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

Authors: Weijun Ma; Wei Liu; Quanxin Zhu; Kaibo Shi
Journal: Complexity
 (2021)

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## 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 thekth 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 1/2<H<1. Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.


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## Body

## 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 indexH∈0,1 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 thekth 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 thekth 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 Section2, 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

Letℝn be the n-dimensional Euclidean space, x be the Euclidean norm of a vector x, ℝ=−∞,+∞ be the set of real numbers, and ℝ+=0,+∞ be the set of positive real numbers. Sℝ+ is the Schwartz space of rapidly decreasing smooth functions on ℝ+, and Sℝ+ is its dual space of tempered distributions.LetΩ,ℱ,ℙ be a complete probability space and L2ℙ be the shorthand notation for L2Ω,ℱ,ℙ.BHt is a fBm with Hurst index H∈0,1, while Bt is a Brownian motion, with both being defined on the complete probability space. Assume that Bt and BHt are independent.Let us define an operatorMH for 0<H<1 on Sℝ+ by(1)MH^fy=CHy1/2−Hf^y,f∈Sℝ+,where f^ represents the Fourier transform of f and(2)CH=2ΓH−12cosπ2H−12−1Γ2H+1sinπH1/2,with Γ denoting the gamma function. LH2ℝ+ is the closure of Sℝ+ with norm(3)fLH2ℝ+2=∫0+∞Mfx2dx,f∈Sℝ+.We takeMfx=fx for H=1/2, the identity map.Definition 1.
( see [25, 37]). Let H be a constant belonging to 0,1. A fBm BH=BHt,t∈ℝ of Hurst index H is a centered Gaussian process with continuous sample paths and covariance(4)RHt,s=EBHtBHs=12t2H+s2H−t−s2H;t,s∈ℝ.
WhenH=1/2, the fBm is a standard Brownian motion.
We recall an Itô formula for the stochastic differential equation with fBm(5)dxt=αtdt+γtdBHt,on t∈ℝ+ with the initial value x0=x0∈ℝ, where αt and γt are deterministic continuous functions with αt∈L1ℝ+;ℝ, γt∈LH2ℝ+;ℝ.
We now present a useful lemma of the fBm.Lemma 1.
(see [25, 35, 36]). Let Qt,xt∈C1,2ℝ+×ℝ;ℝ and the stochastic process xt be described by (5). Assuming that the random variables(6)Qt,xt,∫0t∂Q∂xs,xsdxs,∫0ts2H−1γ2s∂2Q∂x2s,xsdsall belong to L2ℙ for all t∈ℝ+; then,(7)Qt,xt=Q0,x0+∫0t∂Q∂ss,xsds+∫0t∂Q∂xs,xsdxs+H∫0ts2H−1γ2s∂2Q∂x2s,xsds.
We next state the mfBm-Itô lemma, which plays a critical role in what follows.
