# On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Orderα∈(1,2)

Authors: Mohammed M. Matar; Esmail S. Abu Skhail
Journal: Journal of Mathematics
 (2018)

---

## Abstract

We study the Mittag-Leffler and class-K function stability of fractional differential equations with orderα∈(1,2). We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.


---

## Body

## 1. Introduction

Recently, the topic of fractional differential equations has attracted many researchers which plays a main role in various applications. Many areas such as physics, biophysics, aerodynamics, control theory, viscoelasticity, capacitor theory, electrical circuit, description of memory, and hereditary properties (see [1–4]) used the fractional models instead of classical models. Recently, stability of fractional differential systems has attracted many authors for more investigation in this topic [5]. In 1996, Matignon [6] firstly studied the stability of linear fractional differential systems with the Caputo derivative. Since then, many researchers have done further studies on the stability of linear fractional differential systems (see [7, 8]). In nonlinear systems, Lyapunov’s direct method provides a way to analyze the stability as Mittag-Leffler stability of a system without explicitly solving the differential equations (see [9–11]). The method generalizes the idea which shows that the system is stable if there are some Lyapunov function candidates for the system. Lyapunov’s direct method is a sufficient condition to show the stability of the systems, which means that the systems may still be stable even if one cannot find Lyapunov’s function candidate to conclude the system stability property. For the nonlinear fractional differential systems, the stability analysis is much more difficult and only a few are available.Some authors (see [12–14]) studied the following nonlinear fractional differential system:(1)Dct0,tαxt=ft,xt,t∈t0,∞.with initial values x(k)t0=xk,k=1,2,…,n-1, where n-1<α<n. Then, some literatures on the stability of linear fractional differential systems with 0<α<1 have appeared (see [15, 16]). However, not all the fractional differential systems have fractional orders in (0,1). There exist fractional models which have fractional orders lying in (1,2), for example, super-diffusion. In [17], the authors study the stability ofn-dimensional linear fractional differential systems(2)Dt0,tαRLxt=Axt,of order 1<α<2 and initial conditions(3)Dt0,tα-kRLxtt=t0=xk-1,k=1,2.The corresponding perturbed system is(4)Dt0,tαRLxt=Axt+ft,xt,1<α<2,Dt0,tα-kRLxtt=t0=xk-1,k=1,2,which is investigated by authors in [17].For fractional-order Hopfield neural networks in applications to parallel computation and signal processing, it is required that there be a well-defined computable solution for all possible initial states. From a mathematical viewpoint, the stability analysis of a unique equilibrium point is very necessary and valuable. Recently, several important and interesting results for the stability analysis of fractional-order Hopfield neural networks have been gained [4].The marine protected area model [18] can describe the ecological linkage between the reserve and fishing ground by the autonomous linear system(5)x′1t=-m1+D1x1t+D2x2t,x′2t=-m2+D2+Ex2t+D1x1t.Here, mi>0 is the natural mortality (death) rate, Di>0 is the dispersal rate, and E>0 is the harvesting rate. The corresponding fractional behavior of such system may exist in theoretical view as the fractional order α→1±.Motivated by these articles, we investigate the stability of the nonautonomous perturbed fractional semilinear system of the form(6)Dt0,tαxt=Axt+ft,xt,t>t0,xt0=x0,x′t0=x1,where x(t)∈Rn, xk∈Rn(k=0,1), A∈Rn×n, 1<α<2, and f:[t0,∞)×Rn→Rn is a given continuous function such that ft,0=0. The fractional derivative Dt0,tα is either Caputo or Riemann-Liouville.


