Article

Stability and bifurcation analysis on a Logistic model with discrete and distributed delays

Applied Mathematics and Computation (Impact Factor: 1.55). 10/2006; 181(2):1745-1757. DOI: 10.1016/j.amc.2006.03.025
Source: DBLP

ABSTRACT

In this paper, a Logistic model with discrete and distributed delays is investigated. We first consider the stability of the positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions. A numerical simulation for supporting the theoretical analysis is also given.

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Available from: Yongli Song, Dec 14, 2013
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    • "Lotka-Volterra system is one of the most classical and important systems in the field of mathematical biology. Since the word of Volterra, there have been extensively detailed investigations on Lotka-Volterra system including stability, attractivity, persistence, periodic oscillation, bifurcation and chaos (see [1] [2] [3] [4] [5] [6] and the references therein). In particular, the properties of periodic solutions arising from the Hopf bifurcation are of great interest [7] [8] [9] [10]. "

    Full-text · Dataset · Oct 2015
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    • "Lotka-Volterra system is one of the most classical and important systems in the field of mathematical biology. Since the word of Volterra, there have been extensively detailed investigations on Lotka-Volterra system including stability, attractivity, persistence, periodic oscillation, bifurcation and chaos (see [1] [2] [3] [4] [5] [6] and the references therein). In particular, the properties of periodic solutions arising from the Hopf bifurcation are of great interest [7] [8] [9] [10]. "
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    • "When í µí°¹(í µí± ) = í µí»¿(í µí± − í µí¼) and í µí°º(í µí± ) = í µí»¿(í µí± − í µí¼‚), í µí¼, í µí¼‚ ≥ 0, model (2) admits two different discrete time delays; Ruan [7] and Yan and Zhang [11] discussed the stability of the interior equilibrium of model (2) and Hopf bifurcation of nonconstant periodic solutions regarding the sum of two delays í µí¼ and í µí¼‚ as the bifurcation parameter. Furthermore, dynamic effect of intraspecific competition on population dynamics of model (2) is studied in [8] [9]. By assuming that í µí°¹(í µí± ) = í µí»¿(í µí± − í µí¼), í µí¼ ≥ 0, the delay kernel function í µí°º(í µí± ) may take the weak generic kernel function í µí°º(í µí± ) = í µí»¼í µí±’ −í µí»¼í µí± and strong generic kernel function í µí°º(í µí± ) = í µí»¼ 2 í µí± í µí±’ −í µí»¼í µí± (í µí»¼ > 0), where the weak generic kernel implies that the importance of events in the past simply decreases exponentially and the further one looks into the past while the strong generic kernel implies that a particular time in the past is more important than any other [1] [2] [5]. "
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    ABSTRACT: We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.
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