Almost periodic stochastic Beverton-Holt di erence equation with higher delays and with competition between overlapping generations

The paper studies the existence of an almost periodic solution of some system of stochastic Beverton-Holt equation with higher delays and with competition between overlapping generations under some reasonable assumptions.


Introduction
Hereditary systems (or systems with delays or after-e ects) are widely used to model processes in biology. Stochastic di erence equations of order greater than one have received less studies about them, and they are imperative in application where the state (i.e the size of a population) after k steps depends on the previous l + states (l ≥ ). A very important aspect of the qualitative study of the solutions of stochastic di erence equations is their almost periodicity.
In this paper, we investigate the solutions of some system of stochastic Beverton-Holt equation with higher delays and with competition between overlapping generations given by where γ t ∈ ( , ) is the survival rate for each t ∈ Z+, τ is a xed positive integer, and f : Z+ × R τ+ → R+ is the recruitment function given by with K t > being carrying capacity, β i 's are positive real numbers, and µ > is the intrinsic growth rate. We also assume that K t and γ t are random and that the γ t 's are independent and independent of X . This assumption together with Equation(1.1) imply that the sequence {γ t } t∈Z+ is independent of the sequence {X t } t∈Z+ . The paper is organized as follows. In Section 2, we recall a basic theory of almost periodic random sequences on Z+. In Section 3, we apply the techniques developed in Section 2 to nd some su cient conditions for the existence of almost periodic solutions to stochastic Beverton-Holt di erence equation with higher delays and with competition between overlapping generations.

Preliminaries
In this section we establish a basic theory for almost periodic random sequences. To facilitate our task, we rst introduce the notations needed in the sequel. Let (B, · ) be a Banach space and let (Ω, F, P) be a complete probability space. Throughout the rest of the paper, Z+ denotes the set of all nonnegative integers. De ne L (Ω; B) to be the space of all B-valued random variables V such that It is then routine to check that L (Ω; B) is a Banach space when it is equipped with its natural norm · de ned by, V := E V for each V ∈ L (Ω, B). Let X = {X t } t∈Z+ be a sequence of B-valued random variables satisfying E X t < ∞ for each t ∈ Z+. Thus, interchangeably we can, and do, speak of such a sequence as a function, which goes from Z+ into L (Ω; B).
This setting requires the following preliminary de nitions.
De nition 2.1. An L (Ω; B)-valued random sequence X = {X t } t∈Z+ is said to be Bohr almost periodic in mean if for each ε > there exists N (ε) > such that among any N consecutive integers there exists at least an integer p > for which E X t+p − X t < ε, ∀ t ∈ Z+.
An integer p > with the above-mentioned property is called an ε-almost period for X. The collection of all B-valued random sequences X = {X t } t∈Z+ which are Bohr almost periodic in mean is then denoted by AP(Z+; L (Ω; B)). The proofs of the following propositions are straightforward. An important and straightforward consequence of Theorem 2.4 is the next corollary, which pays a key role in the proof of our main result . ., and X N = {X N t } t∈Z+ are N random sequences, which belong to AP(Z+; L (Ω, B)), then for each ε > there exists N (ε) > such that among any N (ε) consecutive integers there exists an integer p > for which for t ∈ Z+ and for j = , , . . . , N.
De nition 2.6. A B-valued random sequence X = {X t } t∈Z+ is said to be almost periodic in probability if for each ε > , and η > there exists N (ε, η) > such that among any N consecutive integers there exists at least an integer p > for which Theorem 2.7. If X is almost periodic in mean, then it is almost periodic in probability and there also exists a constant M > such that E X t ≤ M for all t ∈ Z+. Conversely, if X is almost periodic in probability and the sequence X t , t ∈ Z+ is uniformly integrable, then X is almost periodic in mean.
De nition 2.8. A B-valued random sequence X = {X t } n∈Z+ satis es Bochner's almost sure uniform double sequence criterion if, for every pair of sequences (k ′ i ) and (l ′ i ), there exists a measurable subset Ω ⊂ Ω with P(Ω ) = and there exist subsequences k = (k i ) ⊂ (k ′ i ) and l = (l i ) ⊂ (l ′ i ) respectively, with the same indexes (independent of ω) such that, for every t ∈ Z+, (In this case, Ω depends on the pair of sequences (k ′ i ) and (l ′ i ).
Theorem 2.9. [1] The following properties of X are equivalent: (i) X satis es Bochner's almost sure uniform double sequence criterion.
(ii) X is almost periodic in probability.
The proof of the theorem can be seen in Bedouhene et al. [1] for instance.
Theorem 2.10. [1] If X satis es Bochner's almost sure uniform double sequence criterion and the sequence X t , t ∈ Z+ is uniformly integrable, then X is almost periodic in mean. Here again, the number p will be called an ε-translation of F and the set of all ε-translations of F is denoted by E(ε, F, K). In view of the above, the space AP(Z+; L (Ω; B)) of almost periodic random sequences equipped with the sup norm · ∞ is also a Banach space. We now state the following composition result.
Theorem 2.13. [2] Let F : then for any almost periodic random sequence X = {X t } t∈Z+ , then the L (Ω; B )-valued random sequence Y t = F(t, X t ) is almost periodic in mean.

Main Result
In the rest of the paper we let R+, | · | , R k , · k be the eld of nonnegative real numbers equipped with its absolute value, the k-dimensional space of real numbers equipped with the Euclidean norm, which is de ned by (x , . . . , x k ) k = x + . . . + x k , for all (x , . . . , x k ) ∈ R k , respectively. We begin this section with the following crucial lemma.

Lemma 3.1.
Consider the function f : R τ+ → R+ de ned by where both {K t } t∈Z+ and {γ t } t∈Z+ belong to AP(Z+; L (Ω; R+)) and µ > . Then (i) f is Lipschitz in the following sense: with τ being the time delay; (ii) If X is almost periodic in mean, then the sequence f (t, (X t−τ , X t−τ+ , . . . , X t− , X t )) t∈Z+ is almost periodic in mean.
Then f can be written as follows: for each t ∈ Z+.
We then have Thus, Note that Thus, To prove (ii), we use the fact that {γ t }, {Kn}, and {X t } are almost periodic in mean and make use of Corollary 2.5. We can then choose M > such that E|K t | < M for all t ∈ Z+ and for each ε > there exists a positive integer N (ε) consecutive integers. there exists an integer p > . a common ε-almost periodic for {γ t }, {K t }, and {X t } such that , E|X t+p − X t | ≤ ε µ for all t ∈ Z+, and E|β i X t+p−i+ − β i X t−i+ | ≤ β ε (τ+ )µ , for each i = , . . . , τ + and for all t ∈ Z+.
Using the same notations as before, we now evaluate: Thus, which in turn implies that We now evaluate carefully E A t+p B t+p − A t B t using the hypothesis of independence of random sequence By combining, we obtain We now state our main result. It is clear that Γ is well de ned. Now, let φ, ψ ∈ AP(Z+, L (Ω; R+)) having the same property as X de ned in Eq.(1.1). Since γ t , t ∈ Z+ are independent and independent of φ and ψ, one can easily see that E|γs| × × E f (r, φr−τ , φ r−τ+ , · · · , φ r− , φr) − f (r, (ψr−τ , ψ r−τ+ , · · · , ψ r− , ψr) , and hence letting β = sup n∈Z+ E[γn] and using Lemma 3.1(i) we obtain Hence, Obviously, Γ is a contraction whenever Lβ − β < . In that event, using the Banach xed point theorem it easily follows that Γ has a unique xed point, X, which obviously is the unique mean almost periodic solution of Eq.(1.1).