Abstract
This paper examines the dynamics of the exponential population growth system with mixed fractional Brownian motion. First, we establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mixed fractional Brownian motion. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mixed fractional Brownian motion. Second, we propose two sufficient and necessary conditions for the almost sure exponential stability and the th moment exponential stability of the solution of the constant coefficient exponential population growth system with mixed fractional Brownian motion. Furthermore, we conduct some large deviation analysis of this mixed fractional population growth system. To the best of the authors’ knowledge, this is the first paper to investigate how the Hurst index affects the exponential stability and large deviations in the biological population system. It is interesting that the phenomenon of large deviations always occurs for addressed system when . Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.
1. Introduction
Many scholars recently have paid considerable attention to stochastic differential equations (SDEs), as they can be applied in many fields such as mathematics, physics, mechanics, biology, economics, complex networks, control engineering, multiagent systems, and financial markets [1–8]. However, these applications are largely dependent on the stability of the systems, namely, the long-time asymptotic behavior of the solutions to SDEs. In particular, in biology, engineering, complex networks, and control systems, it is most important to guarantee that the systems are stable, thus highlighting the need to investigate the stability of SDEs.
As is well known, there is a great amount of literature on stability analysis, for instance, see [2, 3, 9–17] and the references therein. It should be pointed out that the works in [2, 3, 9–14, 18–22] only considered the case of SDEs with Markovian noise such as Brownian motion, telegraph noise (or burst noise), Poisson noise, and Lévy noise. However, the stochastic process may fulfil the long-range dependence in some important fields like economics [17, 23–28], neural networks [5, 29], telecommunication networks [23, 26, 30], biology population [31, 32], and so forth.
The fractional Brownian motion (fBm) with the Hurst index is a Gaussian self-similar process with stationary increments. The concept of fBm was introduced by Kolmogorov [33] and Hurst [34] and then investigated in [23], where an integral is defined as the ordinary Brownian motion in the pointwise sense. fBm is one of the most important driving noises for stochastic systems, mainly as a result of its important properties, for example, long-term dependence and self-similarity features. These important properties make fBm have powerful memory effect and great potential applications [5, 16, 17, 24, 25, 27, 29, 31, 32, 35–38]. Unfortunately, because fBm is not a semimartingale or Markov process, the theories of these processes cannot be applied to investigate fBm.
Population systems are often subject to environmental noise [2, 15, 18, 21, 39–41]. In particular, environmental Brownian noise can suppress explosions in the generalized Lotka–Volterra model investigated in [39]. Additionally, a predator-prey model with telegraph noise was considered in [40]. Shaikhet investigated the stability of a stochastic glassy-winged sharpshooter population [15], while Khodabin et al. [41] studied the interpolation solution of the population systems with Brownian motion. It should be noted that references [2, 15, 18, 39–41] only considered the case of population systems driven by Markovian noise. However, with regard to the analysis of the dynamics of population systems with non-Markovian processes, there has been little work in the literature [31, 32]. Therefore, it is significant to further reveal the influence of the non-Markovian process on the dynamics of biological population systems.
In this paper, we consider the dynamics of the exponential population growth system with mixed fractional Brownian motion (mfBm), which is a linear combination of independent Brownian motion and fBm. The mfBm is a non-Markovian, long-range dependent, mixed-self-similar, and correlated process [37, 42]. To the best of our knowledge, except for [31, 32], the extant literature heavily focuses on the dynamics of population systems with Brownian motion.
The almost sure exponential stability and the th moment exponential stability are the most important stability problems. The relationship between them can be described by the theory of large deviations [11]. Several sufficient and necessary conditions for two types of exponential stability of the differential equations with Brownian motion were proposed [12]. However, the aforementioned two types of exponential stability of systems with mfBm have not been studied. Do the Hurst index and mfBm affect the exponential stability of systems? What are the large deviations, which are the rare events in biological population dynamics with mfBm? In this paper, we aim to give a positive answer to the exponential population growth system with mfBm.
