9,544 results
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2. Three Ways of Treating a Linear Delay Differential Equation
- Author
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Sah, Si Mohamed, Rand, Richard H., and Belhaq, Mohamed, editor
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- 2018
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3. Continuous model for the rock–scissors–paper game between bacteriocin producing bacteria
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Neumann, Gunter and Schuster, Stefan
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- 2007
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4. Linear and Nonlinear Damping Effects on the Stability of the Ziegler Column
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Luongo, Angelo, D’Annibale, Francesco, and Belhaq, Mohamed, editor
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- 2015
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5. Hopf Bifurcations in Delayed Rock–Paper–Scissors Replicator Dynamics
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Wesson, Elizabeth and Rand, Richard
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- 2016
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6. On the Use of the Multiple Scale Harmonic Balance Method for Nonlinear Energy Sinks Controlled Systems
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Luongo, Angelo, Zulli, Daniele, and Belhaq, Mohamed, editor
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- 2015
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7. Hopf Bifurcation of Rayleigh Model with Delay
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Sun, Yan, Ma, Wenlian, Yang, Yuhang, editor, Ma, Maode, editor, and Liu, Baoxiang, editor
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- 2013
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8. Hopf Bifurcation in a Three-Stage-Structured Prey-Predator System with Predator Density Dependent
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Li, Shunyi, Xue, Xiangui, Zhao, Maotai, editor, and Sha, Junpin, editor
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- 2012
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9. Study on Effect of Wind Power System Parameters for Hopf Bifurcation Based on Continuation Method
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Ji, Li, XueSong, Zhou, and Wan, Xiaofeng, editor
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- 2011
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10. Comparative Analysis of Solution Methods for Delay Differential Equations in Hematology
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Bencheva, Gergana, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Lirkov, Ivan, editor, Margenov, Svetozar, editor, and Waśniewski, Jerzy, editor
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- 2010
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11. Nonlinear Dynamics of Networks: Applications to Mathematical Music Theory
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Clark, Jonathan Owen, Klouche, Timour, editor, and Noll, Thomas, editor
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- 2009
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12. Temporally-Periodic Solitons of the Parametrically Driven Damped Nonlinear Schrödinger Equation
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Zemlyanaya, E. V., Barashenkov, I. V., Alexeeva, N. V., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Margenov, Svetozar, editor, Vulkov, Lubin G., editor, and Waśniewski, Jerzy, editor
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- 2009
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13. Large Matrices Arising in Traveling Wave Bifurcations
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Simon, Peter L., Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Lirkov, Ivan, editor, Margenov, Svetozar, editor, and Waśniewski, Jerzy, editor
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- 2008
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14. Bifurcation Analysis of Reinforcement Learning Agents in the Selten’s Horse Game
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Lazaric, Alessandro, Munoz de Cote, Enrique, Dercole, Fabio, Restelli, Marcello, Carbonell, Jaime G., editor, Siekmann, J\'org, editor, Tuyls, Karl, editor, Nowe, Ann, editor, Guessoum, Zahia, editor, and Kudenko, Daniel, editor
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- 2008
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15. Reliable Computation of Equilibrium States and Bifurcations in Nonlinear Dynamics
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Gwaltney, C. Ryan, Stadtherr, Mark A., Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Dough, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Dongarra, Jack, editor, Madsen, Kaj, editor, and Waśniewski, Jerzy, editor
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- 2006
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16. Synchronised Behaviour in Three Coupled Faraday Disk Homopolar Dynamos
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Moroz, Irene M., Beig, R., editor, Ehlers, J., editor, Frisch, U., editor, Hepp, K., editor, Hillebrandt, W., editor, Imboden, D., editor, Jaffe, R. L., editor, Kippenhahn, R., editor, Lipowsky, R., editor, v. Löhneysen, H., editor, Ojima, I., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, and Lumley, John L., editor
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- 2001
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17. Computing Periodic Orbits
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Guckenheimer1, John, Beig, R., editor, Ehlers, J., editor, Frisch, U., editor, Hepp, K., editor, Hillebrandt, W., editor, Imboden, D., editor, Jaffe, R. L., editor, Kippenhahn, R., editor, Lipowsky, R., editor, v. Löhneysen, H., editor, Ojima, I., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, and Lumley, John L., editor
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- 2001
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18. Bifurcations and bursting of streaks in the turbulent wall layer
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Aubry, N., Sanghi, S., Moreau, R., editor, Metais, O., editor, and Lesieur, M., editor
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- 1991
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19. Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations
- Author
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Tina Verma and Arvind Kumar Gupta
- Subjects
Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,Evolutionary game theory ,Biodiversity ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Metapopulation ,symbols.namesake ,Transcritical bifurcation ,Evolutionary biology ,Mutation (genetic algorithm) ,symbols ,education ,Evolutionary dynamics ,Mathematics - Abstract
Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.
