Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations

Definition 2.1

([7, 8]) x ∈ C(ℝ, ℝⁿ) is called almost periodic, if for any ϵ > 0, it is possible to find a real number l = l(ϵ) > 0, for any interval with length l(ϵ), there exists a number τ = τ(ϵ) in this interval such that ∥x(t+τ) − x(t)∥ < ϵ, ∀ t ∈ ℝ, where ∥⋅∥ is arbitrary norm of ℝⁿ. τ is called to the ϵ-almost period of x, T(x, ϵ) denotes the set of ϵ-almost periods for x and l(ϵ) is called to the length of the inclusion interval for T(x, ϵ). The collection of those functions is denoted by AP(ℝ, ℝⁿ). Let AP(ℝ): = AP(ℝ, ℝ).

Lemma 2.1

([5]) If x ∈ AP(ℝ) is differentiable, then for ∀ϵ > 0, it follows:

1. there exists ξ_ϵ ∈ [0, +∞) such that x(ξ_ϵ) ∈ [x⁺ − ϵ, x⁺] and x′(ξ_ϵ) = 0;

2. there exists η_ϵ ∈ [0, +∞) such that fx(η_ϵ) ∈ [x⁻, x⁻+ϵ] and x′(η_ϵ) = 0.

Lemma 2.2

([6]) If x ∈ AP(ℝ) is differentiable, for any interval [a, b] with b − a > 0, let ξ, η ∈ [a, b] and

then

Lemma 2.3

([6]) Assume that x ∈ AP(ℝ), there exist ξ ∈ [a, b], ξ ∈ (−∞, a] and ξ ∈ [b, +∞) such that

Lemma 2.4

([6]) Assume that x ∈ AP(ℝ), there exist η ∈ [a, b], η ∈ (−∞, a] and η ∈ [b, +∞) such that

Lemma 2.5

([7]) If f ∈ AP(ℝ) and f¯=m(f)=1T∫0Tf(s)ds>0, then we have

where a is an arbitrary real constant and T₀ > 0 is a constant independent of a.

The method to be used in this paper involves the applications of the continuation theorem of coincidence degree.

Let 𝕏 and 𝕐 be real Banach spaces, L : DomL ⊆ 𝕏 → 𝕐 be a linear mapping and N : 𝕏 → 𝕐 be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if ImL is closed in 𝕐 and dimKerL = codimImL < +∞. If L is a Fredholm mapping of index zero and there exist continuous projectors P : 𝕏 → 𝕏 and Q: 𝕐 → 𝕐 such that ImP = KerL, Ker Q = ImL = Im (I − Q). It follows that L_P = L|_DomL∩KerP : (I − P)𝕏 → ImL is invertible and its inverse is denoted by K_P. If Ω is an open bounded subset of 𝕏, the mapping N will be called L-compact on Ω if QN(Ω) is bounded and K_P(I−Q)N : Ω → 𝕏 is compact. Since Im Q is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL.

Lemma 2.6

([9]) Let Ω ⊆ 𝕏 be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on Ω. If all the following conditions hold:

1. Lx ≠ λ Nx, ∀ x ∈ ∂Ω ∩ DomL, λ ∈ (0, 1);

2. QNx ≠ 0, ∀ x ∈ ∂Ω ∩ KerL;

3. deg{JQN, Ω ∩ KerL, 0} ≠ 0, where deg(⋅, ⋅, ⋅) is the Brouwer degree.

Then Lx = Nx has a solution on Ω ∩ DomL.

For f ∈ AP(ℝ), we denote by

the set of Fourier exponents and the module of f, respectively.

Theorem 3.1

Assume that (H₁) holds. Suppose further that

1. μ > 0 and ν > 0,

then system (1.2) admits at least one positive almost periodic solution.

Proof

Under the invariant transformation (x₁, x₂, x₃)^T = (e^y₁, e^y₂, e^y₃)^T, system (1.2) reduces to

It is easy to see that if system (3.0) has one almost periodic solution (y₁, y₂, y₃)^T, then (x₁, x₂, x₃)^T = (e^y₁, e^y₂, e^y₃)^T is a positive almost periodic solution of system (1.2). Therefore, to complete the proof it suffices to show that system (3.0) has one almost periodic solution.

Take 𝕏 = 𝕐 = 𝕍₁ ⨁ 𝕍₂, where

where

φ = (φ₁, φ₂, φ₃)^T ∈ C([−τ₀, 0], ℝ³), τ₀ = max_{i = 1, 2}{τ⁺, δ⁺, σ⁺}, θ₀ is a given positive constant. Define the norm

then 𝕏 and 𝕐 are Banach spaces with the norm ∥⋅∥. Set

where DomL = {z = (y₁, y₂, y₃)^T ∈ 𝕏: y₁, y₂, y₃ ∈ C¹(ℝ)} and

With these notations, system (3.0) can be written in the form

It is not difficult to verify that KerL = 𝕍₂, ImL = 𝕍₁ is closed in 𝕐 and dim KerL = 3 = codim ImL. Therefore, L is a Fredholm mapping of index zero (see Lemma 2.12 in [6]). Now define two projectors P: 𝕏 → 𝕏 and Q: 𝕐 → 𝕐 as

Then P and Q are continuous projectors such that ImP = KerL and ImL = KerQ = Im(I − Q).