Consider a stochastic differential equation with mfBm(8)dxt=αtdt+βtdBt+γtdBHt,t∈ℝ+,x0=x0,where αt∈L1ℝ+;ℝ, βt∈L2ℝ+;ℝ, and γt∈LH2ℝ+;ℝ.Lemma 2.
LetQt,xt∈C1,2ℝ+×ℝ;ℝ and the stochastic process xt be described by (8). Assume that the random variables(9)Qt,xt,∫0t∂Q∂xs,xsdxs,∫0tβ2s∂2Q∂x2s,xsds,∫0ts2H−1γ2s∂2Q∂x2s,xsdsall belong to L2ℙ for all t∈ℝ+; then,(10)Qt,xt=Q0,x0+∫0t∂Q∂ss,xs+αs∂Q∂xs,xsds+∫0t12β2s+Hs2H−1γ2s∂2Q∂x2s,xsds+∫0tβs∂Q∂xs,xsdBs+∫0tγs∂Q∂xs,xsdBHs.Proof.
Inspired by references [22, 43, 44], we here employ two methods to prove Lemma 2.
Method 1. Using fBm-Itô Lemma 1 to Qt,xt, we have(11)dQt,xt=∂Q∂t+12β2t∂2Q∂x2+Ht2H−1γ2t∂2Q∂x2dt+∂Q∂xdxt,=∂Q∂t+12β2t∂2Q∂x2+Ht2H−1γ2t∂2Q∂x2+αt∂Q∂xdt+βt∂Q∂xdBt+γt∂Q∂xdBHt.
Then, integrating both sides of the above expression (11) from 0 to t gives (10).
Method 2. Using the well-known Taylor expansion formula leads to(12)dQt,xt=∂Q∂tdt+∂Q∂xdxt+12∂2Q∂x2dxt2+odt.
BecauseEBt2=t and EBHt2=t2H, we can derive that(13)dBt2=dt,dBHt2≈VardBHt=EdBHt2=dt2H=2Ht2H−1dt.
Moreover, the Itô formulas and independence of Brownian motion and fBm show that(14)dt⋅dt=dt⋅dBt=dBt⋅dBHt=dt⋅dBHt=0.
Therefore,(15)dxt2=αtdt+βtdBt+γtdBHt2=α2tdt2+β2tdBt2+γ2tdBHt2+2αtβtdtdBt+2αtγtdtdBHt+2βtγtdBtdBHt=β2tdt+2Ht2H−1γ2tdt+odt,where(16)odt=α2tdt2+2αtβtdtdBt+2αtγtdtdBHt+2βtγtdBtdBHt.
By substituting (15) into (12) and letting dt⟶0 yields(17)dQt,xt=∂Q∂tdt+12∂2Q∂x2β2tdt+2Ht2H−1γ2tdt+∂Q∂xαtdt+βtdBt+γtdBHt=∂Q∂tdt+αt∂Q∂xdt+βt∂Q∂xdBt+γt∂Q∂xdBHt+12β2t∂2Q∂x2dt+Ht2H−1γ2t∂2Q∂x2dt.
Hence, integrating both sides of the above expression from 0 tot yields (10).Lemma 3.
Assuming that the conditions of Lemma2 hold, if xt is a solution of system (8), then(18)dQt,xt=∂Q∂tt,xtdt+∂Q∂xt,xtdxt+12∂2Q∂x2t,xtdxt2,where dxt2=dxt⋅dxt is computed by the rules(19)dt⋅dt=dt⋅dBt=dBt⋅dBHt=dt⋅dBHt=0,dBt⋅dBt=dt,dBHt⋅dBHt=2Ht2H−1dt.Remark 1.
Lemmas2 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 dxt2 are slightly different. Moreover, when H=1/2, 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 Lemma2, we will give some useful lemmas for mfBm.Lemma 4.
Assuming thatut and vt are Borel measurable bounded scalar functions defined on ℝ+, then(20)Eexp∫t0tusdBs+vsdBHs=exp∫t0t12u2s+Hs2H−1v2sds,0≤t0≤t.Proof.
Letzt=exp∫t0tusdBs+vsdBHs, and apply Lemma 2.(21)dzt=exp∫t0tusdBs+vsdBHsutdBt+vtdBHt+12exp∫t0tusdBs+vsdBHsu2tdt+Hexp∫t0tusdBs+vsdBHst2H−1v2tdt.
It is easy to obtain(22)zt=zt0+∫t0texp∫t0surdBr+vrdBHrusdBs+vsdBHs+12∫t0texp∫t0surdBr+vrdBHru2sds+H∫t0texp∫t0surdBr+vrdBHrs2H−1v2sds.
