Bifurcation analysis for a regulated logistic growth model

AbstractIn this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay τ passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions

Bifurcations for a predator–prey system with two delays

AbstractIn this paper, a predator–prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838], we may show the global existence of periodic solutions

Steady periodic irrotational blood flow with time-dependent body force

In this paper, we study steady 2D periodic blood flow propagating along blood vessels with free boundary conditions. In particular, we focus on the dynamic behavior of irrotational flows with time dependent body force. An equivalent formulation with fix boundary is developed by utilizing flow force functions. The local bifurcation result is obtained by Crandall-Rabinowitz theorem. The existence of a local $C^1$-curve of small-amplitude solution is strictly proved

Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response

A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

We investigate the dynamics of a delayed neural network model consisting of n identical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory and S1-equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results

The case report of MOG and NMDAR IgG double positive encephalitis treated with subcutaneous ofatumumab

The phenotypic spectrum of myelin oligodendrocyte glycoprotein (MOG)- IgG–associated disorders (MOGAD) has broadened in the past few years, and atypical phenotypes are increasingly recognized. Isolated seizures and MRI-negative brainstem and cerebellar symptoms or encephalitis have rarely been reported as a feature of MOGAD and represent a diagnostic challenge. Meanwhile, the coexistence of MOG IgG and other CNS autoimmune antibodies is infrequent. We report a patient presented with isolated epileptic onset, relapsed with MRI-negative brainstem symptoms and MRI-negative encephalitis. He was positive for MOG IgG throughout the disease course while concomitant NMDAR IgG was not detected positive until second relapse. He showed decreasing response to conventional first-line therapy. The last relapse was during a COVID-19 epidemic with limited inpatient resources. Fortunately, he was ultimately controlled on subcutaneous ofatumumab, a novel fully humanized anti-CD20 mAb. This is the first report about subcutaneous ofatumumab treatment in MOG and NMDAR IgG double positive encephalitis with 12-month follow-up, depicting its potential as a therapeutic option