Dynamics and oscillations of a class of cushing equations including gamma-distributed delays with a gap

Abstract

The stability of dynamical systems in presence of time-delay is a problem of big interest since the presence of a time-delay may induce instabilities, and complex behaviors for the corresponding schemes. In particular, the problem becomes even more difficult in the case when the delays are distributed. Besides, as we all know, many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. Such a class form pseudo almost automorphic functions which a natural generalization of the concept of almost periodicity and almost automorphy. This paper studies the dynamics and oscillations of a class of the Cushing equations including gamma-distributed delays with a gap

x' (t) =-α x( t) +β(t) x( t-τ) + λ ( t) ∫₀^t g( θ ) x( t- θ ) dθ .

In particular, sufficient conditions are obtained to ensure the existence and stability of the unique pseudo almost automorphic solution for the Cushing equation with gamma-distributed delays .