## 2. Preliminaries and Notations

We introduce in this section several definitions and results that will be used later in the sequel.Definition 1 ([19]).
The Riemann-Liouville fractional integral of a functionf∈Ct0,T(orL1) is defined by(7)Iαft=1Γα∫t0tt-sα-1fsds,α>0.Definition 2 ([19]).
The Riemann-Liouville fractional derivative of orderα∈n-1,n,n∈N, is defined by(8)Dt0,tαRLft=ddtnIn-αft,where d/dt is the ordinary differential operator.Theorem 3 ([19]).
Iff∈Ct0,T and α∈n-1,n,n∈N, then the equality(9)IαDt0,tαRLft=ft-∑k=1nfn-αn-kt0Γα-k+1t-t0α-k,t>t0,holds everywhere on t0,T, where fn-αt=In-αft∈Cnt0,T.Definition 4 ([20]).
The Caputo derivative (left-sided) of a functionf∈Cnt0,T, n-1<α≤n, is defined as(10)Dt0,tαcft=In-αfnt,n-1<α<n,fnt,α=n.Lemma 5 ([21]).
Letn-1<α≤n; then(11)IαDt0,tαcft=ft-∑k=0n-1fkt0k!t-t0k.Definition 6.
The Laplace transform of a real functionf is defined as (12)Fs=Lfs=∫0∞e-stftdt.The Laplace transform of the Caputo fractional derivative is given by(13)LDt0,tαcfts=sαLfts-∑k=0n-1sα-k-1fkt0,n-1<α<n.The Laplace transform of the Riemann-Liouville fractional derivative is given by [20](14)LDt0,tαRLfts=sαLfts-∑k=0n-1skDt0,tα-k-1fRLt0,n-1<α<n.The Laplace transform of tα is given by(15)Ltα=Γα+1sα+1,s>0,α>-1.The convolution of the functions f and g is defined as(16)f∗gt=∫0tgt-sfsds.The inverse Laplace transform of the product of the functions f and g is given by(17)L-1fsgst=L-1fst∗L-1gst=∫0tL-1gst-rL-1fsrdr.Lemma 7 ([20]).
The Laplace transform of the Mittag-Leffler function is given by(18)Ltβ-1Eα,β±θtαs=sα-βsα∓θ-1,Res>θ1/α.We conclude that(19)LEα±θtαs=sα-1sα∓θ-1,Res>θ1/α.The Mittag-Leffler function [21] is very important in solving fractional differential equations and in many applications. The exponential function as special case of Mittag-Leffler which is frequently used in the solution of integer order differential equations. The Mittag-Leffler function of one parameter is defined as(20)Eαz=∑k=0∞zkΓαk+1,z∈C,Reα>0.The Mittag-Leffler function of two parameters is defined by(21)Eα,βz=∑k=0∞zkΓαk+β,z∈C,Reα>0,Reβ>0.which both satisfy the fact that Eα,1z=Eαz, and E1z=ez.Lemma 8 ([22]).
The systemDt0,tαxt=Axt is asymptotically stable if arg⁡eigA>απ/2 where 0<α<2, and eig(A) are the eigenvalues of matrix A. The stable and unstable regions for 0<α<1 and 1<α<2 are shown in Figures 1 and 2, respectively.Figure 1Figure 2Remark 9 ([22]).
