Research article

Asymptotic behavior for a class of population dynamics

  • Received: 06 February 2020 Accepted: 19 March 2020 Published: 01 April 2020
  • MSC : 34C25, 34K13, 34K25

  • This paper investigates the asymptotic behavior for a class of n-dimensional population dynamics systems described by delay differential equations. With the help of technique of differential inequality, we show that each solution of the addressed systems tends to a constant vector as t → ∞, which includes many generalizations of Bernfeld-Haddock conjecture. By the way, our results extend some existing literatures.

    Citation: Chuangxia Huang, Luanshan Yang, Jinde Cao. Asymptotic behavior for a class of population dynamics[J]. AIMS Mathematics, 2020, 5(4): 3378-3390. doi: 10.3934/math.2020218

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  • This paper investigates the asymptotic behavior for a class of n-dimensional population dynamics systems described by delay differential equations. With the help of technique of differential inequality, we show that each solution of the addressed systems tends to a constant vector as t → ∞, which includes many generalizations of Bernfeld-Haddock conjecture. By the way, our results extend some existing literatures.


    Recently, many differential equations models arising from population dynamics, biological mathematics and engineer have attracted more and more attention [1,2,3,4,5,6], and the hot topics include asymptotical stability [7,8,9,10,11,12], limit cycles [13,14], bifurcation [15,16,17], and periodic solutions [18,19,20]. At the international conference on nonlinear systems and their applications in 1976, the following conjecture:

    Conjecture. Each solution of the scalar differential equation

    x(t)=x13(t)+x13(tr),r>0 (1.1)

    tends to a constant as t.

    was proposed by Bernfeld and Haddock [21]. Model (1.1) arises from population dynamics of a single species, where x13(t) and x13(tr) always describe the instantaneous mortality rate and the feedback controls depending on the values of the stable variable with respective delay r, respectively.

    Since the seminal works obtained by Jehu [22] and Ding [23] on the above conjecture, variants of the conjecture have been extensively investigated (see, [24,25,26,27,28,29,30,31,32,33,34]). A simple generalization of the conjecture is described by the following autonomous population dynamics model

    x(t)=F(x(t))+G(x(tr)), (1.2)

    where F,GC(R1,R1) describe the instantaneous mortality rate and the feedback controls depending on the values of the stable variable with respective delays r, respectively. The higher dimensional generalizations for compartmental systems were also presented in [24,25,26,27,28,29,30]. Yet the time-varying delays are more realistic than constant delays in population and ecology models, so Liu [35,36] generalized the conjecture as the following non-autonomous population dynamics models with time-varying delays:

    x(t)=p(t)[x13(t)+x13(tr(t))], (1.3)

    and Xiao [37] established the following generalized version

    {x1(t)=γ1(t)[F1(x1(t))+G1(x2(tτ2(t)))],x2(t)=γ2(t)[F2(x2(t))+G2(x1(tτ1(t)))]. (1.4)

    Here, Fi,GiC(R,R), Fi is increasing on R, p,γ1,γ2C(R,(0, +)),i=1,2.

    In addition, the results in above references indicate that the solution of the systems which is bounded tends to a constant solution as t+. There are two main methods to prove them, one is the analysis method of the monotone dynamical system [24,25,29,33,34], the other is the differential inequality analysis technique [23,27,32,35,36,37]. In particular, assume that,

    F1,F2Ω={FC(R,R)F(0)=0},

    where F is strictly increasing on R, and continuous differentiable on R{0}, the author in [37] established the convergence of system (1.1) and generalized the Bernfeld-Haddock conjecture to two-dimensional system. Unfortunately, if FiGi(i=1,2), there is a simple but serious error in proving Theorem 3.1 in [37]. For the convenience of reading, we'll point out the details of this error in Remark 3.1. Furthermore, in order to correct the above error, we take the following n-dimensional non-autonomous population dynamics model with time-varying delays into consideration:

    {x1(t)=γ1(t)[F1(x1(t))+F1(x2(tτ2(t)))],x2(t)=γ2(t)[F2(x2(t))+F2(x3(tτ3(t)))],xi(t)=γi(t)[Fi(xi(t))+Fi(xi+1(tτi+1(t)))],xn1(t)=γn1(t)[Fn1(xn1(t))+Fn1(xn(tτn(t)))],xn(t)=γn(t)[Fn(xn(t))+Fn(x1(tτ1(t)))], (1.5)

    and FiΩ,γi,τiC(R,R+),iJ={1,2,,n}. Evidently, for FiGi(iJ), Eqs 1.1, 1.3, 1.4 and the systems in [35] are special models of system (1.5).

