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Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches

  • This work studies a fractional model in the context of the fractal-fractional operator involving a power law kernel for waste plastic in the ocean consisting of waste plastic material, marine debris, and reprocessing. The novelty of this study lies in utilizing the fractal-fractional framework to analyze the behavior of the proposed model and derive qualitative theoretical results. We investigated the equilibrium points and analyzed their stability using the basic reproduction number. We examined the global stability of possible equilibrium points, as well as their unstable conditions. The sensitivity analysis can also help to better understand the model's dynamics better. The existence and uniqueness of results were studied by utilizing Banach's contraction mapping principle, while Ulam's stability for the proposed model was also investigated. Furthermore, we derived the numerical algorithms by utilizing four different techniques, including the decomposition, Adams-Bashforth, Newton polynomial, and predictor-corrector methods. By providing the input factor values, we illustrated the graphic numerical simulations to enhance our understanding of the waste plastic process and optimize the control strategies. Moreover, we compare ordinary, fractional, and fractal-fractional derivatives in the sense of the Caputo type with the reported real data of 70 years.

    Citation: Chatthai Thaiprayoon, Jutarat Kongson, Weerawat Sudsutad. Dynamics of a fractal-fractional mathematical model for the management of waste plastic in the ocean with four different numerical approaches[J]. AIMS Mathematics, 2025, 10(4): 8827-8872. doi: 10.3934/math.2025405

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  • This work studies a fractional model in the context of the fractal-fractional operator involving a power law kernel for waste plastic in the ocean consisting of waste plastic material, marine debris, and reprocessing. The novelty of this study lies in utilizing the fractal-fractional framework to analyze the behavior of the proposed model and derive qualitative theoretical results. We investigated the equilibrium points and analyzed their stability using the basic reproduction number. We examined the global stability of possible equilibrium points, as well as their unstable conditions. The sensitivity analysis can also help to better understand the model's dynamics better. The existence and uniqueness of results were studied by utilizing Banach's contraction mapping principle, while Ulam's stability for the proposed model was also investigated. Furthermore, we derived the numerical algorithms by utilizing four different techniques, including the decomposition, Adams-Bashforth, Newton polynomial, and predictor-corrector methods. By providing the input factor values, we illustrated the graphic numerical simulations to enhance our understanding of the waste plastic process and optimize the control strategies. Moreover, we compare ordinary, fractional, and fractal-fractional derivatives in the sense of the Caputo type with the reported real data of 70 years.



    The Beverton–Holt recurrence

    xt+1=ηKxtK+(η1)xt, (1.1)

    where η is the proliferation rate and K is the carrying capacity, was derived in [1] in the context of fisheries. The solution of the logistic differential equation evaluated at time T+t0 is used to describe the adult fish population (new generation), and the solution at time t0 represents the juveniles (old generation). The derivation led to a proliferation rate η=erT>1 , where r>0 is the growth rate of the underlying continuous model and T is the time span until adulthood. Equation (1.1) for more general parameters is also known as the Pielou equation [2]. For related work on the Beverton–Holt equation, see [3,4,5,6,7,8,9,10,11,12], and for other related work, see [13,14,15,16].

    In [17], Cushing and Henson investigated the effects of a periodically enforced carrying capacity onto flour beetles, which were modeled by the Beverton–Holt recurrence (1.1). Based on their observations, the authors conjectured that the introduction of a periodic environment on populations, modeled by (1.1), results in the existence of a unique periodic solution. Further, the authors predicted that a periodic environment is deleterious to the population, as the average of the unique periodic solution is bounded above by the mean of the periodic environment. These conclusions were formulated as the first and second Cushing–Henson conjectures.

    In the case of a two-periodic Kt and constant η , the conjectures have been analytically verified in [18]. For higher-order periodic carrying capacities Kt , the conjectures have been the focus of the works [19,20]. A discussion assuming additionally time-dependent growth rates can be found in [21]. The extension of the conjectures to periodic time scales was addressed in [22]. Periodic time scales are time domains such that if t is in the time scale, then so is t+ω , hence requiring an additive time structure. The discrete time setting obeys this additive property and is a special case of a periodic time scale, in contrast to the quantum time setting qN0 , which is not periodic. Periodic time scales are a subset of arbitrary time scales, a theory developed by Stefan Hilger in 1988, that unifies the discrete and continuous theories. Studying the Beverton–Holt model on time scales provides a platform to consider time-dependent time spans until adulthood instead of a constant time span T , as assumed in the derivation of the Beverton–Holt model. Due to a lack of existing periodicity definitions for general time domains, the conjectures remained unsolved for this and other examples of nonperiodic time scales. In [23], in order to extend the study to the quantum time scale, the authors defined periodicity in the quantum setting and discussed the Cushing–Henson conjectures. In the quantum time setting, the Beverton–Holt model reads as

    x(qt)=ηK(t)x(t)K(t)+(η1)x(t)

    with the carrying capacity K:qN0R+:=(0,) ( q>1 ) and proliferation rate η>1 . In [23], the authors proved the existence and global stability of a unique periodic solution for periodic carrying capacities, i.e., for K such that qωK(qωt)=K(t) , confirming the first Cushing–Henson conjecture for the q -Beverton–Holt model. In [24], the authors were able to extend this result to time-dependent proliferation rates. The second Cushing–Henson conjecture, however, only remained true in the quantum time scale with a slight modification as follows.

    Theorem 1.1 (See [23, Theorem 5.6]). The average of the ω -periodic solution ˉx of the q -Beverton–Holt model is strictly lessthan the average of the ω -periodic carrying capacitytimes the constant qλ1λ , i.e.,

    qω1ˉx(t)Δtqλ1λqω1K(t)Δt,

    where λ=1(q1)A , η=11(q1)A , and A>0 .