The perturbed systemDt0,tαxt=Axt+f(t,x(t)) is asymptotically stable if arg⁡eigA>απ/2, and f satisfies the fact that f(t,x)≤γtx,t≥t0, for some γ∈L1.It may be observed that the behavior of Mittag-Leffler function is a relaxation forα<1, it is exponential for α=1, it becomes a damped oscillation for 1<α<2, and it oscillates for α=2. The decay is very fast as  t→0+ and very slow ast→∞. See Figures 3, 4, and 5.Figure 3Figure 4Figure 5Definition 10 ([23]).
The constantxeq is an equilibrium point of fractional differential system(22)Dt0,tαxt=ft,xt,t∈t0,∞,if and only ifft,xeq=Dt0,tαxtxt=xeq, for all t>t0, where Dt0,tα means either Caputo or Riemann-Liouville fractional operator.For convenience, we state all definitions and theorems for stability in Caputo sense when the equilibrium point is the origin ofRn, i.e., xeq=0. There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables. Indeed, suppose that the equilibrium point is y, then consider the change of variable z=x-y. Therefore, the Caputo derivative of z is(23)Dt0,tαcz=Dt0,tαcx-y=ft,x=ft,z+y=gt,y,where gt,0=0, and in the new variable y, the system has an equilibrium point at the origin.Definition 11 ([23]).
The zero solution of system (22) is said to be stable if there exists M>0 such that x(t)≤M, for all t>t0.Definition 12 ([23]).
The zero solution of system (22) is said to be asymptotically stable if x(t)→0, as t→∞.A functionm is Lipschitz function if there exists a positive constant ρ such that(24)mx-my≤ρx-y.Definition 13 ([23]).
LetB⊂Rn be a domain containing the origin. The zero solution of system (22) is said to satisfy the Mittag-Leffler criteria if(25)xt≤mxt0Eα,1-θt-t0αγ,t>t0,where θ≥0, γ>0,m0=0,m(x)≥0, 0<α<1, and m is Lipschitz function on B.Definition 14 ([23]).
LetB⊂Rn be a domain containing the origin. The zero solution of system (22) is said to satisfy the generalized Mittag-Leffler criteria if(26)xt≤mxt0t-t0-δEα,1-δ-θt-t0αγ,t>t0,where θ≥0, γ>0, 0<α<1, -α<δ<1-α, m0=0, m(x)≥0, and m is Lipschitz function on B.Remark 15.
Ifδ=0, the generalized Mittag-Leffler is reduced to Mittag-Leffler criteria. Moreover, the Mittag-Leffler stability implies the asymptotic stability. According to investigations in [17], the above definitions can be applied to the case of 1<α<2 for some systems.Definition 16 ([24]).
A functiong is said to belong to a class-K function if g:[0,∞)→[0,∞) is a continuous function satisfying g0=0, and it is strictly increasing.Definition 17 ([9]).
Letx∗ be an equilibrium point for system (22), and let L:O→R defined on an open set O containing x∗ and L is continuously differentiable on O. Suppose further that(27)Lx∗=0,Lx>0,forx≠x∗.Then x∗ is said to be stable if L˙x=d/dxL(x(t))≤0, on O∖{x∗}.A functionL satisfying the above two conditions is called a Lyapunov function.Lemma 18.