Furthermore, through an easy computation we find that the inverse K_P : ImL → KerP ∩ DomL of L_P has the form

Then QN : 𝕏 → 𝕐 and K_P(I − Q)N : 𝕏 → 𝕏 read

where f(x) is defined by f[x(t)] = ∫0t [Nx(s) − QNx(s)] ds.ąd’ Then N is L-compact on Ω (see Lemma 2.13 in [6]).

In order to apply Lemma 2.6, we need to search for an appropriate open-bounded subset Ω.

Corresponding to the operator equation Lz = λ z, λ ∈ (0, 1), we have

Suppose that z = (y₁, y₂, y₃)^T ∈ DomL ⊆ 𝕏 is a solution of system (3.1) for some λ ∈ (0, 1), where DomL = {z = (y₁, y₂, y₃)^T ∈ 𝕏 : y_i ∈ C¹(ℝ), i = 1, 2, 3}.

There must exist {αn(i):n∈N+} such that

From (H₁) and Lemma 2.5, ∀ t₀ ∈ ℝ, there exists a constant ω ∈ (0, +∞) independent of t₀ such that

For ∀ n₀ ∈ ℕ⁺, we consider αn0(i)−ω,αn0(i) (i = 1, 2, 3), where ω is defined as that in (3.3). By Lemma 2.3, there exist ξi∈αn0(i)−ω,αn0(i),ξ_i∈−∞,αn0(i)−ω and ξ¯i∈αn0(i),+∞ such that

By (3.4), we obtain from system (3.1) that

From the second equation of system (3.5), it follows from (3.3)-(3.4) that

which implies that

Let Ii=s∈ξi,αn0(i):y˙i(s)≥0(i=1,2,3). It follows from the first equation of system (3.1) and (3.3) that

By Lemma 2.2, it follows from (3.6)-(3.7) that

which implies that

In view of (3.2), letting n₀ → +∞ in the above inequality leads to

Similar to the arguments as that in (3.6)-(3.7), we can obtain from the third equation of system (3.5) that

By a similar discussion as that in (3.8), it follows from (3.9) that

On the other hand, from (H₂) and Lemma 2.5, ∀ t₀ ∈ ℝ, there exists a constant l ∈ [ω, +∞) independent of t₀ such that

For ∀ n₀ ∈ ℤ, by Lemma 2.4, there exist η_i ∈ [n₀l, n₀l+l], η_i ∈ (−∞, n₀l] and η_i ∈ [n₀l+l, +∞) such that

By (3.12), we obtain from system (3.1) that

By the first equation of system (3.13), we have from (3.3), (3.11) and (3.12) that

which yields that

Furthermore, by the second equation of system (3.1), it follows that

Then

Obviously, f2− is a constant independent of n₀. So it follows from (3.16) that

Similar to the arguments as that in (3.14)-(3.15), we can obtain from the third equation of systems (3.1) and (3.13) that

and

It follows from (3.18)-(3.19) that

Obviously, f3− is a constant independent of n₀. So it follows from (3.20) that

Finally, there must exist {θ_n : n ∈ ℕ⁺} such that

where y1− = inf_{s ∈ ℝ}y₁(s). For ∀ n₀ ∈ ℕ⁺, we consider [θ_n0 − ω, θ_n0], where ω is defined as that in (3.3). By Lemma 2.4, there exist η₁ ∈ [θ_n0 − ω, θ_n0], η₁ ∈ (−∞, θ_n0 − ω] and η₁ ∈ [θ_n0, +∞) such that

In view of the first equation of system (3.1), we get

which implies from (3.2) and (3.23) that

which yields that

Let J = {s ∈ [η₁, θ_n0]: ẏ₁(s) ≤ 0}. Then we have from the first equation of system (3.1) and (3.2) that

By Lemma 2.2, we get from (3.24)-(3.25) that

So

Letting n₀ → +∞ in the last inequality leads to

Furthermore, by the first equation of system (3.5), we have from (3.3)-(3.4) that

which yields that

And

Similar to the argument as that in (3.8), we obtain from (3.27)-(3.28) that

Set C=∑i=13(|fi−|+|fi+|)+1. Clearly, C is independent of λ ∈ (0, 1). Let Ω = {z ∈ 𝕏 : ∥z∥_𝕏 < C}. Therefore, Ω satisfies condition (a) of Lemma 2.6.

Now we show that condition (b) of Lemma 2.6 holds, i.e., we prove that QNz ≠ 0 for all z = (y₁, y₂, y₃)^T ∈ ∂Ω ∩ KerL = ∂Ω ∩ ℝ³. If it is not true, then there exists at least one constant vector z0=(y10,y20,y30)T ∈ ∂Ω such that

Similar to the argument as that in (3.8), (3.10), (3.17), (3.21), (3.26) and (3.29), it follows that

Then z₀ ∈ Ω ∩ ℝ³. This contradicts the fact that z₀ ∈ ∂Ω. This proves that condition (b) of Lemma 2.6 holds.

Finally, we will show that condition (c) of Lemma 2.6 is satisfied. Let us consider the homotopy

where

From the above discussion it is easy to verify that H(ι, z) ≠ 0 on ∂Ω ∩ KerL, ∀ ι ∈ [0, 1]. Further, Φ z = 0 has a solution: