Miljana Jovanović

Stochastic Gilpin–Ayala competition model with infinite delay

Abstract In this paper we study the stochastic Gilpin–Ayala competition model with an infinite delay. We verify that the environmental noise included in the model does not only provide a positive global solution (there is no explosion in a finite time), but this solution is also stochastically ultimately bounded. We obtain certain asymptotic results regarding a large time behavior.

A Taylor polynomial approach in approximations of solution to pantograph stochastic differential equations with Markovian switching

The subject of this paper are analytic approximate methods for pantograph stochastic differential equations with Markovian switching, as well as their counterparts without Markovian switching. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. In the case with Markovian switching we will present the approximate method based on Taylor approximation of coefficients in two arguments and show that the appropriate approximate solutions converge in the L^p-norm to the solution of the initial equation. Then we will present the other approximate method which deals with Taylor approximation in the first argument. In both cases the closeness between the approximate solution and the solution of the initial equation depends on the number of degrees in Taylor approximations of coefficients, although the presence of the Markov chain affects it. These approximate methods are then adapted to the case without Markovian switching. The first method gives the possibility of proving L^p convergence as well as a.s. convergence of the appropriate sequence of approximate solutions to the solution of the initial equation.

Dynamics of non-autonomous stochastic Gilpin-Ayala competition model with time-varying delays

In this paper the non-autonomous stochastic Gilpin-Ayala competition model with time-dependent delay is considered. We prove existence and uniqueness of the global positive solution and various properties of that solution such as boundedness, moment and pathwise estimation, extinction, non-persistence in time average and weak persistence. In addition, we give a special case of the considered system and impose its properties. All studied features are natural requirements from the biological point of view. Furthermore, examples with numerical simulations are given to illustrate our results.

On perturbed nonlinear Itô type stochastic integrodifferential equations

In this paper we consider the solution of the stochastic nonlinear integrodifferential equation of the Ito type with small perturbations, by comparing it with the solution of the corresponding unperturbed equation of the equal type. We investigate the closeness in the (2m)th moment sense of these solutions on finite fixed intervals or on intervals whose length tends to infinity as small perturbations tend to zero.

An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay

The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in L^p-norm and with probability 1 to the solution of the initial equation. Also, the rate of the L^p convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function.

Neutral stochastic functional differential equations with additive perturbations

The paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations, by comparing it with the solution to the corresponding unperturbed equation of the equal type. We give conditions under which these solutions are close in the (2m)th mean, on finite time-intervals and on intervals whose length tends to infinity as small perturbations tend to zero.

GENERALIZED STOCHASTIC PERTURBATION-DEPENDING DIFFERENTIAL EQUATIONS

The paper is devoted to the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.The paper is devoted to the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.

On a class of backward doubly stochastic differential equations

In this paper, a new class of backward doubly stochastic differential equations is studied. This type of equations has a more general form of the forward Ito integrals compared to the ones which have been studied until now. We conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations, considered earlier in paper [5] by Pardoux and Peng. Some comparison theorems are also given, as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations.

Analysis of non-autonomous stochastic Gompertz model with delay☆

Abstract This paper presents the analysis of behavior of stochastic Gompertz model with delay. We prove existence and uniqueness of the global positive solution of the considered model. Besides, the conditions for species to be persistent are established, as well as the conditions under which population becomes extinct. Finally, numerical illustration with real life example is carried out to confirm our theoretical results.

Perturbed stochastic hereditary differential equations with integral contractors

Abstract The present paper deals with the asymptotic behavior of the solution of the perturbed stochastic hereditary differential equation of the Ito type, by comparing it in the (2 m ) th moment sense with the solution of the appropriate unperturbed equation, on finite intervals or on intervals whose length tends to infinity. The problems are considered by using the concept of a random integral contractor, which includes the Lipschitz condition as a special case.