
The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.
Citation: Xuemin Xue, Xiangtuan Xiong, Yuanxiang Zhang. Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation[J]. AIMS Mathematics, 2021, 6(10): 11425-11448. doi: 10.3934/math.2021662
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The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.
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
x′(t)=−δx(t)+Px(t−τ)e−γx(t−τ) |
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
{Δx1(k)=−a1(k)x1(k)+b1(k)x2(k)+∑nj=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k)), Δx2(k)=−a2(k)x2(k)+b2(k)x1(k)+∑nj=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k)). | (1.1) |
We assume that ai:Z→(0,1), bi:Z→(0,∞), cij:Z→(0,∞), τij:Z→Z+ and γij: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 b1 and b2, respectively.
Because τij (1≤i≤2) have ω-periodicity, we can find the maximum values
¯τi=max1≤j≤n{max1≤k≤ωτij(k)}∈Z+ |
of {τi1(k)}, {τi2(k)}, …, {τin(k)} for i=1,2. Note that 0<ai(k)<1 for k∈Z. Then, the solution x(⋅,ϕ)=(x1(⋅,ϕ1),x2(⋅,ϕ2))T of system (1.1) that satisfies the initial condition
xi(s)=ϕi(s)>0fors∈[−¯τi,0]∩Z | (1.2) |
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).
We discuss the parametric delay difference system
{Δx1(k)=−λa1(k)x1(k)+λb1(k)x2(k)+λ∑nj=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k)), Δx2(k)=−λa2(k)x2(k)+λb2(k)x1(k)+λ∑nj=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k)) | (2.1) |
for each parameter λ∈(0,1). Let a_i=min1≤k≤ωai(k) and ¯bi=max1≤k≤ωbi(k) for i=1,2. Then, an estimation of upper and lower bounds of positive ω-periodic solution of (2.1) can be conducted.
Proposition 2.1. Suppose that
a_1a_2−¯b1¯b2>0 | (2.2) |
and there exists a constant γ>1 such that
n∑j=1cij(k)>γai(k)fork=1,2,…,ωand1≤i≤2. | (2.3) |
Then, every positive ω-periodic solution x=(x1,x2)T of (2.1) is bounded. Specifically,
A1<x1(k)≤B1andA2<x2(k)≤B2fork=1,2,…,ω, |
where
A1≤min{lnγ¯γ1,γB1e−¯γ1B1}andB1=a_2(a_1a_2−¯b1¯b2)e(n∑j=1¯c1jγ_1j+¯b1a_2n∑j=1¯c2jγ_2j), |
A2≤min{lnγ¯γ2,γB2e−¯γ2B2}andB2=a_1(a_1a_2−¯b1¯b2)e(n∑j=1¯c2jγ_2j+¯b2a_1n∑j=1¯c1jγ_1j), |
in which γ_1j=min1≤k≤ωγ1j(k), γ_2j=min1≤k≤ωγ2j(k), ¯c1j=max1≤k≤ωc1j(k), ¯c2j=max1≤k≤ωc2j(k), ¯γ1=max1≤j≤n{max1≤k≤ωγ1j(k)} and ¯γ2=max1≤j≤n{max1≤k≤ωγ2j(k)}.
Remark 1. Note that Ai and Bi are the lower bound and upper bound of xi, respectively. We can verify the fact that Ai<Bi for i=1,2. From the definitions of A1 and A2, we see that
A1≤γB1e−¯γ1B1≤γe¯γ1andA2≤γB2e−¯γ2B2≤γe¯γ2. |
Hence, we obtain
B1>a_2(a_1a_2−¯b1¯b2)en∑j=1¯c1jγ_1j=1/(1−¯b1¯b2a_1a_2)×1a_1en∑j=1¯c1jγ_1j>∑nj=1¯c1ja_11e¯γ1>γe¯γ1≥A1. |
Similarly, it follows that
B2>a_1(a_1a_2−¯b1¯b2)en∑j=1¯c2jγ_2j>γe¯γ2≥A2. |
Proof. Let x=(x1,x2)T be arbitrary positive ω-periodic solution of (2.1) under the initial condition (1.2). For i=1,2, we define
¯xi=max1≤k≤ωxi(k)andx_i=min1≤k≤ωxi(k). |
Then x_i≤xi(k)≤¯xi for k∈Z+. We can rewrite system (2.1) into
{x1(k+1)=(1−λa1(k))x1(k)+λb1(k)x2(k)+λ∑nj=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k)), x2(k+1)=(1−λa2(k))x2(k)+λb2(k)x1(k)+λ∑nj=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k)). | (2.4) |
Taking the maximum on both sides of the first equation of (2.4) in one period, we have
¯x1=max1≤k≤ω{x1(k+1)}≤max1≤k≤ω{(1−λa1(k))x1(k)}+λmax1≤k≤ω{b1(k)x2(k)}+λmax1≤k≤ω{n∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))}≤max1≤k≤ω{(1−λa1(k))}max1≤k≤ω{x1(k)}+λmax1≤k≤ω{b1(k)}max1≤k≤ω{x2(k)}+λmax1≤k≤ω{n∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))}≤(1−λa_1)¯x1+λ¯b1¯x2+λmax1≤k≤ω{n∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))}. |
Similarly, we obtain
¯x2≤(1−λa_2)¯x2+λ¯b2¯x1+λmax1≤k≤ω{n∑j=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k))}. |
Hence, it leads to
¯x1≤¯b1a_1¯x2+1a_1max1≤k≤ω{n∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))}≤¯b1a_1¯x2+1a_1en∑j=1¯c1jγ_1j, | (2.5) |
and
¯x2≤¯b2a_2¯x1+1a_2max1≤k≤ω{n∑j=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k))}≤¯b1a_2¯x1+1a_2en∑j=1¯c2jγ_2j. | (2.6) |
By (2.5) and (2.6), basic computations show that
¯x1≤1/(1−¯b1¯b2a_1a_2)×(1a_1en∑j=1¯c1jr_1j+¯b1a_1a_2en∑j=1¯c2jγ_2j)=a_2(a_1a_2−¯b1¯b2)e(n∑j=1¯c1jγ_1j+¯b1a_2n∑j=1¯c2jγ_2j)=B1, |
¯x2≤1/(1−¯b1¯b2a_1a_2)×(1a_2en∑j=1¯c2jr_2j+¯b2a_1a_2en∑j=1¯c1jγ_1j)=a_1(a_1a_2−¯b1¯b2)e(n∑j=1¯c2jγ_2j+¯b2a_1n∑j=1¯c1jγ_1j)=B2. |
Note that 1−λai(k)>0 for all k∈Z and i=1,2. Multiplying both sides of the two equation of (2.1) by ∏kr=01/(1−λa1(r)) and ∏kr=01/(1−λa2(r)) respectively, we have
x1(k+1)k∏r=011−λa1(r)−x1(k)k−1∏r=011−λa1(r)−λb1(k)x2(k)k∏r=011−λa1(r)=λn∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))k∏r=011−λa1(r), | (2.7) |
and
x2(k+1)k∏r=011−λa2(r)−x2(k)k−1∏r=011−λa2(r)−λb2(k)x1(k)k∏r=011−λa2(r)=λn∑j=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k))k∏r=011−λa2(r). | (2.8) |
