Search

In this paper, a simplified discrete-time SIR model with nonlinear incidence and recovery rates is discussed. Here, using the integral step size and the intervention level as control parameters, we mainly discuss three types of codimension-two bifurcations (fold-flip bifurcation, 1:3 resonance, and 1:4 resonance) of the simplified discrete-time SIR model in detail by bifurcation theory and numerical continuation techniques. Parameter conditions for the occurrence of codimension-two bifurcations are obtained by constructing the corresponding approximate normal form with translation and transformation of several parameters and variables. To further confirm the accuracy of our theoretical analysis, numerical simulations such as phase portraits, bifurcation diagrams, and maximum Lyapunov exponents diagrams are provided. In particular, the coexistence of bistability states is observed by giving local attraction basins diagrams of different fixed points under different integral step sizes. It is possible to more clearly illustrate the model's complex dynamic behavior by combining theoretical analysis and numerical simulation. [ABSTRACT FROM AUTHOR]

This paper presents a two-dimensional simplified Hodgkin–Huxley model under exposure to electric fields. The Hopf bifurcations of the simplified Hodgkin–Huxley model are investigated through qualitative analysis and numerical simulations. A necessary and sufficient condition for the existence of Hopf bifurcations is derived, and the conditions for supercritical and subcritical Hopf bifurcations are obtained. Finally, bifurcation diagrams are given for two parameters, and numerical examples are presented to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model, in which the prey's growth is affected by the additive predation of its potential predators. Due to the additive predation term in prey, the model may exhibit the cases of the strong Allee effect, weak Allee effect and no Allee effect. In each case, the dynamics of global features of the model are investigated. Compared to the well-known Lotka–Volterra type model, the model proposed in this paper exhibits much richer and more complex dynamic behaviors, such as the Allee effect, the sensitivity to the initial conditions caused by the strong Allee effect, the oscillatory behavior and the Hopf and heteroclinic bifurcations. Furthermore, the stability and Hopf bifurcation of the model with the density dependent feedback time delay in prey are investigated. By the normal form method and center manifold theory, the explicit formulas are presented to determine the direction of Hopf bifurcation and the stability and period of Hopf-bifurcating periodic solutions. Theoretical analysis and numerical simulation indicate that the delay may destabilize the model, and cause the Hopf bifurcation not only at the interior equilibrium but also at a boundary equilibrium. [ABSTRACT FROM AUTHOR]

This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p = p # , by investigating stable and unstable manifolds of saddles. When p > p # , both species go extinct, and when p < p # , there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays. [ABSTRACT FROM AUTHOR]