# Global Attractivity of an Integrodifferential Model of Mutualism

Authors: Xiangdong Xie; Fengde Chen; Kun Yang; Yalong Xue
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Sufficient conditions are obtained for the global attractivity of the following integrodifferential model of mutualism:dN1(t)/dt=r1N1(t)[((K1+α1∫0∞J2(s)N2(t-s)ds)‍/(1+∫0∞J2(s)N2(t-s)ds))‍-N1(t)], dN2(t)/dt=r2N2(t)[((K2+α2∫0∞J1(s)N1(t-s)ds)‍/(1+∫0∞J1(s)N1(t-s)ds‍))-N2(t)], where ri,Ki, and αi, i=1,2, are all positive constants. Consider αi>Ki, i=1,2. Consider Ji∈C([0,+∞),[0,+∞)) and ∫0∞‍Ji(s)ds=1, i=1,2. Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones.


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## Body

## 1. Introduction

Mutualism, one of the most important relationships in the theory of ecology, however, was pointed out by Murray as follows [1]: “this area has not been as widely sutdied as the others even though its importance is comparable to that of predator-prey and competition interactions.”Traditional two species Lotka-Volterra take the form(1)dN1dt=r1N1(1-N1K1+b12N2K1),dN2dt=r2N2(1-N2K2+b21N1K2).
Murray [1] gave detail analysis of the phase trajectories for the above system. He also pointed out that the system has certain drawback; one is the sensitivity between unbounded growth and a finite positive steady state. Despite the drawback of the system, since it is the most simple model on the mutualism, scholars incorporated delays to the above system and proposed the following system:
(2)dx1dt=x1(t)[r1-a11x1(t-τ11)+a12x2(t-τ12)],dx2dt=x2(t)[r2+a21x1(t-τ21)-a22x2(t-τ22)].
Chen et al. [2] had given two examples to show that under the assumption a11a22>a12a21, a condition which could ensure the global stability of the system without delay, the system still admits unbounded solution. He and Gopalsamy [3] and Mukherjee [4] tried to investigate the persistent and stability property of the nonautonomous case of above system; however, in their main results, in addition to condition a11a22>a12a21, they further assumed that the density of one of the species should be bounded from above, such an assumption is by no means easy to verify. To overcome this difficulty, Lu et al. [5, 6] and Nakata and Muroya [7] tried to give restriction on the coefficients of the system or restriction on the delay of the system, and some interesting results about the permanence of Lotka-Volterra type mutualism system with delay were obtained. Liu et al. [8] and Lu [9] also investigated the positive periodic solution of the Lotka-Volterra type mutualism model.On the other hand, stimulated by the functional response of the predator-prey system, Wright [10] proposed the following two species mutualism model:
(3)dNdt=r1N[K1-c1N+b1a1M1+a1Th1M],dMdt=r2M[K2-c2N+b2a2N1+a2Th2N].
Obviously, the model could be revised as follows:
(4)dN(t)dt=r1N[K1+α1M1+a1Th1M-c1N],dM(t)dt=r2M[K2+α2N1+a2Th2N-c2M],
where α1=K1+a1Th1, α2=K2+a2Th2, and one could easily see that αi>Ki, i=1,2.It is well known that in a more realistic model the delay effect should be an average over past populations. This results in an equation with a distributed delay or an infinite delay. Based on the model (4), Gopalsamy [11] further proposed the following two species mutualism model:
(5)dN1(t)dt=r1N1[K1+α1∫0∞K2(s)N2(t-s)ds1+∫0∞K2(s)N2(t-s)ds-N1(t)],dN2(t)dt=r2N2[K2+α2∫0∞K1(s)N1(t-s)ds1+∫0∞K1(s)N1(t-s)ds-N2(t)].