Choosing natural numbers k1 and k2 such that
¯τ1≤k1≤¯τ1+ω−1andx1(k1)=x_1, |
¯τ2≤k2≤¯τ2+ω−1andx2(k2)=x_2. |
Summing both sides of (2.7) and (2.8) over k ranging from k1 to k1+ω−1 and k2 to k2+ω−1 respectively, by using xi(ki+ω)=xi(ki)=x_i, we obtain
x_1k1−1∏r=011−λa1(r)(k1+ω−1∏r=k111−λa1(r)−1) =λk1+ω−1∑s=k1((b1(s)x2(s)+n∑j=1c1j(s)x1(s−τ1j(s))e−γ1j(s)x1(s−τ1j(s)))s∏r=011−λa1(r)), |
and
x_2k2−1∏r=011−λa2(r)(k2+ω−1∏r=k211−λa2(r)−1) =λk2+ω−1∑s=k2((b2(s)x1(s)+n∑j=1c2j(s)x2(s−τ2j(s))e−γ2j(s)x2(s−τ2j(s)))s∏r=011−λa2(r)). |
Note that ai (i=1,2) is positive ω-periodic. It follws that
ki+ω−1∏r=ki(1−λai(r))=ω−1∏r=0(1−λai(r)). | (2.9) |
Hence, we obtain
x_1=λ∏k1+ω−1r=0(1−λa1(r))1−∏ω−1r=0(1−λa1(r))(k1+ω−1∑s=k1(b1(s)x2(s)+n∑j=1c1j(s)x1(s−τ1j(s))e−γ1j(s)x1(s−τ1j(s)))s∏r=011−λa1(r))=λ1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1((b1(s)x2(s)+n∑j=1c1j(s)x1(s−τ1j(s))e−γ1j(s)x1(s−τ1j(s)))k1+ω−1∏r=s+1(1−λa1(r))), | (2.10) |
and
x_2=λ1−∏ω−1r=0(1−λa2(r))k2+ω−1∑s=k2((b2(s)x1(s)+n∑j=1c2j(s)x1(s−τ2j(s))e−γ2j(s)x1(s−τ2j(s)))k1+ω−1∏r=s+1(1−λa2(r))). | (2.11) |
Recall that ¯γi=max1≤j≤n{max1≤k≤ω−1γij(k)} for i=1,2. We define f1(u)=ue−¯γ1u and f2(u)=ue−¯γ2u for u≥0. Since x_i≤xi(k)≤¯xi for all k∈Z+, it turns out that
xi(s−τij(s))e−γij(s)xi(s−τij(s))≥min{fi(x_i),fi(¯xi)}fors≥¯τijfori=1,2. |
Note that k1≥¯τ1. By using (2.3) and (2.10), we have
x_1≥λmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1(n∑j=1c1j(s)k1+ω−1∏r=s+1(1−λa1(r)))>λmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1(γa1(s)k1+ω−1∏r=s+1(1−λa1(r)))=γmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1(λa1(s)k1+ω−1∏r=s+1(1−λa1(r)))=γmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1((1−(1−λa1(s)))k1+ω−1∏r=s+1(1−λa1(r)))=γmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))k1+ω−1∑s=k1(k1+ω−1∏r=s+1(1−λa1(r))−k1+ω−1∏r=s(1−λa1(r)))=γmin{f1(x_1),f1(¯x1)}1−∏ω−1r=0(1−λa1(r))(k1+ω−1∏r=k1+ω(1−λa1(r))−k1+ω−1∏r=k1(1−λa1(r))). |
Calculating by the same way, from (2.3) and (2.11), we obtain
x_2=γmin{f2(x_2),f2(¯x2)}1−∏ω−1r=0(1−λa2(r))(k2+ω−1∏r=k2+ω(1−λa2(r))−k2+ω−1∏r=k2(1−λa2(r))). |
Then, it follows from (2.9) that
x_i>γmin{fi(x_i),fi(¯xi)}fori=1,2. | (2.12) |
It is natural to divide the argument into two cases: (ⅰ) fi(x_i)≤fi(¯xi); (ⅱ) fi(x_i)>fi(¯xi).
Case (ⅰ): It follows from (2.12) that x_i>γfi(x_i). Specifically, we have
x_1>γf1(x_1)=γx_1e¯γ1x_1andx_2>γf2(x_2)=γx_2e¯γ2x_2, |
which imply that x_1>lnγ/¯γ1 and x_2>lnγ/¯γ2.