Taking mathematical expectation on both sides of (22) gives(23)Ezt=Ezt0+12∫t0tEzsu2sds+H∫t0tEzss2H−1v2sds,that is,(24)ddtEzt=12u2t+Ht2H−1v2tEzt,Ezt0=1.
Consequently,(25)Ezt=exp∫t0t12u2s+Hs2H−1v2sds.Lemma 5.
Assuming thatk>0, ut and vt are Borel measurable bounded scalar functions defined on ℝ+, and(26)zt=exp∫t0tusdBs+vsdBHs,then(27)Ezkt=expk2∫t0t12u2s+Hs2H−1v2sds,0≤t0≤t.Proof.
By Lemma2, we gain(28)zkt=zkt0+k∫t0tuszksdBs+k∫t0tvszksdBHs+k2∫t0t12u2s+Hs2H−1v2szksds.
Taking mathematical expectation on both sides of (28) gives(29)Ezkt=Ezkt0+k2∫t0t12u2s+Hs2H−1v2sEzksds,that is,(30)ddtEzkt=k212u2t+Ht2H−1v2tEzkt,Ezkt0=1.
Consequently,(31)Ezkt=expk2∫t0t12u2s+Hs2H−1v2sds.Lemma 6.
Assuming thatk>0, ut and vt are Borel measurable bounded scalar functions defined on ℝ+, and(32)zt=exputBt+vtBHt,then(33)Ezkt=expk212u2t+Ht2H−1v2t.Proof.
Using the mfBm-Itô Lemma2 implies(34)zkt=zk0+k∫0tuszksdBs+k∫0tvszksdBHs+k2∫0t12u2s+Hs2H−1v2szksds.
Taking expectation on both sides of (34) yields(35)Ezkt=Ezk0+k2∫0t12u2s+Hs2H−1v2sEzksds,that is,(36)ddtEzkt=k212u2t+Ht2H−1v2tEzkt,Ezk0=1.
Consequently,(37)Ezkt=expk212u2t+Ht2H−1v2t.
Moreover, whenut=u and vt=v (u and v are constants, the same as below), one finds that(38)Ezkt=expk212u2t+Ht2Hv2.Remark 3.
Ifut=0 or vt=0, 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 constantCH>0 such that(39)limsupt⟶+∞BHttHloglogt=CH,and then(40)limsupt⟶+∞BHttHloglogt=CH+ε,which indicates that(41)limsupt⟶+∞BHtt=0,where ϵ>0 is a constant.Remark 4.
The conclusion of Lemma7 also holds when H=1/2 and C1/2=2 (see [2], Theorem 4.2 on page 16), namely, the case of standard Brownian motion.Lemma 8.
Assuming thatut and vt are Borel measurable bounded scalar functions defined on ℝ+ and(42)xt=∫0tusdBs+∫0tvsdBHs,12<H<1,then(43)xt∼ℕ0,∫0tu2sds+H∫0tv2st−s2H−1ds.Proof.
We recall the basic property of fBm (see, e.g., [5], Remark 2.2, and [25], Definition 1.1.1 on page 5):(44)Ext=E∫0tusdBs+E∫0tvsdBHs=0.
It follows from the independence and isometry of Brownian motion and fBm that(45)Varxt=Ext−Ext2=E∫0tusdBs+∫0tvsdBHs2=E∫0tusdBs2+E∫0tvsdBHs2=∫0tu2sds+H2H−1∫0t∫0tvξvηξ−η2H−2dξdη=∫0tu2sds+H2H−1∫0t∫0ξv2sξ−s2H−2dsdξ=∫0tu2sds+H2H−1∫0t∫stv2sξ−s2H−2dξds=∫0tu2sds+H∫0tv2st−s2H−1ds.Remark 5.
