key: cord-0556419-0y48nd54
authors: Tesfay, Almaz; Yuan, Shenglan; Brannan, Daniel Tesfay James
title: Most Probable Dynamics of the Single-Species with Allee Effect under Jump-diffusion Noise
date: 2021-12-14
journal: nan
DOI: nan
sha: ea965bc46077427cc9075f0339958f606ed6d0a2
doc_id: 556419
cord_uid: 0y48nd54

For a stochastic single-species model with the Allee effect, we compute the most probable phase portrait (MPPP) using the non-local Fokker-Planck equation. This bistable stochastic model is driven by multiplicative Levy noise as well as white noise. The two fixed points are stable equilibria and one is the unstable state between them. In order to study the biological behavior of species, we focus on the transition pathways from the extinction state to the upper fixed stable state for the transcription factor activator in a single-species model. By calculating the solution of the non-local Fokker-Planck equation corresponding to the population system of the single-species model, the maximum possible path is obtained, and the corresponding maximum possible stable equilibrium state is determined. We also obtain the Onsager-Machlup (OM) function for the stochastic model and solve the corresponding most probable paths. Our numerical simulation shows that: (i) The maximum of the stationary density function is located at the most probable stable equilibrium state when non-Gaussian noise is presented in the system; (ii) When the initial value increases from extinction state to the upper stable state, the most probable trajectories converge to the maximal likely equilibrium state (maximizer), in our case lies between 9 and 10; (iii) As time goes on, the most probable paths increase to stable state quickly, and remain a nearly constant level, then approach to the upper stable equilibrium state. These numerical experiment findings help researchers for further experimental study, in order to achieve good knowledge about dynamic systems in biology.

Single-species dynamics have always been a core research area in theoretical ecology. Studying singlespecies dynamics enables us to identify the conditions of extinction and the persistence of the species. One of the original and still strong motivations for developing mathematical models is to understand the cause of cycles in particular populations [1] .

Population modeling is also important for species management: managing fisheries for the highest possible sustainable yield, developing recovery plans for species threatened by extinction, and trying to contain or prevent the spread of invasive species [2] [3] [4] .

In literature, one can find several models of the dynamical single-species growth system. Gompertz growth model [5] , Verhulst growth model with or without Allee effect [6] , power law growth model [7] , the interconnections between deterministic and stochastic system [8] , Gilpin-Ayala model [9] are only a few to mention. In this study, we investigate a single-species system focusing on the Verhulst growth model with Allee effect which was developed by Y. Jin [10] .

The dynamics of biological phenomena, particularly that of populations of living beings, besides some clear trends, are frequently influenced by unpredictable components due to the complexity and variability of environmental conditions [11] . Extensive investigations in modeling and analysis of random fluctuations [12, 14] in biological dynamical systems have been on going for long time now. The studies of events in population such as persistence, extinction, and stationary distribution in stochastic single-species models constitute an interesting and important research field. One of the hot issues in population dynamics is developing sufficient condition for the persistence of biological species as mentioned in [15] [16] [17] and the references therein.

The population may be affected by sudden environmental noises [18] . For example, such as severe acute respiratory syndrome (SARS), human immunodeficiency virus (HIV), the smoking habit [19] , and the recent COVID-19 [20] , earthquakes, temperature, and hurricanes. These sudden environmental perturbations may bring substantial social and economic losses. Stochastic single-species model perturbed by Brownian motion has been researched extensively by many authors [21] [22] [23] [24] [25] . However, stochastic extension of population process driven by Gaussian noise cannot explain the aforementioned random and intermittent environmental perturbations. Introducing a Lévy process into the underlying population dynamics would explain the impact of these random jumps [26] . To the best of our knowledge, there have been only few studies conducted so far on dynamical systems in which the noise source is a Lévy process [27, 28, 9] . Introducing the non-Gaussian noise into the biological system to simulate impulses caused by external disturbances (mentioned above) is more appropriate and more near to reality than the white noise. The investigation of the single-species model is still in its infancy even though noisy fluctuations that naturally portray random intermittent jumps. Lévy noise is widely applied in studying natural and man made phenomenon in science, among which we mention, biology [29, 30] , physics [31] , and economics [32, 33] .