Case (ⅱ): Function fi is unimodal and takes the only peak value at 1/¯γi. Also, fi monotonically increases on [0,1/¯γi] and monotonically decreases on [1/¯γi,∞). If ¯xi≤1/1/¯γi, then we see that fi(x_i)≤fi(¯xi)≤fi(1/¯γi), which is a contradiction. Hence, it follows that ¯xi>1/¯γi. Note that ¯xi≤Bi. From (2.12), we obtain
x_1>γf1(¯x1)≥γf1(B1)=γB1e−¯γ1B1 |
and
x_2>γf2(¯x2)≥γf2(B2)=γB2e−¯γ2B2. |
Thus, we estimate
x_1>min{lnγ¯γ1,γB1e−¯γ1B11}≥A1 |
and
x_2>min{lnγ¯γ2,γB2e−¯γ2B22}≥A2. |
Now, it can be concluded that each positive ω-periodic solution x=(x1,x2)T of (2.1) satisfies
A1<x_1≤x1(k)≤¯x1≤B1 |
and
A2<x_2≤x2(k)≤¯x1≤B2 |
for k∈Z+. The proof is complete.
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
(i) dim Ker L=codim Im L<+∞,
(ii) Im L is closed in X.
If L is a Fredholm operator of index zero and P, Q:X→X are continuous projectors satisfying
Im P=Ker LandKer Q=Im L=Im (I−Q), |
where I is the identity operator from X to X, then the restriction LP:Dom L∩Ker P→Im L is invertible and has the inverse KP:Im L→Dom L∩Ker P.
Let N:X→X be a continuous operator and Ω an open bounded subset of X. The operator N is L-compact on ¯Ω if
(i) QN(¯Ω) is bounded,
(ii) KP(I−Q)N:¯Ω→X is compact.
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=λNx satisfies x∉∂Ω for λ∈(0,1);
(ii) QNx≠0 for x∈∂Ω∩Ker L and
deg{QN,Ω∩Ker L,0}≠0. |
Then, Lx=Nx has at least one solution in X∩¯Ω.
Theorem 3.1. Suppose that (2.2) and (2.3) hold. If
∑ωk=1∑nj=1(cij(k)∑ωk=1(ai(k)−bi(k))>1fori=1,2, | (3.1) |
then system (1.1) has at least one positive ω-periodic solution x∗.
Proof. Let X be a set of ω-periodic functions x=(x1,x2)T defined on Z+ and denote the maximum norm ||x||=max{max1≤k≤ω|x1(k)|,max1≤k≤ω|x2(k)|} for any x∈X. Then, X is a Banach space. Moreover, we define
Lx=((Lx)1(k)(Lx)2(k))=(x1(k+1)−x1(k)x2(k+1)−x2(k)), |
and
Nx=((Nx)1(k)(Nx)2(k))=(− a1(k)x1(k)+b1(k)x2(k)+∑nj=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k))− a2(k)x2(k)+b2(k)x1(k)+∑nj=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k))). |
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
Ker L={x∈X:(x1(k),x2(k))T≡(c1,c2)T∈R2}and Im L={x∈X:ω∑k=1x1(k)=ω∑k=1x2(k)=0}. |
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
Px=((Px)1(Px)2)=(1ωω∑k=1x1(k)1ωω∑k=1x2(k)) |
and let Q=P. Then, P and Q are two continuous projectors such that Im P=Ker L and Ker Q=Im L=Im (I−Q).