    The multiplicative constant in Theorem 1.1 can be expressed by

    qλ1λ=(q1)(1+A)(q1)A=(q1)η(q1)+η1η(q1)η1η=qη1η1. (1.2)

    This indicates that the second conjecture does not necessarily hold, and it reveals that the time scale determines the proportionality constant linearly. Given this reformulation of the classical second Cushing–Henson conjecture in the special case of a quantum time scale, we aim to find a general formulation of this conjecture on arbitrary isolated time scales. On any isolated time scale T , we therefore consider the Beverton–Holt model

    xσ=ηKxK+(η1)x, (1.3)

    where η,K:TR+ , η>1 , and where σ(t) is the next time step following t . We can express the recurrence (1.3) as a dynamic equation

    xσxμ=:xΔ=αxσ(1xK)withα=η1μη, (1.4)

    where μ(t) is the distance to the time point following t , formally introduced in Section 2. Equation (1.4) is known as the logistic dynamic equation [25], and it can be transformed equivalently into a linear dynamic equation using the variable substitution u=1/x for x0 , namely

    uΔ=αu+αK. (1.5)

    To study the effects of periodicity on (1.3) on arbitrary time scales, we utilize the recently introduced definition of periodicity in [26]. This new concept was already successfully applied in [27]. This and other useful definitions are stated in Section 2. Section 3 concerns the generalization of the first Cushing–Henson conjecture, discussing the existence of a unique globally asymptotically stable periodic solution. The generalization of the second Cushing–Henson conjecture is addressed in Section 4. The paper is completed in Section 5 with some concluding remarks.

    In this section, we introduce some necessary time scales fundamentals. A time scale T is a closed nonempty subset of R .

    Definition 2.1 (See [28, Definition 1.1]). For tT , the forward jump operator σ:TT is defined by

    σ(t)=inf{sT:s>t}.

    We adopt the convention that inf=supT . If σ(t)>t , then we say that t is right-scattered. If σ(t)=t , then we say that t is right-dense. Similarly, left-scattered and left-dense points are defined. The graininess μ:TR+0 is defined by μ(t):=σ(t)t . We define fσ:TR by fσ:=fσ . If T has a left-scattered maximum M , then we define Tκ=T{M} ; otherwise, Tκ=T .

    In this work, we focus on isolated time scales, i.e., all points are left-scattered and right-scattered. Hence, in what follows, throughout, T refers to an isolated time scale, and the following definitions and results are taking this special time structure already into account, as well as the entire remainder of this paper.

    Definition 2.2 (See [25, Definition 2.25]). A function p:TR is called regressive, denoted by R , provided

    1+μ(t)p(t)0foralltT.

    Moreover, p is called positively regressive, denoted by R+ , provided

    1+μ(t)p(t)>0foralltT.

    Remark 2.3. Assume f:TR and tTκ . Then the delta-derivative of f , denoted by fΔ (see [25, Definition 1.10]), is

    fΔ(t)=f(σ(t))f(t)μ(t).

    For f,g:TR , the product-rule for tTκ (see [25]) reads as

    (fg)Δ=fΔgσ+fgΔ=fΔg+fσgΔ, (2.1)

    and, if g0 , the quotient-rule (see [25]) reads as

    (fg)Δ=fΔgfgΔggσ.

    The delta integral is defined for a,bT with a<b by

    baf(τ)Δτ=τ[a,b)Tμ(τ)f(τ),

    and consequently,

    FΔ=f,ifF(t)=taf(τ)Δτ.

    Using the product rule, we get the integration by parts formula (see [25])

    ba(fΔg)(t)Δt=(fg)(b)(fg)(a)ba(fσgΔ)(t)Δt. (2.2)

    The following circle-plus addition turns (R,) into an Abelian group.

    Definition 2.4 (See [28, p. 13]). Define circle plus and circle minus by

    pq=p+q+μpq,pq=pq1+μqforqR.

    Theorem 2.5 (See [25, Theorem 2.33). Let pR and t0T . Then the initial value problem

    yΔ=p(t)y,y(t0)=1

    possesses a unique solution.

    The unique solution from Theorem 2.5 is called the dynamic exponential function and is denoted by ep(,t0) . On an isolated time scale, the dynamic exponential function for pR is

    ep(t,t0)=s[t0,t)T(1+μ(s)p(s)),t>t0.

    Useful properties of the dynamic exponential function follow. Part 6 can easily be shown using parts 4 and 5. Part 8 is is the content of [25, Theorem 2.39]. Part 9 is in [25, Theorem 2.48(i)], and the remaining parts are from [25, Theorem 2.36].

    Theorem 2.6. If p,qR and t,s,rT , then

    1. e0(t,s)=1 and ep(t,t)=1 ,

    2. epq(t,s)=ep(t,s)eq(t,s) ,

    3. epq(t,s)=ep(t,s)eq(t,s) ,

    4. ep(t,s)=ep(s,t)=1ep(t,s) ,

    5. ep(σ(t),s)=(1+μ(t)p(t))ep(t,s) ,

    6. ep(t,σ(s))=ep(t,s)1+μ(s)p(s) ,

    7. eΔp(,s)=pep(,s) ,

    8. eΔp(s,)=peσp(s,) ,

    9. pR+ implies ep(t,s)>0 , and

    10. the semigroup property holds: ep(t,r)ep(r,s)=ep(t,s) .

    The following result was used in [26, Proof of Theorem 6.2]. A variant of it was also included in [29, Theorem 2.1], see also [30, First formula in the line two lines after (3)]). Here we state it explicitly and include its short proof.

    Lemma 2.7. Let f:TR{0} be delta-differentiable and s,tT . Then

    efΔf(s,t)=f(s)f(t). (2.3)

    Proof. For fixed tT , define w(s):=f(s)f(t) . Since f0 , fΔfR and

    wΔ(s)=fΔ(s)f(t)=f(s)f(t)fΔ(s)f(s)=w(s)fΔ(s)f(s),w(t)=1.

    By Theorem 2.5, the claim follows.

    Theorem 2.8 (See [31]). If f is nonnegative with fR+ , then

    1tsf(τ)Δτef(t,s)exp(tsf(τ)Δτ). (2.4)

    Theorem 2.9. (Variation of Constants, see [28, Theorem 2.77]). If pR , f:TR , t0T , and y0R , then the unique solution of the IVP

    yΔ=p(t)y+f(t),y(t0)=y0

    is given by

    y(t)=ep(t,t0)y0+tt0ep(t,σ(s))f(s)Δs. (2.5)

    As mentioned in the introduction, the definition of periodicity is crucial in the discussion of effects of periodicity. We refer to our recent work [26], where we introduced periodicity on isolated time sales as follows.

