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Research article

Global behavior of a discrete population model

  • Received: 30 December 2023 Revised: 13 March 2024 Accepted: 18 March 2024 Published: 27 March 2024
  • MSC : 39A10

  • In this work, the global behavior of a discrete population model

    {xn+1=αxneyn+β,yn+1=αxn(1eyn),n=0,1,2,,

    is considered, where α(0,1), β(0,+), and the initial value (x0,y0)[0,)×[0,). To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated.

    Citation: Linxia Hu, Yonghong Shen, Xiumei Jia. Global behavior of a discrete population model[J]. AIMS Mathematics, 2024, 9(5): 12128-12143. doi: 10.3934/math.2024592

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  • In this work, the global behavior of a discrete population model

    {xn+1=αxneyn+β,yn+1=αxn(1eyn),n=0,1,2,,

    is considered, where α(0,1), β(0,+), and the initial value (x0,y0)[0,)×[0,). To illustrate the dynamics behavior of this model, the boundedness, periodic character, local stability, bifurcation, and the global asymptotic stability of the solutions are investigated.



    The model

    {xn+1=λxneayn,yn+1=cxn(1eayn),n=0,1,2,, (1.1)

    is used to describe the Nicholson-Bailey host-parasitoid system, where xn and yn represent the densities of host and parasitoid at the nth generation, respectively, a is the searching efficiency of the parasitoid, λ is the host reproductive rate, and c is the average number of viable eggs laid by a parasitoid on a single host. System (1.1) is simple and its positive equilibrium is unstable [4,5,19], which indicates that the parasitoid populations, or both the parasitoid and host populations, will go extinct. Therefore this simple model is unrealistic for any practical applications. Up to now, the model has been developed to describe the population dynamic behavior of a coupled host-parasitoid (or predator-prey) system. The improved models display more various dynamic behaviors such as stability, bifurcation, and chaotic phenomenon, see [1,7,8,9,10,16,17,20,21,23]. For more detailed information, refer to [13,14,15,22,25].

    As mentioned in [18], in many populations it is reasonable to believe that either a refuge exists which isolates some small fraction of the population from density-dependent effects, or that there is a small amount of immigration from outside the system each generation. Therefore, in this paper we consider the system

    {xn+1=αxneyn+β,yn+1=αxn(1eyn),n=0,1,2,, (1.2)

    where

    α(0,1),β(0,+), (1.3)

    and the initial value (x0,y0)[0,+)×[0,+). The parameter α is the host reproductive rate at per generation (in the absence of a parasitoid), and the term β represents a refuge or a constant amount of immigration of hosts from outside the system per generation.

    In [12], Kulenović and Ladas proposed an open problem (Open Problem 6.10.16) asking for investigating the global character of all solutions of system (1.2) with parameters α(0,1) and β(1,+).

    Inspired by the aforementioned open problem, in this paper the boundedness, periodic character, transcritical bifurcation, local asymptotic stability, and global asymptotic stability of system (1.2) are discussed under condition (1.3). Our result partially solves the above open problem.

    The paper is organized as follows:

    Section 1 is the introduction, and Section 2 involves the preliminaries, where some necessary lemmas are presented. Section 3 deals with the boundedness and periodic character of system (1.2). The linearized stability and bifurcation analysis are discussed in Section 4. Section 5 focuses on the global asymptotic stability of the equilibria of system (1.2). Section 6 is the conclusion.

    Prior to commencing the discussion, we present some essential lemmas.

    Lemma 2.1. (A Comparison Result [11]) Assume that α(0,+) and βR. Let {xn}n=0 and {zn}n=0 be sequences of real numbers such that x0z0 and

    {xn+1αxn+β,zn+1=αzn+β,n=0,1,2,.

    Then xnzn for n0.

    The following lemma is proved in [6], which will be applied in analyzing the global attractivity of Eq (2.1). Additionally, one can refer to [3,11,12] for further information.

    Lemma 2.2. Consider the difference equation

    un+1=g(un),n=0,1,2,. (2.1)

    Let I[0,+) be some interval and assume that gC[I,(0,+)] satisfies the following conditions:

    (i) g(u) is non-decreasing in u.

