Positive periodic solutions for discrete Nicholson system with multiple time-varying delays

: Fly communities exhibit rich ecological dynamics, and one of the important inﬂuencing factors is the interaction between species. A discrete Nicholson-type system with multiple time varying delays which considers the mutualism relationship between two ﬂy species is investigated in this paper. Su ﬃ cient conditions for the existence of positive periodic solutions are elucidated. The result is obtained by the well-known continuation theorem of coincidence degree theory. An example is attached to illustrate our result. Moreover, the actual biological descriptions obtained from our main result are explained.


Introduction
Flies are complete metamorphosis insects that contain various species, including Muscidae (houseflies), Calliphoridae (blowflflies) Drosophilae (fruitflies) and Scrcophagidae (fleshflies), etc.The life history of flies can be divided into egg, larva , pre-pupa, pupa and adult stages.Although the life span of flies is only about one month, they are very fertile and multiply rapidly in a short period [1].The feeding habits of flies are very complex.They can feed on a variety of substances, such as human food, animal waste, kitchen scraps and other refuses.It is known to us that flies transmit various pathogens from filth to humans and cause many diseases [2][3][4].On the other hand, flies are also beneficial to medical research, ecosystem food chain and pollen dispersal.Considering medical research, for example, fruit fly Drosophila is of great significance in studying the pathogenesis and therapy of human diseases.The nervous system of Drosophila is much simpler than that of human beings, but it also exhibits complex behavioral characteristics similar to humans [5,6].Therefore, studying fly population dynamics is of crucial importance to both nature and human society.
The study of biological population growth model promotes the development of human society to a great extent.It has important applications in population control, social resource allocation, ecological environment improvement, species protection and human life and health [7][8][9].To understand the population dynamics of the Australian sheep blowfly, Gurney et al. [10] constructed the autonomous delay differential equation based on experimental data [11,12].In this model, x is the density of mature blowflies, δ is the daily mortality rate of adult blowflies, P is the maximum daily spawning rate of female blowflies, τ is the time required for a blowfly to mature from an egg to an adult, 1/γ is the blowfly population size at which the production function f (u) = ue −γu reaches the maximum value.Subsequently, this model and its modified extensions were continually used to describe rich fly dynamics.Environmental changes play an important role in biological systems.The influence of a periodically changing environment on the system is different from that of a constant environment, and it can better facilitate system evolution.Moreover, delay is one of the important factors which can change the dynamical properties and result in more rich and complex dynamics in biological systems [13,14].Many researchers have assumed periodic coefficients and time delays in the system to combine with the periodic changes of the environment [15][16][17][18].For related literature, we refer to [19,20].However, considering the fact that adult flies number is a discrete value that varies daily and the situations where population numbers are small and individual effects are important or dominate, a discrete model would indeed be more realistic to describe the population evolution in discrete time-steps [21][22][23].
Interactions between different species are extremely important for maintaining ecological balance.Such interactions are typically direct or indirect between multiple species, including positive interactions and negative interactions.Among them, the positive interactions can be divided into three categories according to the degree of action: commensalism, protocooperation and mutualism [24,25].In the paper [9], a delay differential Nicholson-type system concerning the mutualism effects with constant coefficients was proposed.The existence, global stability and instability of positive equilibrium were obtained.Based on this system, Zhou [26] and Amster [27] considered periodic Nicholson-type system combined with nonlinear harvesting terms.The main research theme is the existence of positive periodic solutions.Recently, Ossandóna et al. [28] presented a Nicholson-type system with nonlinear density-dependent mortality to describe the dynamics of multiple species, the uniqueness and local exponential stability of the periodic solution are established.However, relatively few studies on discrete dynamical systems have explored the mutualism of flies.In this paper, we consider the mutualism relationship between two fly species and establish a two-dimensional discrete Nicholson system with multiple time-varying delays We assume that a i : Z → (0, 1), b i : Z → (0, ∞), c i j : Z → (0, ∞), τ i j : Z → Z + and γ i j : Z → (0, ∞) are ω-periodic discrete functions for 1 ≤ i ≤ 2 and 1 ≤ j ≤ n.The period ω is a positive integer.Moreover, the interaction rate of second fly specie on first fly species and that of first fly specie on second fly species are represented by b 1 and b 2 , respectively.
Because τ i j (1 ≤ i ≤ 2) have ω-periodicity, we can find the maximum values Then, the solution x(•, φ) = (x 1 (•, φ 1 ), x 2 (•, φ 2 )) T of system (1.1) that satisfies the initial condition is a positive solution.The purpose of this paper is to present sufficient conditions for the existence of positive ω-periodic solution of (1.1).