    Since the non-autonomous delay systems usually do not produce a semiflow, the approach in [23,24,25,27,29,32,33] can not be used to prove the asymptotic behavior of (1.5). In particular, as far as we know, there are no references about the asymptotic behavior of solutions to the n-dimensional system (1.5) involving time-varying delays. On the basis of the above discussions, we expect to propose a novel proof to show a similar conclusion to Bernfeld-Haddock conjecture that every solution of (1.5) tends to a constant vector as t+.

    The paper is organized in following way. We present the initial condition and some preliminary results in Section 2. The boundness and asymptotic behavior of solutions are investigated in Section 3, which are our main results. In the next section, some examples with numerical simulations are carried out to illustrate the validity of the obtained results.

    In this section, we give the initial condition and present the relevant results which will be used in Section 3.

    Denote

    f+=suptRf(t)andf=inftRf(t),

    where f is and continuous bounded function on R. We suppose that

    τmax=max{τ+i:iJ},τmin=min{τi:iJ}>0.

    Define C=ni=1C([τ+i,0],R) as the Banach space equipped with the supremum norm. Moreover, we assume the initial condition

    xi(t0+θ)=φi(θ), θ[τ+i,0], t0R, φ=(φ1,φ2,,φn)C, iJ. (2.1)

    Let x(t;t0,φ)=(x1(t;t0,φ),x2(t;t0,φ),,xn(t;t0,φ)) be the solution of (1.5) with the initial value condition (2.1). And [t0,η(φ)) is the maximal right-interval of existence of x(t;t0,φ).

    Now, it is assumed that GΩ, and we recall the following Lemmas and Propositions.

    Lemma 2.1. (see [34]) For any constant c0,t0 and x0, the system

    {x(t)=G(x(t))+G(c),x(t0)=x0 (2.2)

    has a unique left-hand solution x(t;t0,x0).

    According to Proposition 4 and Proposition 5 in [35], the following results can be achieved:

    Proposition 2.1. Consider the initial value problem

    {x(t)=G(u)+G(a+ε),u(t0)=u0  (u0<a) (2.3)

    where a0 and 0ε|a|2. Then, we suppose that u=u(t;t0,u0) is the solution of (2.3), and the constant β>0. There must exist σ>0 independent of t0 and ε such that

    (a+ε)u(t;t0,u0)σ>0  for  t[t0,t0+β].

    Proposition 2.2. Consider the initial value problem

    {x(t)=G(u)+G(aε),u(t0)=u0  (u0>a) (2.4)

    where a0 and 0ε|a|2. Then, we suppose that u=u(t;t0,u0) is the solution of (2.3), and the constant β>0. There must exist γ>0 independent of t0 and ε such that

    u(t;t0,u0)(aε)γ>0for t[t0,t0+β].

    Lemma 2.2. (see [37]) Let t0,x0R, α>0, H(t,x)C([t0,t0+α]×R, R), and H(t,x) is non-increasing with respect to the x. Then the following differential equation

    {dxdt=H(t,x),x(t0)=x0

    has a unique solution x=x(t) on [t0,t0+α].

    Lemma 2.3. Consider the system (1.5), assume that φC, there is a unique solution x(t;t0,φ) on [t0,+).

    Proof. Let x(t)=x(t;t0,φ), then, for all iJ, we shall prove that there is a unique solution x(t) on [t0,t0+τmin]. Let

    gi(t)=Fi(xˉi(tτˉi(t)))=Fi(φˉi(tτˉi(t)t0)),

    where

    ˉi={i+1,in,1,i=n, (2.5)

    for any t[t0,t0+τmin]. Consider the following differential equation

    {xi(t)=γi(t)[Fi(xi(t))+gi(t)],xi(t0)=φi(0).