Notably, Lemma8 is not suitable for 0<H<1/2, since the kernel ξ−η2H−2 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 system(46)dPtdt=μtPt,P0=P0,where Pt and μt are the size and relative growth rate of the population at time t, respectively. Let μt=θt be a non-random function. Thus, we get that(47)Pt=P0exp∫0tθsds.In a special case, ifθt=θ, we obtain(48)Pt=P0expθt.It might happen thatμt is not completely known but subject to environmental noise. In other words,(49)μt=θt+“noise”.Let “noise”=utWt+vtWHt; then,(50)μt=θt+utWt+vtWHt,where Wt=dBt/dt and WHt=dBHt/dt are one-dimensional Gaussian white noise and fractional Gaussian noise, respectively. Bt and BHt are one-dimensional Brownian motion and fBm, respectively. ut and vt denote the intensities of the noise at t. Thus, this exponential population growth system with mfBm may be described as follows:(51)dPt=θtPtdt+utPtdBt+vtPtdBHt,P0=P0.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 ut=0,vt≠0; it becomes a stochastic exponential population growth system, if ut≠0,vt=0; and it reduces to a deterministic exponential population growth system, if ut=0,vt=0. 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(52)Pt=P0exp∫0tθs−12u2s−Hs2H−1v2sds+∫0tusdBs+∫0tvsdBHs.Proof.
With the help of (51), we deduce(53)∫0tdPsPs=∫0tθsds+∫0tusdBs+∫0tvsdBHs.
Using the mfBm-Itô Lemma2 to lnx, we obtain(54)dlnPt=dPtPt−12u2t+Ht2H−1v2tdt,or(55)∫0tdPsPs=lnPtP0+∫0t12u2s+Hs2H−1v2sds.
By (53) and (55), we know that(56)lnPtP0=∫0tθs−12u2s−Hs2H−1v2sds+∫0tusdBs+∫0tvsdBHs.
Consequently,(57)Pt=P0exp∫0tθs−12u2s−Hs2H−1v2sds+∫0tusdBs+∫0tvsdBHs.
Moreover, whenθt=θ, ut=u and vt=v, one finds that(58)Pt=P0expθ−12u2−12t2H−1v2t+uBt+vBHt.Theorem 2.
Assuming thatP0, Bt and BHt are independent random variables in (51), then(59)EPt=EP0exp∫0tθsds,and(60)VarPt=exp2∫0tθsdsVarP0+E2P0exp∫0tu2s+2Hs2H−1v2sds−E2P0.Proof.
From (52), one obtains(61)EPt=EP0exp∫0tθs−12u2s−Hs2H−1v2sds×Eexp∫0tusdBs+vsdBHs.
Using Lemma4, one knows that(62)Eexp∫0tusdBs+vsdBHs=exp∫0t12u2s+Hs2H−1v2sds.
Substituting (62) into (61) results in(63)EPt=EP0exp∫0tθsds.
Furthermore,(64)EP2t=EP02exp2∫0tθs−12u2s−Hs2H−1v2sds×Eexp2∫0tusdBs+vsdBHs.
According to Lemma5, one has(65)Eexp2∫0tusdBs+vsdBHs=exp4∫0t12u2s+Hs2H−1v2sds,which together with (64) leads to(66)EP2t=EP02exp2∫0tθs+12u2s+Hs2H−1v2sds.
Consequently,(67)VarPt=exp2∫0tθsdsEP02exp∫0tu2s−2Hs2H−1v2sds−E2P0=exp2∫0tθsdsVarP0+E2P0exp∫0tu2s−2Hs2H−1v2sds−E2P0.Corollary 1.
Assuming thatP0 is a non-random variable, then(68)EPt=P0exp∫0tθsds.
In particular, ifθt=θ, we get(69)EPt=P0expθt,which is the same as (48).Corollary 2.
Assuming thatP0 is a non-random variable, θt=θ, ut=u, and vt=v, then(70)VarPt=P02exp2θtexpu2+t2H−1v2t−1.