Under this research, we consider the population dynamics of a single-species growth model with Allee effect perturbed by stable Lévy fluctuations. We also analyze the impact of Lévy noise fluctuation on the system (1). Investigating the impact of noisy fluctuations serves in demonstrating the intricate interactions between the single-species models and their complex surroundings. In here, we study how Allee effects and stochasticity combine to affect population persistence. To find the numerical solutions for the Fokker-Planck equation determined by non-local differential equation with symmetric α-stable Lévy motion, we apply a finite difference method that is studied by [34] .

The most probable phase portrait was first proposed by J. Duan [35, Section 5.3.3] . Cheng [36] obtained the analytical results of the MPPP and showed that the MPPP can give some information about the propagation of stochastic dynamics in 1-dimensional model. Wang [37] studied the stochastic bifurcation by using the qualitative changes of the MPPP to a stochastic system driven by a stable Lévy noise. In [38] , the scholars investigated the most probable trajectories of the tumor system growth with immune surveillance under correlated Gaussian noises, and derived analytical solution of the most probable steady state by using the extremum theory with the local Fokker-Planck equation in the system. The Onsager-Machlup function for a stochastic model driven by non-Gaussian noise and Gaussian noise was established in [39] . The authors also examined the corresponding MPPP of the stochastic dynamical systems. Cheng et al. [40] focused on the impact of white noise and jump stable Lévy noise in a genetic regulatory system. Furthermore, they obtained the most probable transition pathway by minimizing the Onsager-Machlup action functional for the stochastic model under Gaussian noise.

Motivated by the above works, we are interested in studying the most probable trajectories of the singlespecies growth system under the Gaussian and non-Gaussian noises. Consequently, the most probable paths can be introduced to explore the dynamics of the stochastic system. We are able to take the MPPP as a deterministic geometric tool and give the evolution trajectory of high probability events (common events). Therefore in here, our main focus will be investigating how the most probable trajectories escape to the extinction state from the single-species state more quickly.

Consider the following stochastic single-species growth model with Allee effect:

where X t− is the left limit of the population size X t . Intraspecific competition rate γ 3 The attack rate γ 4 Represents the handling time of predator M = s/γ 2 The carrying capacity t Time

Stochastic forceÑ(dt, dy) = N(dt, dy) − ν(dy)dt is a compensated Poisson random measure with associated Poisson random measure N(dt, dy) and intensity measure ν(dy)dt, in which ν(dy) is Lévy measure on a measurable subset Y of (0, ∞) with ν(Y) < ∞.

The following restriction on system (1) is natural for biological meaning:

When ǫ(y) > 0, the perturbation stands for increasing of the species (e.g., planting), while ǫ(y) < 0 stands for decreasing (e.g., harvesting and epidemics).

The main aim of this study is to investigation dynamics of single species biological populations in random environments. We model the evolution of these populations with first order ordinary autonomous differential equations bringing in which the coefficients and inputs are stochastic processes. The two stochastic processes germane to this study are Brownian motion and the Lévy process. Brownian motion describes random fluctuations that are continuous in time; see Subsection 2.1. The Lévy process, of which Brownian motion is a special case, is used to model random fluctuations which may have discontinuities or jumps; see Subsection 2.2. Here, we develop the stochastic single-species model with Allee effect influenced by Gaussian and non-Gaussian noises. Firstly, we review the deterministic model, calculate its equilibrium solutions (fixed points) and describe the behavior of the fixed points. Secondly, we obtain the maximum possible paths, and the corresponding maximum possible stable equilibrium states, that is, attracting the nearby maximum possible paths of the stochastic system (1). We do this by finding the stationary density function which is the solution of the non-local Fokker-Planck equation. To solve the non-local partial differential equation, we use finite difference method that was proposed in [34] . This method help us explore some dynamical behaviors of the single-species system under the impact of non-Gaussian Lévy noise.