    (ii) Equation (2.1) has a unique positive equilibrium ˉuI and the function g(u) satisfies the negative feedback condition:

    (uˉu)(g(u)u)<0for everyuI{ˉu}.

    Then, every positive solution of Eq (2.1) with initial conditions in I converges to ˉu.

    Consider the difference equation

    yn+1=G(yn,yn1),n=0,1,2,. (2.2)

    The following strategy for obtaining global attractivity results of Eq (2.2) is derived from [12], which is also referenced in [2].

    Lemma 2.3. Let [a,b] be an interval of real numbers and assume that G:[a,b]×[a,b][a,b] is a continuous function satisfying the following properties:

    (i) G(x,y) is non-decreasing in x[a,b] for each y[a,b], and G(x,y) is non-increasing in y[a,b] for each x[a,b].

    (ii) If (m,M)[a,b]×[a,b] is a solution of the system

    G(m,M)=m,andG(M,m)=M,

    then m=M.

    Then, Eq (2.2) has a unique equilibrium ˉx, and every solution of Eq (2.2) converges to ˉx.

    Theorem 3.1. Assume that (1.3) holds. Then every nonnegative solution of system (1.2) is bounded and eventually enters an invariant rectangle [β,β1α]×[0,αβ1α].

    Proof. Using (1.2) and noting that 0<eyn1 for yn0, we get

    β<xn+1=αxneyn+β<αxn+β,n=0,1,2,.

    Consider the initial value problem

    zn+1=αzn+β,n=0,1,2,, (3.1)

    with initial value z0=x0. It follows by Lemma 2.1 that

    xnzn,forn=0,1,2,.

    The solution of Eq (3.1) is given by

    zn=αn(z0β1α)+β1α,n1,

    and for n>1,

    zn+1zn=αn(1α)(β1αz0).

    Therefore, the sequence {zn} is decreasing and bounded below by β1α with the initial value z0>β1α, and it is increasing and bounded above by β1α with the initial value z0<β1α, and zn=β1α for n1 with the initial value z0=β1α. Thus, limnzn=β1α. Hence, for every ϵ>0, there is an integer N such that, for n>N,

    xnzn<β1α+ε,

    and so xnβ1α for n>N. Furthermore, when n>N,

    0yn+1=αxn(1eyn)αxnαβ1α,

    holds.

    Set

    M=max{x0,x1,,xN,β1α},L=max{y0,y1,,yN+1,αβ1α}.

    Then

    βxnM,0ynL,forn0.

    Moreover, if (x0,y0)[β,β1α]×[0,αβ1α], then

    βx1=αx0ey0+βαx0+βαβ1α+β=β1α,
    0y1=αx0(1ey0)αx0αβ1α,

    and by using induction, we obtain

    (xn,yn)[β,β1α]×[0,αβ1α]forn0.

    So, the rectangle [β,β1α]×[0,αβ1α] is invariant, which completes the proof.

    Theorem 3.2. Assume that (1.3) holds. Then system (1.2) has no positive prime period-two solution.

    Proof. Assume for the sake of contradiction that

    ,(ξ1,η1),(ξ2,η2),(ξ1,η1),(ξ2,η2),

    is a positive prime period-two solution of system (1.2). Then it should satisfy

    ξ2=αξ1eη1+β,η2=αξ1(1eη1), (3.2)

    and

    ξ1=αξ2eη2+β,η1=αξ2(1eη2). (3.3)

    Clearly, ξ1,ξ2β.

    From (3.2) and (3.3), we derive

    ξ2β=αξ1η2,ξ1β=αξ2η1,

    which are equivalent to

    η2η1=(1+α)(ξ1ξ2). (3.4)

    Thus, ξ1=ξ2η2=η1.

    Moreover, (3.2) and (3.3) imply that ξ1,ξ2>β. This is because, if ξ1=β, then ξ2=0 and η1=η2=0, which is a contradiction. Similarly, if ξ2=β, then ξ1=0 and η2=η1=0, which leads to a contradiction as well.