Letf:[t0,∞)×Rn→Rn be Lipschitz function; then the unique solution of (6) is(28)xt=Eα,1At-t0αx0+t-t0Eα,2At-t0αx1+∫t0tt-τα-1Eα,αAt-ταfτ,xτdτ.Proof.
The existence and uniqueness of solution for (6) can be obtained by applying Banach fixed point Theorem (see [17]). By applying Laplace transform for (6), we have(29)LDct0,tαxts=LAxts+Lft,xts,and, then,(30)sαLxts-sα-1x0-sα-2x1=LAxts+Lft,xts.Set Lft,xts=Fs,xs; then we get(31)Lxtssα-A=sα-1x0+sα-2x1+Fs,xs,which implies(32)Lxts=sα-1sα-A-1x0+sα-2sα-A-1x1+sα-A-1Fs,xs.Now, taking Laplace inverse to both sides and using Lemma 7, we obtain(33)xt=EαAt-t0αx0+t-t0Eα,2At-t0αx1+L-1sα-A-1Fs,xst.By convolution theorem of Laplace transform on the third term of (33), we have(34)L-1sα-A-1Fs,ust=L-1sα-A-1t∗L-1Fs,ust=∫t0tt-τα-1Eα,αAt-ταfτ,xτdτ.Thus, we get the result.Lemma 19.
LetV:[t0,∞)×Rn→R+ be continuously differentiable function. The solution of the fractional differential equation (35)Dt0,tαcVt,xt=-kVt,xtis(36)Vt,xt=V0Eα,1-kt-t0α+V1t-t0Eα,2-kt-t0α,where V0=Vt0,xt0, and V1=V′t0,xt0.Proof.
By taking Laplace transformation to both sides of (35), we get (37)LDt0,tαcVt,xt=-kLVt,xt.Then(38)sαLVt,xt-sα-1V0-sα-2V1=-kLVt,xt.So,(39)LVt,xt=sα-1V0+sα-2V1sα+k=V0sα-1sα+k+V1sα-2sα+k.By applying the inverse Laplace transform L-1 and using Lemma 7, we obtain the solution.Lemma 20.
LetV:t0,∞×Rn→R+ be continuously differentiable function. The solution of the fractional differential equation (40)Dt0,tαRLVt,xt=-Mt,is (41)Vt,xt=V0Γαt-t0α-1+V1Γα-1t-t0α-2-1Γα∫t0tt-sα-1Msdswhere Dt0,tα-mRLVt,xtt=t0+=Vm-1, m=1,2.Proof.
By taking Laplace transform to both sides of (40), we get (42)sαLVt,xt-V0-sV1=-LMt,which implies that(43)LVt,xt=V01sα+V11sα-1-1sαLMt.By taking L-1 for both sides, we obtain (44)Vt,xt=V0Γαt-t0α-1+V1Γα-1t-t0α-2-L-11sα∗Mt=V0Γαt-t0α-1+V1Γα-1t-t0α-2-1Γα∫t0tt-rα-1Mrdr.This finishes the proof.Remark 21.
In Lemma20, if we replace Riemann-Liouville derivative by Caputo derivative, then the solution is(45)Vt,xt=V0+V1t-t0-1Γα∫t0tt-sα-1Msds,where V0=Vt0,xt0, and V1=V′t0,xt0.Lemma 22.
LetV:t0,∞×Rn→R+ be continuously differentiable function. The solution of the fractional differential equation (46)Dt0,tαRLVt,xt=-kVt,xt,k≥0,is(47)Vt,xt=V0t-t0α-1Eα,α-kt-t0α+V1t-t0α-2Eα,α-1-kt-t0α,where Dt0,tα-mRLVt,xtt=t0+=Vm-1m=1,2.Proof.
By taking Laplace transform for both sides of (46), we get (48)LDt0,tαRLVt,xt=-kLVt,xt,and then(49)sαLVt,xt-V0-sV1=-kLVt,xt.So,(50)LVt,xt=V0+sV1sα+k=V01sα+k+V1ssα+k.By taking L-1 and using Lemma 7, we obtain the results.


## 3. Stability Results

In this section, we study the stability of nonlinear fractional differential systems with Caputo derivative by utilizing a Lyapunov-like function, taking into account the relation between asymptotical stability and generalized Mittag-Leffler criteria.The Lyapunov direct method is a sufficient condition to show the stability of systems, which means the system may still be stable even if one cannot find a Lyapunov function candidate to conclude the system stability property.Hereafter, we assume that the Lyapunov functionV:t0,∞×Rn→R+ is continuously differentiable with respect to t, Lipschitz with respect to x, and Vt,0=0.Theorem 23.
Letx=0 be an equilibrium point of system (6) and let V satisfy (51)axb≤Vt,xt,and(52)Dt0,tαRLVt,xt≤0,where x∈Rn, a, and b are positive real numbers. Then, the zero solution is asymptotically stable whenever Dt0,tα-1RLVt,xtt=t0+=0, and Dt0,tα-2RLVt,xtt=t0+≥0.Proof.
Inequality (52) implies the existence of a nonnegative function M such that(53)Dt0,tαRLVt,xt=-Mt.Using Lemma 20, we get(54)Vt,xt=V0Γαt-t0α-1+V1Γα-1t-t0α-2-1Γα∫t0tt-sα-1Msds,where Dt0,tα-mRLVt,xtt=t0+=Vm-1m=1,2. The negativity of the last term of (54) will imply that(55)Vt,xt≤V0Γαt-t0α-1+V1Γα-1t-t0α-2In accordance with (51), we obtain (56)xt≤V0aΓαt-t0α-1+V1aΓα-1t-t0α-21/b.Now, set V0=0; then x(t)→0, as t→∞. Therefore, the zero solution of system (6) is asymptotically stable.Theorem 24.
Letx=0 be an equilibrium point of system (6) and let V satisfy(57)axb≤Vt,xt,(58)Dt0,tαcVt,xt≤-cVt,xt,where x∈Rn, a, b, and c are arbitrary positive constants. Then, the zero solution satisfies the Mittag-Leffler criteria whenever Vt0,xt0≥0, and V′t0,xt0=0.Proof.
Using (58), there exists a nonnegative function M such that(59)Dt0,tαcVt,xt+cVt,xt=-Mt.By Lemma 18 and noting that IαM(t) is nonnegative, we deduce that (60)Vt,xt≤V0Eα,1-ct-t0α+V1t-t0Eα,2-ct-t0α,where V0=Vt0,xt0, and V1=V′t0,xt0=0. Then,(61)Vt,xt≤V0Eα,1-ct-t0α.In accordance with (57), we obtain x(t)≤V0/aEα,1-ct-t0α1/b, for m=V0/a≥0. Then, the zero solution of system (6) satisfies the Mittag-Leffler criteria.Theorem 25.
Letx=0 be an equilibrium point of system (6) and let V satisfy (62)Vt,xt≥axb,and (63)Dt0,tαRLVt,xt≤0,where t≥t0, x∈Rn, a and b are arbitrary positive constant. Then, the zero solution satisfies the generalized Mittag-Leffler criteria whenever Dt0,tα-1RLVt,xtt=t0+=0, and Dt0,tα-2RLVt,xtt=t0+≥0.Proof.
There exists a nonnegative functionM such that(64)Dt0,tαRLVt,xt=-MtBy using Lemma 20, we can deduce as in the proof of proceeding results that(65)Vt,xt≤V0Γαt-t0α-1+V1Γα-1t-t0α-2,where Dt0,tα-mRLVt,xtt=t0+=Vm-1, m=1,2, 1<α<2. Therefore,(66)xt≤V0aΓαt-t0α-1+V1aΓα-1t-t0α-21/b.Now, set V0=0; then(67)xt≤V1aΓα-1t-t0α-21/b≤V0aΓαt-t0α-1Eα,α(0.t-t01/b.Now, set m=V0/aΓ(α-1)≥0, and, therefore, the zero solution of system (6) satisfies the generalized Mittag-Leffler criteria.Theorem 26.
Letx=0be an equilibrium point of system (6). Assume there exists a class-K function g satisfying(68)Vt,xt≥g-1xt,(69)Dt0,tαcVt,xt≤0,t≥t0,(70)supt≥t0⁡gVt0,xt0+V′t0,xt0t-t0≤M,for some nonnegative constant M. Then, the zero solution is stable.Proof.
Using (69), there exists a nonnegative function M such that Dt0,tαcVt,xt=-M(t). Then(71)Vt,xt=V0+V1t-t0-IαMt,where V0=Vt0,xt0, and V1=V′t0,xt0. Now, (68) and (71) imply that(72)g-1xt≤V0+V1t-t0-IαMt≤V0+V1t-t0.Therefore,(73)xt≤gV0+V1t-t0.Then, by (70), we have x(t)≤M,t≥t0. Therefore, the zero solution of system (6) is stable.Theorem 27.
Letx=0 be an equilibrium point of system (6) and assume that there exists a class-K function g satisfying(74)Vt,xt≥g-1xt,(75)Dt0,tαRLVt,xt≤0,where t≥t0. Then, the zero solution is asymptotically stable whenever Dt0,tα-1RLVt,xtt=t0+=0.Proof.
As in Theorem23, we have(76)Vt,xt≤V0Γαt-t0α-1+V1Γα-1t-t0α-2,where Dt0,tα-mRLVt,xtt=t0+=Vm-1m=1,2. Now, inequalities (74) and (76) imply that(77)xt≤gV0Γαt-t0α-1+V1Γα-1t-t0α-2.Therefore x(t)≤gV1/Γα-1t-t0α-2, for V0=0. Hence, the continuity of g implies that x(t)→0 as t→∞. Therefore, the zero solution of system (6) is asymptotically stable.