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- 2021
20. Multiple limit cycles for the continuous model of the rock–scissors–paper game between bacteriocin producing bacteria.
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Daoxiang, Zhang and Yan, Ping
- Subjects
- *
LIMIT cycles , *CONTINUOUS functions , *HOPF bifurcations , *ROCK-paper-scissors (Game) , *BACTERIOCINS - Abstract
In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]
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- 2017
- Full Text
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21. A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock-Paper-Scissors
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Claire M. Postlethwaite and Alastair M. Rucklidge
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Population ,General Physics and Astronomy ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,01 natural sciences ,symbols.namesake ,0101 mathematics ,education ,Quantitative Biology - Populations and Evolution ,Mathematical Physics ,Saddle ,Mathematics ,Hopf bifurcation ,Equilibrium point ,education.field_of_study ,Partial differential equation ,37G15, 34C37, 37C29, 91A22 ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Ode ,Populations and Evolution (q-bio.PE) ,Heteroclinic cycle ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Pattern Formation and Solitons ,010101 applied mathematics ,Ordinary differential equation ,FOS: Biological sciences ,symbols - Abstract
One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures
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- 2019
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22. Evolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population flow
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Junpyo Park
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Hopf bifurcation ,education.field_of_study ,General Mathematics ,Applied Mathematics ,Population ,General Physics and Astronomy ,Robustness (evolution) ,Statistical and Nonlinear Physics ,Fixed point ,symbols.namesake ,symbols ,Outflow ,Statistical physics ,Balanced flow ,Evolutionary dynamics ,education ,Multistability ,Mathematics - Abstract
Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.
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- 2021
23. Multiple limit cycles for the continuous model of the rock–scissors–paper game between bacteriocin producing bacteria
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Ping Yan and Zhang Dao-xiang
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Hopf bifurcation ,Continuous modelling ,Applied Mathematics ,010102 general mathematics ,Heteroclinic cycle ,16. Peace & justice ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,symbols ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Mathematical economics ,Mathematics - Abstract
Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.
- Published
- 2017
24. Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities
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Platkowski, Tadeusz and Zakrzewski, Jan
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- *
LIMIT cycles , *GAME theory , *ROCK-paper-scissors (Game) , *BIFURCATION theory , *POLYMORPHISM (Crystallography) , *EQUILIBRIUM , *MATHEMATICAL models , *MATRICES (Mathematics) - Abstract
Abstract: We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
25. Hopf Bifurcations in Delayed Rock–Paper–Scissors Replicator Dynamics
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Richard H. Rand and Elizabeth Wesson
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Statistics and Probability ,Economics and Econometrics ,Population ,Interval (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Control theory ,Limit cycle ,0103 physical sciences ,Replicator equation ,Applied mathematics ,Limit (mathematics) ,010306 general physics ,education ,Bifurcation ,Mathematics ,Hopf bifurcation ,education.field_of_study ,Applied Mathematics ,Function (mathematics) ,Computer Graphics and Computer-Aided Design ,Computer Science Applications ,Nonlinear Sciences::Chaotic Dynamics ,Computational Mathematics ,Computational Theory and Mathematics ,symbols - Abstract
We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.
- Published
- 2015
26. Nonlinear dynamics with Hopf bifurcations by targeted mutation in the system of rock-paper-scissors metaphor
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Junpyo Park
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Hopf bifurcation ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Biology ,01 natural sciences ,Intraspecific competition ,010305 fluids & plasmas ,symbols.namesake ,Nonlinear system ,Targeted Mutation ,Linear stability analysis ,Evolutionary biology ,0103 physical sciences ,Mutation (genetic algorithm) ,symbols ,010306 general physics ,Biological sciences ,Gene evolution ,Mathematical Physics - Abstract
The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.
