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Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms
Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms✩ Bingwen Liu, Shuhua Gong ∗ College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China
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Article history: Received 21 October 2010 Accepted 13 December 2010 Keywords: Nonlinear density-dependent mortality term Time-varying delays Permanence Nicholson-type delay differential system
abstract In this paper, we study the generalized Nicholson-type delay systems with nonlinear density-dependent mortality terms. Under proper conditions, we establish some criteria to guarantee the permanence of this model. Moreover, we give two examples to illustrate our main results. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction In a classic study of population dynamics, a delay differential equation model: x′ (t ) = −δ x(t ) + Px(t − τ )e−ax(t −τ ) ,
(1.1)
is frequently used, where x(t ) is the size of the population at time t , P is the maximum per capita daily egg production, 1a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time. This equation was introduced by Nicholson [1] to model laboratory fly population. Its dynamics was later studied in [2,3], where this model was referred to as the Nicholsons blowflies equation [2]. Recently, to describe the models of Marine Protected Areas and B-cell Chronic Lymphocytic Leukemia dynamics that belong to the Nicholson-type delay differential systems, Berezansky et al. [4] proposed the following Nicholson-type delay systems:
x′1 (t ) = −a1 x1 (t ) + b1 x2 (t ) + c1 x1 (t − τ )e−x1 (t −τ ) , x′2 (t ) = −a2 x2 (t ) + b2 x1 (t ) + c2 x2 (t − τ )e−x2 (t −τ ) ,
(1.2)
with initial conditions: xi (s) = ϕi (s),
s ∈ [−τ , 0], ϕi (0) > 0,
(1.3)
where ϕi ∈ C ([−τ , 0], [0, +∞)), ai , bi , ci and τ are nonnegative constants, i = 1, 2. According to the new studies in population dynamics (see [5,6]), a linear model of density-dependent mortality will be most accurate for populations at low densities, and marine ecologists are currently in the process of constructing new
✩ This work was supported by the Key Project of Chinese Ministry of Education (Grant No. 210 151), and the Scientific Research Fund of Zhejiang Provincial Education Department of PR China (Grant No. Y200907784). ∗ Corresponding author. Tel.: +86 057383643075. E-mail address: [email protected] (S. Gong).
1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.12.009
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B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
fishery models with nonlinear density-dependent mortality rates. Therefore, Berezansky et al. [5] point out an open problem: How about the dynamic behaviors of the following scalar Nicholson’s blowflies model with a nonlinear density-dependent mortality term: x′ (t ) = −D(x(t )) + Px(t − τ )e−x(t −τ ) ,
(1.4)
where the nonlinear density-dependent mortality term D(x) might have one of the following forms: D(x) = bax or +x D(x) = a − be−x with constants a, b > 0. Now, a corresponding question arises: How about the dynamic behaviors of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. The main purpose of this paper is to give the conditions to ensure the permanence of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, we will consider the following Nicholson-type delay systems with nonlinear density-dependent mortality terms: x′1 (t ) = −D11 (t , x1 (t )) + D12 (t , x2 (t )) + c1 (t ) x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) , x′2 (t ) = −D22 (t , x2 (t )) + D21 (t , x1 (t )) + c2 (t ) x2 (t − τ2 (t ))e−γ2 (t )x2 (t −τ2 (t )) ,
(1.5)
where Dij (t , x) =
aij (t )x bij (t ) + x
or Dij (t , x) = aij (t ) − bij (t )e−x ,
aij , bij , ci , γi : R → (0, +∞) are all continuous functions bounded above and below by positive constants, τi : R → [0, +∞) is a bounded continuous function, ri = supt ∈R τi (t ) > 0, and i, j = 1, 2. Throughout this paper, the set of all (nonnegative) real vectors will be denoted by R(R+ ). Let C = C ([−r1 , 0], R) × C ([−r2 , 0], R) be the continuous functions space equipped with the usual supremum norm ‖ · ‖, and let C+ = C ([−r1 , 0], R+ ) × C ([−r2 , 0], R+ ). For the sake of convenience, we set g + = sup g (t ), t ∈R
g − = inf g (t ), t ∈R
where g is a bounded continuous function defined on R. If xi (t ) is defined on [t0 − ri , σ ) with t0 , σ ∈ R and i = 1, 2, then we define xt ∈ C as xt = (x1t , x2t ) where xit (θ ) = xi (t + θ ) for all θ ∈ [−ri , 0] and i = 1, 2. For any ϕ, ψ ∈ C , we write ϕ ≤ ψ if ψ − ϕ ∈ C+ , ϕ < ψ if ϕ ≤ ψ and ϕ ̸= ψ . Due to the biological interpretation of model (1.5), only positive solutions are meaningful and therefore admissible. Thus, we just consider the admissible initial conditions xt0 = ϕ,
ϕ ∈ C+ and ϕi (0) > 0,
i = 1, 2.
