The Beverton–Hold model on isolated time scales

1.   Introduction

The Beverton–Holt recurrence

where $\eta$ 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 $T+t_0$ is used to describe the adult fish population (new generation), and the solution at time $t_0$ represents the juveniles (old generation). The derivation led to a proliferation rate $\eta = e^{rT} > 1$ , 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 $K_t$ and constant $\eta$ , the conjectures have been analytically verified in ^[18]. For higher-order periodic carrying capacities $K_t$ , 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 $t+\omega$ , 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 $q^{ \mathbb{N}_0}$ , 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

with the carrying capacity $K:q^{ \mathbb{N}_0}\to \mathbb{R}^+: = (0, \infty)$ ( $q > 1$ ) and proliferation rate $\eta > 1$ . 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 $q^\omega K(q^\omega t) = K(t)$ , 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 $\omega$ -periodic solution $\bar{x}$ of the $q$ -Beverton–Holt model is strictly lessthan the average of the $\omega$ -periodic carrying capacitytimes the constant $\frac{q-\lambda}{1-\lambda}$ , i.e.,

where $\lambda = 1-(q-1)A$ , $\eta = \frac{1}{1-(q-1)A}$ , and $A > 0$ .

The multiplicative constant in Theorem 1.1 can be expressed by

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 $\mathbb{T}$ , we therefore consider the Beverton–Holt model

where $\eta, K: \mathbb{T}\to \mathbb{R}^+$ , $\eta > 1$ , and where $\sigma(t)$ is the next time step following $t$ . We can express the recurrence (1.3) as a dynamic equation

where $\mu(t)$ 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 $x\neq 0$ , namely

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.

2.   Time scales fundamentals

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

Definition 2.1 (See ^[28, Definition 1.1]). For $t\in \mathbb{T}$ , the forward jump operator $\sigma: \mathbb{T}\to \mathbb{T}$ is defined by

We adopt the convention that $\inf\emptyset = \sup \mathbb{T}$ . If $\sigma(t) > t$ , then we say that $t$ is right-scattered. If $\sigma(t) = t$ , then we say that $t$ is right-dense. Similarly, left-scattered and left-dense points are defined. The graininess $\mu: \mathbb{T}\to \mathbb{R}_0^+$ is defined by $\mu(t): = \sigma(t)-t$ . We define $f ^{\sigma}: \mathbb{T}\to \mathbb{R}$ by $f ^{\sigma}: = f\circ\sigma$ . If $\mathbb{T}$ has a left-scattered maximum $M$ , then we define $\mathbb{T}^{\kappa} = \mathbb{T}\setminus\{M\}$ ; otherwise, $\mathbb{T}^{\kappa} = \mathbb{T}$ .

In this work, we focus on isolated time scales, i.e., all points are left-scattered and right-scattered. Hence, in what follows, throughout, $\mathbb{T}$ 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 $p: \mathbb{T}\to \mathbb{R}$ is called regressive, denoted by $\mathcal{R}$ , provided

Moreover, $p$ is called positively regressive, denoted by $\mathcal{R}^+$ , provided

Remark 2.3. Assume $f: \mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^{\kappa}$ . Then the delta-derivative of $f$ , denoted by $f ^{\Delta}$ (see ^[25, Definition 1.10]), is

For $f, g: \mathbb{T}\to \mathbb{R}$ , the product-rule for $t\in \mathbb{T}^{\kappa}$ (see ^[25]) reads as

and, if $g\neq 0$ , the quotient-rule (see ^[25]) reads as

The delta integral is defined for $a, b\in \mathbb{T}$ with $a < b$ by

and consequently,

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

The following circle-plus addition turns $(\mathcal{R}, \oplus)$ into an Abelian group.

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

Theorem 2.5 (See ^[25, Theorem 2.33). Let $p\in \mathcal{R}$ and $t_0\in \mathbb{T}$ . Then the initial value problem

possesses a unique solution.

The unique solution from Theorem 2.5 is called the dynamic exponential function and is denoted by $e_p(\cdot, t_0)$ . On an isolated time scale, the dynamic exponential function for $p\in \mathcal{R}$ is

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 $p, q\in \mathcal{R}$ and $t, s, r\in \mathbb{T}$ , then

1. $e_0(t, s) = 1$ and $e_p(t, t) = 1$ ,

2. $e_{p\oplus q}(t, s) = e_p(t, s)e_q(t, s)$ ,

3. $e_{p\ominus q}(t, s) = \frac{e_p(t, s)}{e_q(t, s)}$ ,

4. $e_{\ominus p}(t, s) = e_p(s, t) = \frac{1}{e_p(t, s)}$ ,

5. $e_p(\sigma(t), s) = (1+\mu(t)p(t))e_p(t, s)$ ,

6. $e_p(t, \sigma(s)) = \frac{e_p(t, s)}{1+\mu(s)p(s)}$ ,

7. $e_p ^{\Delta}(\cdot, s) = pe_p(\cdot, s)$ ,

8. $e_p ^{\Delta}(s, \cdot) = -pe_p ^{\sigma}(s, \cdot)$ ,

9. $p\in \mathcal{R}^+$ implies $e_p(t, s) > 0$ , and

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

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 $f: \mathbb{T}\to \mathbb{R}\setminus\{0\}$ be delta-differentiable and $s, t\in \mathbb{T}$ . Then

Proof. For fixed $t\in \mathbb{T}$ , define $w(s): = \frac{f(s)}{f(t)}$ . Since $f\neq 0$ , $\frac{f ^{\Delta}}{f}\in \mathcal{R}$ and

By Theorem 2.5, the claim follows.

Theorem 2.8 (See ^[31]). If $f$ is nonnegative with $-f\in \mathcal{R}^+$ , then

Theorem 2.9. (Variation of Constants, see ^[28, Theorem 2.77]). If $p\in \mathcal{R}$ , $f: \mathbb{T}\to \mathbb{R}$ , $t_0\in \mathbb{T}$ , and $y_0\in \mathbb{R}$ , then the unique solution of the IVP

is given by

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 $\omega\in \mathbb{N}$ . A function $f: \mathbb{T}\to \mathbb{R}$ is called $\omega$ -periodic, denoted by $f\in \mathcal{P}_\omega$ , provided

Example 2.11. If $\mathbb{T} = \mathbb{Z}$ , then $\nu(t) = t+\omega$ , $\nu ^{\Delta}(t) = 1$ , and (2.6) reduces to the known definition of periodicity, $f(t+\omega) = f(t)$ .

Example 2.12. If $\mathbb{T} = q^{ \mathbb{N}_0}$ , then $\nu(t) = q^\omega t$ , $\nu ^{\Delta}(t) = q^\omega$ , and (2.6) reduces to the known definition of periodicity, $q^\omega f(q^\omega t) = f(t)$ , 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 $f, g\in \mathcal{P}_\omega$ , then

Lemma 2.14. (See ^[26, Theorem 5.1]). $f\in \mathcal{P}_1$ iff $\mu f$ is constant.

Lemma 2.15. (See ^[26, Lemma 4.6]). We have $\mathcal{P}_1\subset \mathcal{P}_\omega$ for all $\omega\in \mathbb{N}$ .

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

Theorem 2.17 (Chain Rule, Substitution Rule). For $f: \mathbb{T}\to \mathbb{R}$ , we have

Moreover, if $s, t\in \mathbb{T}$ , then

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

i.e., (2.9) holds.

Theorem 2.18. (See ^[26, Theorem 4.9]). If $p\in \mathcal{P}_\omega\cap \mathcal{R}$ and $t, s\in \mathbb{T}$ , then

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 $f, g: \mathbb{T}\to \mathbb{R}$ and $a, b, c\in \mathbb{T}$ . For $\Phi(t, s): = f(t)g(s)$ , we have

Proof. In what follows, we use the notation

which imply

First,

so (2.11) holds. Next,

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

3.   First Cushing–Henson conjecture

Recall that throughout, $\omega\in \mathbb{N}$ and $\nu = \sigma^\omega$ . Recall also that $\eta > 1$ and

Throughout the remainder of this paper, we assume

We formulate some assumptions.

$\rm ($ A $_{{1}}$ $\rm )$ $\left({\sigma ^{\Delta} \eta}\right)^{\nu} = \sigma ^{\Delta} \eta$ ,

$\rm ($ A $_{{2}}$ $\rm )$ $\frac{K}{ \eta-1}\in \mathcal{P}_\omega$ .

Theorem 3.1 (First Cushing–Henson Conjecture). Let $t_0\in \mathbb{T}$ . Assume $\rm ($ A $_{{1}}$ $\rm )$ and $\rm ($ A $_{{2}}$ $\rm )$ . Define

If $\lambda\neq 0$ , then (1.4) has a unique $\omega$ -periodic solution $\bar{x}$ , given by

If additionally, $\mathbb{T}$ is unbounded above, $\int_{t_0}^\infty \alpha(s)\Delta s = \infty$ , and $\bar{x}$ and $K$ are bounded above, then $\bar{x}$ 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 $x_0 > 0$ , the solution $x$ satisfies $x(t) > 0$ for all $t\in \mathbb{T}$ , $t\geq t_0$ .

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

Lemma 3.2. Consider

$\rm ($ A $_{{3}}$ $\rm )$ $\frac{- \alpha+\frac{1}{\mu}}{\sigma ^{\Delta}}\in \mathcal{P}_\omega$ ,

$\rm ($ A $_{{4}}$ $\rm )$ $\left({\frac{ {\ominus(-\alpha)}}{K}}\right)^\nu = \frac{ {\ominus(-\alpha)}}{K}$ .

$\rm ($ A $_{{5}}$ $\rm )$ $\varphi: = (- \alpha)\ominus\frac{\mu ^{\Delta}}{\mu}\in \mathcal{P}_\omega$ .

Then $\rm ($ A $_{{1}}$ $\rm )$ holds iff $\rm ($ A $_{{3}}$ $\rm )$ holds iff $\rm ($ A $_{{5}}$ $\rm )$ holds, and $\rm ($ A $_{{2}}$ $\rm )$ holds iff $\rm ($ A $_{{4}}$ $\rm )$ holds.

Proof. The three calculations

and (using $1+\mu ^{\Delta} = \sigma ^{\Delta}$ and Lemmas 2.13, 2.14, and 2.15)

complete the proof.

Lemma 3.3. Let $t, s\in \mathbb{T}$ . For $\varphi\in \mathcal{R}$ defined in $\rm ($ A $_{{5}}$ $\rm )$ ,

and

If $\rm ($ A $_{{5}}$ $\rm )$ holds, then

and

Proof. By Theorem 2.6 (part 3) and (2.3), we get (3.5). Using (3.5) with (2.7) yields (3.6). If $\varphi\in \mathcal{P}_\omega$ , then $\ominus \varphi\in \mathcal{P}_\omega$ by Lemma 2.13. Employing (3.5) and (3.6) together with (2.10) (applied to $p = \ominus \varphi$ ) implies (3.7). Theorem 2.6 and (3.7) result in (3.8).

Lemma 3.4. Assume $\rm ($ A $_{{3}}$ $\rm )$ and $\rm ($ A $_{{4}}$ $\rm )$ . Let $t_0\in \mathbb{T}$ and define

Then, for $\lambda$ defined in (3.2), we have

and

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

and hence

Thus,

which shows (3.10). Next,

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 $\bar{x}$ given by (3.3) is an $\omega$ -periodic solution of (1.4). Note that

and thus,

(use also (3.2) in the last equality), so $\bar{x}$ is $\omega$ -periodic. With $\bar{u} = 1/\bar{x}$ , we get

and thus,

so $\bar{u}$ solves (1.5), and thus $\bar{x} = 1/\bar{u}$ solves (1.4). Altogether, $\bar{x}$ is an $\omega$ -periodic solution of (1.4).

Conversely, we assume that $\tilde{x}$ is any $\omega$ -periodic solution of (1.4). Then $\tilde{u} = 1/\tilde{x}$ satisfies (1.5), i.e., $\tilde{u} ^{\Delta}(t) = - \alpha(t)\tilde{u}(t)+\frac{ \alpha(t)}{K(t)}$ . Hence,

(note that (2.5) was applied with $t_0$ replaced by $t$ and $t$ replaced by $\nu(t)$ ), so

which, upon solving for $\tilde{u}(t)$ , results in

i.e.,

To prove the global asymptotic stability of $\bar{x}$ , let $x$ be the unique solution of (1.4) with initial condition $x_0 > 0$ , and let $\bar{x}_0: = \bar{x}(t_0)$ . Since $1/x$ solves (1.5), using (2.5), we get

Thus, we obtain

which tends to zero as $t\to\infty$ because $\alpha > 0$ and $- \alpha\in \mathcal{R}^+$ so that

completing the proof.

Example 3.5. If $\mathbb{T} = \mathbb{Z}$ , then

and, as noted in Example 2.11, periodicity defined in (2.6) is consistent with the classical periodicity definition, i.e., $f$ is $\omega$ -periodic if $f(t+\omega) = f(t)$ for all $t\in \mathbb{Z}$ . In this case, $\rm ($ A $_{{1}}$ $\rm )$ states that

and $\rm ($ A $_{{2}}$ $\rm )$ says that

Together, $\rm ($ A $_{{1}}$ $\rm )$ and $\rm ($ A $_{{2}}$ $\rm )$ are equivalent to

i.e., both $\eta$ and $K$ are $\omega$ -periodic. Next, $\rm ($ A $_{{3}}$ $\rm )$ says that

and $\rm ($ A $_{{4}}$ $\rm )$ states that

Together, $\rm ($ A $_{{3}}$ $\rm )$ and $\rm ($ A $_{{4}}$ $\rm )$ are equivalent to

i.e., both $\alpha$ and $K$ are $\omega$ -periodic. We also note that $\varphi = (- \alpha)\ominus 0 = - \alpha$ , and so $\varphi$ is $\omega$ -periodic if and only if $\alpha$ is $\omega$ -periodic. If $\alpha > 1$ is constant and $K$ is $\omega$ -periodic, then $\rm ($ A $_{{3}}$ $\rm )$ and $\rm ($ A $_{{4}}$ $\rm )$ are satisfied, and $\bar{x}$ from (3.3) is consistent with the unique $\omega$ -periodic solution derived in ^[22]. In that case, $\bar{x}$ and $K$ are bounded, as any periodic function on $\mathbb{Z}$ is bounded, and

Hence, the $\omega$ -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 $\mathbb{T} = \mathbb{Z}$ also contains an extension of the classical first Cushing–Henson conjecture as presented in ^[21], where the authors considered both $K$ and $\alpha$ to be $\omega$ -periodic. Again, in this case, all assumptions of Theorem 3.1 are satisfied, and the unique $\omega$ -periodic solution is globally asymptotically stable.

Example 3.6. If $\mathbb{T} = q^{ \mathbb{N}_0}$ , then

and, as noted in Example 2.12, periodicity defined in (2.6) is consistent with the periodicity definition from ^[23], i.e., $f$ is $\omega$ -periodic if $q^\omega f(q^\omega t) = f(t)$ for all $t\in q^{ \mathbb{N}_0}$ . In this case, $\rm ($ A $_{{1}}$ $\rm )$ states that

and $\rm ($ A $_{{2}}$ $\rm )$ says that

Together, $\rm ($ A $_{{1}}$ $\rm )$ and $\rm ($ A $_{{2}}$ $\rm )$ are equivalent to

i.e., both $\eta/\mu$ and $K$ are $\omega$ -periodic. Next, $\rm ($ A $_{{3}}$ $\rm )$ says that

and $\rm ($ A $_{{4}}$ $\rm )$ states that

Together, $\rm ($ A $_{{3}}$ $\rm )$ and $\rm ($ A $_{{4}}$ $\rm )$ are equivalent to

i.e., both $\alpha$ and $K$ are $\omega$ -periodic. We also note that

and so $\varphi$ is $\omega$ -periodic if and only if $\alpha$ is $\omega$ -periodic. Since these assumptions coincide with the assumptions in ^[24], ^[24, Conjecture 1] is the same as Theorem 3.1 if $\mathbb{T} = q^{ \mathbb{N}_0}$ . We would like to remind the reader that Theorem 3.1 is therefore also a generalization of ^[23, Conjecture 1] that assumes $\alpha$ to be $1$ -periodic.

Examples 3.5 and 3.6 show that assumptions $\rm ($ A $_{{4}}$ $\rm )$ and $\rm ($ A $_{{5}}$ $\rm )$ are equivalent to $\alpha$ and $K$ being $\omega$ -periodic, in the sense of (2.6), if $\mathbb{T} = \mathbb{Z}$ or $\mathbb{T} = q^{ \mathbb{N}_0}$ . One might wonder for which other time scales this observation is true.

Theorem 3.7. Assume $\mathbb{T}$ is such that

Then $\rm ($ A $_{{4}}$ $\rm )$ and $\rm ($ A $_{{5}}$ $\rm )$ hold if and only ifboth $\alpha$ and $K$ are $\omega$ -periodic.

Proof. Assume (3.12). First, assuming $\rm ($ A $_{{4}}$ $\rm )$ and $\rm ($ A $_{{5}}$ $\rm )$ hold, we get from $\rm ($ A $_{{5}}$ $\rm )$ that

due to (3.12) and Lemma 2.13. Hence, using again Lemma 2.13, we obtain $\alpha\in \mathcal{P}_\omega$ . Then, by $\rm ($ A $_{{4}}$ $\rm )$ ,

so $K$ is $\omega$ -periodic. Conversely, assuming both $\alpha$ and $K$ are $\omega$ -periodic, we get by Lemma 2.13 that $- \alpha\in \mathcal{P}_\omega$ , and hence

due to (3.12) and Lemma 2.13. Hence, $\rm ($ A $_{{5}}$ $\rm )$ holds. Moreover,

showing $\rm ($ A $_{{4}}$ $\rm )$ .

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

Example 3.8. Let $\omega = 1$ and assume $\alpha$ and $K$ are $\omega$ -periodic. Let $\tilde{x}$ be an $\omega$ -periodic solution of (1.4). Let $\tilde{u} = 1/\tilde{x}$ . By Lemma 2.14, $\tilde{c}: = \alpha/K$ is constant. Moreover, since $\tilde{x} = 1/\tilde{u}$ satisfies (2.6), we get

Thus,

and hence, due to

we obtain

Hence,

and

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

Example 3.9. Let $\omega = 4$ . For $q > 0$ , consider

where the "empty sum" is by convention zero, i.e., $t_0 = 0$ . Then

Hence (3.12) holds. Let $K_0, K_1, K_2, K_3 > 0$ , $\bar{K}_i = K_{i\rm{mod}4}$ , $a\neq 1$ and define

Clearly, $K\in \mathcal{P}_\omega$ by design, and since $\eta^\nu = \eta$ , we have $\alpha\in \mathcal{P}_\omega$ . By Theorem 3.7, $\rm ($ A $_{{3}}$ $\rm )$ and $\rm ($ A $_{{5}}$ $\rm )$ hold, so that by Theorem 3.1, the unique $4$ -periodic solution is given by

where $\lambda = a^4-1\neq 0$ .

4.   Second Cushing–Henson conjecture

We now bring our attention to the second Cushing–Henson conjecture, which reads for the Beverton–Holt difference equation as follows. If $\eta > 1$ is constant, $K: \mathbb{Z}\to \mathbb{R}^+$ is $\omega$ -periodic, i.e., $K(t+\omega) = K(t)$ for all $t\in \mathbb{Z}$ , then the average of the unique periodic solution $\bar{x}$ of (1.1) is less than or equal (equal iff $K$ is constant) the average of the periodic carrying capacity, i.e.,

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 $\rm ($ A $_{{1}}$ $\rm )$ accordingly for $\omega = 1$ . More specifically, we consider the assumptions

$\rm ($ A $_{{6}}$ $\rm )$ $\left({\sigma ^{\Delta} \eta}\right)^{\sigma} = \sigma ^{\Delta} \eta$ , $\rm ($ A $_{{7}}$ $\rm )$ $\frac{- \alpha+\frac{1}{\mu}}{\sigma ^{\Delta}}\in \mathcal{P}_1$ , $\rm ($ A $_{{8}}$ $\rm )$ $\varphi = (- \alpha)\ominus\frac{\mu ^{\Delta}}{\mu}\in \mathcal{P}_1$ .

Remark 4.1. According to Lemma 3.2, we have

and, by Lemma 2.15,

Assume now any of the conditions $\rm ($ A $_{{6}}$ $\rm )$ , $\rm ($ A $_{{7}}$ $\rm )$ , and $\rm ($ A $_{{8}}$ $\rm )$ . By Remark 2.14,

Moreover, due to (3.4), we have

Because of

and thus, we get that

Theorem 4.2 (Second Cushing–Henson Conjecture). Assume $\rm ($ A $_{{4}}$ $\rm )$ , $\rm ($ A $_{{8}}$ $\rm )$ , and $C<1$ , where $C$ is defined in (4.1). Then the average of the unique $\omega$ -periodic solution $\bar{x}$ of (1.4) is bounded above by

and equality holds iff $\frac{K}{ {\ominus(-\alpha)}}$ 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:

We apply (4.4) with

where we also put

Note that $\rm ($ A $_{{4}}$ $\rm )$ implies (use (2.7))

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

Lemma 4.3. Assume $\rm ($ A $_{{4}}$ $\rm )$ and $\rm ($ A $_{{5}}$ $\rm )$ . Define $\lambda$ by (3.2). We have

and

Moreover, if $\rm ($ A $_{{8}}$ $\rm )$ holds, then

where $D$ is defined in (4.2).

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

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

which shows (4.7). Finally, assuming $\rm ($ A $_{{8}}$ $\rm )$ , we have (4.1) and (4.2). Then

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

where the last equality holds because

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

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

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.,

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

and

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

we can write (4.3) also as

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.

5.   Conclusion

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.

Acknowledgments

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.

Conflict of interest

The authors declare there is no conflict of interest.