In this work, the global behavior of a discrete population model
{xn+1=αxne−yn+β,yn+1=αxn(1−e−yn),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=αxne−yn+β,yn+1=αxn(1−e−yn),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=λxne−ayn,yn+1=cxn(1−e−ayn),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=αxne−yn+β,yn+1=αxn(1−e−yn),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 β∈R. Let {xn}∞n=0 and {zn}∞n=0 be sequences of real numbers such that x0≤z0 and
{xn+1≤αxn+β,zn+1=αzn+β,n=0,1,2,…. |
Then xn≤zn for n≥0.
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 g∈C[I,(0,+∞)] satisfies the following conditions:
(i) g(u) is non-decreasing in u.
(ii) Equation (2.1) has a unique positive equilibrium ˉu∈I and the function g(u) satisfies the negative feedback condition:
(u−ˉu)(g(u)−u)<0for everyu∈I∖{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,yn−1),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,β1−α]×[0,αβ1−α].
Proof. Using (1.2) and noting that 0<e−yn≤1 for yn≥0, we get
β<xn+1=αxne−yn+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
xn≤zn,forn=0,1,2,…. |
The solution of Eq (3.1) is given by
zn=αn(z0−β1−α)+β1−α,n≥1, |
and for n>1,
zn+1−zn=αn(1−α)(β1−α−z0). |
Therefore, the sequence {zn} is decreasing and bounded below by β1−α with the initial value z0>β1−α, and it is increasing and bounded above by β1−α with the initial value z0<β1−α, and zn=β1−α for n≥1 with the initial value z0=β1−α. Thus, limn→∞zn=β1−α. Hence, for every ϵ>0, there is an integer N such that, for n>N,
xn≤zn<β1−α+ε, |
and so xn≤β1−α for n>N. Furthermore, when n>N,
0≤yn+1=αxn(1−e−yn)≤αxn≤αβ1−α, |
holds.
Set
M=max{x0,x1,…,xN,β1−α},L=max{y0,y1,…,yN+1,αβ1−α}. |
Then
β≤xn≤M,0≤yn≤L,forn≥0. |
Moreover, if (x0,y0)∈[β,β1−α]×[0,αβ1−α], then
β≤x1=αx0e−y0+β≤αx0+β≤αβ1−α+β=β1−α, |
0≤y1=αx0(1−e−y0)≤αx0≤αβ1−α, |
and by using induction, we obtain
(xn,yn)∈[β,β1−α]×[0,αβ1−α]forn≥0. |
So, the rectangle [β,β1−α]×[0,αβ1−α] 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=αξ1e−η1+β,η2=αξ1(1−e−η1), | (3.2) |
and
ξ1=αξ2e−η2+β,η1=αξ2(1−e−η2). | (3.3) |
Clearly, ξ1,ξ2≥β.
From (3.2) and (3.3), we derive
ξ2−β=αξ1−η2,ξ1−β=αξ2−η1, |
which are equivalent to
η2−η1=(1+α)(ξ1−ξ2). | (3.4) |
Thus, ξ1=ξ2⟺η2=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
ξ2−βξ1−β=ξ1e−η1ξ2e−η2=ξ1ξ2eη2−η1=ξ1ξ2e(1+α)(ξ1−ξ2), |
and thus
e(1+α)(ξ1−ξ2)=ξ2(ξ2−β)ξ1(ξ1−β), |
which means that
ξ1(ξ1−β)e(1+α)ξ1=ξ2(ξ2−β)e(1+α)ξ2. | (3.5) |
Set
A(t)=t(t−β)e(1+α)t. |
Then
A′(t)=e(1+α)t[(t−β)(t+αt+1)+t], |
from which it follows that A′(t)>0 for t≥β>0, and thus A(t) is strictly increasing in t for t≥β>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 β≤1−αα. Then system (1.2) possesses a unique nonnegative equilibrium ˉEx=(β1−α,0).
(ii) Assume that (1.3) holds and β>1−αα. Then system (1.2) possesses two equilibria: ˉEx=(β1−α,0) and ˉE=(ˉx,ˉy)∈[β,β1−α]×[0,αβ1−α].
Proof. The equilibria of system (1.2) can be obtained by solving the following equations:
{x=αxe−y+β,y=αx(1−e−y). | (4.1) |
Clearly, y=0 is always the solution of the second equation of (4.1), and thus ˉEx=(β1−α,0) is always the equilibrium of system (1.2).