## 4. The Comparison Results on Stability

In this section we discuss the comparison of the stability for the two systems with Caputo and Riemann-Liouville derivatives. It is well known that the comparison method is an effective way in judging the stability of ordinary differential systems. We will discuss similar results on the stability of fractional differential systems by using the comparison method.Consider the semilinear system(78)Dt0,tαRLut=Aut+ht,ut,t>t0,1<α<2,Dt0,tα-kRLut0+=uk-1,k=1,2,where u(t)∈Rn, uk∈Rn(k=0,1), A∈Rn×n, and h:t0,∞×Rn→Rn is a continuous function such that h(t,0)=0. Moreover, h satisfies the Lipschitz condition with respect to u to ensure the existence and uniqueness of solution (78).Applying the Laplace technique as in Lemma18 but using (14), we deduce the following result.Lemma 28.
For1<α<2, the solution of system (78) is(79)ut=t-t0α-1Eα,αAt-t0αu0+t-t0α-2Eα,α-1At-t0αu1+∫t0tt-τα-1Eα,αAt-ταhτ,uτdτ.In the next result, we recall that the set of vectors can be partially ordered if their components are so.Theorem 29.
Letu=0 be an equilibrium of system (78). Assume that g is a class-K function and there exists a function V:t0,∞×Rn→Rn such that Vt,0=0, and for each t≥t0, the following conditions hold:
(i)Vt,ut,≥Vt,xt≥g-1xt, for any solution x of system (6).
(ii)Dt0,tαRLVt,ut-AVt,ut≤0.
(iii)∫t0tt-τα-1Eα,αAt-ταhτ,uτdτ has nonnegative finite components.
(iv)Dt0,tα-kRLut0+=Dt0,tα-kRLVt,utt=t0+,k=1,2.
Moreover, assume thatg is nondecreasing. Then
(1) if the zero solution of system (78) is stable, then the zero solution of system (6) is stable;
(2) if the zero solution of system (78) is asymptotically stable, then the zero solution of the system (6) is asymptotically stable.Proof.
(1) LetM(t) be vector-valued function defined on Rn whose components are nonnegative and by (ii) it satisfies the following:(80)Dt0,tαRLVt,ut=AVt,ut-Mt.As in the proof of Theorem 24, we have(81)Vt,ut=t-t0α-1Eα,αAt-t0αV0+t-t0α-2Eα,α-1At-t0αV1-IαMt≤t-t0α-1Eα,αAt-t0αV0+t-t0α-2Eα,α-1At-t0αV1,where V0=Dt0,tα-1RLVt,utt=t0+, and V1=Dt0,tα-2RLVt,utt=t0+. Using condition (iv), we have u0=V0,andu1=V1; hence condition (iii) implies that(82)Vt,ut≤t-t0α-1Eα,αAt-t0αu0+t-t0α-2Eα,α-1At-t0αu1+∫t0tt-τα-1Eα,αAt-ταhτ,uτdτ=ut.We deduce that(83)Vt,ut≤ut.Now, condition (i) implies that(84)g-1xt≤Vt,xt≤Vt,ut≤ut.Since, the zero solution of system (78) is stable, then there exists M>0 such that(85)g-1xt≤ut≤M,which implies that xt≤gM=L,L>0. Therefore, the zero solution of system (6) is stable.
(2) Let the zero solution of system (78) be asymptotically stable, then limt→∞⁡u(t)=0. Again, using g-1x(t)≤ut, we deduce that 0≤x(t)≤g(ut). With the continuity of g and the condition g0=0, we have limt→∞⁡x(t)=0; then the zero solution of system (6) is asymptotically stable.