- Published
- 2019
27. Bifurcation analysis and chaos control in Zhou's dynamical system
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Aly, E. S., El-Dessoky, M. M., Yassen, M. T., Saleh, E., Aiyashi, M. A., and Msmali, Ahmed Hussein
- Published
- 2022
- Full Text
- View/download PDF
28. Asymptotically stable equilibrium and limit cycles in the Rock–Paper–Scissors game in a population of players with complex personalities
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Jan Zakrzewski and Tadeusz Płatkowski
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Statistics and Probability ,Hopf bifurcation ,Computer Science::Computer Science and Game Theory ,education.field_of_study ,Population ,Normal-form game ,Condensed Matter Physics ,Stability (probability) ,symbols.namesake ,Nonlinear Sciences::Adaptation and Self-Organizing Systems ,Exponential stability ,Stability theory ,symbols ,Quantitative Biology::Populations and Evolution ,Limit (mathematics) ,education ,Mathematical economics ,Bifurcation ,Mathematics - Abstract
We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model.
- Published
- 2011
29. Continuous model for the rock–scissors–paper game between bacteriocin producing bacteria
- Author
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Stefan Schuster and Gunter Neumann
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Hopf bifurcation ,Differential equation ,Continuous modelling ,Applied Mathematics ,Degenerate energy levels ,Heteroclinic cycle ,Numerical Analysis, Computer-Assisted ,Models, Biological ,Agricultural and Biological Sciences (miscellaneous) ,Stability (probability) ,symbols.namesake ,Bacteriocins ,Game Theory ,Biological Clocks ,Control theory ,Modeling and Simulation ,Limit cycle ,Escherichia coli ,symbols ,Biological system ,Bifurcation ,Mathematics - Abstract
In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.
- Published
- 2007
30. Modeling the rock - scissors - paper game between bacteriocin producing bacteria by Lotka-Volterra equations
- Author
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Gunter Neumann and Stefan Schuster
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Hopf bifurcation ,education.field_of_study ,Ecology ,Differential equation ,Applied Mathematics ,Population ,Lotka–Volterra equations ,Heteroclinic cycle ,Fixed point ,symbols.namesake ,Limit cycle ,symbols ,Discrete Mathematics and Combinatorics ,Applied mathematics ,Heteroclinic orbit ,education ,Mathematics - Abstract
In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a R ock- S cissors- P aper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.
- Published
- 2007
31. Limit cycles and the benefits of a short memory in rock-paper-scissors games
- Author
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James Burridge
- Subjects
Hopf bifurcation ,Computer Science::Computer Science and Game Theory ,education.field_of_study ,Recall ,media_common.quotation_subject ,Population ,Short-term memory ,Fixed point ,Asymmetry ,Computer Science::Multiagent Systems ,symbols.namesake ,Bifurcation theory ,symbols ,Limit (mathematics) ,education ,Mathematical economics ,Mathematics ,media_common - Abstract
When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.