However, the author did not investigate the dynamic behaviors of the system.Recently, Li and Xu [12] proposed and studied the following nonautonomous case of the system (5):
(6)dN1(t)dt=r1(t)N1(t)×[K1(t)+α1(t)∫0∞J2(s)N2(t-s)ds1+∫0∞J2(s)N2(t-s)ds-N1(t)],dN2(t)dt=r2(t)N2(t)×[K2(t)+α2(t)∫0∞J1(s)N1(t-s)ds1+∫0∞J1(s)N1(t-s)ds-N2(t)],
where ri, Ki, αi, and σi, i=1,2, are continuous functions bounded above and below by positive constants. Consider αi>Ki, i=1,2. Consider Ji∈C([0,+∞),[0,+∞)) and ∫0∞‍Ji(s)ds=1, i=1,2. Under the assumption that ri, Ki, and αi, i=1,2, are continuous periodic functions with common period ω. αi>Ki, i=1,2, Ji∈C([0,+∞),[0,+∞)) and ∫0∞‍Ji(s)ds=1, i=1,2. By applying the coincidence degree theory, they showed that system (6) admits at least one positive ω-periodic solution. Chen and You [13] argued that the general nonautonomous case is more suitable. Concerned with the persistent property of the system (6), by applying an integral inequality (see Lemma 3 in the next section), they obtained the following result.Theorem A.
The system (6) is always permanent. That is, there exist constants mi, Mi, i=1,2, which are independent of the solution of the system (6), such that
(7)mi≤liminf⁡t→+∞Ni(t)≤limsup⁡t→+∞Ni(t)≤Mi,i=1,2.Such a result is a roughly one, since it only tells us that the solution is finally bounded above and below by positive constants and there is no fine description of the stable or unstable property of the solution, for example, whether the delay of the system could induce the Hopf bifurcation to period solution or not? Does the system admit some kind of chaotic behaviors? Is it difficult to obtain sufficient conditions which ensure the global attractivity of the positive solution? Indeed, to the best of the authors’ knowledge, to this day, still no scholars investigate the stability property of the system (6), which is one of the most important topics in the study of population dynamics. Noting that system (6) is nonautonomous one and for such kind of system, generally speaking, by constructing some suitable Lyapunov functional, one could always obtain some sufficient conditions which ensure the stability of the system; however, the condition is not easy to verify [14]. This motivated us to investigate the stability property of the system (5).From the point of view of biology, in the sequel, we will consider (5) together with the initial conditions
(8)Ni(s)=ϕi(s),s∈(-∞,0],i=1,2,
where ϕi∈BC+ and
(9)BC+={ϕ∈C((-∞,0],[0,+∞)):ϕ(0)>0,ϕbebounded},i=1,2.
From [15], system (5) has a unique positive solution (N1(t),N2(t)) satisfying the initial condition (8).The aim of this paper is, by further developing the analysis technique of Chen and You [13] and de Oca and Vivas [16] and using the differential inequality theory, to obtain a set of sufficient conditions to ensure the global attractivity of the system (5). More precisely, we will prove the following result.Theorem 1.
System (5) admits a unique positive equilibrium (N1*,N2*), which is globally attractive; that is, for any positive solution (N1(t),N2(t)) of system (5) with the initial condition (8), one has
(10)lim⁡t→+∞Ni(t)=Ni*,i=1,2.We will prove this theorem in the next section and then give a brief discussion in Section3. For more works on the mutualism or cooperation system, one could refer to [12–15, 17–21] and the references cited therein.