It can be shown that the restriction LP:Dom L∩Ker P→Im L has the inverse KP:Im L→Dom L∩Ker P given by
KPx=((KPx)1(KPx)2)=(k−1∑s=0x1(s)−1ωω−1∑s=0s∑r=0x1(r)k−1∑s=0x2(s)−1ωω−1∑s=0s∑r=0x2(r)) |
for x=(x1,x2)T∈Im L. In fact, for i=1,2, since
(KPx)i(k+ω)−(KPx)i(k)=k+ω−1∑s=0xi(s)−1ωω−1∑s=0s∑r=0xi(r)−k−1∑s=0xi(s)+1ωω−1∑s=0s∑r=0xi(r)=k+ω−1∑s=kxi(s)=ω−1∑s=0xi(s)=0 |
for all k∈Z+, we see that KPx∈Dom L. Moreover, it follows that
(PKPx)i=1ωω∑k=1KPxi(k)=1ωω∑k=1(k−1∑s=0xi(s)−1ωω−1∑s=0s∑r=0xi(r))=1ω(ω∑k=1k−1∑s=0xi(s)−ωωω−1∑s=0s∑r=0xi(r))=1ω(ω∑k=1k−1∑s=0xi(s)−ω∑k=1k−1∑r=0xi(r))=0. |
Hence, KPx∈Ker P.
For any x∈Im L, one has
(LPKPx)i=(KPx)i(k+1)−(KPx)i(k)=k∑s=0xi(s)−1ωω−1∑s=0s∑r=0xi(r)−k−1∑s=0xi(s)+1ωω−1∑s=0s∑r=0xi(r)=xi(k)=(Ix)i. |
Furthermore, for any x∈Dom L∩Ker P, one has
(KPLPx)i=KP(xi(k+1)−xi(k))=k−1∑s=0(xi(s+1)−xi(s))−1ωω−1∑s=0s∑r=0(xi(r+1)−xi(r))=xi(k)−xi(0)−1ωω−1∑s=0(xi(s+1)−xi(0))=xi(k)−1ωω∑s=1xi(s). |
Since x∈Ker P=Ker Q=Im L, we see that ∑ωs=1xi(s)=0. Hence, (KPLPx)i=xi(k)=(Ix)i. We therefore conclude that KP=L−1P.
We define
Ω={x=(x1,x2)T∈X:A1<x1(k)<B1+1,A2<x2(k)<B2+1} |
and prove that the operator N defined above is L-compact on ¯Ω. We first check that QN(¯Ω) is bounded.
Since x1(k)<B1+1 and x2(k)<B2+1 for k∈Z+, we obtain
(QNx)1=1ωω∑k=1(− a1(k)x1(k)+b1(k)x2(k)+n∑j=1c1j(k)x1(k−τ1j(k))e−γ1j(k)x1(k−τ1j(k)))<1ωω∑k=1(¯b1(B2+1)+1en∑j=1¯c1jγ_1j)=(¯b1(B2+1)+1en∑j=1¯c1jγ_1j), |
and
(QNx)2=1ωω∑k=1(− a2(k)x2(k)+b2(k)x1(k)+n∑j=1c2j(k)x2(k−τ2j(k))e−γ2j(k)x2(k−τ2j(k)))<1ωω∑k=1(¯b2(B1+1)+1en∑j=1¯c2jγ_2j)=(¯b2(B1+1)+1en∑j=1¯c2jγ_2j) |
for x∈¯Ω. Hence, the operator QN is bounded on ¯Ω.
We next show that KP(I−Q)N:¯Ω→X is compact. From the definitions of N, QN and Kp, we obtain
(Kp(I−Q)Nx)1=k−1∑s=0(− a1(s)x1(s)+b1(s)x2(s))+k−1∑s=0(n∑j=1c1j(s)x1(s−τ1j(s))e−γ1j(s)x1(s−τ1j(s)))−(kω−ω+12ω)ω∑s=1(− a1(s)x1(s)+b1(s)x2(s))−(kω−ω+12ω)ω∑s=1(n∑j=1c1j(s)x1(s−τ1j(s))e−γ1j(s)x1(s−τ1j(s)))−1ωω−1∑s=0s∑r=0(− a1(r)x1(r)+b1(r)x2(r))−1ωω−1∑s=0s∑r=0(n∑j=1c1j(r)x1(r−τ1j(r))e−γ1j(r)x1(r−τ1j(r))). |
Meanwhile, we have
(Kp(I−Q)Nx)2=k−1∑s=0(− a2(s)x2(s)+b2(s)x1(s))+k−1∑s=0(n∑j=1c2j(s)x2(s−τ2j(s))e−γ2j(s)x2(s−τ2j(s)))−(kω−ω+12ω)ω∑s=1(− a2(s)x2(s)+b2(s)x1(s))−(kω−ω+12ω)ω∑s=1(n∑j=1c2j(s)x2(s−τ2j(s))e−γ2j(s)x2(s−τ2j(s)))−1ωω−1∑s=0s∑r=0(− a2(r)x2(r)+b2(r)x1(r))−1ωω−1∑s=0s∑r=0(n∑j=1c2j(r)x2(r−τ2j(r))e−γ2j(r)x2(r−τ2j(r))) |
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 KP(I−Q)N(E) is relatively compact.