    Additionally, combining (3.2), (3.3), and (3.4), we can obtain

    ξ2βξ1β=ξ1eη1ξ2eη2=ξ1ξ2eη2η1=ξ1ξ2e(1+α)(ξ1ξ2),

    and thus

    e(1+α)(ξ1ξ2)=ξ2(ξ2β)ξ1(ξ1β),

    which means that

    ξ1(ξ1β)e(1+α)ξ1=ξ2(ξ2β)e(1+α)ξ2. (3.5)

    Set

    A(t)=t(tβ)e(1+α)t.

    Then

    A(t)=e(1+α)t[(tβ)(t+αt+1)+t],

    from which it follows that A(t)>0 for tβ>0, and thus A(t) is strictly increasing in t for tβ>0. So, (3.5) implies that ξ1=ξ2. Therefore, η1=η2, a contradiction.

    The proof is complete.

    Theorem 4.1. (i) Assume that (1.3) holds and β1αα. Then system (1.2) possesses a unique nonnegative equilibrium ˉEx=(β1α,0).

    (ii) Assume that (1.3) holds and β>1αα. Then system (1.2) possesses two equilibria: ˉEx=(β1α,0) and ˉE=(ˉx,ˉy)[β,β1α]×[0,αβ1α].

    Proof. The equilibria of system (1.2) can be obtained by solving the following equations:

    {x=αxey+β,y=αx(1ey). (4.1)

    Clearly, y=0 is always the solution of the second equation of (4.1), and thus ˉEx=(β1α,0) is always the equilibrium of system (1.2).

    From the first equation of (4.1), we get

    x=β1αey,

    and thus

    y=αβ(1ey)1αey,

    or, equivalently,

    yαyeyαβ(1ey)=0. (4.2)

    Let

    ϕ(y)=yαyeyαβ(1ey). (4.3)

    Then, ϕ(0)=0, and ϕ(y)y as y+. Moreover, we have

    ϕ(y)=1αey+αyeyαβey=1ey(ey+αyααβ).

    Let

    ψ(y)=ey+αyααβ. (4.4)

    Then, ψ(y)=ey+α>0, from which it follows that the function ψ(y) is strictly increasing in [0,+).

    (i) When β1αα, ψ(y)>ψ(0)=1ααβ0 with y>0. Consequently, ϕ(y)=1eyψ(y)>0 for y>0, and system (1.2) has no other equilibrium, which implies that conclusion (i) is valid.

    (ii) When β>1αα, ψ(0)=1ααβ<0, and ψ(+)=+. By the continuity of the function ψ(y), there exists a unique root y(0,+) such that

    ψ(y)=0. (4.5)

    Hence, ψ(y)<0 with 0<y<y, and ψ(y)>0 with y>y. Moreover, ϕ(y)<0 with 0<y<y, and ϕ(y)>0 with y>y. It follows that ϕ(y) is decreasing in (0,y), and ϕ(y) is increasing in (y,+). Thus, the function ϕ(y) attains its minimum at y, ϕ(y)<ϕ(0)=0, and by the continuity of the function ϕ(y), equation ϕ(y)=0 has a unique positive root ˉy such that ˉy>y.

    Adding the two equations of system (4.1) yields

    ˉx+ˉy=αˉx+β, (4.6)

    hence

    ˉx=(βˉy)/(1α). (4.7)

    By (4.1) and (4.7), it is easy to obtain that ˉx[β,β1α] and ˉy[0,αβ1α]. Thus, in this case, system (1.2) possesses an additional equilibrium ˉE=(ˉx,ˉy), and conclusion (ii) follows.

    The proof is complete.

    Theorem 4.2. (i) Assume that (1.3) holds. Then the equilibrium ˉEx=(β1α,0) is locally asymptotically stable when β<1αα, is nonhyperbolic when β=1αα, and is unstable (a saddle point) when β>1αα.

    (ii) Assume that (1.3) holds and β>1αα. Then the unique positive equilibrium ˉE is locally asymptotically stable (a sink).

    Proof. Let

    F(x,y)=(f(x,y)g(x,y))=(αxey+βαx(1ey)).