In this paper, first, we will establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mfBm. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mfBm. Second, two sufficient and necessary conditions for the almost sure exponential stability and the th moment exponential stability of the solution of constant coefficient exponential population growth system with mfBm are given. In view of the exponential stability of the system, we investigate the phenomenon of large deviations. In addition, it is indicated that the almost sure exponential stability and instability are clearly different when the Hurst index takes different values.
The rest of the paper is organized as follows. In Section 2, we give some notations and lemmas. In Sections 3 and 4, we present some explicit expressions, mathematical expectations, and variances of the solutions of the exponential population growth system with mfBm. We study the exponential stability in Section 5. In Section 6, we present the phenomenon of large deviations in biological population system. Finally, Section 7 summarizes the conclusions.
2. Preliminaries
Let be the -dimensional Euclidean space, be the Euclidean norm of a vector , be the set of real numbers, and be the set of positive real numbers. is the Schwartz space of rapidly decreasing smooth functions on , and is its dual space of tempered distributions.
Let be a complete probability space and be the shorthand notation for is a fBm with Hurst index , while is a Brownian motion, with both being defined on the complete probability space. Assume that and are independent.
Let us define an operator for on bywhere represents the Fourier transform of andwith denoting the gamma function. is the closure of with norm
We take for , the identity map.
Definition 1. ( see [25, 37]). Let be a constant belonging to . A fBm of Hurst index is a centered Gaussian process with continuous sample paths and covarianceWhen , the fBm is a standard Brownian motion.
We recall an Itô formula for the stochastic differential equation with fBmon with the initial value , where and are deterministic continuous functions with , .
We now present a useful lemma of the fBm.
Lemma 1. (see [25, 35, 36]). Let and the stochastic process be described by (5). Assuming that the random variablesall belong to for all ; then,We next state the mfBm-Itô lemma, which plays a critical role in what follows.
Consider a stochastic differential equation with mfBmwhere , , and .
Lemma 2. Let and the stochastic process be described by (8). Assume that the random variablesall belong to for all ; then,
Proof. Inspired by references [22, 43, 44], we here employ two methods to prove Lemma 2.
Method 1. Using fBm-Itô Lemma 1 to , we haveThen, integrating both sides of the above expression (11) from 0 to gives (10).
Method 2. Using the well-known Taylor expansion formula leads toBecause and , we can derive thatMoreover, the Itô formulas and independence of Brownian motion and fBm show thatTherefore,whereBy substituting (15) into (12) and letting yieldsHence, integrating both sides of the above expression from 0 to yields (10).
Lemma 3. Assuming that the conditions of Lemma 2 hold, if is a solution of system (8), thenwhere is computed by the rules
Remark 1. Lemmas 2 and 3 are called the Itô lemma of mfBm (mfBm-Itô lemma). It follows from (18) that the forms of the Itô formulas for Brownian motion and mfBm are highly unified. However, the computation rules of are slightly different. Moreover, when , system (8) is driven by two independent Brownian motions. Therefore, Lemmas 2 and 3 also hold.
Remark 2. Several versions of the Itô lemma for fBm can be found in the literature (see [26–28, 36, 45–49]). Therefore, Lemma 2 or 3 is an extension of the fBm-Itô lemma with generalizability of those known results.
Based on Lemma 2, we will give some useful lemmas for mfBm.
Lemma 4. Assuming that and are Borel measurable bounded scalar functions defined on , then
Proof. Let , and apply Lemma 2.It is easy to obtainTaking mathematical expectation on both sides of (22) givesthat is,Consequently,
Lemma 5. Assuming that , and are Borel measurable bounded scalar functions defined on , andthen
Proof. By Lemma 2, we gainTaking mathematical expectation on both sides of (28) givesthat is,Consequently,
Lemma 6. Assuming that , and are Borel measurable bounded scalar functions defined on , andthen
Proof. Using the mfBm-Itô Lemma 2 impliesTaking expectation on both sides of (34) yieldsthat is,Consequently,Moreover, when and ( and are constants, the same as below), one finds that
Remark 3. If or , then Lemmas 4–6 are cases of the corresponding fBm or classical Brownian motion theory, respectively. Therefore, Lemmas 4–6 are generalizations of the classical Brownian motion and fBm theories.