An arbitrary ω-periodic solution of (2.1) corresponds one-to-one to a solution of Lx=λNx with parameter λ∈(0,1). Proposition 2.1 displays that each positive solution x=(x1,x2)T of Lx=λNx satisfies that A1<x1≤B1 and A2<x2≤B2. It is obvious that if y=(y1,y2)T∈∂Ω, then y is never a solution of Lx=λNx. Hence, the condition (i) of Lemma 2.2 holds. If x=(x1,x2)T∈∂Ω∩Ker L, then there are four cases to be considered: (1) x=(A1,x2)T, (2) x=(B1+1,x2)T, (3) x=(x1,A2)T, (4) x=(x1,B2+1)T.
Case (1): It follows from x1≡A1 that
(QNx)1=1ωω∑k=1(−A1a1(k)+b1(k)x2(k)+n∑j=1cij(k)A1e−γ1j(k)A1)≥A1ωω∑k=1(−a1(k)+1eA1¯γ1n∑j=1cij(k))>A1ωω∑k=1(−a1(k)+γeA1¯γ1a1(k))=A1ω(γeA1¯γ1−1)ω∑k=1a1(k). |
Since A1≤lnγ/¯γ1, we see that eA1¯γ1≤γ. Hence, (QNx)1>0.
Case (2): Because of x1≡B1+1, we have
(QNx)1=1ωω∑k=1(−(B1+1)a1(k)+b1(k)x2(k)+n∑j=1cij(k)(B1+1)e−γ1j(k)(B1+1))≤1ωω∑k=1(−a_1(B1+1)+¯b1B2+n∑j=1¯c1jeγ_1j)=−a_1(B1+1)+¯b1B2+1en∑j=1¯c1jγ_1j=−a_1−a_1a_2(a_1a_2−¯b1¯b2)e(n∑j=1¯c1jγ_1j+¯b1a_2n∑j=1¯c2jγ_2j)+a_1¯b1(a_1a_2−¯b1¯b2)e(n∑j=1¯c2jγ_2j+¯b2a_1n∑j=1¯c1jγ_1j)+1en∑j=1¯c1jγ_1j=−a_1<0. |
Similarly, we can show that (QNx)2>0 in Case (3) and (QNx)2<0 in Case (4). We therefore conclude that QNx=((QNx)1,(QNx)2)T≠0 for each x∈∂Ω∩Ker L.
Define a continuous operator H:Ω∩Ker L×[0,1]→X by
H(x,μ)=(H1(x,μ)H2(x,μ))=(−μ(Ix1−A1+B12)+(1−μ)(QNx)1−μ(Ix2−A2+B22)+(1−μ)(QNx)2). |
Recall that the elements of ∂Ω∩Ker L are vectors satisfying x=(A1,x2)T, y=(B1+1,y2)T, z=(z1,A2)T and w=(w1,B2+1)T. For x=(A1,x2)T, we can check that
H1(x,μ)=−μ(A1−A1+B12)+(1−μ)(QNx)1=−μ(A1−B12)+(1−μ)(QNx)1>0. |
Moreover,
H1(y,μ)=−μ(B1+1−A1+B12)+(1−μ)(QNy)1=−μ(A1−B1+22)+(1−μ)(QNy)1<0 |
for y=(B1+1,y2)T. 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
deg{QN,Ω∩Ker L,0}=deg{(−Ix1+A1+B12−Ix2+A2+B22),Ω∩Ker L,0}=1≠0. |
Hence, the condition (ⅱ) of Lemma 2.2 holds. Therefore, the equation Lx=Nx 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.