## 4. The Generalized Exponential Population Growth System with mfBm

The symbolsWt=W1t,W2t,…,Wnt, Bt=B1t,B2t,…,Bnt, WHt=W1Ht,W2Ht,…,WnHt, and BHt=B1Ht,B2Ht,…,BnHt denote, respectively, the n-dimensional Gaussian white noise, Brownian motion, fractional Gaussian noise, and fBm.By considering(71)μt=ut+12∑i=1nui2t+∑i=1nvitWit+∑i=1nwitWiHt,where vit and witi=1,…,n are the intensities of the noise at time t and uit denote the error of the growth rate subject to randomly fluctuating environment source i at time t, then we obtain the generalized exponential population growth system with mfBm:(72)dPt=ut+12∑i=1nui2tdt+∑i=1nvitdBit+∑i=1nwitdBiHtPt,P0=P0.Theorem 3.
The explicit solution of system (72) is given by(73)Pt=P0exp∫0tus+12∑i=1nui2s−vi2s−Hs2H−1∑i=1nwi2sds+∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.Proof.
It can be obtained from (72) that(74)∫0tdPsPs=∫0tus+12∑i=1nui2sds+∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.
Applying mfBm-Itô Lemma3 to lnx, we get that(75)dlnPt=dPtPtt−12∑i=1nvi2t+Ht2H−1∑i=1nwi2tdt,or(76)∫0tdPsPs=lnPsP0+∫0t12∑i=1nvi2s+Hs2H−1∑i=1nwi2sds.
According to (74) and (76), we show that(77)lnPtP0=∫0tus+12∑i=1nui2s−vi2s−Hs2H−1∑i=1nwi2sds+∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.
Consequently,(78)Pt=P0exp∫0tus+12∑i=1nui2s−vi2s−Hs2H−1∑i=1nwi2sds+∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.
Moreover, whenut=u, uit=ui, vit=vi, and wit=wii=1,…,n, one finds that(79)Pt=P0expu+12∑i=1nui2−vi2−12t2H−1∑i=1nwi2t+∑i=1nviBit+∑i=1nwiBiHt.Theorem 4.
Assuming thatP0, Bit, and BiHti=1,…,n are independent random variables in system (72), then(80)EPt=EP0exp∫0tus+12∑i=1nui2sds,and(81)VarPt=exp2∫0tus+12∑i=1nui2sds×VarP0+E2P0exp∑i=1n∫0tvi2s+2Hs2H−1wi2sds−E2P0.Proof.
From (73),(82)EPt=EP0exp∫0tus+12∑i=1nui2s−vi2s−Hs2H−1∑i=1nwi2sds×Eexp∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.
Employing Lemma4 and independent of the Brownian motion and fBm, one sees that(83)Eexp∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs=exp∑i=1n∫0t12vi2s+Hs2H−1wi2sds.
Substituting (83) into (82) leads to(84)EPt=EP0exp∫0tus+12∑i=1nui2sds.
Furthermore,(85)EP2t=EP02exp2∫0tus+12∑i=1nui2s−vi2s−Hs2H−1∑i=1nwi2sds×Eexp2∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs.
Applying Lemma5, one derives that(86)Eexp2∑i=1n∫0tvisdBis+∑i=1n∫0twisdBiHs=exp4∑i=1n∫0t12vi2s+Hs2H−1wi2sds.
Hence,(87)EP2t=EP02exp2∫0tus+12∑i=1nui2s+vi2s+Hs2H−1∑i=1nwi2sds.
Therefore,(88)VarPt=exp2∫0tus+12∑i=1nui2sds×EP02exp∑i=1n∫0tvi2s+2Hs2H−1wi2sds−E2P0=exp2∫0tus+12∑i=1nui2sds×VarP0+E2P0exp∑i=1n∫0tvi2s+2Hs2H−1wi2sds−E2P0.Corollary 3.
Assuming thatP0 is a non-random variable, ut=u, uit=ui, vit=vi, and wit=wii=1,…,n, then(89)EPt=P0expu+12∑i=1nui2t,and(90)VarPt=P02exp2u+12∑i=1nui2texp∑i=1nvi2+t2H−1wi2t−1.