Lemma 7 (law of the iterated logarithm for fBm [17, 25, 50]). There exists a suitable constant such thatand thenwhich indicates thatwhere is a constant.
Remark 4. The conclusion of Lemma 7 also holds when and (see [2], Theorem 4.2 on page 16), namely, the case of standard Brownian motion.
Lemma 8. Assuming that and are Borel measurable bounded scalar functions defined on andthen
Proof. We recall the basic property of fBm (see, e.g., [5], Remark 2.2, and [25], Definition 1.1.1 on page 5):It follows from the independence and isometry of Brownian motion and fBm that
Remark 5. Notably, Lemma 8 is not suitable for , since the kernel cannot be integrated over the diagonal.
3. The Exponential Population Growth System with mfBm
In this section, we discuss the exponential population growth system with mfBm.
We consider a simple population growth systemwhere and are the size and relative growth rate of the population at time , respectively. Let be a non-random function. Thus, we get that
In a special case, if , we obtain
It might happen that is not completely known but subject to environmental noise. In other words,
Let “noise” ; then,where and are one-dimensional Gaussian white noise and fractional Gaussian noise, respectively. and are one-dimensional Brownian motion and fBm, respectively. and denote the intensities of the noise at . Thus, this exponential population growth system with mfBm may be described as follows:
Remark 6. The exponential population growth system with mfBm (51) is a non-Markovian process because the stochastic perturbation is a mfBm. Notably system (51) reduces to the fractional exponential population growth system, if ; it becomes a stochastic exponential population growth system, if ; and it reduces to a deterministic exponential population growth system, if . Thus, the exponential population growth system with mfBm (51) includes fractional, stochastic, and deterministic systems as special cases.
Theorem 1. The explicit solution of system (51) is given by
Proof. With the help of (51), we deduceUsing the mfBm-Itô Lemma 2 to , we obtainorBy (53) and (55), we know thatConsequently,Moreover, when , and , one finds that
Theorem 2. Assuming that , and are independent random variables in (51), thenand
Proof. From (52), one obtainsUsing Lemma 4, one knows thatSubstituting (62) into (61) results inFurthermore,According to Lemma 5, one haswhich together with (64) leads toConsequently,
Corollary 1. Assuming that is a non-random variable, thenIn particular, if , we getwhich is the same as (48).
Corollary 2. Assuming that is a non-random variable, , , and , then
4. The Generalized Exponential Population Growth System with mfBm
The symbols , , , and denote, respectively, the -dimensional Gaussian white noise, Brownian motion, fractional Gaussian noise, and fBm.
By consideringwhere and are the intensities of the noise at time and denote the error of the growth rate subject to randomly fluctuating environment source at time , then we obtain the generalized exponential population growth system with mfBm:
Theorem 3. The explicit solution of system (72) is given by
Proof. It can be obtained from (72) thatApplying mfBm-Itô Lemma 3 to , we get thatorAccording to (74) and (76), we show thatConsequently,Moreover, when , , , and , one finds that
Theorem 4. Assuming that , , and are independent random variables in system (72), thenand
Proof. From (73),Employing Lemma 4 and independent of the Brownian motion and fBm, one sees thatSubstituting (83) into (82) leads toFurthermore,Applying Lemma 5, one derives thatHence,Therefore,
Corollary 3. Assuming that is a non-random variable, , , , and , thenand
5. Exponential Stability
For the sake of simplicity, in what follows, we discuss a constant coefficient exponential population growth system with mfBm:where the parameters are defined as before.
represents the solution of system (91) at time , with an initial value at time . We then give the definition of sample (or simply) Lyapunov exponent and th moment Lyapunov exponent for the solution of system (91).