(μ, η)-pseudo almost automorphic solutions of a new class of competitive Lotka-Volterra model with mixed delays

Abstract: In the natural world, competition is an important phenomenon that can manifest in various generalized environments (economy, physics, ecology, biology,...). One of the famous models which is able to represent this concept is the Lotka-Volterra model. A new class of a competitive Lotka-Volterra model with mixed delays and oscillatory coe cients is investigated in this work. Thus, by using the (μ, η)-pseudo almost automorphic functions function class and the Leray-Schauder xed-point theorem, it can be proven that solutions exist. In addition, in such situations, we have a number of species that coexist and all the rest will be extinct. Therefore, the study of permanence becomes unavoidable. Therefore, su cient and new conditions are given in order to establish the permanence of species without using a comparison theorem. By the new Lyapunov function we prove the asymptotic stability for the considered model. Moreover, we investigate the globally exponential stability of the (μ, η)-pseudo almost automorphic solutions. In the end, an example is given to support theoretical result feasibility.


Introduction
The population dynamics is a part of mathematical biology that aims to describe, in terms of mathematical models, the interaction between various types of populations in a given environment (e.g. animal populations in ecology, cell populations in biology, viral populations in epidemiology). These models are governed by evolutionary equations such as di erences, functional equations, partial derivatives or stochastic equations. The interaction between populations generally takes place in a uctuating environment over time. For example, the temperature, humidity and availability of nutrients or water are physical parameters that vary over time with the variation of seasons. In addition, there exist in population dynamics di erent interactions between populations (mutualism, competition, predator-prey, etc.). One of the most famous and important population dynamics model is the Lotka-Volterra ecological population model proposed by Lotka and Volterra. This model was rst proposed to describe the predator-prey relationship in an ecosystem, and soon became well known and formed the basis of many important models in mathematical biology, population dynamics, biotechnology, ecology, etc. Furthermore, it describes several interacting between species. In these approach, the interaction of individuals of the same species is not the only one that modi es the strategy of population dynamics, but also the interaction between individuals of di erent species. For example, many populations may compete to exploit the resources of the environment. This is the case of competition interaction. Along the same vein, the pioneer study of Lotka and Volterra proposed an original equation class in order to model these population dynamics (see [15,24,28,29]), given by where z i (t) is the population size of species i and the derivative z ′ i (t) represent the variation of populations over time, i is the growth rate of species i, and ℘ ij ≥ represents the e ect of the population interaction.
In addition, in [10] the authors employed the above model by considering an important factor, which was the e ect of toxins, and they considered the following system where z i (t) denotes the i-th species population density at time t for the common pool of resources. Terms ϑ z (t)z (t) and ϑ z (t)z (t) denote the toxic substance e ects. Furthermore, in some speci c situations, people might hope for changing the position of an existing solution, with the aim of keeping stability. This is of high importance for controlling the ecology balance. To realize this, we can alter the system structurally through the introduction of some feedback control variables with the goal of stabilizing a population at another solution. Balancing between two sh populations within one closed pond was explained by V. Volterra in 1931 using the theory of feedback. On the other hand, systems of di erential equations with delay argument occupy more than a place of central importance in all areas of science and particularly in the biological sciences. These equations are used as models to describe many physical and biological systems (e.g., population dynamics and epidemiology). In fact, many actual systems have the property of aftere ect, i.e., the future states depend not only on the present, but also on the past history. Aftere ect is believed to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, etc. Moreover, as it is well known, in general, it is not easy to nd the solutions of all linear and nonlinear di erential equations. But, nding solutions becomes more di cult for delay di erential equations compared to the di erential equations without delay. Therefore, it is very important to determine the qualitative behavior of solutions when there is a delay. In contrast, the time delay has an in uence on the dynamics of a population or several interacting species that can arise from a great variety in the system ecology, such as gestation and maturation. In [18], Gopalsamy and Weng considered the following model They discussed the positive equilibrium and the globally asymptotic stability. It should be pointed out that the most utilized concepts to solve the Lotka-Volterra system are periodic or almost periodic. Nevertheless, a lot of phenomena process some regularity, while being non periodic. It is well known that the biological system is frequently deeply disturbed by human exploring actions, such as planting and harvesting, which indicates that it is in general impossible to be explained by periodic or almost periodic situations. On the other hand, the concept of (µ, η)-pseudo almost automorphic periodic can be in fact a natural generalization of periodicity, almost periodicity as well as almost automorphic functions. In this paper, we consider a recent (µ, η)-pseudo almost automorphic functions concept to have a more precise description of our model. Moreover, in [16] the authors used the concept of almost periodicity and studied the following system where v i measures the controllable rate, k i is the control species death rate, r i denotes the control species input, α ij , β ij and γ ij provide an intra-speci c interference nonlinear measure, q i denotes the external input rate, υ ij is the toxic inhibition rate of the i-th species by others and vice versa, ℘ ij and ω ij measure the competition amount between i-th and j-th species (i ≠ j), i denotes the intrinsic exponential growth rate of the i-th species, p i is the control variable, and z i is the i-th species population density. In biology there is a very important concept that ensures the survival of the biological species, which is the concept of permanence which implies the survival of all the species that exist initially. is introduced by Schuster, Sigmund and Wol (1979). The notion of permanence is a mathematical tool that allows both to give a mathematical meaning to the notion of lo ng-term survival of populations in biology, and also to obtain important properties on the asymptotic behavior of solutions. Among the various investigations, the permanence or even persistence is related to the problem of coexistence species has received much attention in recent decades. Further, the notion of permanence a great interested in the study of the Lotka-Volterra system. In [16], the important problem was to study the persistence of the above system by using the comparison theorem. Afterward, they obtained the existence of a solution via the proprieties of almost periodic functions. The authors studied through an appropriate Lyapunov function, the global asymptotic stability.
The solutions of (2) will satisfy the following initial conditions with e = sup t {e ij (t), θ ij , i, j = , , . . . , N}. Inspired by previous studies, our motivation of this paper consists in studying a (µ, η)-pseudo almost automorphic solution of system (2), which has not existed up to now. One of the very serious and interesting topics in the study of di erential equations is that if the force function and/ or the coe cients possesses a speci c property, are we going to nd the same characteristics in the solution under some delay?
Generally, compared with the previous works, the main contributions of our paper are -Our ndings generalize the results in the literature ( [16,25,26]), since the (µ, η)-pseudo almost automorphic functions class contains classes of periodic, almost automorphic, almost periodic, pseudo almost periodic, pseudo almost automorphic and µ-pseudo almost automorphic functions. -Our conditions are di erent from those in ( [16,[25][26][27]), which can be used to prove the existence, permanence and stability of other models. -In the latter studies, the authors proved the existence of almost periodic solutions using properties related to the almost periodic solutions (see [16,[25][26][27]). We prove the existence of the solutions of system(2), by utilising the combination between the Leray-Schauder xed point theorem and the exponential dichotomy, which has been never used for studying the model of Lotka-Volterra. -Permanence is a mathematical tool that allows not only giving a mathematical meaning to the notion of long-term survival of populations in biology, but also obtainning important properties on the asymptotic behavior of solutions ( [15,16,29]). Among the various investigations, permanence or even persistence is related to the problem of coexistence species, which has received much attention in recent decades. However, the study of permanence is dedicated to use the comparison theorem ( see [11,16,[25][26][27]). For this, we prove the permanence of the solutions, without using the comparison theorem. -In this work, by new and su cient conditions, we study such globally asymptotic stability by using a new Lyapunov function, which is completely di erent from that presented in ( [16]). It is worth mention-ing that, in the work of ( [16,[25][26][27]), the authors proved only the asymptotic or attractive stability of the almost periodic solution (not the exponential stability). Besides, we study in this paper the globally exponential stability of our model. The paper is organized as follows: Some characteristics of the (µ, η)-pseudo almost automorphic functions are recalled in section 2. Addition to that, several necessary notations, de nitions, and preliminaries are introduced for ulterior. In section 3, rstly, via exponential dichotomy, our system is transformed into one equation. Secondly, the permanence of the new equation is proven. In section 4, we derive conditions enough for the existence of (µ, η)-pseudo almost automorphic solutions of the new equation within a suited convex set. In section 5, we drive su cient conditions in order to study the global asymptotic and exponential stabilities of the solutions. An illustrative example is presented in section 6.