## 5. Exponential Stability

For the sake of simplicity, in what follows, we discuss a constant coefficient exponential population growth system with mfBm:(91)dPt=θPtdt+uPtdBt+vPtdBHt,P0=P0,where the parameters are defined as before.Pt;t0,P0 represents the solution of system (91) at time t, with an initial value P0 at time t0. We then give the definition of sample (or simply) Lyapunov exponent and kth moment Lyapunov exponent for the solution of system (91).Definition 2.
(see [2, 14]). The two limits,(92)L≔limsupt⟶+∞1tlogPt;t0,P0,and(93)Lk≔limsupt⟶+∞1tlogEPt;t0,P0k,are called the sample Lyapunov exponent and the kth 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)
When0<H<1/2, the trivial solution of system (91) is almost surely exponentially stable if θ≤0 or θ<1/2u2 but unstable if θ>1/2u2.(ii)
WhenH=1/2, the trivial solution of system (91) is almost surely exponentially stable if θ≤0 or θ<1/2u2+v2 but unstable if θ>1/2u2+v2.(iii)
When1/2<H<1, the trivial solution of system (91) is almost surely exponentially stable for all parameters θ,u and v.Proof.
From (41) and (58), we get(94)L=limsupt⟶+∞1tlogPt;0,P0=limsupt⟶+∞1tlogP0expθ−12u2−12t2H−1v2t+uBt+vBHt=limt⟶+∞θ−12u2−12t2H−1v2.
Then, we obtain(95)L=θ−12u2,0<H<12,θ−12u2+v2,H=12,−∞,12<H<1.
Consequently, the trivial solution is almost surely exponentially stable if and only if the sample Lyapunov exponent is negative. Hence, Theorem5 must hold.Remark 7.
Theorem5 shows that the trivial solution of system (91) is almost surely exponentially stable for the arbitrary Hurst index 0<H<1 when θ≤0. This means that the mfBm cannot destabilize the exponential population growth system (Figures 1–6).Figure 1
The trajectories of system (91) with P0=1,H=0.3. (a) θ=1,u=1,v=1. (b) θ=1,u=1,v=2. The trajectories of system (91) are almost surely exponentially unstable for both (a) and (b) in the sense of Theorem 5 (i).Figure 2
The trajectories of system (91) with P0=1,H=0.3. (a) θ=1,u=3,v=2. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (i).Figure 3
The trajectories of system (91) with P0=1,H=0.5. (a) θ=2,u=1,v=1. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially unstable for (a) but stable for (b) in the sense of Theorem 5 (ii).Figure 4
The trajectories of system (91) with P0=1,H=0.5. (a) θ=1,u=2,v=1. (b) θ=1,u=1,v=2. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (ii).Figure 5
The trajectories of system (91) with P0=1,H=0.8. (a) θ=1,u=1,v=1. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (iii).Figure 6
The trajectories of system (91) with P0=1,H=0.8. (a) θ=1,u=1,v=2. (b) θ=1,u=2,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (iii).


### 5.2.kth Moment Exponential Stability

The following theorem gives a necessary and sufficient condition for thekth moment exponential stability of system (91).Theorem 6.
(i)
When0<H<1/2, the trivial solution of system (91) is kth moment exponentially stable if one of the following conditions holds: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2; (3) k=1 and θ<0; or (4) k>1, θ<0, and 1/2k−1u2<−θ. Moreover, it is kth moment exponentially unstable if one of the following conditions is satisfied: (1) 0<k<1 and θ>1/21−ku2; (2) k=1 and θ>0; or (3) k>1 and θ≥0.(ii)
WhenH=1/2, the trivial solution of system (91) is kth moment exponentially stable if one of the following conditions holds: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2+v2; (3) k=1 and θ<0; or (4) k>1, θ<0, and 1/2k−1u2+v2<−θ. Moreover, it is kth moment exponentially unstable if one of the following conditions is satisfied: (1) 0<k<1 and θ>1/21−ku2+v2; (2) k=1 and θ>0; or (3) k>1 and θ≥0.(iii)
When1/2<H<1, the trivial solution of system (91) is kth moment exponentially stable if 0<k<1 or k=1 and θ<0, but it is kth moment exponentially unstable if k>1 or k=1 and θ>0.Proof.
According to Lemma6 and (58), one deduces(96)Lk=limsupt⟶+∞1tlogEPt;0,P0k=limsupt⟶+∞1tlogP0kEexpkθ−12u2−12t2H−1v2t+kuBt+kvBHt=limt⟶+∞kθ+12kk−1u2+12kk−1v2t2H−1.
Therefore, one gets(97)Lk=kθ+12kk−1u2,0<H<12,kθ+12kk−1u2+v2,H=12,−∞,12<H<1and0<k<1,θ,12<H<1andk=1,+∞,12<H<1andk>1.
The trivial solution iskth moment exponentially stable if and only if the kth moment Lyapunp>ov exponent is negative. Consequently, the required assertion (iii) follows immediately from (97). When H=1/2, by considering Lk=kθ+1/2kk−1u2+v2<0, we can get the following four cases: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2+v2; (3) k=1 and θ<0; and (4) k>1, θ<0 and 1/2k−1u2+v2<−θ. Moreover, by considering Lk=kθ+1/2kk−1u2+v2>0, we can obtain the following three cases: (1) 0<k<1 and θ>1/21−ku2+v2; (2) k=1 and θ>0; and (3) k>1 and θ≥0. Hence (ii) must hold. Similar to the proof in (ii), one can easily deduce (i).


