Dynamics of almost periodic solutions for a discrete Fox harvesting model with feedback control
Advances in Difference Equations volume 2012, Article number: 157 (2012)
We consider the following discrete Fox harvesting model with feedback control of the form
Under the assumptions of almost periodicity of the coefficients, sufficient conditions are established for the existence and uniformly asymptotical stability of almost periodic solutions of this model. The persistence as well as the boundedness of solutions of the above system are discussed prior to presenting the main result. Examples are provided to illustrate the effectiveness of the proposed results.
where is the size of the population, is the per-capita harvesting rate and is the per-capita fecundity rate. Let and be defined in the form
then equation (1) becomes
where is a variable harvesting rate, is an intrinsic factor and is a varying environmental carrying capacity. The positive parameter r is referred to as an interaction parameter [1, 3, 4]. Indeed, if then intra-specific competition is high, whereas if , then the competition is low. For , equation (2) reduces to the classical Gompertzian model with harvesting [2, 5]. Equation (2) is called a Fox surplus production model that has been used to build up certain prediction models such as microbial growth model, demographic model and fisheries model. This equation is considered to be an efficient alternative to the well known r-logistic model. Specifically, the Fox model is more appropriate upon describing lower population density; we refer the reader to [1, 3, 4, 6–10] and, in particular, to the recent paper  for more information.
Ecosystems in the real world are continuously disturbed by unpredictable forces which can result in changing some biological parameters such as survival rates. In ecology, a question of practical interest is whether or not an ecosystem can withstand these unpredictable disturbances which persist for a finite period of time. In the language of control theory, we call these disturbance functions a control variable. In , Gopalsamy and Weng introduced a model with feedback controls in which the control variables satisfy a certain differential equation. The next years have witnessed the appearance of many papers regarding the study of ecosystems with feedback control; see for instance [13–17].
In the last years, many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous counterparts, especially when the populations have no overlapping generations. It is also known that the discrete models can provide more efficient computational methods for numerical simulations [18–20]. By applying the same method used in , one can derive the discrete analogue of (2) as follows:
One of the most important behaviors of solutions which has been the main object of investigations among authors is the periodic behavior of solutions [22–29]. To consider periodic environmental factors acting on a population model, it is natural to study the model subject to periodic coefficients. Indeed, the assumption of periodicity of the parameters in the model is a way of incorporating the time-dependent variability of the environment (e.g., seasonal effects of weather, food supplies, mating habits and harvesting). On the other hand, upon considering long-term dynamical behavior, it has been found that the periodic parameters often turn out to experience some perturbations that may lead to a change in character. Thus, the investigation of almost periodic behavior is considered to be in more accordance with reality; see the remarkable monographs [30–32] and the recent contributions [33–48].
Motivated by the above justifications, we consider the following discrete Fox harvesting model with feedback control in the form
where is the control variable and is the forward difference . Under the assumptions of almost periodicity of coefficients of system (4), we shall study the existence and uniformly asymptotical stability of almost periodic solutions for system (4). The persistence as well as the boundedness of solutions of system (4) are discussed prior to presenting the main result. To the best of author’s observation, no paper has been published in the literature regarding the dynamics of almost periodic solutions of system (4). Thus, the result of this paper is essentially different and presents a new approach.
The remaining part of this paper is organized as follows. In Section 2, some preliminary definitions along with essential lemmas are given. Section 9 discusses the persistence and boundedness of solutions of system (4). In Section 30, sufficient conditions are established to investigate the existence and uniformly asymptotical stability of almost periodic solutions of the said system. Section 5 provides some numerical examples to illustrate the feasibility of our theoretical results.
Let , , and be the sets of real, nonnegative real, integer and nonnegative integer numbers respectively. For any bounded sequence on , we define
Throughout the remainder of this paper, we assume the following condition:
(H.1) , , , , and are bounded nonnegative almost periodic sequences such that
Due to certain biological reasons, we restrict our attention to positive solutions of system (4). Thus, we consider system (4) together with the following initial conditions:
One can easily figure out that the solutions of system (4) with the initial conditions (6) are defined and remain positive for all .
Definition 1 
A sequence is called an almost periodic sequence if the ϵ-translation set is a relatively dense set in for all , that is, for any there is a constant such that in any interval of length there exists a number such that the inequality
is satisfied for all .
Definition 2 
Let where D is an open set in . Then is said to be almost periodic in n uniformly for , or uniformly almost periodic for short, if for any and any compact set , there exists a positive integer such that any interval of length contains an integer τ for which
for all and . The number τ is called the ϵ-translation for .
Lemma 1 
is an almost periodic sequence if and only if for any sequence , there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also an almost periodic sequence.
Consider the following almost periodic difference system:
where , and is almost periodic in n uniformly for and is continuous in x. The product system of (7) is in the form
Our approach is based on the following lemma.
Lemma 2 
Suppose that there exists a Lyapunov functional defined for , , and satisfying the following conditions:
, where with ;
where is a constant;
where is a constant and .
Moreover, if there exists a solution of system (8) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (8) which satisfies . In particular, if is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of system (8) of period ω.
3 Persistence and boundedness
In this section, we prove every solution of system (4) is persistent. In addition to this, we prove that there exists a bounded solution for (4).
Definition 3 System (4) is said to be persistent if there are positive constants , , and such that
for each positive solution of (4).