    Definition 2.10 (See [26, Definition 4.1]). Let ωN . A function f:TR is called ω -periodic, denoted by fPω , provided

    νΔfν=f,whereν=σωandfν=fν. (2.6)

    Example 2.11. If T=Z , then ν(t)=t+ω , νΔ(t)=1 , and (2.6) reduces to the known definition of periodicity, f(t+ω)=f(t) .

    Example 2.12. If T=qN0 , then ν(t)=qωt , νΔ(t)=qω , and (2.6) reduces to the known definition of periodicity, qωf(qωt)=f(t) , which was introduced in [23].

    Now, from our recent paper [26], we collect some results, supplemented by some new tools (substitution rule and change of order of integration formula), that will be used in the remainder of this study.

    Lemma 2.13. (See [26, Theorem 5.6 and Corollary 5.8]). If f,gPω , then

    f+g,fg,f,fgPω.

    Lemma 2.14. (See [26, Theorem 5.1]). fP1 iff μf is constant.

    Lemma 2.15. (See [26, Lemma 4.6]). We have P1Pω for all ωN .

    Lemma 2.16. (See [26, Lemma 3.1]). We have the formula

    μνΔ=μν. (2.7)

    Theorem 2.17 (Chain Rule, Substitution Rule). For f:TR , we have

    FΔν=νΔfνf,ifFν(t)=ν(t)tf(τ)Δτ. (2.8)

    Moreover, if s,tT , then

    ν(t)ν(s)f(τ)Δτ=tsνΔ(τ)f(ν(τ))Δτ. (2.9)

    Proof. Equation (2.8) is the content of [26, Lemma 3.8]. Using (2.8), we get

    ν(t)ν(s)f(τ)Δτ=Fν(t)Fν(s)+tsf(τ)Δτ=ts(FΔν(τ)+f(τ))Δτ=tsνΔ(τ)fν(τ)Δτ,

    i.e., (2.9) holds.

    Theorem 2.18. (See [26, Theorem 4.9]). If pPωR and t,sT , then

    ep(ν(t),t)=ep(ν(s),s)andep(ν(t),ν(s))=ep(t,s). (2.10)

    To conclude this section, we include a result on how to change the order of integration in a double integral. The final formula in the following theorem will be needed in our proof of the second Cushing–Henson conjecture in Section 4, while the other formulas are included for future reference.

    Theorem 2.19. Let f,g:TR and a,b,cT . For Φ(t,s):=f(t)g(s) , we have

    batcΦ(t,s)ΔsΔt=baacΦ(t,s)ΔtΔs+babσ(s)Φ(t,s)ΔtΔs, (2.11)
    baν(t)cΦ(t,s)ΔsΔt=baν(a)cΦ(t,s)ΔtΔs+ν(b)ν(a)bν1(σ(s))Φ(t,s)ΔtΔs, (2.12)
    baν(t)tΦ(t,s)ΔsΔt=baν(a)aΦ(t,s)ΔtΔsbabσ(s)Φ(t,s)ΔtΔs+ν(b)ν(a)bν1(σ(s))Φ(t,s)ΔtΔs, (2.13)
    ν(a)aν(t)tΦ(t,s)ΔsΔt=ν(a)aσ(s)aΦ(t,s)ΔtΔs+ν(ν(a))ν(a)ν(a)ν1(σ(s))Φ(t,s)ΔtΔs. (2.14)

    Proof. In what follows, we use the notation

    F(t):=btf(s)Δs,G(t):=tcg(s)Δs,Gν(t):=ν(t)tg(s)Δs,ψ(t):=g(t)Fν1(σ(t)),

    which imply

    F(b)=0,FΔ=f,GΔ=g,GΔν(2.8)=νΔgνg.

    First,

    batcΦ(t,s)ΔsΔt(2.2)=baFΔ(t)G(t)Δt(2.2)=F(b)G(b)F(a)G(a)baF(σ(s))GΔ(s)Δs,

    so (2.11) holds. Next,

    baν(t)tΦ(t,s)ΔsΔt(2.2)=baFΔ(t)Gν(t)Δt(2.2)=F(b)Gν(b)F(a)Gν(a)baF(σ(s))GΔν(s)Δs(2.2)=F(a)Gν(a)baνΔ(s)ψν(s)ΔsbaF(σ(s))g(s)Δs(2.9)=F(a)Gν(a)ν(b)ν(a)ψ(s)ΔsbaF(σ(s))g(s)Δs

    shows (2.13). Finally, (2.12) follows by adding (2.11) and (2.13), while (2.14) is the same as (2.13) with b=ν(a) .

    Recall that throughout, ωN and ν=σω . Recall also that η>1 and

    α:=η1μη=(η1μ),i.e.,η=1+μ((α))=11μα. (3.1)

    Throughout the remainder of this paper, we assume

    η:T(1,),α,K:T(0,),αR+.

    We formulate some assumptions.

    ( A 1 ) (σΔη)ν=σΔη ,

    ( A 2 ) Kη1Pω .

    Theorem 3.1 (First Cushing–Henson Conjecture). Let t0T . Assume ( A 1 ) and ( A 2 ) . Define

    λ:=νΔ(t0)e(α)(ν(t0),t0)1. (3.2)

    If λ0 , then (1.4) has a unique ω -periodic solution ˉx , given by

    ˉx(t)=λν(t)te(α)(σ(s),t)α(s)K(s)Δs. (3.3)

    If additionally, T is unbounded above, t0α(s)Δs= , and ˉx and K are bounded above, then ˉx is globally asymptotically stablefor solutions with positive initial conditions.

    Given the structure of (1.3), we immediately obtain that solutions remain positive for positive initial conditions, i.e., for x0>0 , the solution x satisfies x(t)>0 for all tT , tt0 .

    Before proving Theorem 3.1, we give a series of auxiliary results.

    Lemma 3.2. Consider

    ( A 3 ) α+1μσΔPω ,

    ( A 4 ) ((α)K)ν=(α)K .

    ( A 5 ) φ:=(α)μΔμPω .

    Then ( A 1 ) holds iff ( A 3 ) holds iff ( A 5 ) holds, and ( A 2 ) holds iff ( A 4 ) holds.

