Narrow Results

An eco-epidemiological predator–prey system containing two separate prey patches has been assessed in this paper with disease in prey population involving ratio-dependent functional responses and incidence rates. The model dynamics is studied by employing environmental perturbations to develop more genuine as well as realistic dynamics. This comprehensive study offers analytical insights into the positivity and uniform boundedness of the model system. Through numerical investigations, the study reveals the occurrence of local bifurcations, bistable regions, hysteresis phenomena, and parametric regions where saddle-node and transcritical bifurcation curves exist. Utilizing appropriate Lyapunov functions, it is demonstrated that a unique globally positive solution emerges from positive initial values. It has also been proved that the suggested model is stochastically ultimate bounded. Afterward, certain sufficient criteria illustrate the extermination of disease and persistence in mean. Notably, stochastic perturbations are shown to potentially impede disease propagation, suggesting strategies for dynamically controlling the spread of the illness. Additionally, the study derives a set of requirements for the existence of an ergodic stationary distribution. Numerous numerical simulations, conducted using MATLAB and MATCONT, effectively illustrate the theoretical findings.

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.

In this work we study a wide class of symmetric control systems that has the Chua's circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua's circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three'parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system.

First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

In this work we analyze what happens when the generalized conditions given in [Balibrea et al., 2008], which produce the appearance of local bifurcations of continuous dynamical systems, fail. As a result, we are able to find out some situations of local stability in the presence of nonhyperbolic equilibria.

We investigate the effects of time delay and piecewise-linear threshold policy harvesting for a delayed predator–prey model. It is the first time that Holling response function of type III and the present threshold policy harvesting are associated with time delay. The trajectories of our delayed system are bounded; the stability of each equilibrium is analyzed with and without delay; there are local bifurcations as saddle-node bifurcation and Hopf bifurcation; optimal harvesting is also investigated. Numerical simulations are provided in order to illustrate each result.

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system.

First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.