(1.6)
We write xt (t0 , ϕ)(x(t ; t0 , ϕ)) for a solution of the initial value problems (1.5) and (1.6). Also, let [t0 , η(ϕ)) be the maximal right-interval of existence of xt (t0 , ϕ). The remaining part of this paper is organized as follows. In Sections 2 and 3, we shall derive new sufficient conditions for checking the permanence of model (1.5). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections. 2. Permanence of Nicholson-type delay systems with Dij (t , x) =
aij (t )x bij (t )+x
(i, j = 1, 2)
Theorem 2.1. Assume that the following conditions are satisfied. c1+ c2+ − + + min{a− , 11 , a22 } > a12 + a21 + − + eγ1 eγ2− sup
aii (t )
t ∈R bii (t )ci (t )
< 1,
(2.1)
i = 1, 2.
Then, the system (1.5) and (1.6) with Dij (t , x) =
(2.2) aij (t )x bij (t )+x
(i, j = 1, 2) is permanent.
B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
1933
Proof. Let x(t ) = x(t ; t0 , ϕ). First, under the conditions (2.1) and (2.2) we show that x(t ) for all t ∈ [t0 , η(ϕ)) is bounded, and η(ϕ) → +∞. In view of ϕ ∈ C+ , using Theorem 5.2.1 in [7, p. 81], we have xt (t0 , ϕ) ∈ C+ for all t ∈ [t0 , η(ϕ)). From (1.5) and the fact that a(t )x b(t ) + x
a(t )x
≤
for all t ∈ R, x ≥ 0,
b(t )
we get x′1 (t ) = −
≥−
a11 (t )x1 (t ) b11 (t ) + x1 (t ) a11 (t )x1 (t ) b11 (t )
+
a12 (t )x2 (t ) b12 (t ) + x2 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t ))
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) .
(2.3)
In view of x1 (t0 ) = ϕ1 (0) > 0, integrating (2.3) from t0 to t, we have x1 (t ) ≥ e
−
a11 (u) t0 b11 (u) du
t
x1 (t0 ) + e
−
a11 (u) t0 b11 (u) du
∫
t
t
a11 (v) t0 b11 (v) dv
s
e
c1 (s)x1 (s − τ1 (s))e−γ1 (s)x1 (s−τ1 (s)) ds
t0
> 0,
for all t ∈ [t0 , η(ϕ)).
x2 (t ) > 0,
for all t ∈ [t0 , η(ϕ)).
Similarly
It follows that xi (t ) > 0
for all t ∈ [t0 , η(ϕ)), i = 1, 2.
Set y(t ) = x1 (t ) + x2 (t ),
where t ∈ [t0 − r , η(ϕ)), r = min{r1 , r2 }.
Notice that max xe−x = x≥0
1 e
and
α1 + α2 α1 α2 ≤ + , where αi > 0, 1 ≤ i ≤ 6. α3 + α4 + α5 + α6 α3 + α4 α5 + α6
We obtain y′ (t ) = −
− ≤− ≤−
a11 (t )x1 (t ) b11 (t ) + x1 (t ) a22 (t )x2 (t )
+
a12 (t )x2 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t ))
b12 (t ) + x2 (t ) a21 (t )x1 (t )
b22 (t ) + x2 (t ) b21 (t ) + x1 (t ) a11 (t )x1 (t ) + a22 (t )x2 (t ) b11 (t ) + b22 (t ) + x1 (t ) + x2 (t ) − a− 11 x1 (t ) + a22 x2 (t )
b11 (t ) + b22 (t ) + x1 (t ) + x2 (t ) min{a11 , a22 }y(t ) −
≤−
+
+ c2 (t )x2 (t − τ2 (t ))e−γ2 (t )x2 (t −τ2 (t ))
+ a12 (t ) + a21 (t ) + + + a+ 12 + a21 +
−
+
b11 (t ) + b22 (t ) + y(t )
+ + a+ 12 + a21 +
c1
eγ1
−
c1+ eγ1−
c1 (t )
+
eγ1 (t )
+
c2 (t ) eγ2 (t )
c2+ eγ2−
+
+
c2
eγ2−
.
