Introducing a scavenger onto a predator prey model.
We consider a stochastic delay differential equation with exponentially stable drift and diffusion driven by a general Lévy process. The diffusion coefficient is assumed to be locally Lipschitz and bounded. Under a mild condition on the large jumps of the Lévy process, we show existence of an invariant measure. Main tools in our proof are a variation-of-constants formula and a stability theorem in our context, which are of independent interest.
We consider the stochastic differential equation (1) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup describing the evolution of measures along trajectories and vice versa.
The response of an oscillator to a small amplitude periodic excitation is discussed. In particular, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder. When such an invariant torus exists, some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbed system on the torus. In particular, the perturbed solutions in the basin of attraction of the invariant torus are...
In this communication we state and analyze the new inverse problems in the theory of differential equations related to the construction of an analytic planar verctor field from a given, finite number of solutions, trajectories or partial integrals.Likewise, we study the problem of determining a stationary complex analytic vector field Γ from a given, finite subset of terms in the formal power series (...).