    Proof. The three calculations

    μ{νΔ(α+1μσΔ)να+1μσΔ}(2.7)=μ{νΔ(1μαμσΔ)ν1μαμσΔ}(2.7)=(1μασΔ)ν1μασΔ(3.1)=1(σΔη)ν1σΔη,
    μ{νΔ(Kη1)νKη1}(3.1)=μ{νΔ(Kμ((α)))νKμ((α))}(2.7)=(K(α))νK(α),

    and (using 1+μΔ=σΔ and Lemmas 2.13, 2.14, and 2.15)

    α+1μσΔφ=1μP1Pω (3.4)

    complete the proof.

    Lemma 3.3. Let t,sT . For φR defined in ( A 5 ) ,

    eφ(t,s)=eα(t,s)μ(s)μ(t) (3.5)

    and

    eφ(ν(t),t)=νΔ(t)e(α)(ν(t),t). (3.6)

    If ( A 5 ) holds, then

    νΔ(t)e(α)(ν(t),t)=νΔ(s)e(α)(ν(s),s) (3.7)

    and

    νΔ(t)e(α)(ν(t),ν(s))=νΔ(s)e(α)(t,s). (3.8)

    Proof. By Theorem 2.6 (part 3) and (2.3), we get (3.5). Using (3.5) with (2.7) yields (3.6). If φPω , then φPω by Lemma 2.13. Employing (3.5) and (3.6) together with (2.10) (applied to p=φ ) implies (3.7). Theorem 2.6 and (3.7) result in (3.8).

    Lemma 3.4. Assume ( A 3 ) and ( A 4 ) . Let t0T and define

    Hν(t):=ν(t)th(s)Δswithh(t):=e(α)(σ(t),t0)α(t)K(t). (3.9)

    Then, for λ defined in (3.2), we have

    HΔν(t)=λh(t) (3.10)

    and

    Hνν(t)=(λ+1)Hν(t). (3.11)

    Proof. First, by Theorem 2.6 (part 5), we have

    h(t)=e(α)(t,t0)α(t)(1μ(t)α(t))K(t)=((α)K)(t)e(α)(t,t0),

    and hence

    hν(t)(A4)=((α)K)ν(t)e(α)(ν(t),t0)(A4)=((α)K)(t)e(α)(ν(t),t0)=e(α)(ν(t),t)h(t).

    Thus,

    HΔν(t)(2.8)=νΔ(t)hν(t)h(t)(3.7)=(λ+11)h(t)=λh(t),

    which shows (3.10). Next,

    Hνν(t)(3.10)=Hν(ν(t))=Hν(t)+ν(t)tHΔν(s)Δs(3.10)=Hν(t)+λν(t)th(s)Δs(3.10)=Hν(t)+λHν(t)=(λ+1)Hν(t)

    proves (3.11).

    With Lemmas 3.3 and 3.4, we now have sufficient machinery to prove Theorem 3.1.

    Proof of Theorem 3.1. In a first step, we show that ˉx given by (3.3) is an ω -periodic solution of (1.4). Note that

    ˉx(t)(3.9)=λeα(t,t0)Hν(t),

    and thus,

    νΔ(t)ˉxν(t)(3.11)=λνΔ(t)eα(ν(t),t0)Hνν(t)(3.11)=λνΔ(t)eα(ν(t),t0)(λ+1)Hν(t)(3.11)=νΔ(t)eα(ν(t),t)(λ+1)ˉx(t)(3.7)=ˉx(t)

    (use also (3.2) in the last equality), so ˉx is ω -periodic. With ˉu=1/ˉx , we get

    ˉu(t)=1λeα(t,t0)Hν(t),

    and thus,

    ˉuΔ(t)(2.1)=1λ{eα(σ(t),t0)HΔν(t)α(t)eα(t,t0)Hν(t)}(3.10)=eα(σ(t),t0)h(t)α(t)ˉu(t)(3.9)=α(t)K(t)α(t)ˉu(t),

    so ˉu solves (1.5), and thus ˉx=1/ˉu solves (1.4). Altogether, ˉx is an ω -periodic solution of (1.4).

    Conversely, we assume that ˜x is any ω -periodic solution of (1.4). Then ˜u=1/˜x satisfies (1.5), i.e., ˜uΔ(t)=α(t)˜u(t)+α(t)K(t) . Hence,

    νΔ(t)˜u(t)=νΔ(t)˜x(t)=1˜xν(t)=˜uν(t)=˜u(ν(t))(2.5)=eα(ν(t),t)˜u(t)+ν(t)teα(ν(t),σ(s))α(s)K(s)Δs=eα(ν(t),t){˜u(t)+eα(t,t0)ν(t)te(α)(σ(s),t0)α(s)K(s)Δs}=eα(ν(t),t){˜u(t)+eα(t,t0)Hν(t)}

    (note that (2.5) was applied with t0 replaced by t and t replaced by ν(t) ), so

    (1+λ)˜u(t)(3.7)=νΔ(t)e(α)(ν(t),t)˜u(t)(3.7)=˜u(t)+eα(t,t0)Hν(t),

    which, upon solving for ˜u(t) , results in

    ˜u(t)=eα(t,t0)Hν(t)λ,

    i.e.,

    ˜x(t)=1˜u(t)=λeα(t,t0)Hν(t)=ˉx(t).

    To prove the global asymptotic stability of ˉx , let x be the unique solution of (1.4) with initial condition x0>0 , and let ˉx0:=ˉx(t0) . Since 1/x solves (1.5), using (2.5), we get

    x(t)=x0eα(t,t0)(1+x0tt0h(s)Δs).

    Thus, we obtain

    x(t)ˉx(t)=x0eα(t,t0)(1+x0tt0h(s)Δs)ˉx0eα(t,t0)(1+ˉx0tt0h(s)Δs)=x0ˉx0eα(t,t0)(1+x0tt0h(s)Δs)(1+ˉx0tt0h(s)Δs)=(x0ˉx0)ˉx(t)ˉx0(1+x0tt0h(s)Δs),

    which tends to zero as t because α>0 and αR+ so that

    1+x0tt0h(s)Δs(2.4)1+x0Ktt0eα(t0,σ(s))α(s)Δs(2.4)=1+x0K(eα(t0,t)1)(2.4)=1x0K+x0Keα(t0,t)(2.4)1x0K+x0K(1t0tα(s)Δs)(2.4)=1+x0Ktt0α(s)Δsast,

    completing the proof.

