• Title/Summary/Keyword: Bifurcating

Search Result 32, Processing Time 0.019 seconds

STABILITY AND BIFURCATION ANALYSIS OF A LOTKA-VOLTERRA MODEL WITH TIME DELAYS

  • Xu, Changjin;Liao, Maoxin
    • Journal of applied mathematics & informatics
    • /
    • v.29 no.1_2
    • /
    • pp.1-22
    • /
    • 2011
  • In this paper, a Lotka-Volterra model with time delays is considered. A set of sufficient conditions for the existence of Hopf bifurcation are obtained via analyzing the associated characteristic transcendental equation. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form method and center manifold theory. Finally, the main results are illustrated by some numerical simulations.

BIFURCATION ANALYSIS OF A DELAYED EPIDEMIC MODEL WITH DIFFUSION

  • Xu, Changjin;Liao, Maoxin
    • Communications of the Korean Mathematical Society
    • /
    • v.26 no.2
    • /
    • pp.321-338
    • /
    • 2011
  • In this paper, a class of delayed epidemic model with diffusion is investigated. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation are also carried out to support our analytical findings. Finally, biological explanations and main conclusions are given.

DYNAMICS OF A MODIFIED HOLLING-TANNER PREDATOR-PREY MODEL WITH DIFFUSION

  • SAMBATH, M.;BALACHANDRAN, K.;JUNG, IL HYO
    • Journal of the Korean Society for Industrial and Applied Mathematics
    • /
    • v.23 no.2
    • /
    • pp.139-155
    • /
    • 2019
  • In this paper, we study the asymptotic behavior and Hopf bifurcation of the modified Holling-Tanner models for the predator-prey interactions in the absence of diffusion. Further the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated. Diffusion driven instability of the positive equilibrium solutions and Turing instability region regarding the parameters are established. Finally we illustrate the theoretical results with some numerical examples.

On the Normal Mode Dynamics of a Pendulum Absorber (정규모우드 방법을 활용한 진자형 흡진기의 비선형 동역학에 관한 연구)

  • 심재구;박철희
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
    • /
    • 1996.04a
    • /
    • pp.177-183
    • /
    • 1996
  • By utilizing the concept of normal modes, nonlinear dynamics is studied on pendulum dynamic absorber. When the spring mode loses the stability in undamped free system, a dynamic two-well potential is formed in Poincare map. A procedure is formulated to compute the forced responses associated with bifurcating mode and predict double saddle-loop phenomenon. It is found that quasiperiodic motion and stable periodic motion coexist in some parameter ranges, and only periodic motions or rotation of pendulum with chaotic fluctuation are observed in other ranges.

  • PDF

STABILITY AND BIFURCATION ANALYSIS FOR A TWO-COMPETITOR/ONE-PREY SYSTEM WITH TWO DELAYS

  • Cui, Guo-Hu;Yany, Xiang-Ping
    • Journal of the Korean Mathematical Society
    • /
    • v.48 no.6
    • /
    • pp.1225-1248
    • /
    • 2011
  • The present paper is concerned with a two-competitor/oneprey population system with Holling type-II functional response and two discrete delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium and existence of local Hopf bifurcations are investigated. Particularly, by applying the normal form theory and the center manifold reduction for functional differential equations (FDEs) explicit formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are derived. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.

BIFURCATION ANALYSIS OF A DELAYED PREDATOR-PREY MODEL OF PREY MIGRATION AND PREDATOR SWITCHING

  • Xu, Changjin;Tang, Xianhua;Liao, Maoxin
    • Bulletin of the Korean Mathematical Society
    • /
    • v.50 no.2
    • /
    • pp.353-373
    • /
    • 2013
  • In this paper, a class of delayed predator-prey models of prey migration and predator switching is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.

DYNAMIC BEHAVIOR OF A PREDATOR-PREY MODEL WITH STAGE STRUCTURE AND DISTRIBUTED DELAY

  • Zhou, Xueyong
    • Journal of applied mathematics & informatics
    • /
    • v.28 no.1_2
    • /
    • pp.193-207
    • /
    • 2010
  • In this paper, a predator-prey model with stage structure and distributed delay is investigated. Mathematical analyses of the model equation with regard to boundedness of solutions, nature of equilibria, permanence, extinction and stability are performed. By the comparison theorem, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Taking the product of the per-capita rate of predation and the rate of conversing prey into predator as the bifurcating parameter, we prove that there exists a threshold value beyond which the positive equilibrium bifurcates towards a periodic solution.

EFFECT OF FEAR ON A MODIFIED LESLI-GOWER PREDATOR-PREY ECO-EPIDEMIOLOGICAL MODEL WITH DISEASE IN PREDATOR

  • PAL, A.K.
    • Journal of applied mathematics & informatics
    • /
    • v.38 no.5_6
    • /
    • pp.375-406
    • /
    • 2020
  • The anti-predator factor due to fear of predator in eco- epidemiological models has a great importance and cannot be evaded. The present paper consists of a modified Lesli-Gower predator-prey model with contagious disease in the predator population only and also consider the fear effect in the prey population. Boundedness and positivity have been studied to ensure the eco-epidemiological model is well-behaved. The existence and stability conditions of all possible equilibria of the model have been studied thoroughly. Considering the fear constant as bifurcating parameter, the conditions for the existence of limit cycle under which the system admits a Hopf bifurcation are investigated. The detailed study for direction of Hopf bifurcation have been derived with the use of both the normal form and the central manifold theory. We observe that the increasing fear constant, not only reduce the prey density, but also stabilize the system from unstable to stable focus by excluding the existence of periodic solutions.

POSITIVE SOLUTIONS OF A REACTION-DIFFUSION SYSTEM WITH DIRICHLET BOUNDARY CONDITION

  • Ma, Zhan-Ping;Yao, Shao-Wen
    • Bulletin of the Korean Mathematical Society
    • /
    • v.57 no.3
    • /
    • pp.677-690
    • /
    • 2020
  • In this article, we study a reaction-diffusion system with homogeneous Dirichlet boundary conditions, which describing a three-species food chain model. Under some conditions, the predator-prey subsystem (u1 ≡ 0) has a unique positive solution (${\bar{u_2}}$, ${\bar{u_3}}$). By using the birth rate of the prey r1 as a bifurcation parameter, a connected set of positive solutions of our system bifurcating from semi-trivial solution set (r1, (0, ${\bar{u_2}}$, ${\bar{u_3}}$)) is obtained. Results are obtained by the use of degree theory in cones and sub and super solution techniques.

A Stage-Structured Predator-Prey System with Time Delay and Beddington-DeAngelis Functional Response

  • Wang, Lingshu;Xu, Rui;Feng, Guanghui
    • Kyungpook Mathematical Journal
    • /
    • v.49 no.4
    • /
    • pp.605-618
    • /
    • 2009
  • A stage-structured predator-prey system with time delay and Beddington-DeAngelis functional response is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.