Dynamics of constrained differential delay equations

Stochastic differential delay equations of population dynamics

In this paper we stochastically perturb the delay Lotka–Volterra model ẋ(t)= diag(x1(t), . . . , xn(t))[A(x(t)− x̄)+B(x(t − τ )− x̄)] into the stochastic delay differential equation (SDDE) dx(t)= diag(x1(t), . . . , xn(t)){[A(x(t)− x̄)+B(x(t − τ )− x̄)]dt + σ (x(t)− x̄)dw(t)}. The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially ...

Global dynamics of delay differential equations

In this survey paper the delay differential equation ẋ(t) = −μx(t) + g(x(t− 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely understood only for a small class of nonlinearities. Open problems are formulated.

Delay Differential Equations in Single Species Dynamics

Periodicity in a System of Differential Equations with Finite Delay

The existence and uniqueness of a periodic solution of the system of differential equations d dt x(t) = A(t)x(t − ) are proved. In particular the Krasnoselskii’s fixed point theorem and the contraction mapping principle are used in the analysis. In addition, the notion of fundamental matrix solution coupled with Floquet theory is also employed.  

Delay Differential Equations

where x t ( ) x (t ) and 0. Observe that xt ( ) with 0 represents a portion of the solution trajectory in a recent past. Here, f is a functional operator that takes a time input and a continuous function xt ( ) with 0 and generates a real number (dx (t )/dt ) as its output. A well-known example of a delay differential equation is the Hutchinson equation, or the discrete delay logistic equation,...