A stochastic model for internal HIV dynamics

In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. Numerical simulations conclude the paper.     

Asymptotic dynamics of a deterministic and stochastic predator–prey model with disease in the prey species

In this paper, we discuss a predator–prey model with the Beddington–DeAngelis functional response of predators and a disease in the prey species. At first we study permanence and global stability of a positive equilibrium for the deterministic version of the model. Then we include a stochastic perturbation of the white noise type. We analyse the influence of this stochastic perturbation on the systems and prove that the positive equilibrium is also globally asymptotically stable in this case. The key point of our analysis is to choose appropriate Lyapunov functionals. We point out the differences between the deterministic and stochastic versions of the model. Copyright © 2013 John Wiley & Sons, Ltd.     

Complex dynamics in a stochastic internal HIV model

In this paper, we analyze a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling. By analyzing the Lyapunov exponent, singular boundary and probability density, some new criteria ensuring stochastic stability, D-bifurcation and P-bifurcation for stochastic internal HIV model are obtained, respectively. Numerical simulation results are given to support the theoretical predictions.     

Interpolation solution in generalized stochastic exponential population growth model

In this paper, first we consider model of exponential population growth, then we assume that the growth rate at time t is not completely definite and it depends on some random environment effects. For this case the stochastic exponential population growth model is introduced. Also we assume that the growth rate at time t depends on many different random environment effect, for this case the generalized stochastic exponential population growth model is introduced. The expectations and variances of solutions are obtained. For a case study, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals in each year.     

A stochastic model for interactions of hot gases with cloud droplets and raindrops

In this paper we analyze a stochastic model for interactions of hot gases with cloud droplets and raindrops. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic hot gases modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any hot gas dynamics model. We also carry out analysis on the stochastically ultimate boundedness, extinction and stability of the hot gases model.     

Remarks on stochastic permanence of population models

This paper examines the asymptotic behavior of the stochastic Lotka–Volterra model under Markovian switching. We show that the stochastic Lotka–Volterra model is stochastically permanent. Moreover, we give another type of stochastic permanence, and consider the relationship of two types of stochastic permanence under white noise perturbation.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.     

An integral representation of elasticity and sensitivity for stochastic volatility models

This paper presents a generic probabilistic approach to study elasticities and sensitivities of financial quantities under stochastic volatility models. We describe the shock elasticity, the quantile sensitivity and the vega value of cash flows with respect to perturbation of the volatility function of the model. The main contribution is to establish explicit formulae for these elasticities and sensitivities based on a novel application of the exponential measure change technique in Palmowski and Rolski (Bernoulli 8(6):767–785 2002). We carry out explicit calculations for the Heston model and the 3/2 stochastic volatility model, and derive explicit expressions in terms of model parameters.     

Euler schemes and large deviations for stochastic Volterra equations with singular kernels

In this paper, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation. Then, the large deviation estimates of small perturbation to equations of this type are obtained. We finally apply them to SDEs with the kernel of fractional Brownian motion with Hurst parameter H∈(0,1).     

Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise

A stochastic differential equation modelling a Marchuk’s model is investigated. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. Firstly, the stochastic Marchuk’s model has been simplified by applying stochastic center manifold and stochastic average theory. Secondly, by using Lyapunov exponent and singular boundary theory, we analyze the local stochastic stability and global stochastic stability for stochastic Marchuk’s model, respectively. Thirdly, we explore the stochastic bifurcation of the stochastic Marchuk’s model according to invariant measure and stationary probability density. Some new criteria ensuring stochastic pitchfork bifurcation and P-bifurcation for stochastic Marchuk’s model are obtained, respectively.     

On a logistic growth model with predation and a power‐type diffusion coefficient: I. Existence of solutions and extinction criteria

We consider a logistic growth model with a predation term and a stochastic perturbation yielding constant elasticity of variance. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, namely, linear growth and the Lipschitz condition. Nevertheless, for any positive initial condition, we prove that a solution exists and is unique up to the first time it hits zero. Additionally, we provide alternative criteria for population extinction depending on the choice of parameters. More precisely, we provide criteria that guarantee the following: (i) population extinction with positive probability for a set of initial conditions with positive Lebesgue measure; (ii) exponentially fast population extinction with full probability for any positive initial condition; and (iii) population extinction in finite time with full probability for any positive initial condition. Copyright © 2015 John Wiley & Sons, Ltd.     

Freidlin-Wentzell's large deviations for stochastic evolution equations

We prove a Freidlin-Wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic porous medium equations and stochastic reaction-diffusion equations with polynomial growth zero order term and p-Laplacian second order term.     

Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis

We consider a model of a supply chain consisting of n production facilities in tandem and producing a single product class. External demand is met from the finished goods inventory maintained in front of the most downstream facility (stage 1); unsatisfied demand is backlogged. We adopt a base-stock production policy at each stage of the supply chain, according to which the facility at stage i produces if inventory falls below a certain level w _i and idles otherwise. We seek to optimize the hedging vector w=(w ₁,...,w _n ) to minimize expected inventory costs at all stages subject to maintaining the stockout probability at stage 1 below a prescribed level (service level constraint). We make rather general modeling assumptions on demand and production processes that include autocorrelated stochastic processes. We solve this stochastic optimization problem by combining analytical (large deviations) and sample path-based (perturbation analysis) techniques. We demonstrate that there is a natural synergy between these two approaches.     

Stochastic population dynamics under regime switching

In this paper we will develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We will show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded.     

Long term behaviors of stochastic single-species growth models in a polluted environment

Thresholds for extinction and persistence are important for assessing the risk of mortality in systems exposed to toxicant. In this paper, three single-species models with random perturbation in a polluted environment are proposed and investigated. One is the generalized logistic model and the other two are the stochastic resource–consumer models of Leslie and Gallopin. For each model, the survival threshold is obtained in some cases. In general, each threshold is determined by intensity of the random noise, the mean stress measure in organisms, the population intrinsic growth rate and the stress response rate.     

On Models for an Epidemic

We consider deterministic and stochastic models for the progressof malaria. A simple model takes the form of a set of delaydifferential equations in which a small parameter multipliesthe highest derivatives and the delays. A solution is soughtas a series in powers of the small parameter, using the specialmethods appropriate to this type of singular perturbation problem.We then consider the threshold problem for the primary host. The stochastic model also presents a singular perturbation problem,and the asymptotic method is extended appropriately. This enablesus to examine the distribution of the duration of a simple epidemic,and the stochastic equivalent of the threshold problem.     

Effect of environmental fluctuation on a detritus based ecosystem

Present paper deals with the stochastic perturbation analysis on a detritus based three dimensional food-chain in presence of gestation delays and recycling delay. We have perturbed some demographic parameters by white noise and coloured noise and then extensive numerical simulations are performed to understand the effect of fluctuating environment on the dynamics of the model system for different values of forcing intensities. We have explained how stochastic perturbation terms can be introduced in the model system. Mathematical analysis reveals the fact that the internal dynamics have no ability to suppress the environmental stochasticity and rhythmic oscillation does not persist in presence of environmental driving forces rather oscillate in a irregular fashion.     

Analysis of a stochastic predator-prey system with foraging arena scheme

ABSTRACT

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results.     

Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice

In this paper, we study the existence of traveling waves of a delayed population model with age-structure on a 2-dimensional spatial lattice when the maturation time r is relatively small. Under the assumption that the birth function b satisfies the bistable condition without requiring monotonicity, we prove the persistence of traveling wavefronts by means of a perturbation argument based on the existing results on the asymptotic autonomous system and the Fredholm alternative theory.