For each t ∈ [t0 − r , η(ϕ)), we define M (t ) = max{ξ : ξ ≤ t , y(ξ ) =
max y(s)}.
t0 −r ≤s≤t
Now we are in a position to prove that y(t ) is bounded on [t0 , η(ϕ)). In the contrary case, observe that M (t ) → η(ϕ) as t → η(ϕ), we have lim y(M (t )) = +∞.
t →η(ϕ)
But y(M (t )) = maxt0 −r ≤s≤t y(s), and so y′ (M (t )) ≥ 0 for all M (t ) > t0 . Thus, 0 ≤ y′ (M (t ))
≤−
− min{a− 11 , a22 }y(M (t ))
b11 (M (t )) + b22 (M (t )) + y(M (t ))
+ + a+ 12 + a21 +
c1+ eγ1−
+
c2+ eγ2−
,
for all M (t ) > t0 ,
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B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
which yields − min{a− 11 , a22 }y(M (t ))
b11 (M (t )) + b22 (M (t )) + y(M (t ))
+ ≤ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
,
for all M (t ) > t0 .
(2.4)
Noting that the continuities and boundedness of the functions bij (t ), i, j = 1, 2, we can select a sequence {Tn }+∞ n=1 such that lim Tn = η(ϕ),
lim y(M (Tn )) = +∞,
n→+∞
n→+∞
lim bij (M (Tn )) = b∗ij .
n→+∞
(2.5)
In view of (2.4), we get − min{a− 11 , a22 }y(M (Tn ))
b11 (M (Tn )) + b22 (M (Tn )) + y(M (Tn ))
+ ≤ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
.
(2.6)
Letting n → +∞, (2.5) and (2.6) imply that c1+ c2+ − + + , min{a− 11 , a22 } ≤ a12 + a21 + − + eγ1 eγ2− which contradicts with (2.1). This implies that y(t ) is bounded on [t0 , η(ϕ)). From Theorem 2.3.1 in [8], we easily obtain η(ϕ) = +∞. Thus, every solution x(t ; t0 , ϕ) of (1.5) and (1.6) is positive and bounded on [t0 , +∞). So there exist two positive constants K1 and K2 such that 0 < x1 (t ) ≤ K1 ,
0 < x2 (t ) ≤ K2
for all t > t0 .
It follows that lim sup x1 (t ) ≤ K1 , t →+∞
lim sup x2 (t ) ≤ K2 .
(2.7)
t →+∞
We next prove that there exist two positive constants k1 and k2 such that lim inf x1 (t ) ≥ k1 , t →+∞
lim inf x2 (t ) ≥ k2 .
(2.8)
t →+∞
From the first equation of system (1.5), we have x′1 (t ) ≥ −
a11 (t )x1 (t ) b11 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) ,
x1 (t ) > 0,
(2.9)
where t ∈ [t0 , +∞). Suppose the first expression of (2.8) does not hold, that is, lim inf x1 (t ) = 0. t →+∞
For each t ≥ t0 , we define
θ (t ) = max{ξ : ξ ≤ t , x1 (ξ ) = min x1 (s)}. t0 ≤s≤t
Observe that θ (t ) → +∞ as t → +∞ and that lim x1 (θ (t )) = 0.