This study is organized as follows. In the second section, we recall the definitions of the one dimensional Brownian motion B t and symmetric α-stable Lévy motion L α t . In the third section, we discuss the formulation and analysis of the deterministic model of the single-species model (3) with Allee effect . In the fourth section, we explain the analysis of the stochastic single-species model with Allee effect. We also review the definition of the Onsager-Machlup function and most probable phase portraits in Subsection 4.1 and 4.2, respectively. The numerical results and the biological implication of our experimental findings are given in the fifth section. We conclude our research by giving a brief summary in the last section.

Under this section, we define the one-dimensional Brownian motion starting at time t = 0 to be a process B t and α-stable Lévy motion L α t , which constitute a class of stochastic processes that have independent and stationary increments as defined below. Throughout this study, we denote R + = (0, ∞), R = (−∞, ∞), and X t ∈ R + , for t ≥ 0.

Brownian motion B t (also called Wiener process), [41, P. 15] , [42, 43] is a one-dimensional stochastic process, defined on complete probability space (Ω, F, F t , P) that also has independent and stationary increments. Brownian motion B t satisfies the following conditions: (i) B t has continuous paths and its paths are nowhere differentiable almost surely; (ii) B t has stationary increments, i.e., B t − B s is normally distributed with mean 0 and variance t − s for any 0 ≤ s ≤ t; (iii) The process starts at the origin, i.e., B 0 = 0 almost surely; (iv) B t has independent increments, B t − B s , for s < t, is independent of the past.

Lévy motions L t are a class of non-Gaussian stochastic processes. A Lévy motion L t having values in R is defined by a drift coefficientb ∈ R,Q ≥ 0 and a Borel measure ν defined on R \ {0}. The triplet (b,Q, ν) is the so-called generating triplet of Lévy motion L t . A Lévy motion can be written as linear combination of time t, a Brownian motion B t and a pure jumping process [35, 44, 45] , i.e., L t can be expressed as

is the jump measure, and B t is the independent standard Brownian motion.

The Lévy-Khinchin formula for Lévy motion has a specific form for its characteristic function.

A stable distribution S α (θ, β, γ) is the distribution for a stable random variable, where the stability index α ∈ (0, 2), the skewness β ∈ (0, ∞), the shift γ ∈ (−∞, ∞), and scale index θ ≥ 0. A α-stable Lévy motion L α t [13, 46, 47] is a non-Gaussian stochastic process satisfying

t has stochastically continuous sample paths, i.e., for 0 ≤ s ≤ t and δ > 0, the probability P(

In the case of a one-dimensional isotropic α-stable Lévy motion, the Lévy triplet has the drift vector b = 0 and the diffusionQ = 0. In this study, we focus on jump process with a specific size in generating triplet (0, 0, ν α ) for the random variable S α which can be defined by

, and Γ is the Gamma function.

A special case of α-stable Lévy motion is Brownian motion when α = 2. Poisson process, α-stable process, compound Poisson process, etc. are also examples of Lévy processes [44] .

The deterministic form of the model (1) without noise is given as

This system can be written as dX t dt = − dU(X) dX , where U(X) is the potential function given by

The single species model (2) with Allee effect has equilibrium points X 1 = 0 and

If β < 1, then the equilibrium states of system (2) are X 1 = 0, an extinction equilibrium;

, a lower unstable equilibrium;

, an upper stable equilibrium. stable state X 1 = 0, and unstable state

For simplicity and convenience of discussion, we choose the parameters

For β < 1, the extinction state, X 1 = 0, and the equilibrium solution X 3 are stable but X 2 is unstable. Figure 1(b) shows that when the value of attack rate γ 3 increases, the unstable state X 2 and stable state X 3 get more close to each other, and become one solution and finally disappear indicating the occurrence of the saddle-node bifurcation.