## 5. Applications

The Hopfield neural networks with constant external inputs are described by [4](86)D0,t1+cxit=-aixit+∑j=1nbijfjxjt+Ii,where i=1,2,…,n and n is the number of units in a neural network, xi(t) is the state of the ith unit at time t, fj denotes the activation function of the jth neuron, bij denotes the constant connection weight of the jth neuron on the ith neuron, ai>0 represents the rate with which the ith neuron resets its potential to the resting state when disconnected from the network, Ii denotes the constant external inputs, and 1+ means the approach of the fractional order α to classical first-order derivative from right side. System (6) can be easily obtained by letting x=xin×1, A=-aijn×n, where aij=ai, for i=j and aij=0, for i≠j, and f=fin×1, where fit,xt=∑j=1nbijfjxjt+Ii,i=1,2,…,n.Example 30.
Consider the scalar Hopfield neural networks system(87)D0,tαcxt=-xt+ft,xt,t≥0,1<α<2,for x0=0=x′(0). Assume that the Lyapunov function isVt,xt=xt and that ft,xt≤0, ft,0=0; then,(88)D0,tαcVt,xt=-Vt,xt+ft,xt≤-Vt,xt.Thus, the conditions of Theorem 24 hold with a=b=c=1. Therefore, the zero solution of system (87) satisfies the Mittag-Leffler criteria.To illustrate the biological applications [18], we consider two habitat areas, with a fish population dispersing between the two areas, whilst fishing takes place only in region 2, with region 1 established as a no-fishing zone. To describe the ecological linkage between the reserve and fishing ground we propose the fractional alternative model of the autonomous linear system(89)D0,tαcx1t=-m1+D1x1t+D2x2t,D0,tαcx2t=-m2+D2+Ex2t+D1x1t.Here, mi>0 is the natural mortality (death) rate, Di>0 is the dispersal rate, and E>0 is the harvesting rate. If we assume that x=[x1,x2]T, and A=-m1+D1D2D1-m2+D2+E, we obtain the autonomous system(90)D0,tαcxt=Axt.The corresponding nonautonomous system can be modeled as (6).Example 31.
Consider the system(91)D0,t1.5RLx1t=-x1t+0.5x2t,D0,t1.5RLx2t=x1t-x2t+e-tx2tx1t+1,for t>0, and x0=0, x′0=1. Let A=-10.50.5-1∈R2×2, and ft,x(t)=0e-tx2t/x1t+1∈R2, t≥0; then system (91) can be transformed as D0,t1.5RLxt=Axt+f(t,xt), where xt=x1tx2t∈R2. The function f is Lipschitz function which ensures the existence of a solution x(t) given by Lemma 28, and ft,0=0 ensures the existence of zero solution for (91). The argument of eigenvalues of A is π; then the solution of (91) is stable (see [17], Theorem 5.1). Therefore, we can always find a constant M>0, such that M-2t-4≤x(t)≤M, for any t>0. Choose the Lyapunov candidate as Vt,x=-Mt2x; then Vt,0=0,D0,t1.5RLVt,x=-MΓ(3)/Γ(1.5)tx≤0, and Dt0,t0.5RLVt,xt=0+=-MΓ(3)/Γ(2.5)t3xt=0+=0. If gt=t2,t≥0, then g-1t=t. It is obvious now that Vt,xt2=M2t4x2≥x. Hence all hypotheses of Theorem 27 are satisfied; then system (91) is asymptotically stable.



---

*Source: 1723481-2018-08-05.xml*