## 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)
When0<H<1/2, the trivial solution of system (91) is almost surely exponentially stable if θ≤0 or θ<1/2u2 but unstable if θ>1/2u2.(ii)
WhenH=1/2, the trivial solution of system (91) is almost surely exponentially stable if θ≤0 or θ<1/2u2+v2 but unstable if θ>1/2u2+v2.(iii)
When1/2<H<1, the trivial solution of system (91) is almost surely exponentially stable for all parameters θ,u and v.Proof.
From (41) and (58), we get(94)L=limsupt⟶+∞1tlogPt;0,P0=limsupt⟶+∞1tlogP0expθ−12u2−12t2H−1v2t+uBt+vBHt=limt⟶+∞θ−12u2−12t2H−1v2.
Then, we obtain(95)L=θ−12u2,0<H<12,θ−12u2+v2,H=12,−∞,12<H<1.
Consequently, the trivial solution is almost surely exponentially stable if and only if the sample Lyapunov exponent is negative. Hence, Theorem5 must hold.Remark 7.
Theorem5 shows that the trivial solution of system (91) is almost surely exponentially stable for the arbitrary Hurst index 0<H<1 when θ≤0. This means that the mfBm cannot destabilize the exponential population growth system (Figures 1–6).Figure 1
The trajectories of system (91) with P0=1,H=0.3. (a) θ=1,u=1,v=1. (b) θ=1,u=1,v=2. The trajectories of system (91) are almost surely exponentially unstable for both (a) and (b) in the sense of Theorem 5 (i).Figure 2
The trajectories of system (91) with P0=1,H=0.3. (a) θ=1,u=3,v=2. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (i).Figure 3
The trajectories of system (91) with P0=1,H=0.5. (a) θ=2,u=1,v=1. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially unstable for (a) but stable for (b) in the sense of Theorem 5 (ii).Figure 4
The trajectories of system (91) with P0=1,H=0.5. (a) θ=1,u=2,v=1. (b) θ=1,u=1,v=2. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (ii).Figure 5
The trajectories of system (91) with P0=1,H=0.8. (a) θ=1,u=1,v=1. (b) θ=−1,u=1,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (iii).Figure 6
The trajectories of system (91) with P0=1,H=0.8. (a) θ=1,u=1,v=2. (b) θ=1,u=2,v=1. The trajectories of system (91) are almost surely exponentially stable for both (a) and (b) in the sense of Theorem 5 (iii).