Priori bounds for parametric system and auxiliary lemma
We discuss the parametric delay difference system for each parameter λ ∈ (0, 1).Let a i = min 1≤k≤ω a i (k) and b i = max 1≤k≤ω b i (k) for i = 1, 2. Then, an estimation of upper and lower bounds of positive ω-periodic solution of (2.1) can be conducted.
and there exists a constant γ > 1 such that n j=1 c i j (k) > γ a i (k) for k = 1, 2, . . ., ω and 1 ≤ i ≤ 2. (2.3) Then, every positive ω-periodic solution x = (x 1 , x 2 ) T of (2.1) is bounded.Specifically, where Remark 1.Note that A i and B i are the lower bound and upper bound of x i , respectively.We can verify the fact that A i < B i for i = 1, 2. From the definitions of A 1 and A 2 , we see that Hence, we obtain Similarly, it follows that Proof.Let x = (x 1 , x 2 ) T be arbitrary positive ω-periodic solution of (2.1) under the initial condition (1.2).For i = 1, 2, we define (2.4) Taking the maximum on both sides of the first equation of (2.4) in one period, we have Similarly, we obtain Hence, it leads to and .

Electronic Research Archive
Volume 31, Issue 11, 6982-6999.and Now, it can be concluded that each positive ω-periodic solution x = (x 1 , x 2 ) T of (2.1) satisfies and Suppose that X is a Banach space and L : Dom L ⊂ X → X is a linear operator.The operator L is called a Fredholm operator of index zero if If L is a Fredholm operator of index zero and P, Q : X → X are continuous projectors satisfying where I is the identity operator from X to X, then the restriction L P : Dom L∩Ker P → Im L is invertible and has the inverse K P : Im L → Dom L ∩ Ker P.
Let N : X → X be a continuous operator and Ω an open bounded subset of X.The operator We present the continuation theorem of coincidence degree theory (for example, see [29,30]) as follows: Lemma 2.2.Let L : Dom L ⊂ X → X be a Fredholm operator of index zero and let N : X → X be L-compact on Ω. Suppose that (i) every solution x of Lx = λN x satisfies x ∂Ω for λ ∈ (0, 1); (ii) QN x 0 for x ∈ ∂Ω ∩ Ker L and deg QN, Ω ∩ Ker L, 0 0.
Then, Lx = N x has at least one solution in X ∩ Ω.
Proof.Let X be a set of ω-periodic functions x = (x 1 , x 2 ) T defined on Z + and denote the maximum norm ||x|| = max{max 1≤k≤ω |x 1 (k)|, max 1≤k≤ω |x 2 (k)|} for any x ∈ X.Then, X is a Banach space.Moreover, we define and . It is not difficult to show that L is a linear operator from X to X and N is a continuous operator from X to X.
From the definition of L, we see that .
It turns out that dim Ker L = 2 = codim Im L < +∞ and Im L is closed in X.Thus, L is a Fredholm operator of index zero.We define P : X → X by and let Q = P.Then, P and Q are two continuous projectors such that Im P = Ker L and Ker It can be shown that the restriction L P : Dom L ∩ Ker P → Im L has the inverse K P : Im L → Dom L ∩ Ker P given by In fact, for i = 1, 2, since for all k ∈ Z + , we see that K P x ∈ Dom L .Moreover, it follows that Hence, K P x ∈ Ker P.
For any x ∈ Im L, one has Furthermore, for any x ∈ Dom L ∩ Ker P, one has Since x ∈ Ker P = Ker Q = Im L, we see that ω s=1 x i (s) = 0. Hence, (K P L P x) i = x i (k) = (I x) i .We therefore conclude that K P = L −1 P .We define and prove that the operator N defined above is L-compact on Ω.We first check that QN(Ω) is bounded. Since for x ∈ Ω.Hence, the operator QN is bounded on Ω.We next show that K P (I − Q)N : Ω → X is compact.From the definitions of N, QN and K p , we obtain Meanwhile, we have for x ∈ X.For any bounded subset E ⊂ Ω ⊂ X, it is a subspace of a finite dimensional Banach space X.Hence, E is closed, and