Consider the delay difference system
{Δx1(k)=−a1(k)x1(k)+b1(k)x2(k)+c11(k)x1(k−1)e−γ11(k)x1(k−1)+c12(k)x1(k−1)e−γ12(k)x1(k−1),Δx2(k)=−a2(k)x2(k)+b2(k)x1(k)+c21(k)x2(k−4)e−γ21(k)x2(k−4)+c22(k)x2(k−4)e−γ22(k)x2(k−4). |
Here, we assume that
a1(k)={1/2ifk=1,2/5ifk=2,1/4ifk=3,1/5ifk=4,a2(k)={3/4ifk=1,3/5ifk=2,1/2ifk=3,5/6ifk=4, |
b1(k)={1/5ifk=1,1/4ifk=2,1/7ifk=3,1/6ifk=4,b2(k)={1/20ifk=1,1/12ifk=2,1/24ifk=3,1/18ifk=4, |
c11(k)={1/2ifk=1,3/4ifk=2,1/3ifk=3,2/3ifk=4,c12(k)={5/6ifk=1,4/5ifk=2,2/5ifk=3,1/6ifk=4,c21(k)={7/8ifk=1,4/5ifk=2,2/3ifk=3,6/7ifk=4,c22(k)={1/4ifk=1,1/2ifk=2,1/10ifk=3,20/21ifk=4, |
γ11(k)={3ifk=1,1ifk=2,1.5ifk=3,2ifk=4,γ12(k)={10ifk=1,4ifk=2,3ifk=3,5ifk=4,γ21(k)={5ifk=1,2ifk=2,1ifk=3,2.5ifk=4,γ22(k)={2ifk=1,1.5ifk=2,8ifk=3,3ifk=4. |
In addition, ai(k)=ai(k+4), bi(k)=bi(k+4), cij(k)=cij(k+4) and γij(k)=γij(k+4) for k∈Z, i=1,2 and j=1,2. Theorem 3.1 shows that the system has at least one positive 4-periodic solution.
It is clear that ω=4, ai, bi, cij, γij and τij (1≤i≤2,1≤j≤2) are ω-periodic discrete functions satisfying 0<ai(k)<1, 0<bi(k)<1, cij(k)>0 and γij(k)>0 for k∈Z+. Since a_1=1/5, a_1=1/2, ¯b1=1/4 and ¯b2=1/12, we see that
a_1a_1−¯b1¯b2=15×12−14×112=19240>0. |
Hence, condition (2.2) is satisfied. Let γ=11/10>1. Then, we can easily check condition (2.3)
(c11(k)+c12(k))>γa1(k)and(c21(k)+c22(k))>γa2(k) |
for k=1,2,3,4. Moreover, it can be calculated that
4∑k=1(c11(k)+c12(k))4∑k=1(a1(k)−b1(k))=1869248>1and4∑k=1(c21(k)+c22(k))4∑k=1(a2(k)−b2(k))=221106181>1. |
Namely, condition (3.1) holds. Therefore, from Theorem 3.1, it turns out that the system has at least one positive 4-periodic solution.
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 (3.1) of Theorem 3.1 can be rewritten into the form
ω∑k=1(n∑j=1(c1j(k)+b1(k))>ω∑k=1a1(k)andω∑k=1(n∑j=1(c2j(k)+b2(k))>ω∑k=1a2(k). |
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).
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The paper is supported by College Students Innovations Special Project funded by Northeast Forestry University of China (Grant No. 202210225156) and Fundamental Research Funds for the Central Universities of China (Grant No. 41422003).
The authors declare that there is no conflicts of interest.
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