- Published
- 2015
32. Nonlinear dynamics of the rock-paper-scissors game with mutations
- Author
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Danielle F. P. Toupo and Steven H. Strogatz
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Physics ,Hopf bifurcation ,Mutation rate ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,State (functional analysis) ,Quantitative Biology::Genomics ,Stability (probability) ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Nonlinear system ,symbols.namesake ,Control theory ,Limit cycle ,Mutation (genetic algorithm) ,FOS: Mathematics ,symbols ,Quantitative Biology::Populations and Evolution ,Limit (mathematics) ,Statistical physics ,Mathematics - Dynamical Systems ,Adaptation and Self-Organizing Systems (nlin.AO) - Abstract
We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game., 6 pages, 5 figures
- Published
- 2015
33. Bifurcation, Catastrophe, and Turbulence
- Author
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Zeeman, E. C., Hilton, Peter J., editor, and Young, Gail S., editor
- Published
- 1982
- Full Text
- View/download PDF
34. Characterization of spiraling patterns in spatial rock-paper-scissors games
- Author
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Alastair M. Rucklidge, Bartosz Szczesny, and Mauro Mobilia
- Subjects
Population Dynamics ,FOS: Physical sciences ,Ecological and Environmental Phenomena ,Metapopulation ,Pattern Formation and Solitons (nlin.PS) ,Quantitative Biology - Quantitative Methods ,symbols.namesake ,Spatio-Temporal Analysis ,Mutation Rate ,Quantitative Biology::Populations and Evolution ,Statistical physics ,Quantitative Biology - Populations and Evolution ,Quantitative Methods (q-bio.QM) ,Condensed Matter - Statistical Mechanics ,Phase diagram ,Mathematics ,Hopf bifurcation ,Statistical Mechanics (cond-mat.stat-mech) ,Stochastic process ,Numerical analysis ,Populations and Evolution (q-bio.PE) ,Biodiversity ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear system ,Nonlinear Dynamics ,FOS: Biological sciences ,symbols - Abstract
The spatio-temporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatio-temporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair-exchange and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away far from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allows us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility., 16 two-column pages, 16 figures
- Published
- 2014
35. Oscillatory dynamics in rock–paper–scissors games with mutations
- Author
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Mauro Mobilia
- Subjects
Mutation rate ,Population Dynamics ,Fixed point ,Parameter space ,01 natural sciences ,010305 fluids & plasmas ,Quantitative Biology::Populations and Evolution ,Statistical physics ,Limit (mathematics) ,Mathematics ,Reproduction ,Applied Mathematics ,Biodiversity ,General Medicine ,Biological Evolution ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Markov Chains ,Amplitude ,Modeling and Simulation ,symbols ,General Agricultural and Biological Sciences ,Adaptation and Self-Organizing Systems (nlin.AO) ,Algorithms ,Statistics and Probability ,Physics - Physics and Society ,FOS: Physical sciences ,Physics and Society (physics.soc-ph) ,Models, Biological ,General Biochemistry, Genetics and Molecular Biology ,symbols.namesake ,Game Theory ,Control theory ,Limit cycle ,0103 physical sciences ,Computer Simulation ,Selection, Genetic ,Quantitative Biology - Populations and Evolution ,010306 general physics ,Condensed Matter - Statistical Mechanics ,Ecosystem ,Hopf bifurcation ,Stochastic Processes ,General Immunology and Microbiology ,Statistical Mechanics (cond-mat.stat-mech) ,Stochastic process ,Populations and Evolution (q-bio.PE) ,Kinetics ,Nonlinear Dynamics ,FOS: Biological sciences ,Mutation ,Linear Models - Abstract
We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude., Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biology
- Published
- 2010
36. Dynamic analysis and bifurcation control of a delayed fractional-order eco-epidemiological migratory bird model with fear effect.
- Author
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Song, Caihong and Li, Ning
- Subjects
MIGRATORY birds ,INFECTIOUS disease transmission ,COST control ,HOPF bifurcations ,PSYCHOLOGICAL feedback ,COMPUTER simulation - Abstract
In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. Hopf bifurcation in a delayed prey–predator model with prey refuge involving fear effect.
- Author
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Parwaliya, Ankit, Singh, Anuraj, and Kumar, Ajay
- Subjects
PREDATION ,HOPF bifurcations ,COMPUTER simulation ,EQUILIBRIUM ,FEAR in animals - Abstract
This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Fractional-order delayed Ross–Macdonald model for malaria transmission
- Author
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Cui, Xinshu, Xue, Dingyu, and Li, Tingxue
- Subjects
Original Paper ,Local stability ,Control and Systems Engineering ,Applied Mathematics ,Mechanical Engineering ,Aerospace Engineering ,Ocean Engineering ,Fractional-order ,Hopf bifurcation ,Electrical and Electronic Engineering ,Incubation periods ,Malaria - Abstract
This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.