The critical value of attack rate γ c = 2.67 of the deterministic single-species system (2) with Allee effect is obtained by solving the equation U(X 1 ) = U(X 3 ). This value is an indication to the transition phenomena between the unstable and stable state for for deterministic single-species growth model. The steady state (extinction state) X 1 is stable if γ 3 > γ c , and the steady state X 3 exhibits the stability property for γ 3 < γ c .

In this section, we discuss the behavior of the solution of the system presented in Eq. (1). Firstly, we recall the definition of the Onsager-Machlup function for the stochastic differential equation driven by jump noise. This helps to measure OM induced by the jump process. Secondly, we examine the corresponding most probable paths. Finally, we put the numerical experiment findings using finite difference method [34] . Hence, the numerical solution of the stochastic model provides useful information for understanding the dynamical behavior of the system.

The Onsager-Machlup functional defines a probability density for a stochastic process in which the probability density is estimated implicitly, and it is similar to the Lagrangian of a dynamical system. It can be used for purposes of reweighting and sampling trajectories, as well as for determining the most probable trajectory based on variational arguments. The most probable transition pathway can be obtained by minimizing the Onsager-Machlup function. The whole procedure enables us to detect the dynamics of the most probable path.

The stochastic single-species system (1) with Allee effect, as proved by Jin [10] , has a unique global and positive solution with the initial condition X 0 = x 0 . The jump-diffusion process X t is adapted and càdlàg; see Figure 2 . Denote the space of càdlàg paths starting at x 0 of a solution process X = {X t , t ≥ 0} of (1) by

This space equipped with Skorokhod's J 1 -topology generated by the metric d R + is a Polish space [55] . For functions x 1 , x 2 ∈ D x 0 , define

for every t, s ≥ 0 and someλ ∈ Λ R + ,

We consider the corresponding jump-diffusion process X t (ω) := ω(t), t ∈ [0, T ] defined on the canonical probability space (R [0,T ] , B(R) [0,T ] , P T ). Since the paths of X are càdlàg, we identify X t on the space 

Hence, D T x 0 is a Banach space. We are interested in finding the most probable tube of X t , so it is meaningful to determine the probability that paths lie within the closed tube

It is worthwhile to note that this probability can be calculated or estimated by using the induced measure in function space [39] . Define the measure µ X on B(R) [0,T ] induced by the solution process X t for the model

Using the induced measure, once an ε > 0 is given, we are able to investigate the probabilities of tubes of the same 'thickness' for all z ∈ D T x 0 by

when K(z, ε) ∈ B(R) [0,T ] . As the tube K(z, ε) depends on the choice of a function z(t), we have to look for that function z(t) which maximizes Eq. (4). If we restrict ourselves on differentiable functions z(t), then the following definition makes sense [49] .

Definition 1. Let ε > 0 be given. Consider a tube surrounding a reference path z(t). If for ε sufficiently small we estimate the probability of the solution process X t , t ∈ [0, T ] lying in this tube in the form : 

Our main result about the expression of the OM function for a jump-diffusion process is present in the following theorem. Theorem 1. For the stochastic system (1) with the jump measure satisfying Y ǫ(y)ν(dy) < ∞, the Onsager-Machlup function [39] is characterized, up to an additive constant, by:

x 0 is a differentiable function. The contribution of pure jump Lévy noise to the OM function is the third term. When the jump measure is absent, we cover the OM function for the case of diffusion. In terms of OM function, the measure of tube K(z, ε) defined as (3) can be approximated as follows:

where Y c t is defined by The proof of this theorem is given in [39, Theorem 4.1] .

In Gaussian noise case (ǫ = 0), the stochastic single-species model (1) becomes

We apply Lamperti transforms for solving multiplicative noise SDE [35, Example 6.48] . This method helps us to transform the multiplicative noise into additive noise. Because numerically solving an additive-noise SDE is usually easier than solving a SDE in Eq. (5).

