Permanence in Multispecies Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Lemma 2.1. Suppose that there is a positive constant ω such that

and function is bounded on t ∈ R₊ and μ ∈ [0, ω]. Then we have

• (a)

  there exist positive constants m and M such that

  $$ (2.9)
  for any positive solution x(t) of system (2.6);

• (b)

  lim _t→∞(x⁽¹⁾(t) − x⁽²⁾(t)) = 0 for any two positive solutions x⁽¹⁾(t) and x⁽²⁾(t) of system (2.6).

Lemma 2.2. If assumption (H₂) holds, then there exist constants d > 0 and D > 0 such that for any t₂ ≥ t₁ ≥ 0

Theorem 3.1. Suppose that assumptions (H₁)-(H₂) hold. If there exist positive constants ω_i such that for each 1 ≤ i ≤ n:

and the functions are bounded for all t ∈ R₊ and μ ∈ [0, ω_i]. Then the system (1.1) is permanent, that is, there are positive constants γ > 0 and M > 0 such that for any positive solution x(t) = (x₁(t), x₂(t), …, x_n(t)) of system (1.1).

Proof. Let x(t) = (x₁(t), x₂(t), …, x_n(t)) be any positive solution of system (1.1). We first prove that the components x_i  (i = 1,2, …, n) of system (1.1) are bounded. From assumption (H₁) and the ith equation of system (1.1), we have

by the comparison theorem of impulsive differential equation, we have where y_i(t) is the solution of (3.1) with initial value y_i(0) = x_i(0). From the condition (3.2), we directly have Hence, from conclusion (a) of Lemma 2.1, we can obtain a constant M_i1 > 0, and there is a T_i1 > 0 such that y_i(t) < M_i1 for all t ≥ T_i1. Let M = max _1≤i≤n{M_i1} and T₁ = max _1≤i≤n{T_i1}, we have Hence, we finally have

Claim 1. There is a constant η > 0 such that limsup _t→∞x_i(t) > η  (i = 1,2, …, n) for any positive solution x(t) = (x₁(t), x₂(t), …, x_n(t)) of system (1.1). In fact, if Claim 1 is not true, then there is an integer k ∈ {1,2, …, n} and a positive solution x(t) = (x₁(t), x₂(t), …, x_n(t)) of system (1.1) such that

Hence, there is a constant T₄ > T₃ such that On the other hand, by (3.13) there is a T₅ ≥ T₄ such that where i = 1,2, …, n and i ≠ k. By (3.11) and (3.16), we obtain for all t ≥ T₅ + τ. Thus, from (3.12) we finally obtain lim _t→∞x_k(t) = ∞, which lead to a contradiction.

Claim 2. There is a constant γ > 0 such that liminf _t→∞x_i(t) > γ  (i = 1,2, …, n) for any positive solution of system (1.1).

If Claim 2 is not true, then there is an integer k ∈ {1,2, …, n} and a sequence of initial function {ϕ_m} ⊂ C₊[−τ, 0] such that

where constant η is given in Claim 1. By Claim 1, for every m there are two time sequences $$ and $$, satisfying: such that From the above proof, there is a constant T^(m) ≥ T₂ such that x_i(t, ϕ_m) < M  (i = 1,2, …, n) for all t ≥ T^(m). Further, there is an integer $$ such that $$ for all $$. From (3.11) and lemma 2.2., we can obtain where $$. Consequently, from (3.20) we have By (3.12), there is a large enough P > 0 such that for all t ≥ T₂, a ≥ P and a ∈ [lw_i, (l + 1)w_i) and i = 1,2, …, n, then, we obtain where $$. So, we choose L = 2 + (r₂w_i/ε) such that for all l > L, we have From (3.23), there is an integer N₀ such that for any m > N₀ and $$, we have where constant Q > P + τ.

So, when m > N₀ and $$, for any $$, from (3.11), (3.21), (3.25), and (3.26) we can obtain

Consequently, from (3.20) and (3.25) it follows This leads to a contradiction. Therefore, Claim 2 is true. This completes the proof.

When system (1.1) degenerates into the periodic case, then we can assume that there is a constant ω > 0 and an integer q > 0 such that a_i(t + ω) = a_i(t), b_i(t + ω) = b_i(t), a_ij(t + ω) = a_ij(t), t_k+q = t_k + ω and h_ik+q = h_ik for all t ∈ R₊, k = 1,2, … and i, j = 1,2, …, n. From Remarks  2.3 and  2.4 in [1], we can see the fixed positive solution x_j0 of system (3.1) can be chosen to be the ω-periodic solution of system (3.1). Therefore, as a consequence of Theorem 3.1. we have the following result.

Corollary 3.2. Suppose that system (1.1) is ω-periodic and for each i = 1,2, …, n,

Then, system (1.1) is permanent.