- Published
- 2022
39. A 2-D Discrete Cubic Chaotic Mapping with Symmetry
- Author
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M. Mammeri
- Subjects
Nonlinear Sciences::Chaotic Dynamics ,Physics ,Hopf bifurcation ,symbols.namesake ,Chaotic dynamical systems ,Dynamical systems theory ,Physical phenomena ,Short paper ,symbols ,Lyapunov exponent ,Symmetry breaking ,Bifurcation diagram ,Mathematical physics - Abstract
In the theoretical research of chaotic dynamical system, the different type of bifurcations is a very interesting powerful tool for analyzing the qualitative behavior of chaotic dynamical system; this short paper is devoted to analysis of a simple 2-D symmetry discrete chaotic map with quadratic and cubic nonlinearities. The dynamical behaviors of the map are investigated by mathematical analysis and simulated numerically using package of Matlab . We compute numerically the bifurcation diagram and largest Lyapunov exponent and phase portraits. The research results indicate that there are interesting nonlinear physical phenomena in this simple 2-D symmetry discrete cubic map, such as symmetry bifurcation, Hopf bifurcation, symmetry breaking bifurcation and identical symmetric attractors. The important nonlinear physical phenomena obtained in this paper would benefit the study of the cubic chaotic map and the development of the theory of chaotic discrete dynamical systems. En la investigación teórica de los sistemas dinámicos caóticos, los diferentes tipos de bifurcaciones son una herramienta poderosa muy interesante para analizar el comportamiento cualitativo de los sistemas dinámicos caóticos; este breve artículo está dedicado al análisis de un mapa caótico discreto de simetría bidimensional simple con no linealidades cuadráticas y cúbicas. Los comportamientos dinámicos del mapa se investigan mediante análisis matemático y se simulan numéricamente utilizando el paquete de Matlab . Calculamos numéricamente el diagrama de bifurcación y el mayor exponente de Lyapunov y los retratos de fase. Los resultados de la investigación indican que existen interesantes fenómenos físicos no lineales en este sencillo mapa cúbico discreto de simetría 2-D, como la bifurcación de simetría, la bifurcación de Hopf, la bifurcación de ruptura de simetría y los atractores simétricos idénticos. Los importantes fenómenos físicos no lineales obtenidos en este trabajo beneficiarían el estudio del mapa cúbico caótico y el desarrollo de la teoría de los sistemas dinámicos discretos caóticos.
- Published
- 2021
40. Mathematical derivation and mechanism analysis of beta oscillations in a cortex-pallidum model.
- Author
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Xu, Minbo, Hu, Bing, Wang, Zhizhi, Zhu, Luyao, Lin, Jiahui, and Wang, Dingjiang
- Abstract
In this paper, we develop a new cortex-pallidum model to study the origin mechanism of Parkinson's oscillations in the cortex. In contrast to many previous models, the globus pallidus internal (GPi) and externa (GPe) both exert direct inhibitory feedback to the cortex. Using Hopf bifurcation analysis, two new critical conditions for oscillations, which can include the self-feedback projection of GPe, are obtained. In this paper, we find that the average discharge rate (ADR) is an important marker of oscillations, which can divide Hopf bifurcations into two types that can uniformly be used to explain the oscillation mechanism. Interestingly, the ADR of the cortex first increases and then decreases with increasing coupling weights that are projected to the GPe. Regarding the Hopf bifurcation critical conditions, the quantitative relationship between the inhibitory projection and excitatory projection to the GPe is monotonically increasing; in contrast, the relationship between different coupling weights in the cortex is monotonically decreasing. In general, the oscillation amplitude is the lowest near the bifurcation points and reaches the maximum value with the evolution of oscillations. The GPe is an effective target for deep brain stimulation to alleviate oscillations in the cortex. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Detections of bifurcation in a fractional-order Cohen-Grossberg neural network with multiple delays.
- Author
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Huang, Chengdai, Mo, Shansong, and Cao, Jinde
- Abstract
The dynamics of integer-order Cohen-Grossberg neural networks with time delays has lately drawn tremendous attention. It reveals that fractional calculus plays a crucial role on influencing the dynamical behaviors of neural networks (NNs). This paper deals with the problem of the stability and bifurcation of fractional-order Cohen-Grossberg neural networks (FOCGNNs) with two different leakage delay and communication delay. The bifurcation results with regard to leakage delay are firstly gained. Then, communication delay is viewed as a bifurcation parameter to detect the critical values of bifurcations for the addressed FOCGNN, and the communication delay induced-bifurcation conditions are procured. We further discover that fractional orders can enlarge (reduce) stability regions of the addressed FOCGNN. Furthermore, we discover that, for the same system parameters, the convergence time to the equilibrium point of FONN is shorter (longer) than that of integer-order NNs. In this paper, the present methodology to handle the characteristic equation with triple transcendental terms in delayed FOCGNNs is concise, neoteric and flexible in contrast with the prior mechanisms owing to skillfully keeping away from the intricate classified discussions. Eventually, the developed analytic results are nicely showcased by the simulation examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Modeling Excitable Cells with Memristors.
