Existence of periodic solutions in predator–prey and competition dynamic systems

In this paper, we systematically explore the periodicity of some dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator–prey systems and competition systems) in mathematical biology governed by differential equations and difference equations. Easily verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models when the time scale is chosen as or , respectively. The main approach is based on a continuation theorem in coincidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on general time scales.     

KAM tori of Hamiltonian perturbations of 1D linear beam equations

In this paper, one-dimensional (1D) nonlinear beam equations

     

Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

We are interested in the following class of equations:

     

Global asymptotic stability of a family of difference equations

We consider the family of difference equations of the form

     

Construction of highly stable parallel two-step Runge–Kutta methods for delay differential equations

It is shown that any A-stable two-step Runge–Kutta method of order and stage order for ordinary differential equations can be extended to the P-stable method of uniform order for delay differential equations.     

Invariants of two- and three-dimensional hyperbolic equations

We consider linear hyperbolic equations of the form

     

Distributed order calculus and equations of ultraslow diffusion

We consider equations of the form

     

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form