Assumptions, de nitions and lemmas
De nition 1. [7] A continuous function G : R → R can be considered as almost automorphic if for every sequence (en) n∈N there exists an (rn) n∈N ⊂ (en) n∈N subsequence such that exists for all t ∈ R, and are well de ned for each t ∈ R.
Let AA(R, R) basically the collection of all almost automorphic functions from R to R. Theorem 1. [6] Let H ∈ AA U (R × X, Y) ( see [8]). As a result, a superposition operator N H actually determined as follows: Let D be a Lebesgue σ-eld of R, J denote a set of every positive measure η on D satisfying η(R) = ∞ and De nition 2. [13] For µ, η ∈ J, the µ and η measures can be considered as equivalent if there exist constants c, d > and a bounded interval J ⊂ R such that Now, the concept of ergodicity is to be introduced.

De nition 4. [13] Let µ, η ∈ J. A function D ∈ BC(R, R N ) can be considered as (µ, η)-pseudo almost automorphic if it is expressed by D
Now, the next hypothesis (A1) is considered: For all w ∈ R we can actually have ϵ > besides one bounded interval J in a certain way that µ({c + w : c ∈ P}) ≤ ϵµ(P) and this is in case P ∈ D ful lls P ∩ J = ∅.
Throughout this paper, the following notations will be adapted, for an H : R → R+ continuous function: It is also assumed that the following conditions (A ) − (A ) holds, where (A2) There exists a continuous function λ : R → R+ as follows

Permanence of the solutions
Lemma 2. Let consider assumptions (A3) hold. For every ≤ i ≤ N, the following equation has a unique solution (Ψz i )(t), given as follows Proof. Since for all ≤ i ≤ N, k i > , then the linear equation admits on R an exponential dichotomy. Moreover, equation (4) has got one special formal solution Besides, the study of system (2) is equivalent to the following equation Next, in order to show the existence, permanence and stability of the (µ, η)-pseudo almost automorphic of equation (2), we need only to study the system (5).
When we calculate an upper right Φ(t) derivative, we can get the following with ι being any positive constant. As a matter of fact, for any R > , if Φ(t) ≥ R for all t ≥ R, then D + Φ ≤ −ιΦ for all t ≥ R. As a result, there will be a contradiction. Hence, there should exist a J > R in a certain way that As a consequence, Φ(t) has a maximum with J ≤ t ≤ J at least. Let consider, Φ(t) attaining its maximum at J (J < J < J ). Thus, D + Φ(t) > , for J − δ < t < J , for δ is a positive constant, which contradicts what follows

Lemma 3. Under (A3)-(A4), and
there exists R > such that for all i = , . . . , N, and for all t ≥ J , we have Proof. De ne G(t) = min ≤i≤N z i (t). For each provided t > , we actually have one i( ≤ i ≤ N) as follows Next, it must be proven that G(t) ≥ R , for t ≥ J . By contradiction, we consider G(t) < R for t ≥ −e, for that reason Integrating (6) over [J , t) leads to By hypothesis (A5), for t → ∞ we have e L(t−J ) → ∞, which contradicts the fact that G(t) ≤ R , for t ≥ J .
By the comparability theorem, we have

Existence of the (µ, η)-pseudo almost automorphic solutions
Some results are established in this section as regards (µ, η)-pseudo almost automorphic solutions existence Remark 5. One of the our contributions is the presentation of some new proofs that relate to the following Lemma.  (t)).
Since κ ij (., s), z j (.) are almost automorphic, it is possible to extract for each sequence of real numbers (jn) a subsequence (ln) in a certain manner that for n ≥ N, we have for all ι ∈ [−θ ij , ], and lim n→∞ z j (t + ln + ι) = z j (t + ι), lim n→∞ z j (t + ln + ι) = z j (t + ι) (8) uniformly for t ∈ R. From (7) and (8), there exist a positive integer N, in a certain way that On the other hand, It is proven in the same approach that, for all t ∈ R lim n→∞J j (t − ln) = G j (t), which proves that G j (.) ∈ AA(R, R+).
Proof. For every ≤ i ≤ N, function i : s → i (s) is (µ, η)-pseudo almost automorphic. Then, for every ≤ i ≤ N, i can be expressed as with i (.) ∈ AA(R, R+) and i ∈ ε(R, R+, µ, η). As a result, Let consider (pn) a sequence of real numbers. Since i , i are almost automorphic functions, we are capable of extracting a subsequence (an) of (pn) in a certain degree that Afterwards, Thus, Hence, we have θ ∈] , [ in a certain way Using Lebesgue dominated convergence theorem, we are able to obtain lim n→∞ (Ξ i )(t + an) = (ΞX i )(t).
The same approach proves that for every t ∈ R It remains to show χ = χ = .
By hypothesis (A3), we get Consequently, Then Ξ is a mapping in Ω into itself.
Step2: V is equi-continuous. In fact, t → i (t), i (t) are continuous, then for each ε > , we can have δ > , in a certain way that for |h| < δ, we get (z (t), . . . , z N (t)) be another solution of system (5) and ζ = (ζ , . . . , ζ N ) be a positive vector, such that ι = min ≤i≤N ζ i . The Lyapunov function is de ned thusly The upper right derivative D + V(.) of V(.) given Consequently, solution Z * (t) of system (5) is stable in Lyapunov sense. In fact, integrating (10) over [T , t] leads to which gives Then, lim t→∞ |z α ii i (t) − (z * i ) α ii (t)| = . This complete the proof.
. Exponential stability of the (µ, η)-pseudo almost automorphic solutions Since Z(t), Z * (t) are solutions of system (5) in region Ω, then Z(t), Z * (t) satis es Consequently, solution Z * (t) of system (5) can be considered stable in Lyapunov sense. Furthermore, Therefore, solution Z * (t) of system (5) can be global exponential stable.

Conclusion
The (µ, η)-pseudo almost automorphic functions consist of a bigger class of functions, so a highly complex behavior can be expressed for these functions. Within this work, a Lotka-Volterra N-species Gilpin-Ayala type competitive model with feedback control has been investigated. Applying the Leray-Schauder Alternative theorem and the exponential dichotomy, we obtain new adequate conditions for the existence, permanence and globally exponential stability and asymptotic stability of (µ, η)-pseudo almost automorphic solutions. After that, an example has been provided to show the e ectiveness of the achieved results. As a matter of fact, we can apply the method of this paper with the purpose of studying other mathematical, economical and biological models.