In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equat... more In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of [Formula: see text]. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.
ABSTRACT In this paper, we derive and analyze both analytically and numerically a mathematical mo... more ABSTRACT In this paper, we derive and analyze both analytically and numerically a mathematical model for three interacting populations. These take the form of a herbivore, a plant, and a pollinator. The full model is a nonlinear reaction–diffusion–advection system, which is derived on the basis of a series of plausible and widely supported ecological hypothesis. The study we present here deals with the conditions for the coexistence of the three interacting species. The analysis is carried out in two stages. For the homogeneous case, the mathematical model reduces to a nonlinear three-dimensional autonomous ODE system, which, as the ecological parameters change, exhibits different dynamical behaviors, namely a limit cycle and a positive attractor. The non-homogeneous case is studied mainly by means of numerical simulations of the full model defined on a rectangular region and considering appropriate initial and boundary conditions. Our results strongly suggest the stabilizing role played by the herbivore population which, in turn means that the introduction of this population into the mutualistic pollinator–plant interaction, favors the coexistence of the three interacting species.
The authors consider a predator-prey model with different types of Holling functional responses s... more The authors consider a predator-prey model with different types of Holling functional responses such as general sigmoid functions; special emphasis is laid on ratio-dependent responses where the denominator is a quadratic function of the prey. The authors investigate the existence of a center, of a limit cycle, of Hopf bifurcation, and, in some cases, results on the global phase portrait in the Poincaré disc are also derived.
The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index... more The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non–algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also
ABSTRACT In the study of the black holes with a Higgs field appears in a natural way the Lotka-Vo... more ABSTRACT In the study of the black holes with a Higgs field appears in a natural way the Lotka-Volterra differential system (x) over dot = x(y - 1), (y) over dot = y(1 + y - 2x(2) - z(2)), (z) over dot = zy, in R-3. Here we provide the qualitative analysis of the flow of this system describing the alpha-limit set and the omega-limit set of all orbits of this system in the whole Poincare ball, i.e. we identify R-3 with the interior of the unit ball of R-3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity.
ABSTRACT We consider a two parameter family of maps which are obtained as composition of two logi... more ABSTRACT We consider a two parameter family of maps which are obtained as composition of two logistic maps. We say that a map presents a coexistence of dynamics if it has two attractors. In this article, we give conditions on the parameters to have coexistence of dynamics and we give a topological description of the parameter space.
The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index... more The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non–algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also
In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equat... more In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of [Formula: see text]. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.
ABSTRACT In this paper, we derive and analyze both analytically and numerically a mathematical mo... more ABSTRACT In this paper, we derive and analyze both analytically and numerically a mathematical model for three interacting populations. These take the form of a herbivore, a plant, and a pollinator. The full model is a nonlinear reaction–diffusion–advection system, which is derived on the basis of a series of plausible and widely supported ecological hypothesis. The study we present here deals with the conditions for the coexistence of the three interacting species. The analysis is carried out in two stages. For the homogeneous case, the mathematical model reduces to a nonlinear three-dimensional autonomous ODE system, which, as the ecological parameters change, exhibits different dynamical behaviors, namely a limit cycle and a positive attractor. The non-homogeneous case is studied mainly by means of numerical simulations of the full model defined on a rectangular region and considering appropriate initial and boundary conditions. Our results strongly suggest the stabilizing role played by the herbivore population which, in turn means that the introduction of this population into the mutualistic pollinator–plant interaction, favors the coexistence of the three interacting species.
The authors consider a predator-prey model with different types of Holling functional responses s... more The authors consider a predator-prey model with different types of Holling functional responses such as general sigmoid functions; special emphasis is laid on ratio-dependent responses where the denominator is a quadratic function of the prey. The authors investigate the existence of a center, of a limit cycle, of Hopf bifurcation, and, in some cases, results on the global phase portrait in the Poincaré disc are also derived.
The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index... more The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non–algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also
ABSTRACT In the study of the black holes with a Higgs field appears in a natural way the Lotka-Vo... more ABSTRACT In the study of the black holes with a Higgs field appears in a natural way the Lotka-Volterra differential system (x) over dot = x(y - 1), (y) over dot = y(1 + y - 2x(2) - z(2)), (z) over dot = zy, in R-3. Here we provide the qualitative analysis of the flow of this system describing the alpha-limit set and the omega-limit set of all orbits of this system in the whole Poincare ball, i.e. we identify R-3 with the interior of the unit ball of R-3 centered at the origin and we extend analytically this flow to its boundary, i.e. to the infinity.
ABSTRACT We consider a two parameter family of maps which are obtained as composition of two logi... more ABSTRACT We consider a two parameter family of maps which are obtained as composition of two logistic maps. We say that a map presents a coexistence of dynamics if it has two attractors. In this article, we give conditions on the parameters to have coexistence of dynamics and we give a topological description of the parameter space.
The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index... more The signature formula of Eisenbud–Levine and Khimshiashvili for computing the Poincaré–Hopf index of a real analytic vector field at an algebraically isolated singularity is well known. We present in this paper an algebraic formula which allows to compute the index in the non–algebraically isolated case when the complex zeros associated to the complexified vector field have codimension one. We also
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