From the first equation of (4.1), we get
x=β1−αe−y, |
and thus
y=αβ(1−e−y)1−αe−y, |
or, equivalently,
y−αye−y−αβ(1−e−y)=0. | (4.2) |
Let
ϕ(y)=y−αye−y−αβ(1−e−y). | (4.3) |
Then, ϕ(0)=0, and ϕ(y)∼y as y→+∞. Moreover, we have
ϕ′(y)=1−αe−y+αye−y−αβe−y=1ey(ey+αy−α−αβ). |
Let
ψ(y)=ey+αy−α−αβ. | (4.4) |
Then, ψ′(y)=ey+α>0, from which it follows that the function ψ(y) is strictly increasing in [0,+∞).
(i) When β≤1−αα, ψ(y)>ψ(0)=1−α−αβ≥0 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>1−αα, ψ(0)=1−α−αβ<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=(β−ˉy)/(1−α). | (4.7) |
By (4.1) and (4.7), it is easy to obtain that ˉx∈[β,β1−α] and ˉy∈[0,αβ1−α]. 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=(β1−α,0) is locally asymptotically stable when β<1−αα, is nonhyperbolic when β=1−αα, and is unstable (a saddle point) when β>1−αα.
(ii) Assume that (1.3) holds and β>1−αα. Then the unique positive equilibrium ˉE is locally asymptotically stable (a sink).
Proof. Let
F(x,y)=(f(x,y)g(x,y))=(αxe−y+βαx(1−e−y)). |
By simple calculation, we have
∂f∂x=αe−y,∂f∂y=−αxe−y,∂g∂x=α(1−e−y),∂g∂y=αxe−y. |
(i) The Jacobian matrix of F evaluated at ˉEx is given by
JF(ˉEx)=(α−αβ1−α0αβ1−α), |
and its eigenvalues are
λ1=α,λ2=αβ1−α. |
Notice that α∈(0,1), so 0<λ1<1, and 0<λ2<1 with β<1−αα, λ2=1 with β=1−αα, and λ2>1 with β>1−αα, which means that result (i) follows.
(ii) The Jacobian matrix of F evaluated at ˉE is given by
JF(ˉE)=(αe−ˉy−αˉxe−ˉyα(1−e−ˉy)αˉxe−ˉy), |
and its characteristic equation is
λ2−pλ+q=0, |
where p=αe−ˉy(1+ˉx), q=α2ˉxe−ˉy.
Since the second equation of (4.1) implies that ˉx=ˉy/(α(1−e−ˉy)), it can be concluded that
0<q=α2ˉxe−ˉy=αˉx⋅αe−ˉy=ˉy1−e−ˉy⋅αe−ˉy=αˉyeˉy−1<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+αˉy−α−αβ>ψ(y∗)=0, |
where y∗ is the minimum point of ϕ(y). Thus,
1+αˉye−ˉy−αe−ˉy−αβe−ˉy>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,
ˉy−αˉye−ˉy+αβe−ˉy−αβ=0. | (4.9) |
Adding (4.8) and (4.9) yields
1+ˉy−αe−ˉy−αβ>0. | (4.10) |
From (4.6), we have
ˉy=β−(1−α)ˉx, | (4.11) |
and from the first equation of system (4.1), we have
e−ˉy=ˉx−βαˉx. | (4.12) |
Substituting (4.11) and (4.12) into (4.10) yields
1+β−(1−α)ˉx−ˉx−βˉx−αβ>0, |
from which it follows that
α(ˉx−β)>ˉx−β−βˉx. |
Applying (4.12), we have q=α2ˉxe−ˉy=α(ˉx−β) and
|p|=αe−ˉy(1+ˉx)=ˉx−βˉx(1+ˉx)=1+ˉx−β−βˉx<1+α(ˉx−β)=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=1−αα, the equilibrium ˉEx=(β1−α,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 1−αα. In fact, in this case, a transcritical bifurcation takes place at ˉEx.
Theorem 4.3. Assume that (1.3) holds and let β∗=1−αα. Then system (1.2) undergoes a transcritical bifurcation at ˉEx when the parameter β passes through the critical value β∗.