- Author
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Sah, Maheshwar, Ascoli, Alon, Tetzlaff, Ronald, Rajamani, Vetriveeran, and Budhathoki, Ram Kaji
- Subjects
MEMRISTORS ,POTASSIUM channels ,VOLTAGE-gated ion channels ,ION channels ,BIOLOGICAL membranes ,BIOLOGICAL systems ,POTASSIUM ions - Abstract
This paper presents an in-depth analysis of an excitable membrane of a biological system by proposing a novel approach that the cells of the excitable membrane can be modeled as the networks of memristors. We provide compelling evidence from the Chay neuron model that the state-independent mixed ion channel is a nonlinear resistor, while the state-dependent voltage-sensitive potassium ion channel and calcium-sensitive potassium ion channel function as generic memristors from the perspective of electrical circuit theory. The mechanisms that give rise to periodic oscillation, aperiodic (chaotic) oscillation, spikes, and bursting in an excitable cell are also analyzed via a small-signal model, a pole-zero diagram of admittance functions, local activity, the edge of chaos, and the Hopf bifurcation theorem. It is also proved that the zeros of the admittance functions are equivalent to the eigen values of the Jacobian matrix, and the presence of the positive real parts of the eigen values between the two bifurcation points lead to the generation of complicated electrical signals in an excitable membrane. The innovative concepts outlined in this paper pave the way for a deeper understanding of the dynamic behavior of excitable cells, offering potent tools for simulating and exploring the fundamental characteristics of biological neurons. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Bogdanov–Takens Bifurcation of Kermack–McKendrick Model with Nonlinear Contact Rates Caused by Multiple Exposures.
- Author
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Li, Jun and Ma, Mingju
- Subjects
HOPF bifurcations ,LIMIT cycles ,BIFURCATION diagrams ,PHASE diagrams ,DEATH rate - Abstract
In this paper, we consider the influence of a nonlinear contact rate caused by multiple contacts in classical SIR model. In this paper, we unversal unfolding a nilpotent cusp singularity in such systems through normal form theory, we reveal that the system undergoes a Bogdanov-Takens bifurcation with codimension 2. During the bifurcation process, numerous lower codimension bifurcations may emerge simultaneously, such as saddle-node and Hopf bifurcations with codimension 1. Finally, employing the Matcont and Phase Plane software, we construct bifurcation diagrams and topological phase portraits. Additionally, we emphasize the role of symmetry in our analysis. By considering the inherent symmetries in the system, we provide a more comprehensive understanding of the dynamical behavior. Our findings suggest that if this occurrence rate is applied to the SIR model, it would yield different dynamical phenomena compared to those obtained by reducing a 3-dimensional dynamical model to a planar system by neglecting the disease mortality rate, which results in a stable nilpotent cusp singularity with codimension 2. We found that in SIR models with the same occurrence rate, both stable and unstable Bogdanov-Takens bifurcations occur, meaning both stable and unstable limit cycles appear in this system. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Bifurcation Analysis of a Non-Linear Vehicle Model Under Wet Surface Road Condition.
- Author
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Kumar, Abhay, Verma, Suresh Kant, and Dheer, Dharmendra Kumar
- Subjects
ACCIDENT prevention ,BIFURCATION theory ,HOPF bifurcations ,TRAFFIC accidents ,EQUILIBRIUM - Abstract
The vehicles are prone to accidents during cornering on a wet or low friction coefficient roads if the longitudinal velocity (V
x ) and steering angle (δ) are increased beyond a certain limit. Therefore, it is of major concern to analyze the behaviour and define the stability boundary of the vehicle for such scenarios. In this paper, stability analysis of a 2 degrees of freedom nonlinear bicycle model replicating a car model including lateral (sideslip angle β) and yaw (yaw rate r) dynamics only operating on a wet surface road has been performed. The stability is analysed by utilizing the phase plane method and bifurcation analysis. The obtained converging and diverging nature of the trajectories (β, r) depicts the stable and unstable equilibrium points in the phase plane. The movement of these points results in the transition of the stability known as bifurcation due to the change in the control parameters (Vx , δ). The Matcont/Matlab is utilized to obtain the bifurcation diagrams and the nature of bifurcations. The obtained results show that a saddle node (SNB) and a subcritical Hopf bifurcation (HB) exists for steering angle (±0.08 rad) and higher than (±0.08 rad) with Vx = (10 - 40) m/s respectively. The SNB and HB denotes the spinning of the vehicle and sliding of the vehicle respectively, thus generating a unstable behaviour. A stability boundary is obtained representing the stable and unstable range of parameters. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