## 5.2.kth Moment Exponential Stability

The following theorem gives a necessary and sufficient condition for thekth moment exponential stability of system (91).Theorem 6.
(i)
When0<H<1/2, the trivial solution of system (91) is kth moment exponentially stable if one of the following conditions holds: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2; (3) k=1 and θ<0; or (4) k>1, θ<0, and 1/2k−1u2<−θ. Moreover, it is kth moment exponentially unstable if one of the following conditions is satisfied: (1) 0<k<1 and θ>1/21−ku2; (2) k=1 and θ>0; or (3) k>1 and θ≥0.(ii)
WhenH=1/2, the trivial solution of system (91) is kth moment exponentially stable if one of the following conditions holds: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2+v2; (3) k=1 and θ<0; or (4) k>1, θ<0, and 1/2k−1u2+v2<−θ. Moreover, it is kth moment exponentially unstable if one of the following conditions is satisfied: (1) 0<k<1 and θ>1/21−ku2+v2; (2) k=1 and θ>0; or (3) k>1 and θ≥0.(iii)
When1/2<H<1, the trivial solution of system (91) is kth moment exponentially stable if 0<k<1 or k=1 and θ<0, but it is kth moment exponentially unstable if k>1 or k=1 and θ>0.Proof.
According to Lemma6 and (58), one deduces(96)Lk=limsupt⟶+∞1tlogEPt;0,P0k=limsupt⟶+∞1tlogP0kEexpkθ−12u2−12t2H−1v2t+kuBt+kvBHt=limt⟶+∞kθ+12kk−1u2+12kk−1v2t2H−1.
Therefore, one gets(97)Lk=kθ+12kk−1u2,0<H<12,kθ+12kk−1u2+v2,H=12,−∞,12<H<1and0<k<1,θ,12<H<1andk=1,+∞,12<H<1andk>1.
The trivial solution iskth moment exponentially stable if and only if the kth moment Lyapunp>ov exponent is negative. Consequently, the required assertion (iii) follows immediately from (97). When H=1/2, by considering Lk=kθ+1/2kk−1u2+v2<0, we can get the following four cases: (1) 0<k<1 and θ≤0; (2) 0<k<1 and θ<1/21−ku2+v2; (3) k=1 and θ<0; and (4) k>1, θ<0 and 1/2k−1u2+v2<−θ. Moreover, by considering Lk=kθ+1/2kk−1u2+v2>0, we can obtain the following three cases: (1) 0<k<1 and θ>1/21−ku2+v2; (2) k=1 and θ>0; and (3) k>1 and θ≥0. 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 kth moment exponential stability. In general, for sufficiently large k>0, even for stable systems, the kth 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 L<0 and Lk>0 for sufficiently large k>0. When L<0, for P0≠0, Pt;t0,P0⟶0 (t⟶+∞ ) almost surely, and the random variable Pmax=supPt;t0,P0:t≥t0 is almost surely finite. In fact, if Pmax≥Pt;t0,P0>0, then EPmaxk<0 for all k≤0, and EPmaxk is big for sufficiently large k>0, resulting in the kth moment exponential instability, namely, the kth moment Lyapunov exponent Lk>0.We now show the large deviations of system (91).Theorem 7.
(i)
When0<H<1/2, one of the following conditions fulfils the conditions L=θ−u2/2<0 and Lk=kθ+1/2kk−1u2>0:(1)
0<k<1 and 1/21−ku2<θ<1/2u2(2)
k=1 and 0<θ<1/2u2(3)
k>1 and 0≤θ<1/2u2Thus, it holds for some sufficiently largek when 0≤θ<1/2u2. Therefore, the trivial solution of system (91) has large deviations for some special parameters.(ii)
WhenH=1/2, one of the following conditions fulfils the conditions L=θ−1/2u2+v2<0 and Lk=kθ+1/2kk−1u2+v2>0:(1)
0<k<1 and 1/21−ku2+v2<θ<1/2u2+v2(2)
k=1 and 0<θ<1/2u2+v2(3)
k>1 and. 0≤θ<1/2u2+v2Thus, it holds for some sufficiently largek when 0≤θ<1/2u2+v2. Therefore, the trivial solution of system (91) has large deviations for some special parameters.(iii)
When1/2<H<1, we have L=−∞ and Lk=−∞, θ, +∞ for 0<k<1, k=1, k>1, respectively. Therefore, the conditions L<0 and Lk>0 hold for all parameters θ,u, v, and k>1. Consequently, large deviations occur for all parameters θ,u, and v provided that 1/2<H<1.Proof.
Thus, we can prove this theorem using the expressions ofL and Lk.Remark 8.
For the exponential population growth system with fBm (91) (i.e., u=0), if 0<H<1/2, then L=θ and Lk=kθ, which implies that L and Lk have the same sign for all k>0. Thus, the trivial solution of the system is almost surely exponentially stable and kth moment exponentially stable for all k>0, if θ<0. 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 H=1/2 and 1/2<H<1, 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 thekth moment exponential stability of system (91). This relation is tight for small k; however, for sufficiently large k, L<0 and Lk>0. In other words, the following phenomenon may happen: for fixed t, Pt;t0,P0⟶0t⟶+∞ with sample Lyapunov exponent L with probability one, while for Pt;t0,P0 is still large with small probability (see [17], Remark 5.2). This very rare event makes EPt;t0,P0k big for sufficiently large k and gives rise to the kth moment exponential instability in system (91) due to the kth moment Lyapunov exponent Lk>0, 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 thekth 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 1/2<H<1 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.



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