therefore E is compact.By a straightforward calculation, it can be proven that K P (I − Q)N(E) is relatively compact.An arbitrary ω-periodic solution of (2.1) corresponds one-to-one to a solution of Lx = λN x with parameter λ ∈ (0, 1).Proposition 2.1 displays that each positive solution x = (x 1 , x 2 ) T of Lx = λN x satisfies that A 1 < x 1 ≤ B 1 and A 2 < x 2 ≤ B 2 .It is obvious that if y = (y 1 , y 2 ) T ∈ ∂Ω, then y is never a solution of Lx = λN x.Hence, the condition (i) of Lemma 2.2 holds.If x = (x 1 , x 2 ) T ∈ ∂Ω ∩ Ker L, then there are four cases to be considered: (1) Case (1): It follows from Since A 1 ≤ ln γ/γ 1 , we see that e A 1 γ 1 ≤ γ.Hence, (QN x) 1 > 0. Case (2): Because of x 1 ≡ B 1 + 1, we have Similarly, we can show that (QN x) 2 > 0 in Case (3) and (QN x) 2 < 0 in Case (4).We therefore conclude that QN x = ((QN x) 1 , (QN x) 2 ) T 0 for each x ∈ ∂Ω ∩ Ker L. Define a continuous operator H : Recall that the elements of ∂Ω∩Ker L are vectors satisfying Moreover, Hence, H(x, µ) 0 and H(y, µ) 0. By similar computations, we have H(z, µ) 0 and H(w, µ) 0. Therefore, we see that H(x, µ) 0 for (x, µ) ∈ ∂Ω ∩ Ker L × [0, 1].Thus, H is a homotopic mapping.Using the homotopy invariance, we have Hence, the condition (ii) of Lemma 2.2 holds.Therefore, the equation Lx = N x has at least one solution located in X ∩ Ω.Thus, from Lemma 2.2, we obtain that there is a positive ω-periodic solution of system (1.1).The proof is now complete.

Existence of positive 4-periodic solution
Consider the delay difference system 4) .
for k = 1, 2, 3, 4.Moreover, it can be calculated that Namely, condition (3.1) holds.Therefore, from Theorem 3.1, it turns out that the system has at least one positive 4-periodic solution.
Figure 1.Graphs of three arbitrary positive solutions of system.The numerical simulations show that there is a positive 4-periodic solution and this positive 4-periodic solution is locally asymptotically stable.

Conclusions
A discrete Nicholson system that describles the dynamics of two fly species is studied in this paper.The system considers the mutualism effect between fly species.Continuation theorem of coincidence degree theory is used effectively to seek sufficient conditions for the existence of a positive periodic solution.It is easy to check whether these sufficient conditions hold or not by using coefficients.The positive periodic solution indicates a cycle change in the adult fly populations.From the obtained result, we found that mutualistic interactions between species plays an important role in adult flies populations.But the increase in the flies populations resulting from maximum cumulative mutualism effect only should be less than the death of the flies populations because there is the natural generation of flies populations.Moreover, to avoid species extinction and maintain the coexistence of two fly species in a mutually beneficial environment, we see that (i) the adult fly population produced by maximum daily spawning should exceed a constant multiple of dead fly population for each fly species, and the multiple is greater than constant 1 and (ii) the total population growth must be maintained more than the population loss for each fly species.In fact, the third sufficient condition The left side of each inequality represents the production of one fly species in a period under the mutualism influence of another, and the right side represents the death of that species in a period.Hence, statement (ii).

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.