    By simple calculation, we have

    fx=αey,fy=αxey,gx=α(1ey),gy=αxey.

    (i) The Jacobian matrix of F evaluated at ˉEx is given by

    JF(ˉEx)=(ααβ1α0αβ1α),

    and its eigenvalues are

    λ1=α,λ2=αβ1α.

    Notice that α(0,1), so 0<λ1<1, and 0<λ2<1 with β<1αα, λ2=1 with β=1αα, and λ2>1 with β>1αα, which means that result (i) follows.

    (ii) The Jacobian matrix of F evaluated at ˉE is given by

    JF(ˉE)=(αeˉyαˉxeˉyα(1eˉy)αˉxeˉy),

    and its characteristic equation is

    λ2pλ+q=0,

    where p=αeˉy(1+ˉx), q=α2ˉxeˉy.

    Since the second equation of (4.1) implies that ˉx=ˉy/(α(1eˉy)), it can be concluded that

    0<q=α2ˉxeˉy=αˉxαeˉy=ˉy1eˉyαeˉy=αˉyeˉy1<αˉyˉy=α<1.

    Moreover, noticing that the function ψ(y) defined by (4.4) is strictly increasing in (0,+) and that ˉy>y, we can utilize (4.5) to derive

    ψ(ˉy)=eˉy+αˉyααβ>ψ(y)=0,

    where y is the minimum point of ϕ(y). Thus,

    1+αˉyeˉyαeˉyαβeˉy>0. (4.8)

    In addition, the fact that ˉy is the root of the function ϕ(y) given by (4.3) implies that ϕ(ˉy)=0, namely,

    ˉyαˉyeˉy+αβeˉyαβ=0. (4.9)

    Adding (4.8) and (4.9) yields

    1+ˉyαeˉyαβ>0. (4.10)

    From (4.6), we have

    ˉy=β(1α)ˉx, (4.11)

    and from the first equation of system (4.1), we have

    eˉy=ˉxβαˉx. (4.12)

    Substituting (4.11) and (4.12) into (4.10) yields

    1+β(1α)ˉxˉxβˉxαβ>0,

    from which it follows that

    α(ˉxβ)>ˉxββˉx.

    Applying (4.12), we have q=α2ˉxeˉy=α(ˉxβ) and

    |p|=αeˉy(1+ˉx)=ˉxβˉx(1+ˉx)=1+ˉxββˉx<1+α(ˉxβ)=1+q<2.

    By the Schur-Cohn criterion, we obtain that ˉE=(ˉx,ˉy) is locally asymptotically stable.

    The proof is complete.

    When parameters α and β satisfy the condition β=1αα, the equilibrium ˉEx=(β1α,0) is non-hyperbolic with eigenvalue λ2=1. This indicates a bifurcation probably occurs as the parameter β varies and goes through the critical value 1αα. In fact, in this case, a transcritical bifurcation takes place at ˉEx.

    Theorem 4.3. Assume that (1.3) holds and let β=1αα. Then system (1.2) undergoes a transcritical bifurcation at ˉEx when the parameter β passes through the critical value β.

    Proof. Letting un=xnβ1α, vn=yn shifts the equilibrium ˉEx to the origin, and tranforms the system (1.2) into

    {un+1=αunevn+αβ1αevnαβ1α,vn+1=αun(1evn)αβ1αevn+αβ1α,n=0,1,2,. (4.13)

    Define τ=ββ as a small perturbation around β with 0<|τ|1. Then, the map of system (4.13) can be expressed as:

    (uvτ)(αuev+ev+ατ1αevατ1α1αu(1ev)evατ1αev+ατ1α+1τ). (4.14)

    Expanding (4.14) in a Taylor series at (u,v,τ)=(0,0,0) gives

    (uvτ)(α10010001)(uvτ)+(F1(u,v,τ)G1(u,v,τ)0), (4.15)

    where

    F1(u,v,τ)=12v2αuvα1αvτ16v3+12αuv2+α2(1α)v2τ+O(3),
    G1(u,v,τ)=12v2+αuv+α1αvτ+16v312αuv2α2(1α)v2τ+O(3),

    and O(3) is the sum of all remainder terms with a frequency greater than 3.

