forth global existence, non-negativity, and long-term behavior. Subsequently, to handle the extinction problem cost-effectively, Pontryagin's principle is employed to construct the optimal control, which is then numerically solved using an iterative forward–backward method. We numerically explored the impact of the Allee effect on cod survival within the original model and its two extended versions (i) stochastic and (ii) reaction-diffusion, to thoroughly understand the possible consequences wherein a population has cannibalistic tendencies. The numerical comparison between the non-Allee and Allee models (Ordinary, Stochastic, Reaction-Diffusion) reveals that the Allee effect may significantly promote recovery and benefit the cannibalistic population. We adopted a partial rank correlation coefficient (PRCC) to conduct a global sensitivity analysis to estimate the most sensitive parameters responsible for cod prevalence.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140974365","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}
José Geiser Villavicencio-Pulido, I. Barradas, C. Nila-Luévano
Many infections are transmitted by direct contacts. Usually one single direct contact is needed to transmit the required minimum infectious load. Most models describe contagions by single contacts using a term of the type mass action law. However, modelling infections that are transmitted after the susceptible individual had contact with several sources of infection requires more than mass action law terms. We call additive multiple contacts those that do not produce infection by themselves, but can produce infection if they happen simultaneously. We are interested in understanding the role played by R0 missing the mark in infections in which the minimum infectious load is reached not only by single contacts but also by additive multiple contacts. We propose different mathematical models describing not only infections by one single contact but also by additive multiple contacts. We show all models have the same value of R0, but correspond to different epidemiological mechanisms. Two models show contagions by additive multiple contacts and a third one shows reduction of infections by some saturation process which is not captured by R0. This shows that trying to control the epidemics by controlling R0 could be unsufficient or in some cases waste resources.
{"title":"Additive multiple contacts and saturation phenomena in epidemiological models are not detected by $R_0$","authors":"José Geiser Villavicencio-Pulido, I. Barradas, C. Nila-Luévano","doi":"10.1051/mmnp/2024006","DOIUrl":"https://doi.org/10.1051/mmnp/2024006","url":null,"abstract":"Many infections are transmitted by direct contacts. Usually one single direct contact is needed to transmit the required minimum infectious load. Most models describe contagions by single contacts using a term of the type mass action law. However, modelling infections that are transmitted after the susceptible individual had contact with several sources of infection requires more than mass action law terms. We call additive multiple contacts those that do not produce infection by themselves, but can produce infection if they happen simultaneously. We are interested in understanding the role played by R0 missing the mark in infections in which the minimum infectious load is reached not only by single contacts but also by additive multiple contacts. We propose different mathematical models describing not only infections by one single contact but also by additive multiple contacts. We show all models have the same value of R0, but correspond to different epidemiological mechanisms. Two models show contagions by additive multiple contacts and a third one shows reduction of infections by some saturation process which is not captured by R0. This shows that trying to control the epidemics by controlling R0 could be unsufficient or in some cases waste resources.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140353088","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}
Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be $y$ or $z$ independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.
{"title":"(3+1)-dimensional Gardner equation deformed from (1+1)-dimensional Gardner equation and its conservation law","authors":"Guiming Jin, Xueping Cheng, Jianan Wang","doi":"10.1051/mmnp/2024004","DOIUrl":"https://doi.org/10.1051/mmnp/2024004","url":null,"abstract":"Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be $y$ or $z$ independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140254782","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, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G∗ of system (4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions’periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.
本文考虑了一个具有恐惧效应和霍林 III 型功能响应的延迟扩散捕食者-猎物系统，并对该系统施加了诺伊曼边界条件。首先，我们探讨了唯一正常数稳态的稳定性和局部霍普夫分岔的存在性。然后通过比较原理和迭代法得到系统 (4) 的全局吸引域 G∗。通过构建 Lyapunov 函数，我们研究了周期解的周期均匀有界性。最后，我们通过吴的全局霍普夫分岔定理证明了周期解的全局连续性。此外，我们还给出了一些支持分析结果的数值模拟。
{"title":"Global Hopf bifurcation of a delayed diffusive Gause-type predator-prey system with the fear effect and Holling type III functional response","authors":"Qian Zhang, Ming Liu, Xiaofeng Xu","doi":"10.1051/mmnp/2024003","DOIUrl":"https://doi.org/10.1051/mmnp/2024003","url":null,"abstract":"In this paper, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G∗ of system (4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions’periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139773758","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, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G∗ of system (4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions’periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.
本文考虑了一个具有恐惧效应和霍林 III 型功能响应的延迟扩散捕食者-猎物系统，并对该系统施加了诺伊曼边界条件。首先，我们探讨了唯一正常数稳态的稳定性和局部霍普夫分岔的存在性。然后通过比较原理和迭代法得到系统 (4) 的全局吸引域 G∗。通过构建 Lyapunov 函数，我们研究了周期解的周期均匀有界性。最后，我们通过吴的全局霍普夫分岔定理证明了周期解的全局连续性。此外，我们还给出了一些支持分析结果的数值模拟。
{"title":"Global Hopf bifurcation of a delayed diffusive Gause-type predator-prey system with the fear effect and Holling type III functional response","authors":"Qian Zhang, Ming Liu, Xiaofeng Xu","doi":"10.1051/mmnp/2024003","DOIUrl":"https://doi.org/10.1051/mmnp/2024003","url":null,"abstract":"In this paper, a delayed diffusive predator-prey system with the fear effect and Holling type III functional response is considered, and Neumann boundary condition is imposed on this system. First, we explore the stability of the unique positive constant steady state and the existence of local Hopf bifurcation. Then the global attraction domain G∗ of system (4) is obtained by the comparison principle and the iterative method. Through constructing the Lyapunov function, we investigate uniform boundedness of periodic solutions’periods. Finally, we prove the global continuation of periodic solutions by the global Hopf bifurcation theorem of Wu. Moreover, some numerical simulations that support the analysis results are given.","PeriodicalId":18285,"journal":{"name":"Mathematical Modelling of Natural Phenomena","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139833369","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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