Proof. Letting un=xn−β1−α, vn=yn shifts the equilibrium ˉEx to the origin, and tranforms the system (1.2) into
{un+1=αune−vn+αβ1−αe−vn−αβ1−α,vn+1=αun(1−e−vn)−αβ1−αe−vn+αβ1−α,n=0,1,2,…. | (4.13) |
Define τ=β−β∗ as a small perturbation around β∗ with 0<|τ|≪1. Then, the map of system (4.13) can be expressed as:
(uvτ)↦(αue−v+e−v+ατ1−αe−v−ατ1−α−1αu(1−e−v)−e−v−ατ1−αe−v+ατ1−α+1τ). | (4.14) |
Expanding (4.14) in a Taylor series at (u,v,τ)=(0,0,0) gives
(uvτ)↦(α−10010001)(uvτ)+(F1(u,v,τ)G1(u,v,τ)0), | (4.15) |
where
F1(u,v,τ)=12v2−αuv−α1−αvτ−16v3+12αuv2+α2(1−α)v2τ+O(3), |
G1(u,v,τ)=−12v2+αuv+α1−αvτ+16v3−12αuv2−α2(1−α)v2τ+O(3), |
and O(3) is the sum of all remainder terms with a frequency greater than 3.
Let
T=(1100α−10001) |
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(1−α2)Y2−α(α−1)XY+αYω+16(α−1)2(2α+1)Y3+12α(α−1)2XY2−12α(α−1)Y2ω+O(3),G2(X,Y,ω)=12(α2−1)Y2+α(α−1)XY−αYω−16(α−1)2(2α+1)Y3−12α(α−1)2XY2+12α(α−1)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∗:Y↦Y+12(α2−1)Y2−αYω+O(2). |
Since
F∗(0,0)=0,∂F∗∂Y|(0,0)=1, |
∂F∗∂ω|(0,0)=0,∂2F∗∂Y∂ω|(0,0)=−α≠0,∂2F∗∂Y2|(0,0)=α2−1≠0, |
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(1−e−un),n=0,1,2,…, | (5.1) |
with A∈(0,+∞) and the initial value u0∈[0,+∞).
Lemma 5.1. When A≤1, 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(1−e−u)u∈(0,+∞). |
Let
h(u)=A(1−e−u)−u. | (5.2) |
Then, h(0)=0, h(+∞)=−∞, and h′(u)=Ae−u−1, h″(u)=−Ae−u<0.
When A≤1, h′(u)<h′(0)=A−1≤0 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 A≤1. 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 n≥0, and the result follows. Given u0>0, then un>0 for n≥1, and
un+1=A(1−e−un)<Aun≤un, |
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 limn→∞un=0.
(ii) Let g(u)=A(1−e−u). 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,
(u−ˉu)(g(u)−u)<0foru∈(0,∞)∖{ˉu}, |
and condition (ii) in Lemma 2.2 is satisfied. It follows that limn→∞un=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 n≥1, and thus system (1.2) becomes
xn+1=αxn+β,n=1,2,…, |
and
xn=αnx0+β1−α(1−αn)→β1−α,asn→∞, |
since α∈(0,1), finishing the proof.
Theorem 5.4. Assume that (1.3) holds and β≤1−αα. 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 xn≤β1−α for n>N. Using the second equation of system (1.2), we get
yn+1=αxn(1−e−yn)≤αβ1−α(1−e−yn). | (5.3) |
Noticing that, in this case αβ1−α≤1 and applying 5.2 (i), we obtain that every nonnegative solution of the difference equation
˜yn+1=αβ1−α(1−e−˜yn),n=0,1,2,…, |
converges to zero. Using the boundedness of the subsequence {yn}, (5.3) yields
0≤liminfn→∞yn+1≤limsupn→∞yn+1≤limn→∞˜yn+1=0, |
from which it follows that limn→∞yn=0 and limn→∞xn=β1−α. 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<1−αα. 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 n≥0.
Theorem 5.6. Assume that (1.3) holds and β≥1+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 x0y0≠0, then yn>0 for n≥1.
From the second equation of system (1.2), we get
xn=yn+1α(1−e−yn),n=0,1,2,…, |
then
yn+2α(1−e−yn+1)=yn+11−e−yne−yn+β, |
or, equivalently,
yn+2=αyn+1(1−e−yn+1)e−yn1−e−yn+αβ(1−e−yn+1),n=0,1,2,…, | (5.4) |
which is a second-order difference equation with initial values y1=αx0(1−e−y0), y0>0.