    Let

    T=(1100α10001)

    be an invertible matrix. Through the variable transformation

    (uvτ)=T(XYω),

    the map (4.15) is transformed into the form

    (XYω)(α00010001)(XYω)+(F2(X,Y,ω)G2(X,Y,ω)0), (4.16)

    where

    F2(X,Y,ω)=12(1α2)Y2α(α1)XY+αYω+16(α1)2(2α+1)Y3+12α(α1)2XY212α(α1)Y2ω+O(3),G2(X,Y,ω)=12(α21)Y2+α(α1)XYαYω16(α1)2(2α+1)Y312α(α1)2XY2+12α(α1)Y2ω+O(3).

    By the center manifold Theorem 2.1.4 in [24], for the map (4.16), there exists a center manifold that can be locally represented in the form:

    Wc(0,0)={(X,Y,ω)R3|X=h(Y,ω),|Y|<δ,|ω|<δ,h(0,0)=0,Dh(0,0)=0},

    for δ sufficiently small. Suppose that the center manifold has the representation

    X=h(Y,ω)=m1Y2+m2Yω+m3ω2+O(2).

    Then, it satisfies

    N(h(Y,ω))=h(Y+G2(h(Y,ω),Y,ω),ω)[αh(Y,ω)+F2(h(Y,ω),Y,ω)]=0,

    where O(2) represents the sum of all remainder terms with a frequency greater than 2. Hence,

    m1Y2+m2Yω+m3ω2=αm1Y2+αm2Yω+αm3ω2+O(2). (4.17)

    Comparing the corresponding coefficients of terms in Eq (4.17), we have

    m1=0,m2=0,m3=0,

    so the map (4.16) on the center manifold can be written as

    F:YY+12(α21)Y2αYω+O(2).

    Since

    F(0,0)=0,FY|(0,0)=1,
    Fω|(0,0)=0,2FYω|(0,0)=α0,2FY2|(0,0)=α210,

    therefore a transcritical bifurcation takes place at the equilibrium (Y,ω)=(0,0) of the map (4.16), implying that, as the parameter β changes and passes through the critical value β, system (1.2) undergoes a transcritical bifurcation at ˉEx.

    The proof is complete.

    In view of Lemma 4.2, to deal with the global asymptotic stability of Ex and ˉE, it is sufficient to solve its global attractivity.

    Consider the difference equation

    un+1=A(1eun),n=0,1,2,, (5.1)

    with A(0,+) and the initial value u0[0,+).

    Lemma 5.1. When A1, Eq (5.1) possesses a unique equilibrium zero, and when A>1, an additional positive equilibrium ˉu emerges satisfying ˉu>lnA.

    Proof. Clearly, zero is always an equilibrium of Eq (5.1). The positive equilibrium can be obtained by solving the equation

    u=A(1eu)u(0,+).

    Let

    h(u)=A(1eu)u. (5.2)

    Then, h(0)=0, h(+)=, and h(u)=Aeu1, h(u)=Aeu<0.

    When A1, h(u)<h(0)=A10 for u>0, and thus Eq (5.1) has a unique equilibrium, namely zero.

    When A>1, the function h(u) attains its maximum at u=lnA. Hence, by the continuity of h(u), there exists a unique ˉu such that h(ˉu)=0, namely Eq (5.1) has a unique positive equilibrium ˉu which satisfies ˉu>lnA and h(u)>0 for 0<u<ˉu, and h(u)<0 for u>ˉu.

    Lemma 5.2. (i) Assume that A1. Then every nonnegative solution of Eq (5.1) converges to the zero equilibrium.

    (ii) Assume that A>1. Then every positive solution of Eq (5.1) converges to the unique positive equilibrium ˉu.

    Proof. (i) Clearly, un=0 with u0=0 for n0, and the result follows. Given u0>0, then un>0 for n1, and

    un+1=A(1eun)<Aunun,

    from which it follows by induction that the sequence {un} is strictly decreasing and bounded below by zero, so it is convergent. Since, in this case Eq (5.1) has a unique equilibrium zero, hence limnun=0.