## 2. Proof of the Main Result

Now let us state several lemmas which will be useful in the proving of main result.Lemma 2.
System (5) admits a unique positive equilibrium (N1*,N2*).Proof.
The positive equilibrium of the system (5) satisfies the following equation:
(11)K1+α1N21+N2-N1=0,K2+α2N11+N1-N2=0.
System (11) admits a unique positive solution (N1*,N2*), where
(12)N1*=-A2+A22-4A1A32A1,N2*=-B2+B22-4B1B32B1,A1=1+α2,A2=K2-K1-α2α1+1,A3=-α1K2-K1,B1=1+α1,B2=K1-K2-α1α2+1,B3=-K2-α2K1.
This ends the proof of Lemma 2.Following Lemma3 is Lemma 3 of de Oca and Vivas [16].Lemma 3.
Letx:R→R be a bounded nonnegative continuous function, and let k:[0,+∞)→(0,+∞) be a continuous kernel such that ∫0∞‍k(s)ds=1. Then
(13)liminf⁡t→+∞x(t)≤liminf⁡t→+∞∫-∞tk(t-s)x(s)ds≤limsup⁡t→+∞∫-∞tk(t-s)x(s)ds≤limsup⁡t→+∞x(t).As a direct corollary of Lemma 2.2 of Chen [22], we have the following lemma.Lemma 4.
Ifa>0, b>0, and x˙≥x(b-ax), when t≥0 and x(0)>0, one has
(14)liminf⁡t→+∞x(t)≥ba.
If a>0, b>0 and x˙≤x(b-ax), when t≥0 and x(0)>0, one has
(15)limsup⁡t→+∞x(t)≤ba.Now we are in the position of proving the main result of this paper.Proof of Theorem1.
Let(N1(t),N2(t)) be any positive solution of the system (5) with initial condition (8). Similarly to the analysis of (11)–(17) in [13], from the first equation of the system (5) it follows that
(16)dN1(t)dt≤N1(t)[r1(K1+α1)-r1N1(t)].
Thus, as a direct corollary of Lemma 4, according to (16), one has
(17)limsup⁡t→+∞N1(t)≤K1+α1,
and so, from Lemma 3 we have
(18)limsup⁡t→+∞∫0∞J1(s)N1(t-s)ds≤K1+α1.
Hence, for enough small ε>0, it follows from (17) and (18) that there exists a T1′>0 such that
(19)N1(t)<K1+α1+ε≝M1(1),∫0∞J1(s)N1(t-s)ds=∫-∞tJ1(t-s)N1(s)ds≤K1+α1+ε≝M1(1)fort>T1′.
Similarly, for above ε>0, it follows from the second equation of the system (5) that there exists a T1>T1′ such that
(20)N2(t)<K2+α2+ε≝M2(1),∫0∞J2(s)N2(t-s)ds<K2+α2+ε≝M2(1)fort>T1.
Noting that the function g1(x)=((K1+α1x)/(1+x))(α1>K1,x≥0) is a strictly increasing function, hence, (20) together with the first equation of the system (5) implies
(21)dN1(t)dt<N1(t)[r1(K1+α1M2(1))1+M2(1)-r1N1(t)]fort>T1.
Therefore, by Lemma 4, we have
(22)limsup⁡t→+∞N1(t)≤K1+α1M2(1)1+M2(1).
Thus, from Lemma 3 we have
(23)limsup⁡t→+∞∫0∞J1(s)N1(t-s)ds≤K1+α1M2(1)1+M2(1).
That is, for ε>0 defined by (19), there exists a T2′>T1 such that
(24)N1(t)<K1+α1M2(1)1+M2(1)+ε2≝M1(2)>0,∫0∞J1(s)N1(t-s)ds<K1+α1M2(1)1+M2(1)+ε2≝M1(2)>0fort>T2′.
Similarly to the analysis of (22)–(24), from (19) and the second equation of the system (5), there exists a T2>T2′ such that
(25)N2(t)<K2+α2M1(1)1+M1(1)+ε2≝M2(2)>0,∫0∞J2(s)N2(t-s)ds<K2+α2M1(1)1+M1(1)+ε2≝M2(2)>0fort>T2.
Since the function g1(x)=((K1+α1x)/(1+x))(α1>K1,x≥0) is a strictly increasing function, one could easily see that g(x)≥g(0)=K1, and so, from the first equation of the system (5), it follows that
(26)dN1(t)dt≥N1(t)[r1K1-r1N1(t)].
Thus, as a direct corollary of Lemma 4, according to (25), one has
(27)liminf⁡t→+∞N1(t)≥K1,
and so, from Lemma 3, we have
(28)liminf⁡t→+∞∫0∞J1(s)N1(t-s)ds≥K1.
Hence, for enough small ε>0(ε<(1/2)min⁡{K1,K2}), it follows from (27) and (28) that there exists a T3′>0 such that
(29)N1(t)>K1-ε≝m1(1),∫0∞J1(s)N1(t-s)ds≥K1-ε≝m1(1)fort>T3′.
Similarly, for above ε>0, it follows from the second equation of the system (5) that there exists a T3>T3′ such that
(30)N2(t)>K2-ε≝m2(1),∫0∞J2(s)N2(t-s)ds>K2-ε≝m2(1)fort>T3.