    Example 3.5. If T=Z , then

    σ(t)=t+1,σΔ(t)=1,ν(t)=t+ω,νΔ(t)=1,μ(t)=1,μΔ(t)=0

    and, as noted in Example 2.11, periodicity defined in (2.6) is consistent with the classical periodicity definition, i.e., f is ω -periodic if f(t+ω)=f(t) for all tZ . In this case, ( A 1 ) states that

    η(t+ω)=η(t),

    and ( A 2 ) says that

    K(t+ω)η(t+ω)1=K(t)η(t)1.

    Together, ( A 1 ) and ( A 2 ) are equivalent to

    η(t+ω)=η(t)andK(t+ω)=K(t),

    i.e., both η and K are ω -periodic. Next, ( A 3 ) says that

    α(t+ω)+1=α(t)+1,

    and ( A 4 ) states that

    α(t+ω)1α(t+ω)1K(t+ω)=α(t)1α(t)1K(t).

    Together, ( A 3 ) and ( A 4 ) are equivalent to

    α(t+ω)=α(t)andK(t+ω)=K(t),

    i.e., both α and K are ω -periodic. We also note that φ=(α)0=α , and so φ is ω -periodic if and only if α is ω -periodic. If α>1 is constant and K is ω -periodic, then ( A 3 ) and ( A 4 ) are satisfied, and ˉx from (3.3) is consistent with the unique ω -periodic solution derived in [22]. In that case, ˉx and K are bounded, as any periodic function on Z is bounded, and

    tt0αΔs=α(tt0)ast.

    Hence, the ω -periodic solution is globally asymptotically stable for solutions with positive initial conditions. The classical first Cushing–Henson conjecture is therefore a special case of Theorem 3.1. Further, Theorem 3.1 for T=Z also contains an extension of the classical first Cushing–Henson conjecture as presented in [21], where the authors considered both K and α to be ω -periodic. Again, in this case, all assumptions of Theorem 3.1 are satisfied, and the unique ω -periodic solution is globally asymptotically stable.

    Example 3.6. If T=qN0 , then

    σ(t)=qt,σΔ(t)=q,ν(t)=qωt,νΔ(t)=qω,μ(t)=(q1)t,μΔ(t)=q1

    and, as noted in Example 2.12, periodicity defined in (2.6) is consistent with the periodicity definition from [23], i.e., f is ω -periodic if qωf(qωt)=f(t) for all tqN0 . In this case, ( A 1 ) states that

    qη(qωt)=qη(t),

    and ( A 2 ) says that

    qωK(qωt)η(qωt)1=K(t)η(t)1.

    Together, ( A 1 ) and ( A 2 ) are equivalent to

    η(qωt)=η(t)andqωK(qωt)=K(t),

    i.e., both η/μ and K are ω -periodic. Next, ( A 3 ) says that

    qωα(qωt)+1(q1)qωtq=α(t)+1(q1)tq,

    and ( A 4 ) states that

    α(qωt)1(q1)qωtα(qωt)1K(qωt)=α(t)1(q1)tα(t)1K(t).

    Together, ( A 3 ) and ( A 4 ) are equivalent to

    qωα(qωt)=α(t)andqωK(qω)=K(t),

    i.e., both α and K are ω -periodic. We also note that

    φ(t)=α(t)1t1+(q1)tt=α(t)+1tq,

    and so φ is ω -periodic if and only if α is ω -periodic. Since these assumptions coincide with the assumptions in [24], [24, Conjecture 1] is the same as Theorem 3.1 if T=qN0 . We would like to remind the reader that Theorem 3.1 is therefore also a generalization of [23, Conjecture 1] that assumes α to be 1 -periodic.

    Examples 3.5 and 3.6 show that assumptions ( A 4 ) and ( A 5 ) are equivalent to α and K being ω -periodic, in the sense of (2.6), if T=Z or T=qN0 . One might wonder for which other time scales this observation is true.

    Theorem 3.7. Assume T is such that

    μΔμPω,i.e.,μΔν=μΔ. (3.12)

    Then ( A 4 ) and ( A 5 ) hold if and only ifboth α and K are ω -periodic.

    Proof. Assume (3.12). First, assuming ( A 4 ) and ( A 5 ) hold, we get from ( A 5 ) that

    α=φμΔμPω

    due to (3.12) and Lemma 2.13. Hence, using again Lemma 2.13, we obtain αPω . Then, by ( A 4 ) ,

    (α)K=((α)K)ν=νΔνΔ((α)K)ν=νΔ((α))ννΔKν=(α)νΔKν,

    so K is ω -periodic. Conversely, assuming both α and K are ω -periodic, we get by Lemma 2.13 that αPω , and hence

    φ=(α)μΔμPω

    due to (3.12) and Lemma 2.13. Hence, ( A 5 ) holds. Moreover,

    ((α)K)ν=νΔνΔ((α)K)ν=νΔ((α))ννΔKν=(α)K,

    showing ( A 4 ) .

    However, if (3.12) does not hold, then, assuming α and K are ω -periodic instead of ( A 4 ) and ( A 5 ) , there does not even exist an ω -periodic solution of (1.4) in general. This can be verified easily with ω=1 according to the following example.

    Example 3.8. Let ω=1 and assume α and K are ω -periodic. Let ˜x be an ω -periodic solution of (1.4). Let ˜u=1/˜x . By Lemma 2.14, ˜c:=α/K is constant. Moreover, since ˜x=1/˜u satisfies (2.6), we get

    σΔ˜u=˜uσ. (3.13)

    Thus,

    ˜uΔ=˜uσ˜uμ(3.13)=σΔ˜u˜uμ=μΔμ˜u,

    and hence, due to

    0(1.5)=˜uΔ+α˜u˜c=(μΔμ+α)˜u˜c,

    we obtain

    ˜u=μ˜cμΔ+μα. (3.14)

    Hence,

    ˜uσ(3.14)=μσ˜cμΔσ+μσασ(2.7)=μσΔ˜cμΔσ+μσΔασ(2.6)=μσΔ˜cμΔσ+μα

    and

    σΔ˜u(3.14)=μσΔ˜cμΔ+μα.