    (ii) Let g(u)=A(1eu). Observing that the function g(u) is increasing for u>0, and using the properties of the function h(u) defined by (5.2), we obtain

    g(u)=h(u)+u>uwith0<u<ˉu

    and

    g(u)=h(u)+u<uwithu>ˉu.

    Hence,

    (uˉu)(g(u)u)<0foru(0,){ˉu},

    and condition (ii) in Lemma 2.2 is satisfied. It follows that limnun=ˉu with u0>0.

    We now start the discussion of our main results.

    Theorem 5.3. Every solution {(xn,yn)} of system (1.2) with x0y0=0 converges to ˉEx.

    Proof. Notice that y1=0 with x0=0, so it is sufficient to discuss the case that y0=0. Obviously, in this case, yn=0 for n1, and thus system (1.2) becomes

    xn+1=αxn+β,n=1,2,,

    and

    xn=αnx0+β1α(1αn)β1α,asn,

    since α(0,1), finishing the proof.

    Theorem 5.4. Assume that (1.3) holds and β1αα. Then the unique equilibrium ˉEx of system (1.2) is a global attractor of all nonnegative solutions.

    Proof. Let {(xn,yn)} be a nonnegative solution of system (1.2). Then, from Theorem 3.1, the subsequence {xn} is eventually bounded and thus there exists an integer N such that xnβ1α for n>N. Using the second equation of system (1.2), we get

    yn+1=αxn(1eyn)αβ1α(1eyn). (5.3)

    Noticing that, in this case αβ1α1 and applying 5.2 (i), we obtain that every nonnegative solution of the difference equation

    ˜yn+1=αβ1α(1e˜yn),n=0,1,2,,

    converges to zero. Using the boundedness of the subsequence {yn}, (5.3) yields

    0liminfnyn+1limsupnyn+1limn˜yn+1=0,

    from which it follows that limnyn=0 and limnxn=β1α. Thus, limn(xn,yn)=ˉEx.

    The proof is complete.

    In view of Theorems 5.4 and 4.2 (i), we have the following result:

    Theorem 5.5. Assume that (1.3) holds and β<1αα. Then the unique equilibrium ˉEx of system (1.2) is globally asymptotically stable.

    Next, we deal with the global asymptotic stability of the unique positive equilibrium ˉE. We will provide a sufficient condition for ˉE to be globally asymptotically stable with respect to all positive solutions {(xn,yn)} of system (1.2). The positive solution we talk about here means a solution of system (1.2) satisfying xn,yn>0 for n0.

    Theorem 5.6. Assume that (1.3) holds and β1+αα. Then the unique positive equilibrium ˉE of system (1.2) is a global attractor of all positive solutions.

    Proof. Let {(xn,yn)} be a solution of system (1.2) with x0y00, then yn>0 for n1.

    From the second equation of system (1.2), we get

    xn=yn+1α(1eyn),n=0,1,2,,

    then

    yn+2α(1eyn+1)=yn+11eyneyn+β,

    or, equivalently,

    yn+2=αyn+1(1eyn+1)eyn1eyn+αβ(1eyn+1),n=0,1,2,, (5.4)

    which is a second-order difference equation with initial values y1=αx0(1ey0), y0>0.

    Clearly, the equilibrium of Eq (5.4) is not equal to zero and it must satisfy the equation

    yαyeyαβ(1ey)=0,

    which is the equation defined by (4.2). Hence, Eq (5.4) has a unique positive equilibrium, namely ˉy.

    Equation (5.4) implies that

    yn+1αβ(1eyn),n=1,2,. (5.5)

    If β1+αα, then αβ>1. By utilizing Lemma 5.2 (ii), it can be concluded that every positive solution of the difference equation

    ˜yn+1=αβ(1e˜yn),n=1,2,,

    converges to its positive equilibrium, denoted by ˜y, and by Lemma 5.1, ˜y>lnαβ. Hence, for ϵ=˜ylnαβ>0, there exists an integer N such that ˜yn>˜yϵ=lnαβ for n>N. Further, yn˜yn>lnαβ>0 for n>N. Therefore,

    lim infnyn+1lnαβ>0.