Noting that the function g1(x)=((K1+α1x)/(1+x))(α1>K1) is a strictly increasing function, hence, (30) together with the first equation of the system (5) implies
(31)dN1(t)dt>N1(t)[r1(K1+α1m2(1))1+m2(1)-r1N1(t)]fort>T3.
Therefore, by Lemma 4, we have
(32)liminf⁡t→+∞N1(t)≥K1+α1m2(1)1+m2(1).
Thus, from Lemma 3 we have
(33)liminf⁡t→+∞∫0∞J1(s)N1(t-s)ds≥K1+α1m2(1)1+m2(1).
That is, there exists a T4′>T3 such that
(34)N1(t)>K1+α1m2(1)1+m2(1)-ε2≝m1(2)>0,∫0∞J1(s)N1(t-s)ds>K1+α1m2(1)1+m2(1)-ε2≝m1(2)>0fort>T4′.
Similarly to the analysis of (32)–(34), from (28) and the second equation of the system (5), there exists a T4>T4′ such that
(35)N2(t)>K2+α2m1(1)1+m1(1)-ε2≝m2(2)>0,∫0∞J2(s)N2(t-s)ds>K2+α2m1(1)1+m1(1)-ε2≝m2(2)>0fort>T4.
One could easily see that
(36)M1(2)=K1+α1M2(1)1+M2(1)+ε2<K1+α1+ε=M1(1);M2(2)=K2+α2M1(1)1+M1(1)+ε2<K2+α2+ε=M2(1);m1(2)=K1+α1m2(1)1+m2(1)-ε2>K1-ε=m1(1);m2(2)=K2+α2m1(1)1+m1(1)-ε2>K2-ε=m2(1).
Repeating the above procedure, we get four sequences Mi(n), mi(n), i=1,2, n=1,2,…, such that for n≥2(37)M1(n)=K1+α1M2(n-1)1+M2(n-1)+εn;M2(n)=K2+α2M1(n-1)1+M1(n-1)+εn;m1(n)=K1+α1m2(n-1)1+m2(n-1)-εn;m2(n)=K2+α2m1(n-1)1+m1(n-1)-εn.
Obviously,
(38)mi(n)<Ni(t)<Mi(n),fort≥T2n,i=1,2.
We claim that sequences Mi(n), i=1,2 are nonincreasing and sequences mi(n), i=1,2 are nondecreasing. To prove this claim, we will carry out by induction. Firstly, from (36) we have
(39)Mi(2)<Mi(1),mi(2)>mi(1),i=1,2.
Let us assume now that our claim is true for n; that is,
(40)Mi(n)<Mi(n-1),mi(n)>mi(n-1),i=1,2.
Again from the strict increasing of the function gi(x)=((Ki+αix)/(1+x))(αi>Ki,i=1,2), we immediately obtain
(41)M1(n+1)=K1+α1M2(n)1+M2(n)+εn+1<K1+α1M2(n-1)1+M2(n-1)+εn=M1(n);M2(n+1)=K2+α2M1(n)1+M1(n)+εn+1<K2+α2M1(n-1)1+M1(n-1)+εn=M2(n);m1(n+1)=K1+α1m2(n)1+m2(n)-εn+1>K1+α1m2(n-1)1+m2(n-1)-εn=m1(n);m2(n+1)=K2+α2m1(n)1+m1(n)-εn+1>K2+α2m1(n-1)1+m1(n-1)-εn=m2(n).
Therefore,
(42)lim⁡t→+∞Mi(n)=N¯i,lim⁡t→+∞mi(n)=N_i,i=1,2.
Letting n→+∞ in (37), we obtain
(43)N¯1=K1+α1N¯21+N¯2,N¯2=K2+α2N¯11+N¯1,N_1=K1+α1N_21+N_2,N_2=K2+α2N_11+N_1
and (43) shows that (N¯1,N¯2) and (N_1,N_2) are solutions of (11). By Lemma 2, (11) has a unique positive solution E*(N1*,N2*). Hence, we conclude that
(44)N¯i=N_i=Ni*,i=1,2;
that is,
(45)lim⁡t→+∞Ni(t)=Ni*i=1,2.
Thus, the unique interior equilibrium E*(N1*,N2*) is globally attractive. This completes the proof of Theorem 1.


## 3. Discussion

As was pointed out in the introduction section, for Lotka-Volterra type mutualism system, conditions which ensure the globally stability of the system are not so easily verified [3, 4]. In this paper, we study the stability property of the integrodifferential model of mutualism (5). By applying the iterative technique, we obtain a set of sufficient conditions which guarantee the global attractivity of the coexistence equilibrium. Our result (Theorem 1) shows that the conditions which ensure the permanence of the system are enough to ensure the global attractivity of the system. The new finding of this paper is that we found the iterative bound of the solution, such a finding is not detected by Chen and You [13]. Based on this finding, it is possible to obtain the subtle result about the stability of the system. We mention here that we did not consider the delay in the intraspecific competition; whether such kind of delay could induce bifurcation or not is still unknown; we leave this for future study.



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