Clearly, the equilibrium of Eq (5.4) is not equal to zero and it must satisfy the equation
y−αye−y−αβ(1−e−y)=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+1≥αβ(1−e−yn),n=1,2,…. | (5.5) |
If β≥1+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(1−e−˜yn),n=1,2,…, |
converges to its positive equilibrium, denoted by ˜y, and by Lemma 5.1, ˜y>lnαβ. Hence, for ϵ=˜y−lnαβ>0, there exists an integer N such that ˜yn>˜y−ϵ=lnαβ for n>N. Further, yn≥˜yn>lnαβ>0 for n>N. Therefore,
lim infn→∞yn+1≥lnαβ>0. |
In view of Theorem 3.1, it follows that every positive solution of Eq (5.4) eventually enters an invariant interval [lnαβ,αβ1−α]⊂[0,αβ1−α], and ˉy∈[lnαβ,αβ1−α] is unique.
Set
G(u,v)=αu(1−e−u)e−v1−e−v+αβ(1−e−u), |
then G is increasing in u for v>0, and is decreasing in v for u>0.
Let (m,M)∈[lnαβ,αβ1−α]×[lnαβ,αβ1−α] be a solution of the following system:
{m=αm(1−e−m)e−M1−e−M+αβ(1−e−m),M=αM(1−e−M)e−m1−e−m+αβ(1−e−M). |
Then we have
11−e−m−αβm=αe−M1−e−M, | (5.6) |
11−e−M−αβM=αe−m1−e−m. | (5.7) |
Adding (5.6) and (5.7) yields
11−e−m−αβm+αe−m1−e−m=11−e−M−αβM+αe−M1−e−M, |
which is equivalent to
em+αem−1−αβm=eM+αeM−1−αβM. | (5.8) |
Consider the function
I(t)=et+αet−1−αβt,t∈[lnαβ,αβ1−α]. |
To prove that m=M, it is sufficient to show that the function I(t) is injective on the interval [lnαβ,αβ1−α] under the condition that β≥1+αα. Simple computation shows that
I′(t)=−(1+α)et(et−1)2+αβt2=1t2(et−1)2[αβ(et−1)2−(1+α)t2et]≥1+αt2(et−1)2[(et−1)2−t2et]. |
Let
J(t)=(et−1)2−t2et, |
then
J′(t)=2et(et−1−t−12t2)>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,αβ1−α]. 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 limn→∞yn=ˉy and limn→∞xn=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 β≥1+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 β≤1−αα, then this system possesses a unique equilibrium ˉEx. It is globally asymptotically stable for β<1−αα, and as parameter β varies and passes through the critical value 1−αα, this system experiences a transcritical bifurcation at ˉEx. If β>1−αα, then this system possesses two equilibria, ˉEx and ˉE, where ˉEx is unstable and ˉE is locally asymptotically stable. Finally, a sufficient condition β≥1+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 1−αα, 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.
[1] |
J. R. Beddington, C. A. Free, J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58–60. http://dx.doi.org/10.1038/255058a0 doi: 10.1038/255058a0
![]() |
[2] | E. Camouzis, G. Ladas, Dynamics of third-Order rational difference equations with open problems and conjectures, Boca Raton: Chapman & Hall/CRC, 2008. |
[3] | E. A. Grove, G. M. Kent, R. Levins, G. Ladas, S. Valicenti, Global stability in some population models, Poznan: Gordon and Breach, 2000,149–176. |
[4] |
M. P. Hassell, R. M. May, Stability in insect host-parasite models, J. Anim. Ecol., 42 (1973), 693–726. http://dx.doi.org/10.2307/3133 doi: 10.2307/3133
![]() |
[5] | M. P. Hassell, The dynamics of arthropod predator-prey systems, Princeton University Press, 1978. |
[6] | M. L. J. Hautus, A. Emerson, T. S. Bolis, Solution to problem E2721, Amer. Math. Monthly, 86 (1979), 865–866. |
[7] |
S. B. Hsu, M. C. Li, W. S. Liu, M. Malkin, Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss, Discrete Cont. Dyn. S., 9 (2003), 1465–1492. http://dx.doi.org/10.3934/dcds.2003.9.1465 doi: 10.3934/dcds.2003.9.1465
![]() |
[8] |
W. T. Jamieson, J. Reis, Global behavior for the classical Nicholson-Bailey model, J. Math. Anal. Appl., 461 (2018), 492–499. https://doi.org/10.1016/j.jmaa.2017.12.071 doi: 10.1016/j.jmaa.2017.12.071
