In this paper, we consider the dynamics of a delayed diffusive predator-prey model with herd behavior and hyperbolic mortality under Neumann boundary conditions. Firstly, by analyzing the characteristic equations in detail and taking the delay as a bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations induced by delay are investigated. Then, appl...

In this work we study the periodic orbits which bifurcate from all zero-Hopf bifurcations that an arbitrary Kolmogorov system of degree 3 in R3 can exhibit. The main tool used is averaging theory.

This note presents closed-form formulas for determining the critical points of general ndimensional differential equations. The formulas do not require commutating the eigenvalues of the Jacobian of a system. Based on the Hurwitz criterion, explicit necessary and sufficient conditions are obtained. Particular attention is focused on Hopf and double Hopf bifurcations. A model of induction machin...

A symmetric BAM neural network model with delay is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. The existence of multiple periodic solutions is established using a symmetric Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the AmericanMathema...

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that generalized Hopf bifurcations are shifted and turned into generalized Neimark-Sacker points by general one-step methods. We analyze the effect of discretizations methods on the emanating Hopf curve. In particular, we obtain estimates of the discretized eigenvalues along this curve. A detai...

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form me...

In this paper, a diffusive Leslie-type predator-prey model is investigated. The existence of a global positive solution, persistence, stability of the equilibria and Hopf bifurcation are studied respectively. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. Finally, our theoretical result...

We consider an ecological model consisting of two species predators competing for their common prey with explicit interference competition. With a proper rescaling, the is portrayed as singularly perturbed system one fast (prey dynamics) and slow variables (dynamics predators). The exhibits variety rich interesting dynamics, including, but not limited to mixed-mode ...

In this paper we study the limit cycles bifurcating from a nonisolated zero-Hopf equilibrium of a differential system in R3. The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a z...

Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement...