Existence results for some generalized Sigmoid Beverton-Holt models in time scales

The dynamics of Beverton-Holt models, with or without survival rates, have been the subject of substantial investigations, see, e.g., [11,14,21,22]. In particular, dichotomy techniques were used by Diagana et al. [14] to study and obtain sufficient conditions for the existence of a globally attracting almost periodic solution to the Beverton-Holt model with survival rates when both the survival rate ( ) ↦ t γ t and the carrying capacity


Introduction
The dynamics of Beverton-Holt models, with or without survival rates, have been the subject of substantial investigations, see, e.g., [11,14,21,22]. In particular, dichotomy techniques were used by Diagana et al. [14] to study and obtain sufficient conditions for the existence of a globally attracting almost periodic solution to the Beverton-Holt model with survival rates when both the survival rate ( ) ↦ t γ t and the carrying capacity Recall that the Sigmoid Beverton-Holt model has numerous applications, one of which is to record the expected population density for future generations depending on the current generation's population number at a certain point in time. It is also used to predict insurance premiums, forecast future changes in natural populations, and determine appropriate fishing rates to handle the problem of dwindling stock sizes in the fishing industry [11,12,17,20,26].
The Sigmoid Beverton-Holt model without survival rate equation (1.1) was introduced in the literature by Thompson [26] in the case ( ) = a t a for all ∈ + t . Bohner et al. [4] studied the Beverton-Holt model by using a newly introduced definition of periodicity on time scales. For other definitions of periodicity on time scale see [1,8]. In [6,7], the authors considered an inherent periodic growth rate of the Beverton-Holt equation in the first article and an inherent periodic growth rate of the Beverton-Holt q-difference equation in the second article. All these works [2,5,17] consider periodically forced model parameters of the classical Beverton-Holt model.
In this article, it goes back to studying generalized versions of these models in time scales in the case when the parameters are almost periodic.
Consider the generalized Sigmoid Beverton-Holt model without survival rates (1.3) and the generalized Sigmoid Beverton-Holt model with survival rates, where ( ) x t represents the population size of the generation ∈ + t , ( ) ↦ + γ : 0 ,1 is the survival rate, and the coefficients involved ( ( )) ≥ a t t 0 , ( ( )) ≥ b t t 0 , ( ( )) ≥ c t t 0 , and ( ( )) ≥ δ t t 0 are strictly positive bounded sequences. Obviously, the Sigmoid Beverton-Holt model in equation Experiments have shown that an almost periodicity is a more accurate way to describe many natural occurrences than periodicity [18]. In this article, we obtain the existence of a globally attractive almost periodic solution to both equations (1.3) and (1.4) in time scales by making extensive use of tools arising in time scales theory as well as the fixed-point principle. The results of this article generalize, to some extent, various existing results in the literature including some of those obtained by Diagana et al. [14].
This article is organized as follows. In Section 2, we present some preliminary and essential background in the theory of time scales. Moreover, the notion of almost periodicity in this setting is recalled, which can also be found in [10,23,24]. In Section 3, we reformulate equations (1.3) and (1.4) in the time scales realm, and next study the existence and uniqueness of a globally attractive almost periodic solution in the cases when we are in the presence of survival rates or not.

Preliminaries
In this section, we recall the necessary background upon the theory of time scales [9,10,19] and some properties of almost periodic functions needed in the sequel. , the backward jump operator ↦ ρ : , and the graininess function , sup , and .
In this definition, we consider ∅ = inf sup and ∅ = sup inf , where ∅ stands for the empty set.
be a function and ∈ t κ , where is the number (provided it exists) with the property that given any > ε 0, there is a neighborhood U of t (i.e., is said to be right-dense continuous (or rd-continuous) provided it is continuous at right-dense points in and its left-sided limits exist at left-dense points in .
In particular if ∈ t 0 , then the function F defined by We define the improper integral ( ) ∫ +∞ f t t Δ a by the limit (provided it exists) We say, in this case, that the improper integral converges. If this limit does not exist, we say that the improper integral diverges. Similarly if Definition 2.6.
[10] Let ↦ p : be a function. We say that: We denote, respectively, the set of all regressive and rd-continuous functions from into and the set of all positively regressive and rd-continuous functions from into by , , , then we have: Obviously if T is an almost periodic time scale, then = −∞ inf and = +∞ sup .
Lemma 2.1. [25] Let be a time scale almost periodic, ∈ t and ∈ τ Π. Then we have: Definition 2.9. [16,23] Let be an almost periodic time scale. A function is called an almost periodic function if the ε-translation set of f is a relatively dense set in for all > ε 0; that is, for any given > ε 0, there exists a constant ( ) > l ε 0 such that each interval of length ( ) l ε contains a ( ) τ ε belonging to { } E ε f , such that for all ∈ t , In this case, ( ) τ ε is called the ε-translation number of f and ( ) l ε is called the inclusion length of { } E ε f , . The set of all almost periodic functions from into is denoted by ( ) AP , .
is a relatively dense set in for all > ε 0 and for each compact subset S of . We say differently for any given > ε 0 and each compact subset S of , there exists a constant ( , .
In this case, The set of all almost periodic functions in ∈ t uniformly for ∈ x is denoted by , .
is almost periodic.
Let us state the Gronwall's inequality.
In the sequel, we consider to be an almost periodic time scale and is either a closed interval or + such that for any bounded function ↦ w : , we use the notations:

Generalized Sigmoid Beverton-Holt equation without survival rates
We begin this subsection with the following lemma which treats the boundedness of the solution to equation Proof. By using induction, one can easily see that for any ∈ + t , we have ( ) > x t 0. Now, given that > c 0 and that ( ) > x t 0, we deduce that for all ∈ + t , Moreover, we assume that ( ) ≥ x t m, and then we obtain for all ∈ + t , Thus, we have for all ∈ + t , , and finally, we conclude that Let us begin by reformulating equation (1.3) as follows: , we obtain the time scales version of equation (1.3). Consequently, it goes back to studying the dynamic of the following equation: , , where In the rest of this subsection, Lemma 3.1 motivates us to assume in an arbitrary time scale that any solution to equation We equip with the classical absolute value. Obviously, the pair ( ||) ⋅ , forms a complete metric space.
, then the function f is Lipschitz with the Lipschitz constant L, for all ∈ x y , and for all ∈ t , , let ( ) = k z e z ; thus, according to the mean value theorem (see [9], 1 . According the mean value theorem, there exists s τ , between x and y such that x y x y s ln ln ln ln .

Δ Δ
Since ∈ x y , and using the inequality ( ) Hence, we conclude that Finally, the function f is Lipschitz with the Lipschitz constant Consequently, for all ∈ t and all ∈ x X, we obtain

H t τ H t e t τ σ s s s e t σ s s s e t τ σ r τ r τ r e t σ s s s
On the other hand by substituting = + s θ τ, we obtain Consequently, we conclude that ( ) ∈ τ E ε H , Γ , and then, the function H Γ is almost periodic. □ Consider the integral operator ϕ defined for all function ↦ ν : and for all ∈ t by According to the Lemma 3.3 and the Theorem 2.4, we deduce that the function is almost periodic. Therefore, by Lemma 3.4, the function ( ) ϕ ν is almost periodic, and hence, the operator ϕ is well defined.
For all ( ) ∈ ν ν AP , , 1 2 , Thus, the operator ϕ, which maps ( ( ) ∥∥ ) ⋅ ∞ AP , , into itself, is contracting, and so by the Banach fixedpoint theorem [13], we deduce that there exists a unique In addition, for all ∈ t s , , ≥ t s, we obtain  Finally, we deduce that function * x is the unique almost periodic solution to equation , so we obtain , we obtain By using Lemma 2.2 of Gronwall inequality, we have , , Consequently, for all .
x is a globally attractive almost periodic solution to equation (3.3). □ Inspired by [15,Example 4.2], we give the following example to illustrate the results of this section: Moreover, it is clear that those functions are strictly positive almost periodic satisfying for all ∈ t ,

Generalized Sigmoid Beverton-Holt equation with survival rate
0, we suppose that there exists an > m 0 satisfying for all ∈ + t We begin this subsection with the following lemma, which treats the boundedness of the solution to equation (1.4) in the particular time scale = + . where Proof. By using induction, one can easily see that for any ∈ + t , we have ( ) > x t 0. Now, given that > c 0, we deduce that for all ∈ + t , . According to equation (3.5), we obtain for all x 0 . Hence, using equation (3.5), we obtain for all Therefore, we conclude for all , we obtain the time scale version of equation (1.4). Consequently, it goes back to studying the dynamic of the following equation: , , where . We equip with the classical absolute value. Obviously, the pair ( ||) ′ ⋅ , forms a complete metric space.
Lemma 3.6. If > c 0 and < γ 1, then the function g is Lipschitz with the Lipschitz constant L 1 , for all ∈ ′ x z , and all ∈ t , Proof. According to the Lemma 3.2, f is Lipschitz with the Lipschitz constant ′ L , then we have , , , , and ( ( )) ∈ γ AP , 0, 1 . If > c 0, > b 0 and < γ 1, then the function g is almost periodic in ∈ t uniformly for ∈ ′ z .
Proof. According to the Lemma 3.3, we have that the function f is almost periodic in ∈ t uniformly for ∈ ′ z , and since γ is almost periodic, it follows that for each ′ > ε ε , 0 such that , , . Consequently, for all ∈ t and all ∈ ′ x , we obtain g t z f t τ z γ t τ z f t z γ t z f t τ z f t z γ t τ γ t z ε ε M ε , , , Finally, we conclude that the function g is almost periodic uniformly for ∈ ′ z . □ Finally, we conclude that function * x is the unique almost periodic solution to equation (3.7). □   Finally, if ≤ < μ 0 1 and according to Theorems 3.3 and 3.4, then we conclude that equation (3.7) has an unique globally attractive almost periodic solution.

Concluding remark
In this article, we determined that the appropriate time scale is = + . The general case appears to be exceedingly difficult, and as a result, it will be left as an open problem that will be investigated at some point in the future.