45. Multi-modal Swarm Coordination via Hopf Bifurcations.
- Author
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Baxevani, Kleio and Tanner, Herbert G.
- Abstract
This paper outlines a methodology for the construction of vector fields that can enable a multi-robot system moving on the plane to generate multiple dynamical behaviors by adjusting a single scalar parameter. This parameter essentially triggers a Hopf bifurcation in an underlying time-varying dynamical system that steers a robotic swarm. This way, the swarm can exhibit a variety of behaviors that arise from the same set of continuous differential equations. Other approaches to bifurcation-based swarm coordination rely on agent interaction which cannot be realized if the swarm members cannot sense or communicate with one another. The contribution of this paper is to offer an alternative method for steering minimally instrumented multi-robot collectives with a control strategy that can realize a multitude of dynamical behaviors without switching their constituent equations. Through this approach, analytical solutions for the bifurcation parameter are provided, even for more complex cases that are described in the literature, along with the process to apply this theory in a multi-agent setup. The theoretical predictions are confirmed via simulation and experimental results with the latter also demonstrating real-world applicability. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. Complex dynamics of an epidemic model with saturated media coverage and recovery
- Author
-
Yanni Xiao and Tangjuan Li
- Subjects
Original Paper ,Computer science ,Applied Mathematics ,Mechanical Engineering ,Nonlinear recovery function ,Aerospace Engineering ,Media coverage ,Ocean Engineering ,computer.software_genre ,Basic reproduction number ,Complex dynamics ,Mathematical model ,Saddle-node bifurcation ,Media impact ,Control and Systems Engineering ,Data mining ,Hopf bifurcation ,Electrical and Electronic Engineering ,Epidemic model ,computer - Abstract
During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals' response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.
- Published
- 2022
47. Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study
- Author
-
S. N. Raw, B P Sarangi, P. Mishra, and B. Tiwari
- Subjects
Patter formulation ,Computer science ,General Mathematics ,General Physics and Astronomy ,Lyapunov exponent ,01 natural sciences ,Stability (probability) ,symbols.namesake ,Quantitative Biology::Populations and Evolution ,Statistical physics ,0101 mathematics ,Bifurcation ,Hopf bifurcation ,Computer simulation ,Phase portrait ,Turing instability ,010102 general mathematics ,Time evolution ,General Chemistry ,Function (mathematics) ,010101 applied mathematics ,symbols ,Chaos ,Mutual interference ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.
- Published
- 2021
48. Study on the Strong Nonlinear Dynamics of Nonlocal Nanobeam Under Time-Delayed Feedback Using Homotopy Analysis Method
- Author
-
Li, Jia-Xuan, Yan, Yan, Wang, Wen-Quan, and Wu, Feng-Xia
- Published
- 2024
- Full Text
- View/download PDF
49. Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity
- Author
-
Archana Ojha, Nilesh Kumar Thakur, and Smriti Chandra Srivastava
- Subjects
General Mathematics ,Population ,Chaotic ,General Physics and Astronomy ,01 natural sciences ,Stability (probability) ,Zooplankton ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Carrying capacity ,Quantitative Biology::Populations and Evolution ,education ,010301 acoustics ,Mathematics ,Equilibrium point ,Hopf bifurcation ,education.field_of_study ,Toxicity ,fungi ,General Chemistry ,Plankton ,System dynamics ,Local stability ,Hopf-bifurcation ,symbols ,General Earth and Planetary Sciences ,Chaos ,General Agricultural and Biological Sciences ,Biological system ,Time delay ,Research Paper - Abstract
In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.
- Published
- 2021
50. Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects.
- Author
-
Essongo, Frank Eric, Mvogo, Alain, and Ben-Bolie, Germain Hubert
- Abstract
Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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