    According to Lemma 2.2, there is a unique solution xi(t) on [t0,t0+τmin] for all iJ, and x(t) exists and is unique on [t0,t0+τmin]. Hence, it is obvious that there is a unique solution x(t) on [t0+τmin,t0+2τmin],[t0+2τmin,t0+3τmin],. We now complete the proof of Lemma 2.3.

    Theorem 3.1. x(t)=x(t;t0,φ) is bounded and tends to a constant vector as t+.

    Proof. For convenience, we label

    vi(t)=maxtτmaxstxi(s),ui(t)=mintτmaxstxi(s), tt0+τmax, iJ,
    v(t)=max{vi(t):iJ}, M={t|t[t0+τmax,+),v(t)=xi(t) for some iJ},

    and

    u(t)=min{ui(t):iJ}, T={t|t[t0+τmax,+),u(t)=xi(t) for some iJ}.

    Then, in order to get the boundness of x(t), we first prove D+v(t)0 for all tt0+τmax. The proof of this part is divided into the following two cases:

    Case 1. If t[t0+τmax,+)M, then there exist i0J and t[tτmax,t) satisfing that

    v(t)=vi0(t)=maxtτmaxstxi0(s)=xi0(t)>max{xi(t):iJ}.

    Since xi(t) are continuous, we choose a constant 0<δ<τmax to satisfy that

    xi(s)<xi0(t),   s[t,t+δ],iJ,

    which follows that

    xi(s)xi0(t)=maxtτmaxstxi0(s)=vi0(t)=v(t),   s[tτmax,t+δ],iJ,

    and hence, for all h(0,δ),

    v(t+h)=max{maxt+hτmaxst+hxi(s):iJ}max{maxtτmaxst+δxi(s):iJ}maxtτmaxstxi0(s)=vi0(t)=v(t). 

    Then, the following results can be obtained:

    D+v(t)=lim suph0+v(t+h)v(t)h0.

    Case 2. If tM, one can pick i0J such that

    v(t)=vi0(t)=xi0(t)=maxtτmaxstxi0(s). (3.1)

    Then, (1.5) and (2.7) give us

    0xi0(t)=γi0(t)[Fi0(xi0(t))+Fi0(xˉi0(tτˉi0(t)))]γi0(t)[Fi0(xi0(t))+Fi0(xi0(t))]=0.

    Let ρ=12 τmin. When v(s)=xi0(s) for all s(t,t+ρ], we have

    D+v(t)=lim suph0+v(t+h)v(t)h=lim suph0+v(t+h)xi0(t)h=lim suph0+xi0(t+h)xi0(t)h=xi0(t)=0,

    where 0<h<ρ.

    On the other hand, if there exists s1(t,t+ρ] such that v(s1)>xi0(s1), it suffices to deal with the following two categories () and ().

    () If v(s1)=vi0(s1)=maxs1τmaxss1xi0(s), then, we can choose a constant ˜t[s1τmax,s1) such that

    v(s1)=xi0(˜t)=maxs1τmaxss1xi0(s).

    Noting that tτmax<s1τmaxt+ρτmax<t<s1, we gain

    xi0(˜t)xi0(t)=v(t)=vi0(t)=maxtτmaxstxi0(s).

    We claim

    xi0(˜t)=xi0(t). (3.2)

    Otherwise, xi0(˜t)>xi0(t). Then it is not hard to obtain that t<˜t<s1 and

    0xi0(˜t)=γi0(˜t)[Fi0(xi0(˜t))+Fi0(xˉi0(˜tτˉi0(˜t)))],

    which follows

    xˉi0(˜tτˉi0(˜t))xi0(˜t).

    In combination with tτmaxtτˉi0(˜t)<˜tτˉi0(˜t)<˜tρ<t<s1, we obtain

    xi0(˜t)xˉi0(˜tτˉi0(˜t))maxtτmaxstxˉi0(s)v(t)=xi0(t),

    which leads to a contradiction and suggests that the above claim is true. Then

    maxtτmaxss1xi0(s)=xi0(t),

    which together with tτmax<s1τmaxt+ρτmax<t<s1, and

     vi(t)vi0(t), vi(s1)vi0(s1), for all iJ,

    hereafter, we obtain

    maxtτmaxss1xi0(s)=xi0(t)=v(t), v(t+h)=xi0(t),   0<h<s1t,

    and hence

    D+v(t)=lim suph0+v(t+h)v(t)h=lim suph0+xi0(t)xi0(t)h=0.