Assume g ∈ C 2 (R) and define Y t = g = ln(X t ). Then the new SDE has the following form:

where

and

The most probable transition pathway Z m (t) can be obtained the following least action principle

where the integrand function (Onsager-Machlup function) [40] is given by Since the most probable transition path for a stochastic single-species model is the minimizer of the Onsager-Machlup action functional, denoted by Z m (t), thus Eq. (7) satisfies the following Euler-Lagrange equation

The most probable transition pathway Z m (t) of system (6) is given bÿ

To solve two-point boundary value problem in Eq. (9), we apply the shooting method proposed in [50] .

As the solution of the Fokker-Planck equation, the probability density function p(X, t) is a surface in the (X, t, p)-space. For a given time t, the maximizer X m (t) for p(X, t) (X m (t) = max X∈(0,∞) p(X, t)) shows the most probable (i.e., maximal likely) location of this orbit at time t. The orbit traced out by X m (t) is called a most probable orbit starting at x 0 . Thus, the deterministic orbit X m (t) follows the top ridge of the surface in the (X, t, p)-space as time goes on.

The Fokker-Planck equation describes the time evolution of the probability density function, but it can be solved analytically only in special cases. We are interested in the steady-state probability distribution (equilibrium distribution), and express the stationary solution of the non-local Fokker-Planck equation. This makes the estimate of the most probable phase portrait numerically and algorithmically possible in Lévy noise case.

Let f : R → R be a smooth function. Assume that the solution X t of system (1) has a conditional probability density p(X, t|x 0 , 0) [51] [52] [53] [54] . For convenience, we drop the initial condition and simply denote it by p(X, t). On one hand, x 0 =0.55 x 0 =10

x 0 =11 

On the other hand, by virtue of Itô's formula,

Taking expectation on both sides of (10), we gain

Noting that the infinitesimal generator of the solution X t for system (1) is

The equation (11) is rewritten as As a result, the Fokker-Planck equation for the system (1) for the solution process

To simulate the non-local Fokker-Planck equation (13), we apply a numerical finite difference method given in Gao et al. [34] .

If the Lévy motion is replaced by Brownian motion, then local Fokker-Planck equation has the following form:

The stationary probability density function p s (X) of Eq. (14) can be solved by

or equivalently,

Due to the complexity of stationary probability density solution, we take the extrema of the stationary probability density function (spdf) located at X = x s directly. In other words, the spdf satisfies ∂ X ( p s (x s )) = 0. Since, p s (x s ) 0, Eq. (16) becomes

Eq. (17) is completely different from the equilibrium state of deterministic model (2) because of the presence of noise with λ term. The numerical solution of Eq. (17) is plotted in Figure 3 (b).

In order to make the readers understand our results more better, we perform some numerical simulations to illustrate our theoretical results. The numerical simulations which are very useful in the study of real examples of population, are given by the following finite difference method [34] . In the present section, we define bifurcation time as the time between the changes in number of maximal likely equilibrium states. It is a time scale for the birth of a new most probable stable equilibrium state. We also show the intervals where there exist one or two maximal likely stable equilibrium states, the value of the equilibrium states, and the point where the number of metastable states of the stochastic single-species model. Since, the numerical solutions of a model depend on the values of all its deterministic parameters and noise intensities. Here, we discuss the effect of these parameters on the investigated system. For simplicity, we choose four maximum likely pathways together with the initial conditions. The initial conditions are selected in different intervals.

While we plot the above Figures, we fix the deterministic parameters s = 1, γ 2 = 0.1, γ 3 = 2.67, γ 4 = 1/γ 3 , the noise intensity ǫ = 0.5, and the stability index α = 1.5.

The potential function denoted by U(X) in Figure 1 (a) has two stable steady states X 1 and X 3 , and an unstable steady state X 2 for β < 1. This function has a maximum value at the unstable equilibrium solution X 2 . At the stable fixed points X 1 and X 3 , the potential function attains its minima. For the value of β > 1, the system (2) has only one equilibrium point, which is the trivial point X 1 = 0.