(2.10)
t →+∞
However, x1 (θ (t )) = mint0 ≤s≤t x1 (s), and so x′1 (θ (t )) ≤ 0, where θ (t ) > t0 . According to (2.9), we have 0 ≥ x′1 (θ (t ))
≥−
a11 (θ (t ))x1 (θ (t )) b11 (θ (t ))
+ c1 (θ (t ))x1 (θ (t ) − τ1 (θ (t )))e−γ1 (θ (t ))x1 (θ (t )−τ1 (θ (t ))) ,
which is equivalent to a11 (θ (t )) b11 (θ (t ))
x1 (θ (t )) ≥ c1 (θ (t ))x1 (θ (t ) − τ1 (θ (t )))e−γ1 (θ (t ))x1 (θ (t )−τ1 (θ (t ))) ,
(2.11)
where θ (t ) > t0 . This, together with (2.10), implies that lim x1 (θ (t ) − τ1 (θ (t ))) = 0.
t →+∞
(2.12)
B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
1935
Now we select a sequence {tn }+∞ n=1 such that
lim x1 (θ (tn )) = 0,
lim tn = +∞,
n→+∞
n→+∞
lim b11 (θ (tn )) = b11 ,
n→+∞
lim a11 (θ (tn )) = a∗11 ,
n→+∞
lim c1 (θ (tn )) = c1 ,
∗
∗
n→+∞
lim γ1 (θ (tn )) = γ1∗ .
(2.13)
n→+∞
Thus, we get a11 (θ (tn ))
≥ c1 (θ (tn ))e−γ1 (θ(tn ))x1 (θ(tn )−τ1 (θ(tn ))) ,
b11 (θ (tn ))
where θ (tn ) > 0.
(2.14)
Letting n → +∞, from (2.12)–(2.14) we know that sup t ∈R
a11 (t ) b11 (t )c1 (t )
≥ lim
n→+∞
a11 (θ (tn )) b11 (θ (tn ))c1 (θ (tn ))
=
a∗11 b∗11 c1∗
≥ 1,
which contradicts with (2.2). Hence, the first inequality of (2.8) holds and similarly the second one does. Combining (2.7) and (2.8) the whole proof of Theorem 2.1 is complete. 3. Permanence of Nicholson-type delay systems with Dij (t , x) = aij (t ) − bij (t )e−x (i, j = 1, 2) Theorem 3.1. Suppose − − + a+ 11 − b11 < a12 − b12 ,
− − + a+ 22 − b22 < a21 − b21
(3.1)
and + a+ 12 + c1
1 eγ1
−
< a− 11 ,
+ a+ 21 + c2
1 eγ2−
< a− 22
(3.2)
are satisfied. Then, system (1.5) and (1.6) with Dij (t , x) = aij (t ) − bij (t )e−x (i, j = 1, 2) is permanent. Proof. Let x(t ) = x(t ; t0 , ϕ). We first claim that xi (t ) > 0,
for all t ∈ (t0 , η(ϕ)), i = 1, 2.
(3.3)
Contrarily, one of the following cases must occur. Case 1: There exists t1 ∈ (t0 , η(ϕ)) such that x1 (t1 ) = 0,
xi (t ) > 0
for all t ∈ (t0 , t1 ), and i = 1, 2.
Case 2: There exists t2 ∈ (t0 , η(ϕ)) such that x2 (t2 ) = 0,
xi (t ) > 0
for all t ∈ (t0 , t2 ), and i = 1, 2.
If Case 1 holds, we have 0 ≥ x′1 (t1 )
= −a11 (t1 ) + b11 (t1 )e−x1 (t1 ) + a12 (t1 ) − b12 (t1 )e−x2 (t1 ) + c1 (t1 )x1 (t1 − τ1 (t1 ))e−γ1 (t1 )x1 (t1 −τ1 (t1 )) ≥ −a11 (t1 ) + b11 (t1 ) + a12 (t1 ) − b12 (t1 ) − + − ≥ −a+ 11 + b11 − b12 + a12 . − − + It follows that a+ 11 − b11 ≥ a12 − b12 which contradicts with the first inequality of (3.1). Similarly we can prove that Case 2 do not occur. This implies that (3.3) holds. For all t ∈ [t0 , η(ϕ)), we define
m(t ) = max{ξ : ξ ≤ t , x1 (ξ ) =
max
t0 −r1 ≤s≤t
x1 (s)}.