    () If there exists ˜iJ,˜ii0 such that

    v(s1)=v˜i(s1)=maxs1τmaxss1x˜i(s)>vi0(s1),

    then, we can find a constant t1[s1τmax,s1] such that

    v(s1)=x˜i(t1)=maxs1τmaxss1x˜i(s)>vi0(s1)xi0(t). (3.3)

    If s1τmaxt1t, and from (3.1), we have

    v(t)=vi0(t)=maxtτmaxstxi0(s)x˜i(t1),

    which contradicts with (3.3). Therefore, t<t1s1 as well as

    0x˜i(t1)=γ˜i(t1)[F˜i(x˜i(t1))+F˜i(x˜i+1(t1τ˜i+1(t1)))],

    yields

    x˜i+1(t1τ˜i+1(t1))x˜i(t1)>xi0(t). (3.4)

    From (3.1) and the fact that tτmax<t1τ˜i+1t1τmint+ρτmin<t, one can see that

    xi0(t)x˜i+1(t1τ˜i+1(t1)), (3.5)

    which contradicts with (3.4). Therefore, category () does not hold, and we can draw that D+v(t)0 for all tt0+τmax from the proof of the above two cases.

    Accordingly, from the definitions of u(t) and T, by using a similar argument as that adopted above, one can evidence that D+u(t)0,tt0+τmax. Overall, we know that u is non-decreasing and v is non-increasing on [t0+τmax,+). Now, the boundness of the x(t;t0,φ) is proved.

    Next, we prove the convergence of x(t). Let li=lim inft+xi(t;t0,φ),Li=lim supt+xi(t;t0,φ),iJ.

    From the boundedness of x, we can obtain

    limt+v(t)=P,limt+u(t)=D, and  PLiliD,iJ.

    Clearly, it is need to show Li=li for all iJ. Combined with the above discussions, we just need to prove that Li>li for all iJ does not hold. We just consider the case L1>l1, and the remainder of the argument is analogous for iJ{1}. Suppose that, on the contrary, L1>l1. It follows that D<P, one of P and D is not equal to 0. Thus, we suppose that P0 and another case is similar. For ˉH(l1,L1)(D,P), we can choose t0>t0+τmax and {σm}+m=1[t0+τmax,+) such that

    x1(σm)=ˉH,  limm+σm=+,  and  xi(t)P+|P|2 for all t[t0,+),iJ.

    From the fact that v(t) is a monotone function and εm=v(σm)P0 ( as m+), we can presume that, for any positive integer m,

    F1(P)F1(v(σm))=F1(P+εm),0εm|P|2.

    Since

    v(σm)v(t)xi(t),  t[σm,σm+(n+1)τmax],iJ,

    and

    F1(x1(t))+F1(v(σm))0,  t[σm,σm+(n+1)τmax],

    one can find that, for all t[σm,σm+(n+1)τmax],

    x1(t)=γ1(t)[F1(x1(t))+F1(x2(tτ2(t)))]γ1(t)[F1(x1(t))+F1(v(σm))]γ+1[F1(x1(t))+F1(P+εm)]. (3.6)

    Let z(t)=z(t;σm,εm) be the solution of following system

    z(t)=γ+[F1(z(t))+F1(P+εm)], z(σm)=ˉH. (3.7)

    Note that ˉH<P, on the basis of Proposition 2.1, we get

    P+εmz(t;σm,εm)μ>0, t[σm,σm+(n+1)τmax],

    where μ is unconcerned with σm and εm. With the help of (3.6) and (3.7), we can take a constant α(0,μ) such that

    x1(t)z(t)<P+εmα, t[σm,σm+(n+1)τmax], (3.8)
    v1(s)=maxsτmaxtsx1(t)<P+εmα, s[σm+τmax,σm+(n+1)τmax],

    and

    v1(σm+2τmax)v1(σm+τmax)<P+εmα. (3.9)

    For s[σm+2τmax,σm+(n+1)τmax], according to the fact that

    vn(s)=maxsτmaxtsxn(t), (3.10)

    it follows that there exists tn[sτmax,s][σm+τmax,σm+(n+1)τmax] such that

    vn(s)=xn(tn)=maxsτmaxtsxn(t)

    and

    0xn(tn)=γn(tn)[Fn(xn(tn))+Fn(x1(tnτ1(tn)))],

    therefore

    xn(tn)x1(tnτ1(tn))<P+εmα,

    and

    vn(σm+kτmax)<P+εmα,for all k=2,3,,(n+1).