![]() |
[9] |
S. R. J. Jang, J. L. Yu, Discrete-time host-parasitoid models with pest control, J. Biol. Dynam., 6 (2012), 718–739. http://dx.doi.org/10.1080/17513758.2012.700074 doi: 10.1080/17513758.2012.700074
![]() |
[10] |
S. Kapçak, S. Elaydi, Ü. Ufuktepe, Stability of a predator-prey model with refuge effect, J. Differ. Equ. Appl., 22 (2016), 989–1004. http://dx.doi.org/10.1080/10236198.2016.1170823 doi: 10.1080/10236198.2016.1170823
![]() |
[11] | V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with application, Kluwer Academic Publishers, 1993. |
[12] | M. R. S. Kulenović, G. Ladas, Dynamics of second order rational difference equations with open problems and conjectures, Boca Raton: Chapman & Hall/CRC, 2001. |
[13] |
T. H. Li, X. T. Zhang, C. R. Zhang, Pattern dynamics analysis of a space-time discrete spruce budworm model, Chaos, Soliton. Fract., 179 (2024), 114423. http://dx.doi.org/10.1016/j.chaos.2023.114423 doi: 10.1016/j.chaos.2023.114423
![]() |
[14] | W. T. Li, H. R. Sun, X. X. Yan, The asymptotic behavior of a higher order delay nonlinear-difference-equations, Indian J. Pure Appl. Math., 34 (2003), 1431–1441. |
[15] |
W. T. Li, H. R. Sun, Dynamics of a rational difference equation, Appl. Math. Comput., 163 (2005), 577–591. http://dx.doi.org/10.1016/j.amc.2004.04.002 doi: 10.1016/j.amc.2004.04.002
![]() |
[16] | X. Y. Li, Y. Q. Liu, Transcritical bifurcation and flip bifurcation of a new discrete ratio-dependent predator-prey system, Qual. Theory Dyn. Syst., 21 (2022). http://dx.doi.org/10.1007/s12346-022-00646-2 |
[17] |
X. Y. Li, X. M. Shao, Flip bifurcation and Neimark-Sacker bifurcation in a discrete predator-prey model with Michaelis-Menten functional response, Electron. Res. Arch., 31 (2023), 37–57. http://dx.doi.org/10.3934/era.2023003 doi: 10.3934/era.2023003
![]() |
[18] |
H. I. McCallum, Effects of immigration on chaotic popution dynamics, J. Theor. Biol., 154 (1992), 277–284. http://dx.doi.org/10.1016/S0022-5193(05)80170-5 doi: 10.1016/S0022-5193(05)80170-5
![]() |
[19] | J. D. Murray, Mathematical Biology II, New York: Springer-Verlag, 1989. https://doi.org/10.1007/b98869 |
[20] |
A. J. Nicholson, An outline of the dynamics of animal populations, Aust. J. Zool., 2 (1954), 9–65. http://dx.doi.org/10.1071/ZO9540009 doi: 10.1071/ZO9540009
![]() |
[21] |
S. Sinha, S. Parthasarathy, Unusual dynamics of extinction in a simple ecological model, Proc. Natl. Acad. Sci. USA, 93 (1996), 1504–1508. http://dx.doi.org/10.1073/pnas.93.4.1504 doi: 10.1073/pnas.93.4.1504
![]() |
[22] |
S. Stević, Solvability and representations of the general solutions to some nonlinear difference equations of second order, AIMS Math., 8 (2023), 15148–15165. http://dx.doi.org/10.3934/math.2023773 doi: 10.3934/math.2023773
![]() |
[23] |
V. Weide, M. C. Varriale, F. M. Hilker, Hydra effect and paradox of enrichment in discrete-time predator-prey models, Math. Biosci., 310 (2019), 120–127. https://doi.org/10.1016/j.mbs.2018.12.010 doi: 10.1016/j.mbs.2018.12.010
![]() |
[24] | S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, New York: Springer-Verlag, 1990. https://doi.org/10.1007/b97481 |
[25] |
X. T. Zhang, C. R. Zhang, Y. Z. Zhou, Pattern dynamics analysis of a time-space discrete FitzHugh-Nagumo (FHN) model based on coupled map lattices, Comput. Math. Appl., 157 (2024), 92–123. http://dx.doi.org/10.1016/j.camwa.2023.12.030 doi: 10.1016/j.camwa.2023.12.030
![]() |