Existence and Stability of Positive Periodic Solutions for a Neutral Multispecies Logarithmic Population Model with Feedback Control and Impulse

Definition 1. A function N_i : R → (0, ∞) (i = 1,2, …, n) is said to be a positive solution of (1), if the following conditions are satisfied:

• (a)

  N_i(t) is absolutely continuous on each (t_k, t_k+1);

• (b)

  for each k ∈ Z₊, $$ and $$ exist, and $$;

• (c)

  N_i(t) satisfies the first equation of (1) for almost everywhere (for short a.e.) in [0, ∞]∖{t_k} and satisfies $$ for t = t_k,  k ∈ Z₊ = {1,2, …}.

Definition 2. System (1) is said to be globally attractive, if there exists a positive solution (N_i(t), u_i(t)) of (1) such that $$, $$, for any other positive solution $$ of the system (1).

Lemma 3. $$ is the positive invariable region of the system (1).

Proof. In view of biological population, we obtain N_i(0) > 0, u_i(0) > 0. By the system (1), we have

where Then the solution of the system (1) is positive.

Under the above hypotheses (H₁)–(H₃), we consider the neutral nonimpulsive system:

where By a solution (y_i(t), u_i(t)) of (4), it means an absolutely continuous function (y_i(t), u_i(t)) defined on [−τ, 0] that satisfies (4) a.e., for t ≥ 0, and y(ξ) = φ(ξ), y^′(ξ) = φ^′(ξ) on [−τ, 0].

Lemma 4. Suppose that (H₁)–(H₄) hold. Then

• (i)

  if (y_i(t), u_i(t)) is a solution of (4) on [−τ, +∞), then $$ is a solution of (1) on [−τ, +∞),

• (ii)

  if (N_i(t), u_i(t)) is a solution of (1) on [−τ, +∞), then $$ is a solution of (4) on [−τ, +∞).

Proof. (i) It is easy to see that $$ is absolutely continuous on every interval (t_k, t_k+1], t ≠ t_k,  k = 1,2, …,

On the other hand, for any t = t_k, k = 1,2, …, Thus It follows from (6)–(8) that (N_i(t), u_i(t)) is a solution of (1).

(ii) Since $$ is absolutely continuous on every interval (t_k, t_k+1], t ≠ t_k, k = 1,2, …, and in view of (8), it follows that for any k = 1,2, …,

which implies that y_i(t) is continuous on [−τ, +∞). It is easy to prove that y_i(t) is absolutely continuous on [−τ, +∞). Similar to the proof of (i), we can check that $$ are solutions of (4) on [−τ, +∞). The proof of Lemma 4 is completed.

Lemma 5. (y_i(t), u_i(t)) is a ω-periodic solution of (4) if and only if y_i(t) is a ω-periodic solution of the following system:

where and G_i(t, s) is defined by (3).

Proof. The proof of Lemma 5 is similar to that of Lemma 2.2 in [2], and we omit the details here.

Lemma 6. Assume that u(t), τ(t) are all continuously differentiable ω-periodic functions and a(t) is a nonnegative continuous ω-periodic function such that $$; then

where c(t) = b(t)/(1 − τ^′(t)).

Proof. As

Denote $$; then from a(t) ≥ 0, $$, it follows that m < 1. Also, when t ≥ s without loss of generality, we may assume that s + nω ≤ t ≤ s + (n + 1)ω; thus Therefore, and so from (13)–(15) it follows that The proof Lemma 6 is complete.

Theorem 7. In addition to (H₁)–(H₃), assume further that there exist positive constants h_i  (i = 1,2, …, n) and a positive constant M < 1 such that

•  

  H₄ $$.

Then, system (1) has a unique ω-periodic solution with strictly positive components, where Δ_i(t) is defined by (18).

Proof. From the above analysis, to finish the proof of Theorem 7, it is enough to prove under the conditions of Theorem 7 that system (17) has a unique ω-periodic solution. Let

under the norm ∥u∥ = max _1≤i≤nmax _t∈[0,ω]{|u_i(t)|}, Ω is a Banach space. For any u(t) = (u₁(t), …, u_n(t)) ^T ∈ Ω, we consider the periodic solution x_u(t) of periodic differential equation Since A_ii(t) > 0, we know that the linear system of system (20) admits exponential dichotomies on R, and so system (20) has a unique periodic solution x_u(t), which can be expressed as where its proof is similar to that of Theorem 1 in [18]; here we omit it.

Now, by using Lemma 6, x_iu(t) can also be expressed as

where Now we define mapping T : Ω → Ω, Tu(t) = x_u(t). Following this we will prove that T is a contraction mapping; that is, there exists a constant β ∈ (0,1), such that ∥Tu − Tv∥ ≤ β∥u − v∥, for all u, v ∈ Ω. In fact, for any u(t) = (u₁(t), …, u_n(t)) ^T and v(t) = (v₁(t), …, v_n(t)) ^T, we have Hence, It follows from (H₄) that ∥Tu − Tv∥ ≤ ∥u − v∥ for all u, v ∈ Ω. That is, T is a contraction mapping. Hence, there exists a unique fixed point $$; that is, Tx^*(t) = x^*(t). Therefore, x^*(t) is the unique periodic solution of system (17). It follows from (1), (4), (10), and (17) that $$ is the unique positive periodic solution of system (1). The proof of Theorem 7 is completed.

Theorem 8. In addition to (H₁)–(H₄), suppose further that the following condition holds:

•  

  H₅ $$, as t → +∞, i = 1,2, …, n.

Then system (1) has a unique periodic solution which is globally attractive.

Proof. Let $$ be the unique positive periodic solution of system (1), whose existence and uniqueness are guaranteed by Theorem 7, and let N(t) = (N₁(t), N₂(t), …, N_n(t)) ^T be any other solution of system (1). Let $$, $$; then, similar to (17), we have

Let $$; then Multiply both sides of (29) with $$, and then integrate from 0 to t to obtain then Let $$; we see that Substituting (32) into (31), we get therefore, we have where Δ_i(t) is defined by (18). From (H₄), we have From (H₅), we have thus, $$, as t → +∞, i = 1,2, …, n. Hence, the positive ω-periodic solution of (17) is globally attractive; accordingly, $$, $$ as  t → +∞, i = 1,2, …, n, and by Definition 2, the positive ω-periodic solution of (1) is globally attractive. The proof of Theorem 8 is completed.

Remark 9. If e_i(t) = f_i(t) = α_i(t) = β_i(t) = ϑ_i(t) = 0,  θ_ik + 1 = 0,  i = 1,2, …, n,  k = 1,2, …, then system (1) is studied by [3]. Hence, Theorems 7 and 8 generalize the corresponding results in [3].

Remark 10. If θ_ik + 1 = 0, i = 1,2, …, n, k = 1,2, …, then system (1) is studied by [4]. Hence Theorems 7 and 8 also generalize the corresponding results in [4].

Corollary 11. In addition to conditions (H₁)–(H₃), assume further that there exists a positive constant M < 1 such that

•  

  H₆ $$.

Then, (37) has a unique ω-periodic solution with strictly positive components, where Δ(t) is defined by (38).

Corollary 12. In addition to conditions (H₁)–(H₃) and (H₆), suppose further that the following condition holds:

•  

  H₇ $$ as t → +∞.

Then, system (37) has a unique periodic solution which is globally attractive.

Remark 13. The results in the work show that by means of appropriate impulsive perturbations and feedback control we can control the dynamics of these equations.