In Figure 1(b) , we plot the equilibrium states versus attack rate γ 3 . For β < 1, there exist two stable equilibrium states X 1 and X 3 and one unstable equilibrium state X 2 . While β > 1, X 1 = 0 is the only equilibrium state that is stable. Thus, the parameter β = 1 is bifurcation parameter value.

The distance between the unstable X 2 and the stable X 3 equilibrium points becomes very small when β approaches to 1. This indicates that the expected time to extinction may be too short; see Figure 1 (b). Figure 2 displays the numerical simulation of the stochastic single-species model (1) with Allee effect when it is persistent or extinct at different value of initial condition x 0 . This figure proves that the solutions of the stochastic system are positive, and extinction species occurs when the initial condition is less than X 2 ; see Figure 2 (b). While the initial condition is greater than the value of X 2 , there is stochastic persistence.

In Figure 3 (a), we plot the most probable transition pathways, denoted by Z m (t) of system (2.2) for different values of λ. This Figure tells us that as time increases, the most probable Z m (t) increases to stable state X 3 quickly, and remains a nearly constant level, then decays (approaches) to the high stable equilibrium state. Figure 3 (b) demonstrates the curves for the most probable steady state x s of the stochastic singlespecies model with Allee effect driven by Gaussian noise at different values of the Gaussian noise intensity λ. The steady state curves exhibit a bi-stability in the interval (a 1 , , a 2 ). For γ 3 > a 1 , the stable steady state stays at the extinction state. While for γ 3 < a 1 , it remains at the stable equilibrium state. Because of the presence of noise with λ term, the numerical result in Figure 3 (b) is completely different from the numerical result in Figure 1 

The most probable trajectories of the stochastic single-species (1) with Allee effect are plotted in Figures  4 and 5 (a) . Here the values of the noise intensities λ = 0 and ǫ = 0.5, respectively, and stability index α = 1.5, and we choose the interval D = (0, 15). These Figures evolve as initial value x 0 changes, and they tell us that the maximal likely equilibrium state (maximizer) X m (t) which lies between 9 and 10 at the bifurcation time 1.13. In other words, the maximizer in high concentration is between 9 and 10, it is different from the deterministic equilibrium stable solution X 3 = 9.0846 due to the impact of the noises. Figure 5 (a) tells us that the most probable growth state is attracted to the maximal likely equilibrium state in the extinction state, and then it leads to the maximal likely equilibrium state in the high concentration as time goes on. Figure 5 (b) plots the MPPP for different initial values x 0 . From this figure, we observe that the maximum value of the stationary density function p(X, t) is located at the maximum likely stable state X m (t) = 9.0846 with the initial condition p(X, 0) = 40 π e −40(X−x 0 ) 2 . As the initial condition x 0 increases, the peak point of the stationary density function p(X, t) increases. This exhibits extinction of the species may not happen, and the high peak occurs at the maximum likely stable state X m (t).

A function which summarizes about the behavior of the dynamics of a continuous stochastic process is known as the Onsager-Machlup function.

In the present work, we have studied the Onsager-Machlup functional and most probable phase portraits for the stochastic model for single-species population growth with strong Allee effects driven by Lévy noise. We have focused on the effect of different values of the initial condition on the MPPP of the system. Small disturbances may cause a transition between the extinction stable state X 1 and the upper equilibrium state X 3 , thus we have used a deterministic quantity, namely the maximal likely trajectory to analyze the transition phenomena in a jump stochastic environment.

In order to find the most likely pathways in transition phenomena, we have calculated the most probable paths of our stochastic differential equation in (1) using the stationary density function of the non-local Fokker-Planck equation which is associated with a non-local partial differential equation. We have investigated on the effect of deterministic parameters, noise intensities and the domain size on the FPE, i.e., we have studied the dependence of the probability density on the initial condition x 0 . Our finding displays that the maximum of the stationary density function is located at the most probable stable equilibrium state X m .

In conclusion, the most probable path has been used as an indicator that helps the researcher to understand the dynamics of the single-species model based on the evolution of the probability density function over time.

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This work was partly supported by the NSFC grant 12001213.