Asymptotic properties of stochastic population dynamics

In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population

Permanence and almost periodic solution of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales

In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate the main results.Comment: 31page

Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.Comment: Published at https://doi.org/10.15559/18-VMSTA110 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations

In this paper, under a local Lipschitz condition and a monotonicity condition, the problems on the existence and uniqueness theorem as well as the almost surely asymptotic behavior for the global solution of highly nonlinear stochastic differential equations with time-varying delay and Markovian switching are discussed by using the Lyapunov function and some stochastic analysis techniques. Two integral lemmas are firstly established to overcome the difficulty stemming from the coexistence of the stochastic perturbation and the time-varying delay. Then, without any redundant restrictive condition on the time-varying delay, by utilizing the integral inequality, the exponential stability in pth(p ≥ 1)-moment for such equations is investigated. By employing the nonnegative semi-martingale convergence theorem, the almost sure exponential stability is analyzed. Finally, two examples are given to show the usefulness of the results obtained.National Natural Science Foundation of ChinaNatural Science Foundation of Jiangxi Province of ChinaFoundation of Jiangxi Provincial Educations of ChinaMinisterio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

The Stationary Distribution and Extinction of Generalized Multispecies Stochastic Lotka-Volterra Predator-Prey System

This paper is concerned with the existence of stationary distribution and extinction for multispecies stochastic Lotka-Volterra predator-prey system. The contributions of this paper are as follows. (a) By using Lyapunov methods, the sufficient conditions on existence of stationary distribution and extinction are established. (b) By using the space decomposition technique and the continuity of probability, weaker conditions on extinction of the system are obtained. Finally, a numerical experiment is conducted to validate the theoretical findings

The SIS epidemic model with Markovian switching

Population systems are often subject to environmental noise. Motivated by Takeuchi et al. (2006), we will discuss in this paper the effect of telegraph noise on the well-known SIS epidemic model. We establish the explicit solution of the stochastic SIS epidemic model, which is useful in performing computer simulations. We also establish the conditions for extinction and persistence for the stochastic SIS epidemic model and compare these with the corresponding conditions for the deterministic SIS epidemic model. We first prove these results for a two-state Markov chain and then generalise them to a finite state space Markov chain. Computer simulations based on the explicit solution and the Euler--Maruyama scheme are performed to illustrate our theory. We include a more realistic example using appropriate parameter values for the spread of Streptococcus pneumoniae in children