We assume the following condition:
Lemma 3 Let (H.1), (H.2) hold. Then, every solution of system (4) satisfies
Proof Let be a solution of (4). To prove that , we consider two cases:
Case I. There exists such that . By the first equation of (4), we have
which implies that
We claim that for . Indeed, if there is an integer such that and is the least integer between and such that , then and which implies that . This is a contradiction. This proves the claim.
Case II. Let for . Then, exists and equals . Taking the limit of the first equation in (4), we have
Hence . This proves the claim.
Now, we prove that . For any , there exists a large enough integer such that for . By the second equation of (4), we get
Since , we can find a positive number d such that . Thus, by using Stolz’s theorem, we obtain
By the arbitrariness of ε, we obtain . The proof of Lemma 3 is complete. □
Lemma 4 Let (H.1), (H.2) hold. Then every solution of system (4) satisfies
Proof Let be a solution of (4). By virtue of Lemma 3, one can figure out that for any which satisfies , there exists such that
To prove that , we consider two cases:
Case I. There exists such that . We observe that for , we have
For , we get
which implies that
It follows that
We claim that
For the sake of contradiction, assume that there exists such that . Then . Let be the smallest integer such that . Then . The above arguments imply that which is a contradiction. This proves the claim.
Case II. Let for all . Then, exists and it is equal to . Taking the limit of the first equation of (4), we have
Hence and . This proves the claim.
By applying the same arguments followed in the proof of Lemma 3, one can easily show that . The proof of Lemma 4 is complete. □
The results of Lemma 3 and Lemma 4 can be concluded in the following theorem:
Theorem 1 Let (H.1), (H.2) hold. Then system (4) is persistent.
Let Ω be the set of all solutions of system (4) satisfying and for all . By virtue of Theorem 1, it should be noted that Ω is an invariant set of system (4).
In view of Lemma 2, we need to show that there exists a bounded solution of system (4). The following result proves the existence of such a solution.
Theorem 2 Let (H.1), (H.2) hold. Then .
Proof By the almost periodicity of , , , , and , there exists an integer valued sequence with as such that , , , , , as . Let ε be an arbitrary small positive number. It follows from Lemma 3 and Lemma 4 that there exists a positive integer such that
Let and for , . For any positive integer q, it is easy to see that there exist sequences and such that the sequences and have subsequences, denoted by and again, converging on any finite interval of as , respectively. Thus, we have sequences and such that
Therefore, the system
We can easily see that is a solution of system (4) and , for . Since ε is arbitrary, it follows that , for . This completes the proof. □
4 The main result
Let , where . Then, it is easy to find out that . The following inequalities hold:
Theorem 3 Let (H.1), (H.2) hold. Suppose further that
(H.3) for , where
Then, there exists a unique uniformly asymptotically stable almost periodic solution of system (4) which satisfies and for all .
Proof Let . In view of system (4), we have
By the result of Theorem 2, it follows that system (13) has a bounded solution satisfying
Hence, and , where and .
For , we define the norm . Suppose that and are any two solutions of system (13) defined on , then and where and .
Consider the product system of (13)
Construct a Lyapunov function defined on as follows:
It is easy to see that the norm and the norm are equivalent, that is, there exist two constants and such that
Let such that and . Thus, condition (i) of Lemma 2 is satisfied.
where , and . Therefore, condition (ii) of Lemma 2 is satisfied.
Finally, we calculate the difference along system (14). Indeed,
In view of system (15), we observe that
Substituting (16) and (17) back in (15), we obtain
By applying the Mean Value Theorem, we have
where , lie between and . Substituting (19) and (20) back in (18), we get
By virtue of (H.1), (11) and (12), we observe that
Substituting (22)-(29) back in (21), we obtain
By virtue of the condition that , assumption (iii) of Lemma 2 is satisfied. Thus, we conclude that there exists a unique uniformly asymptotically stable almost periodic solution of system (13) which satisfies and for all . It follows that there exists a unique uniformly asymptotically stable almost periodic solution of system (4) which satisfies and for all . □
Assume the following condition:
(H.4) , , , , and are bounded nonnegative periodic sequences of period ω.
Corollary 1 Let (H.2)-(H.4) hold. Then system (4) has a unique uniformly asymptotically stable periodic solution of period ω.
5 Some examples
Example 1 Consider the following system:
where , , , , , , . By calculation, we find , and , . Therefore, . One can easily check the validity of conditions (H.1)-(H.3). Thus, by Theorem 1 and Theorem 3, system (30) is persistent and has a unique uniformly asymptotically stable almost periodic solution.
Example 2 Consider the following system:
where , , , , , , . By calculation, we find , and , . Therefore, . One can easily check the validity of conditions (H.1)-(H.3). Thus, by Theorem 1 and Theorem 3, system (31) is persistent and has a unique uniformly asymptotically stable almost periodic solution.
Example 3 Consider the following system:
where , , , , , , . By calculation, we find , and , . Therefore, . One can easily check the validity of conditions (H.2)-(H.4). Thus, by Theorem 1 and Corollary 1, system (32) is persistent and has a unique uniformly asymptotically stable periodic solution.
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The author would like to express his sincere thanks to the editor Prof. Dr. Elena Braverman for handling the paper during the reviewing process and to the referees for suggesting some corrections that helped making the contents of the paper more accurate.
The author declares that he has no competing interests.
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Alzabut, J. Dynamics of almost periodic solutions for a discrete Fox harvesting model with feedback control. Adv Differ Equ 2012, 157 (2012). https://doi.org/10.1186/1687-1847-2012-157