    For s[σm+3τmax,σm+(n+1)τmax], in view of the fact that

    vn1(s)=maxsτmaxtsxn1(t), (3.11)

    one can choose tn1[sτmax,s][σm+2τmax,σm+(n+1)τmax] satisfying

    vn1(s)=xn1(tn1)=maxsτmaxtsxn1(t)

    and

    0xn1(tn1)=γn1(tn1)[Fn1(xn1(tn1))+Fn(x1(tn1τn1(tn1)))],

    which implies that

    xn1(tn)xn(tn1τn1(tn1))<P+εmα,

    and

    vn1(σm+kτmax)<P+εmα,for all k=3,4,,(n+1).

    Similarly,

    vj(σm+kτmax)<P+εmα,for all k=nj+2,,(n+1),and j=2,3,,n2.

    Consequently,

    v(σm+nτmax)=max{vi(σm+nτmax):iJ}<P+εmα.

    However, we know that limm+v(σm+τmax)=limt+v(t)=P, which leads to a contradiction. Therefore L1=l1.

    Finally, based on the above analysis, we obtain Li=li,iJ and complete the proof of Theorem 3.1.

    Remark 3.1. As mentioned before, there are some mistakes in the proof of Theorem 3.1 in [37]. The author briefly explained the proof of D+u(t)0, but under the premise of GiFi, it is impossible to get the conclusion of D+u(t)0 in the same way as proving D+v(t)0. And for system (1.5), proving D+v(t)0 is similar to D+u(t)0.

    In fact, if γi(t)=1, τi=τ, for different values of n, many generalizations of Bernfeld-Haddock conjecture are special cases of (1.5). Thus, this paper is a more extensive generalization.

    Remark 3.2. To some extent, for different dimensions and delays, many systems mentioned above are special cases of n-dimensional non-autonomous differential equations with time-varying delays, which means that this article not only points out the errors in previous results, but also generalizes it.

    In this section, we give two examples of satisfying population dynamics system (1.5), for different n, we have

    {x1(t)=(1+2cos2t)[x131(t)+x132(t(1+|sint|))],x2(t)=(1+3cos4t)[x352(t)+x351(t(1+|cost|))],xt0=φC([2,0],R)×C([2,0],R), (4.1)

    and

    {x1(t)=(1+5cos2t)[x131(t)+x132(t(1+|sint|))],x2(t)=(1+2cos4t)[x352(t)+x353(t(1+|cost|))],x3(t)=(1+2cos2t)[x133(t)+x134(t(1+|sint|))],x4(t)=(1+cos4t)[x354(t)+x351(t(1+|cost|))],xt0=φC([2,0],R)×C([2,0],R)×C([2,0],R)×C([2,0],R). (4.2)

    As Theorem 3.1 and Remark 3.1 mentioned, one can obtain that every solution of the Eqs 4.1 and 4.2 tends to a constant vector as t. The curves in Figures 1 and 2 make it easy to see that our conclusion is correct.

    Figure 1.  φ(s)=(3sins,3sins),(5sins,5sins),(7sins,7sins),s[2,0], the solutions x(t) of (4.1).
    Figure 2.  φ(s)=(1+2coss,1+2coss),(5sins,5sins),s[2,0], the solutions x(t) of (4.2).

    This paper investigates the asymptotic behavior for a class of n-dimensional population dynamics systems described by delay differential equations. With the help of technique of differential inequality, we show that each solution of the addressed systems tends to a constant vector as t, which includes many generalizations of Bernfeld-Haddock conjecture. Our results extend some existing literatures. However, if we generalize the conjecture to a differential equation with impulses, can we have a similar conclusion? It is an interesting and meaningful work, we leave it as an open problem.

    The authors would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of our original paper. This work was supported by the National Natural Science Foundation of China (Nos. 11971076, 51839002)

    We confirm that we have no conflict of interest.



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