    Thus, with (3.13), we obtain (3.12).

    Example 3.9. Let ω=4 . For q>0 , consider

    T={tm:mN0},wheretm=m1i=0q(1)iformN0,

    where the "empty sum" is by convention zero, i.e., t0=0 . Then

    σ(tm)=tm+1,μ(tm)=tm+1tm=q(1)m,σΔ(tm)=σ(tm+1)σ(tm)μ(tm)=μ(tm+1)μ(tm)=q2(1)m+1,μΔ(tm)=σΔ(tm)1=q2(1)m+11.

    Hence (3.12) holds. Let K0,K1,K2,K3>0 , ˉKi=Kimod4 , a1 and define

    η(tm)=aq2(1)m,K(tm)=q(1)m+1ˉKm.

    Clearly, KPω by design, and since ην=η , we have αPω . By Theorem 3.7, ( A 3 ) and ( A 5 ) hold, so that by Theorem 3.1, the unique 4 -periodic solution is given by

    ˉx(tm)=λ(η(tm)1)ˉKm+ˉKm+2ˉKmˉKm+2+η(tm)(η(σ(tm))1)ˉKm+1+ˉKm+3ˉKm+1ˉKm+3,

    where λ=a410 .

    We now bring our attention to the second Cushing–Henson conjecture, which reads for the Beverton–Holt difference equation as follows. If η>1 is constant, K:ZR+ is ω -periodic, i.e., K(t+ω)=K(t) for all tZ , then the average of the unique periodic solution ˉx of (1.1) is less than or equal (equal iff K is constant) the average of the periodic carrying capacity, i.e.,

    1ωω1t=0ˉx(t)1ωω1t=0K(t).

    Biologically, this inequality is interpreted as deleterious effect of a periodic environment to the population. In order to extend this result to isolated time scales, we aim to find an upper bound for the average of the unique periodic solution. Similar to the discrete case, where the second Cushing–Henson conjecture assumed a constant proliferation rate, we adjust ( A 1 ) accordingly for ω=1 . More specifically, we consider the assumptions

    ( A 6 ) (σΔη)σ=σΔη , ( A 7 ) α+1μσΔP1 , ( A 8 ) φ=(α)μΔμP1 .

    Remark 4.1. According to Lemma 3.2, we have

    (A6) holds iff (A7) holds iff (A8) holds, 

    and, by Lemma 2.15,

     any of (A6),(A7),(A8) implies any of (A1),(A3),(A5)

    Assume now any of the conditions ( A 6 ) , ( A 7 ) , and ( A 8 ) . By Remark 2.14,

    C:=μα+1μσΔ=1μασΔisconstant. (4.1)

    Moreover, due to (3.4), we have

    μφ=C1andthusμ(φ)=μφ1+μφ=1CC=:D. (4.2)

    Because of

    eφ(ν(t0),t0)=τ[t0,ν(t0))T(1+μ(τ)(φ)(τ))=(1+1CC)ω=1Cω,

    and thus, we get that

    λ(3.2)=νΔ(t0)e(α)(ν(t0),t0)1(3.6)=eφ(ν(t0),t0)1=1Cω1.

    Theorem 4.2 (Second Cushing–Henson Conjecture). Assume ( A 4 ) , ( A 8 ) , and C<1 , where C is defined in (4.1). Then the average of the unique ω -periodic solution ˉx of (1.4) is bounded above by

    1ων(t0)t0ˉx(t)Δt1ων(t0)t01C1σΔ(t)CK(t)Δt, (4.3)

    and equality holds iff K(α) is constant.

    The central tool in the proof of Theorem 4.2 is the following generalized Jensen inequality from [32, Theorem 2.2], which reads for the strictly convex function 1/z as follows:

    baw(s)Δsbaw(s)v(s)Δsbaw(s)v(s)Δsbaw(s)Δswithw>0. (4.4)

    We apply (4.4) with

    wt(s):=φ(s)eφ(t,σ(s))>0andvt(s):=μ(t)˜β(s)(φ)(s),

    where we also put

    ˜β:=(α)μKandβ:=1μ2˜β=Kμ((α)).

    Note that ( A 4 ) implies (use (2.7))

    β,˜βPω. (4.5)

    Before proving Theorem 4.2, we offer the following auxiliary result.

    Lemma 4.3. Assume ( A 4 ) and ( A 5 ) . Define λ by (3.2). We have

    ν(t)twt(s)Δs=λ (4.6)

    and

    wt(s)vt(s)=e(α)(σ(s),t)α(s)K(s). (4.7)

    Moreover, if ( A 8 ) holds, then

    wt(s)vt(s)=Dβ(s)φ(t)eφ(t,σ(s)), (4.8)

    where D is defined in (4.2).

    Proof. First, we use Theorem 2.6 (part 8) to integrate

    ν(t)twt(s)Δs=ν(t)tφ(s)eφ(t,σ(s))Δs=eφ(t,ν(t))1=λ,

    where we also used (3.6), (3.7), and (3.2). This proves (4.6). Next, using (3.5), we get

    wt(s)vt(s)=φ(s)eφ(t,σ(s))μ(t)˜β(s)(φ)(s)=eφ(t,s)μ(t)˜β(s)=eα(t,s)μ(s)˜β(s)=eα(t,s)((α))(s)K(s)=e(α)(σ(s),t)α(s)K(s),

    which shows (4.7). Finally, assuming ( A 8 ) , we have (4.1) and (4.2). Then

    wt(s)vt(s)=φ(s)eφ(t,σ(s))μ(t)˜β(s)(φ)(s)=φ(s)(φ)(s)μ(t)μ2(s)β(s)eφ(t,σ(s))=μ(s)φ(s)μ(s)(φ)(s)β(s)eφ(t,σ(s))μ(t)=μ(t)φ(t)μ(s)(φ)(s)β(s)eφ(t,σ(s))μ(t)=μ(s)(φ)(s)β(s)φ(t)eφ(t,σ(s))=Dβ(s)φ(t)eφ(t,σ(s))

    shows (4.8).

    We can now bring our attention to the proof of the second Cushing–Henson conjecture on isolated time scales.

    Proof of Theorem 4.2. We apply the generalized Jensen inequality (4.4) on time scales in the single forthcoming calculation to estimate

    ν(t0)t0ˉx(t)Δt(3.3)=ν(t0)t0λν(t)te(α)(σ(s),t)α(s)K(s)ΔsΔt(4.6)=ν(t0)t0ν(t)twt(s)Δsν(t)te(α)(σ(s),t)α(s)K(s)ΔsΔt(4.7)=ν(t0)t0ν(t)twt(s)Δsν(t)twt(s)vt(s)ΔsΔt(4.4)ν(t0)t0ν(t)twt(s)vt(s)Δsν(t)twt(s)ΔsΔt(4.6)=1λν(t0)t0ν(t)twt(s)vt(s)ΔsΔt(2.14)=1λ{ν(t0)t0σ(s)t0wt(s)vt(s)ΔtΔs+ν(ν(t0))ν(t0)ν(t0)ν1(σ(s))wt(s)vt(s)ΔtΔs}(4.8)=Dλ{ν(t0)t0β(s)σ(s)t0φ(t)eφ(t,σ(s))ΔtΔs+ν(ν(t0))ν(t0)β(s)ν(t0)ν1(σ(s))φ(t)eφ(t,σ(s))ΔtΔs}(4.8)=Dλ{ν(t0)t0β(s)(eφ(t0,σ(s))1)Δs+ν(ν(t0))ν(t0)β(s)(eφ(ν1(σ(s)),σ(s))eφ(ν(t0),σ(s)))Δs}(2.9)=Dλ{ν(t0)t0β(s)(eφ(t0,σ(s))1)Δs+ν(t0)t0νΔ(s)βν(s)(eφ(σ(s),σ(ν(s)))eφ(ν(t0),σ(ν(s))))Δs}(2.10)=Dλ{ν(t0)t0β(s)(eφ(t0,σ(s))1)Δs+ν(t0)t0νΔ(s)βν(s)(eφ(t0,ν(t0))eφ(t0,σ(s)))Δs}(4.5)=Dλ{ν(t0)t0β(s)(eφ(t0,σ(s))1)Δs+ν(t0)t0β(s)(eφ(t0,ν(t0))eφ(t0,σ(s)))Δs}(4.8)=Dλν(t0)t0β(s)(eφ(t0,ν(t0))1)Δs(3.6)=Dν(t0)t0β(s)Δs=ν(t0)t01C1σΔ(s)CK(s)Δs,

    where the last equality holds because

    D\beta(s) \stackrel{(4.2)}{ = } \frac{(1-C)(1-\mu(s) \alpha(s))K(s)}{C\mu(s) \alpha(s)} \stackrel{(4.1)}{ = } \frac{(1-C)\sigma ^{\Delta} C}{C(1-\sigma ^{\Delta} C)}.

    Note also, that due to the strict convexity of 1/z (see [32, Theorem 2.2]), equality holds in (4.4) if and only if v_t(s) is independent of s , which means \beta\in \mathcal{P}_1 .

    Remark 4.4. If \rm ( A _{{4}} \rm ) and \beta\in \mathcal{P}_1 hold, then the unique periodic solution of (1.4) is D\beta , where D is given by (4.2). In detail, the unique periodic solution is

    \begin{equation} \frac{cD}{\mu(t)}, \quad{\rm{where}}\quad c: = \left({\frac{K}{ {\ominus(-\alpha)}}}\right)(t) \quad{\rm{{is\; constant.}}}\quad \end{equation} (4.9)

    We can check (4.9) in two simple ways, namely by calculating it from (3.3), i.e.,

    \begin{align*} \bar{x}(t) \stackrel{(3.3)}{ = }& \frac{\lambda} { \int_t^{\nu(t)}e_ {\ominus(-\alpha)}(\sigma(s), t) \frac{ \alpha(s)}{K(s)}\Delta s}\\ \stackrel{\phantom{(3.3)}}{ = }& \frac{\lambda} { \int_t^{\nu(t)}e_{- \alpha}(t, s) \left({\frac{ {\ominus(-\alpha)}}{K}}\right)(s)\Delta s} = \frac{\lambda c} { \int_t^{\nu(t)}e_{- \alpha}(t, s)\Delta s}\\ \stackrel{(3.5)}{ = }& \frac{\lambda c} { \int_t^{\nu(t)}e_ \varphi(t, s)\frac{\mu(t)}{\mu(s)}\Delta s} \stackrel{\phantom{(3.5)}}{ = } \frac{\lambda c} { \int_t^{\nu(t)}e_ \varphi(t, s) \frac{\mu(t)(\ominus \varphi)(s)}{\mu(s)(\ominus \varphi)(s)}\Delta s}\\ \stackrel{(4.2)}{ = }& \frac{\lambda cD} {\mu(t) \int_t^{\nu(t)}e_{\ominus \varphi}(s, t)(\ominus \varphi)(s)\Delta s} \stackrel{(3.6)}{ = } \frac{cD}{\mu(t)}, \end{align*}

    or by directly checking that it is 1 -periodic (this is clear from Lemma 2.14) and verifying that it solves (1.4), i.e.,

    \begin{align*} & \alpha\left({\frac{cD}{\mu}}\right) ^{\sigma}\left({1-\frac{cD}{\mu K}}\right) -\left({\frac{cD}{\mu}}\right) ^{\Delta}\\ & = \frac{ \alpha cD}{\mu ^{\sigma}}\left({1-\frac{D(1-\mu \alpha)}{\mu \alpha}}\right) +\frac{cD\mu ^{\Delta}}{\mu\mu ^{\sigma}}\\ & = \frac{cD}{\mu\mu ^{\sigma}}\left({\mu \alpha-D(1-\mu \alpha)+\sigma ^{\Delta}-1}\right)\\ & = \frac{cD}{\mu\mu ^{\sigma}}\left({(\mu \alpha-1)(1+D)+\sigma ^{\Delta}}\right)\\ & = \frac{cD}{\mu\mu ^{\sigma}}\left({-C(1+D)+1}\right)\sigma ^{\Delta} \stackrel{(4.2)}{ = }0. \end{align*}

    It can also be verified easily that for (4.9), the inequality (4.3) becomes an equality with cD on both sides:

    \frac{1}{\omega}\int_{t_0}^{\nu(t_0)}\frac{cD}{\mu(t)}\Delta t = cD

    and

    \begin{align*} &\frac{1}{\omega}\int_{t_0}^{\nu(t_0)} \frac{1-C}{\frac{1}{\sigma ^{\Delta}(t)}-C}K(t)\Delta t\\ & = \frac{1}{\omega}\int_{t_0}^{\nu(t_0)} \frac{(1-C)c \alpha(t)} {\left({\frac{1}{\sigma ^{\Delta}(t)}-C}\right)(1-\mu(t) \alpha(t))}\Delta t\\ & = \frac{1}{\omega}\int_{t_0}^{\nu(t_0)} \frac{(1-C)c \alpha(t)} {\left({\frac{1}{\sigma ^{\Delta}(t)}-C}\right)C\sigma ^{\Delta}}\Delta t = \frac{1}{\omega}\int_{t_0}^{\nu(t_0)} \frac{(1-C)c \alpha(t)}{C-C^2\sigma ^{\Delta}(t)}\Delta t\\ & = \frac{1}{\omega}\int_{t_0}^{\nu(t_0)} \frac{(1-C)c \alpha(t)}{C\mu(t) \alpha(t)}\Delta t = \frac{cD}{\omega}\int_{t_0}^{\nu(t_0)}\frac{\Delta t}{\mu(t)} = cD. \end{align*}

    Remark 4.5. For all isolated time scales such that \sigma ^{\Delta} = c is constant for some c\in \mathbb{R} , Theorem 4.2 implies that the average of the unique periodic solution is less than or equal to the average of the carrying capacity multiplied by the constant \frac{ \eta c-1}{ \eta-1} . If \mathbb{T} = \mathbb{Z} , then c = 1 , and the classical second Cushing–Henson conjecture is retrieved. If \mathbb{T} = q^{ \mathbb{N}_0} , then c = q , and Theorem 4.2 is consistent with the second Cushing–Henson conjecture formulated in [23], see also (1.2). The inequality (4.3) reveals that the upper bound is increasing in \sigma ^{\Delta} . Since

    \frac{1-C}{\frac{1}{\sigma ^{\Delta}}-C} = 1+\frac{1-\frac{1}{\sigma ^{\Delta}}}{\frac{1}{\sigma ^{\Delta}}-C} = 1+\frac{\sigma ^{\Delta}-1}{1-C\sigma ^{\Delta}} \stackrel{(4.1)}{ = } 1+\frac{\mu ^{\Delta}}{\mu \alpha},

    we can write (4.3) also as

    \int_{t_0}^{\nu(t_0)}\bar{x}(t)\Delta t \leq\int_{t_0}^{\nu(t_0)} \left({1+\frac{\mu ^{\Delta}(t)}{\mu(t) \alpha(s)}}\right)K(t)\Delta t.

    Hence if \mu ^{\Delta}(s) < 0 for all s\in[t_0, \nu(t_0)) , then the upper bound for the average periodic solution is smaller than the average of the carrying capacities, suggesting an even stronger negative effect of periodicity onto the population compared to the classical case. If \mu ^{\Delta}(s) > 0 for all s\in[t_0, \nu(t_0)) , then the second Cushing–Henson conjecture does not necessarily hold as the multiplicative factor exceeds one. The inequality (4.3) exposes the effects of the time structure onto the upper bound of the mean periodic population.

    In this work, we studied the Beverton–Holt model on arbitrary isolated time scales with time-dependent coefficients. Using the recently formulated periodicity concept for isolated time scales allows to address the Cushing–Henson conjectures for nonperiodic time scales. After an introduction in Section 1 and some preliminaries in Section 2, in Section 3, we provided conditions for the existence and uniqueness of a globally asymptotically stable periodic solution. This generalizes the first Cushing–Henson conjecture to an arbitrary isolated time scale. The provided theorem, when applied to the special case of the discrete time domain \mathbb{Z} , coincides with results in existing literature. The presented conditions for existence and uniqueness of the periodic solution and its global asymptotic stability are equivalent to the conditions in the first conjecture presented in [21]. It therefore generalizes the classical formulation of the first Cushing–Henson conjecture. We also showed that our result is consistent [24, Conjecture 1] in the special case of a quantum time scale. A special subcase in this time scale was discussed in [23]. As we outlined, Theorem 3.1 contains these works as special cases. In Section 4, we focused on the discussion of the second Cushing–Henson conjecture on arbitrary isolated time scales. In the classical case, when \mathbb{T} = \mathbb{Z} , the conjecture concerns the effects of a periodic environment under constant proliferation rate, mathematically formulated by an upper bound of the average periodic solution. The derived upper bound in Theorem 4.2 is, in contrast to the classical case, a weighted average dependent on changes of the time scale. This highlights that the second Cushing–Henson conjecture does not necessarily hold in general, and its statement depends on the change of the time scale. If the time scale changes with a constant rate, that is, \sigma ^{\Delta} is constant, then the average of the periodic solution is bounded by a factor times the average of the carrying capacity. Examples of this special case contain the discrete and the quantum time scale. For both of these time scales, Theorem 4.2 is consistent with existing formulations of the second Cushing–Henson conjecture in [18,21,23]. Our results complement work in [22], where the authors consider the Cushing–Henson conjectures for the Beverton–Holt model on periodic time scales. In contrast to isolated time scales, \omega -periodic time scales assume t+\omega\in \mathbb{T} for all t\in \mathbb{T} . Since periodic and isolated time scales intersect but neither is a subset of the other, modifications of the conjectures remain unknown on arbitrary time scales. We highlight that this work is an application of the new definition of periodicity on isolated time scales, defined in [26]. The introduced method of the application of periodicity on isolated time scales can now be extended to other models, such as delay Beverton–Holt models. In fact, in [33,34,35], the Cushing–Henson conjectures for different delay Beverton–Holt models in the discrete case are discussed. The tools used in this paper can be used to establish these results on an arbitrary isolated time scale.

    The authors would like to thank the three anonymous referees and the handling editor for many useful comments and suggestions, leading to a substantial improvement of the presentation of this article. The second author acknowledges partial support by the project UnB DPI/DPG - 03/2020 and CNPq grant 307582/2018-3.

    The authors declare there is no conflict of interest.



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