Vectored Immunoprophylaxis and Cell-to-Cell Transmission in HIV Dynamics

We consider local and global bifurcations in a HIV model with cell-to-cell transmission and vectored immunoprophylaxis. Both theoretical and numerical analyses are conducted to explore various dynamical behaviors including backward bifurcation, Hopf bifurcation, homoclinic bifurcation, Bogdanov–Takens bifurcation, hysteresis and isola bifurcation. The isola bifurcation of periodic orbits was first detected numerically in HIV model, which means that there is a parameter interval with the same oscillations. It is shown that the effect of vectored immunoprophylaxis in this model is the main cause of the periodic symptoms of HIV disease. Moreover, it is shown that the increase of cell-to-cell transmission may be the main factor causing Hopf bifurcation to disappear, and thus eliminating oscillation behavior. Also, several patterns of dynamical behaviors are found in different parameter intervals including the bistability.

Experimentally Accessible Orbits Near a Bykov Cycle

This paper reports numerical experiments done on a two-parameter family of vector fields which unfold an attracting heteroclinic cycle linking two saddle-foci. We investigated both local and global bifurcations due to symmetry breaking in order to detect either hyperbolic or chaotic dynamics. Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is still out of reach, using a combination of theoretical tools and computer simulations we have uncovered some complex patterns. We have selected suitable initial conditions to analyze the bifurcation diagrams, and regarding these solutions we have located: (a) an open domain of parameters with regular dynamics; (b) infinitely many parabolic-type curves associated to homoclinic Shilnikov cycles which act as organizing centers; (c) a crisis region related to the destruction or creation of chaotic attractors; (d) a large Lebesgue measure set of parameters where chaotic regimes are dominant, though sinks and chaotic attractors may coexist, and in whose complement we observe shrimps.

Rich Bifurcation Structure of Prey–Predator Model Induced by the Allee Effect in the Growth of Generalist Predator

Prey–predator models are building blocks for many food-chain and food-web models in theoretical population biology. These models can be divided into two groups depending on the nature of predators, namely, specialist predator and generalist predator. Generalist predators can survive in the absence of prey but specialist predators go to extinction. Prey–predator models with specialist predator and Allee effect in prey growth have been investigated by several researchers and various types of interesting dynamics have been reported. In this paper, we consider a prey–predator model with generalist predator subject to Allee effect in predator’s growth rate. In general, a prey–predator system with saturating functional response can be destabilized due to the increase of the carrying capacity of prey which is known as paradox of enrichment. In our model with Allee effect in predator growth, we have shown that increase in carrying capacity of prey helps the populations to survive in a coexistence steady state. The considered model is capable of producing bistable dynamics for a reasonable range of parameter values. The complete dynamics of the system are quite rich and all possible local and global bifurcations are studied to understand the dynamics of the model. Analytical results are verified with numerical examples and successive bifurcations are identified with the help of bifurcation diagrams.

Embodied robots driven by self-organized environmental feedback

Which kind of complex behavior may arise from self-organizing principles? We investigate this question for the case of snake-like robots composed of passively coupled segments, with every segment containing two wheels actuated separately by a single neuron. The robot is self-organized both on the level of the individual wheels and with respect to inter-wheel coordination, which arises exclusively from the mechanical coupling of the individual wheels and segments. For the individual wheel, the generating principle proposed results in locomotive states that correspond to self-organized limit cycles of the sensorimotor loop. Our robot interacts with the environment by monitoring the state of its actuators, that is, via propriosensation. External sensors are absent. In a structured environment the robot shows complex emergent behavior that includes pushing movable blocks around, reversing direction when hitting a wall, and turning when climbing a slope. On flat grounds the robot wiggles in a snake-like manner, when moving at higher velocities. We also investigate the emergence of motor primitives, namely, the route to locomotion, which is characterized by a series of local and global bifurcations in terms of dynamical system theory.

Allee’s Effect Bifurcation in Generalized Logistic Maps

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

Harmonically Excited Delay Equation for Machine Tool Vibrations

A machining tool can be subject to different kinds of excitations. The forcing may have external sources (such as rotating imbalance or misalignment of the workpiece) or it can arise from the cutting process itself (e.g. chip formation). We investigate the classical tool vibration model which is a delay-differential equation with a quadratic and cubic nonlinearity and periodic forcing. The method of multiple scales was used to derive the slow-flow equations. The resonance curves of the system are similar to those for the Duffing-equation, having a hardening characteristic. Stability analysis for the fixed points of the slow-flow equations was performed. Local and global bifurcations were studied and illustrated with phase portraits and direct numerical integration of the original equation. Subcritical Hopf, saddle-node and heteroclinic bifurcations were found.

Численный бифуркационный анализ математических моделей с запаздыванием по времени с использованием пакета программ DDE-BIFTOOL

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and making predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modeled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.