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1
Content available remote

Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models

100%
Wang X., Li Z.
Open Mathematics
|
2007
|
tom 5
|
nr 2
397-414
EN
In this paper, we discuss the special diffusive hematopoiesis model $$\frac{{\partial P(t,x)}}{{\partial t}} = \Delta P(t,x) - \gamma P(t,x) + \frac{{\beta P(t - \tau ,x)}}{{1 + P^n (t - \tau ,x)}}$$ with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.
2
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Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

100%
Xu C., Liao M., He X.
International Journal of Applied Mathematics and Computer Science
|
2011
|
tom 21
|
nr 1
97-107
EN
In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.
3
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Hopf bifurcations in a three-species food chain system with multiple delays

88%
Xie X., Zhang W.
Open Mathematics
|
2017
|
tom 15
|
nr 1
508-519
EN
This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.
4
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Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response

88%
Chen Q., Teng Z., Hu Z.
International Journal of Applied Mathematics and Computer Science
|
2013
|
tom 23
|
nr 2
247-261
EN
The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.
5
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Periodic dynamics in a model of immune system

75%
Bodnar M., Foryś U.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 1
113-126
EN
The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.
6
Content available remote

Logistic equations in tumour growth modelling

63%
Foryś U., Marciniak-Czochra A.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 3
317-325
EN
The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in the paper. The results are illustrated using computer simulations.
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