Dynamics of the Exponential Population Growth System with Mixed Fractional Brownian Motion

Ma, Weijun; Liu, Wei; Zhu, Quanxin; Shi, Kaibo

doi:https://doi.org/10.1155/2021/5079147

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Behaviour Analysis, Complexity and Control of Networked Dynamical Systems

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Research Article | Open Access

Volume 2021 | Article ID 5079147 | https://doi.org/10.1155/2021/5079147

Dynamics of the Exponential Population Growth System with Mixed Fractional Brownian Motion

Weijun Ma,¹Wei Liu,²Quanxin Zhu,³and Kaibo Shi⁴

Academic Editor: Hou-Sheng Su

Received05 Sept 2021

Accepted04 Oct 2021

Published30 Dec 2021

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).

Definition 2. (see [2, 14]). The two limits,andare called the sample Lyapunov exponent and the th moment Lyapunov exponent, respectively.

5.1. The Almost Sure Exponential Stability

The following theorem presents a necessary and sufficient condition for system (91) to be almost surely exponentially stable.

Theorem 5. (i)When , the trivial solution of system (91) is almost surely exponentially stable if or but unstable if .(ii)When , the trivial solution of system (91) is almost surely exponentially stable if or but unstable if .(iii)When , the trivial solution of system (91) is almost surely exponentially stable for all parameters and .

Proof. From (41) and (58), we getThen, we obtainConsequently, the trivial solution is almost surely exponentially stable if and only if the sample Lyapunov exponent is negative. Hence, Theorem 5 must hold.

Remark 7. Theorem 5 shows that the trivial solution of system (91) is almost surely exponentially stable for the arbitrary Hurst index when . This means that the mfBm cannot destabilize the exponential population growth system (Figures 1–6).

5.2. th Moment Exponential Stability

The following theorem gives a necessary and sufficient condition for the th moment exponential stability of system (91).

Theorem 6. (i)When , the trivial solution of system (91) is th moment exponentially stable if one of the following conditions holds: (1) and ; (2) and ; (3) and ; or (4) , , and . Moreover, it is th moment exponentially unstable if one of the following conditions is satisfied: (1) and ; (2) and ; or (3) and .(ii)When , the trivial solution of system (91) is th moment exponentially stable if one of the following conditions holds: (1) and ; (2) and ; (3) and ; or (4) , , and . Moreover, it is th moment exponentially unstable if one of the following conditions is satisfied: (1) and ; (2) and ; or (3) and .(iii)When , the trivial solution of system (91) is th moment exponentially stable if or and , but it is th moment exponentially unstable if or and .

Proof. According to Lemma 6 and (58), one deducesTherefore, one getsThe trivial solution is th moment exponentially stable if and only if the th moment Lyapunp>ov exponent is negative. Consequently, the required assertion (iii) follows immediately from (97). When , by considering , we can get the following four cases: (1) and ; (2) and ; (3) and ; and (4) , and . Moreover, by considering , we can obtain the following three cases: (1) and ; (2) and ; and (3) and . Hence (ii) must hold. Similar to the proof in (ii), one can easily deduce (i).

6. The Phenomenon of Large Deviations in Biological Population System

Large deviations [51] involve the probability of rare events in random processes. Such events can have dramatic consequences despite their scarcity, especially if one takes into account the behavior of systems in geology [52], population [53], and climate science [54] on long time scales. These large deviations are considered to be the link between the almost sure exponential stability and the th moment exponential stability. In general, for sufficiently large , even for stable systems, the th moment Lyapunov exponent is often positive due to the large deviations. Therefore, it is important to explore the phenomenon of large deviations in biological population systems such that and for sufficiently large . When , for , ( ) almost surely, and the random variable is almost surely finite. In fact, if , then for all , and is big for sufficiently large , resulting in the th moment exponential instability, namely, the th moment Lyapunov exponent .

We now show the large deviations of system (91).

Theorem 7. (i)When , one of the following conditions fulfils the conditions and :(1) and (2) and (3) and Thus, it holds for some sufficiently large when . Therefore, the trivial solution of system (91) has large deviations for some special parameters.(ii)When , one of the following conditions fulfils the conditions and :(1) and (2) and (3) and. Thus, it holds for some sufficiently large when . Therefore, the trivial solution of system (91) has large deviations for some special parameters.(iii)When , we have and , , for , , , respectively. Therefore, the conditions and hold for all parameters , , and . Consequently, large deviations occur for all parameters , and provided that .

Proof. Thus, we can prove this theorem using the expressions of and .

Remark 8. For the exponential population growth system with fBm (91) (i.e., ), if , then and , which implies that and have the same sign for all . Thus, the trivial solution of the system is almost surely exponentially stable and th moment exponentially stable for all , if . Thus, there are no large deviations of the trivial solution for the exponential population growth system with fBm, which is different from case (i). However, if and , the conclusions are similar to that in cases (ii) and (iii), respectively. Therefore, Theorems 5, 6, and 7 are extensions of those reported in [17, 38, 55].

Remark 9. Large deviations are the most concise formulas of the relation between the almost sure exponential stability and the th moment exponential stability of system (91). This relation is tight for small ; however, for sufficiently large , and . In other words, the following phenomenon may happen: for fixed , with sample Lyapunov exponent with probability one, while for is still large with small probability (see [17], Remark 5.2). This very rare event makes big for sufficiently large and gives rise to the th moment exponential instability in system (91) due to the th moment Lyapunov exponent , which is a very interesting phenomenon in biological population systems.

7. Conclusion

This paper investigates the dynamics of the exponential population growth system with mfBm. We first present some lemmas of the SDEs with mfBm. Then, we offer some explicit expressions, mathematical expectations, and variances of the solutions of the exponential population growth system with mfBm. Furthermore, we provide two necessary and sufficient conditions that prove the exponential stability in the th moment and the almost sure stability of the constant coefficient exponential population growth system with mfBm. We also obtain the large deviations for the system, showing that the Hurst index affects the exponential stability and the large deviations. Notably, the solution of the suggested system of large deviations always exists when due to the long-range dependence, which is an important property of the fBm. Finally, the methods and lemmas presented in this paper can be applied to stochastic networked systems with mfBm, such as multiagent systems, complex networks, and neural networks, to study the dynamics of consensus, synchronization, and stability.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally.

Acknowledgments

This study was supported in part by the Ningxia Key Research and Development Program (Introduction of Talents Project) (2018BEB04029 and 2020BEB04039), Ningxia Natural Science Foundation (2019AAC03029), and National Natural Science Foundation of China (62063028, 12001016, 62173139, and 61773217).

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Copyright © 2021 Weijun Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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