    In view of Theorem 3.1, it follows that every positive solution of Eq (5.4) eventually enters an invariant interval [lnαβ,αβ1α][0,αβ1α], and ˉy[lnαβ,αβ1α] is unique.

    Set

    G(u,v)=αu(1eu)ev1ev+αβ(1eu),

    then G is increasing in u for v>0, and is decreasing in v for u>0.

    Let (m,M)[lnαβ,αβ1α]×[lnαβ,αβ1α] be a solution of the following system:

    {m=αm(1em)eM1eM+αβ(1em),M=αM(1eM)em1em+αβ(1eM).

    Then we have

    11emαβm=αeM1eM, (5.6)
    11eMαβM=αem1em. (5.7)

    Adding (5.6) and (5.7) yields

    11emαβm+αem1em=11eMαβM+αeM1eM,

    which is equivalent to

    em+αem1αβm=eM+αeM1αβM. (5.8)

    Consider the function

    I(t)=et+αet1αβt,t[lnαβ,αβ1α].

    To prove that m=M, it is sufficient to show that the function I(t) is injective on the interval [lnαβ,αβ1α] under the condition that β1+αα. Simple computation shows that

    I(t)=(1+α)et(et1)2+αβt2=1t2(et1)2[αβ(et1)2(1+α)t2et]1+αt2(et1)2[(et1)2t2et].

    Let

    J(t)=(et1)2t2et,

    then

    J(t)=2et(et1t12t2)>0fort>0,

    and so, for t>0,

    J(t)>J(0)=0.

    Therefore, I(t)>0 for t>0, which implies that the function I(t) is strictly increasing on the interval [lnαβ,αβ1α]. Thus, equality (5.8) yields m=M. By applying Lemma 2.3, we get that every positive solution of Eq (5.4) converges to ˉy.

    Consequently, every positive solution of system (1.2) satisfies limnyn=ˉy and limnxn=ˉx, and so limn(xn,yn)=ˉE.

    The proof is complete.

    In view of Theorems 5.6 and 4.2 (ii), we have the following result:

    Theorem 5.7. Assume that (1.3) holds and β1+αα. Then the unique positive equilibrium ˉE of system (1.2) is globally asymptotically stable.

    In this work, the global behavior of a discrete population model (1.2) is considered with the conditions α(0,1), β(0,+). It is shown that, for all α(0,1) and β(0,+), every nonnegative solution of this system is bounded and there is no positive prime period-two solution. However, the existence of equilibria, the local stability, bifurcation, and the global asymptotic stability depend upon the parameters α,β. Specifically, if β1αα, then this system possesses a unique equilibrium ˉEx. It is globally asymptotically stable for β<1αα, and as parameter β varies and passes through the critical value 1αα, this system experiences a transcritical bifurcation at ˉEx. If β>1αα, then this system possesses two equilibria, ˉEx and ˉE, where ˉEx is unstable and ˉE is locally asymptotically stable. Finally, a sufficient condition β1+αα is established, under which ˉE is globally asymptotically stable.

    The research result indicates that the use of refuge or external immigration of hosts can contribute to stabilizing the system. If the level of the use of refuge or external immigration of hosts per generation remains at or below the threshold 1αα, the parasitoids will go extinction for all initial populations. Once this threshold is surpassed, the extinct equilibrium ˉEx loses its stability and the stable coexistence equilibrium ˉE=(ˉx,ˉy) emerges. Specifically, maintaining the level at or above 1+αα guarantees the hosts and the parasitoids will eventually coexist at a steady density (ˉx,ˉy) for all initial populations. Therefore, it is essential to keep enough of a level of refuge or external immigration of hosts for the long-term survival and stability of this system.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the National Natural Science Foundation of China (No. 11701425, 12261078). The second author acknowledges the support of the Longyuan Youth Talent Project of Gansu Province.

    All authors declare that there are no conflicts of interests regarding the publication of this paper.



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