We now show that x1 (t ) is bounded for all t ∈ [t0 , η(ϕ)). In the contrary case, observe that m(t ) → η(ϕ) as t → η(ϕ), we have lim x1 (m(t )) = +∞.
(3.4)
t →η(ϕ)
But x1 (m(t )) = maxt0 −r1 ≤s≤t x1 (s), and so x′1 (m(t )) ≥ 0 for all m(t ) > t0 . Thus 0 ≤ x′1 (m(t ))
≤ −a11 (m(t )) + b11 (m(t ))e−x1 (m(t )) + a12 (m(t )) + c1+
1 eγ1−
for all m(t ) > t0 .
(3.5)
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B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
Letting t → η(ϕ), (3.5) implies that + a+ 12 + c1
1 eγ1−
≥ a− 11 ,
which contradicts with the first inequality of (3.2). This shows that x1 (t ) > 0 and similarly x2 (t ) is bounded for all t ∈ [t0 , η(ϕ)). From Theorem 2.3.1 in [8], we easily obtain η(ϕ) = +∞. So there exist two positive constants L1 and L2 such that 0 < x1 (t ) ≤ L1 ,
0 < x2 (t ) ≤ L2
for all t > t0 .
It follows that lim sup x1 (t ) ≤ L1 , t →+∞
lim sup x2 (t ) ≤ L2 .
(3.6)
t →+∞
In what follows, we prove that there exist two positive constants l1 and l2 such that lim inf x1 (t ) ≥ l1 ,
lim inf x2 (t ) ≥ l2 .
t →+∞
(3.7)
t →+∞
Suppose the first expression of (3.7) does not hold, that is, lim inf x1 (t ) = 0. t →+∞
For each t ≥ t0 , we define
ω(t ) = max{ξ : ξ ≤ t , x1 (ξ ) = min x1 (s)}. t0 ≤s≤t
Observe that ω(t ) → +∞ as t → +∞ and that lim x1 (ω(t )) = 0.
(3.8)
t →+∞
However, x1 (ω(t )) = mint0 ≤s≤t x1 (s), and so x′1 (ω(t )) ≤ 0, where ω(t ) > t0 . Then 0 ≥ x′1 (ω(t ))
≥ −a11 (ω(t )) + b11 (ω(t ))e−x1 (ω(t )) + a12 (ω(t )) − b12 (ω(t ))e−x2 (ω(t )) + ≥ −a11 (ω(t )) + b11 (ω(t ))e−x1 (ω(t )) + a− 12 − b12 ,
where ω(t ) > t0 .
(3.9)
− − + Letting t → +∞, (3.9) implies that a+ 11 − b11 ≥ a12 − b12 , which contradicts with the first inequality of (3.1). Similarly we can prove that the second inequality of (3.2) holds. This ends the proof of Theorem 3.1.
4. Two examples In this section we present two examples to illustrate our results. Example 4.1. Consider the following Nicholson-type delay system:
√ √ (8 + | sin 2t |)x1 (t ) (1 + | cos 2t |)x2 (t ) ′ + x1 (t ) = − t t 12 + t 2 + + x1 (t ) 2 + t2+ + x2 ( t ) 1 1 2 | arctan t | −(4+| arctan t |)x1 (t −2e| arctan t | ) + (1 + cos t )x1 (t − 2e )e , √ √ ′ (7 + | sin 2t |)x2 (t ) (2 + | cos 2t |)x1 (t ) x2 ( t ) = − + t t + x2 (t ) + x1 ( t ) 10 + t 2 + 3 + t2+ 1 1 2 | arctan 2t | −(2+| arctan t |)x1 (t −2e| arctan 2t | ) + (1 + sin t )x1 (t − 2e )e .
(4.1)
π − + + + − + − 2 Obviously, a− 11 = 8, a22 = 7, a12 = 2, a21 = 3, c1 = 2, γ1 = 4, c2 = 2, γ2 = 2, r1 = supt ∈R τ1 (t ) = 2e , r2 = π supt ∈R τ2 (t ) = 2e 2 . So
− min{a− 11 , a22 } = 7,
+ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
=5+
3 2e
and sup t ∈R
a11 (t ) b11 (t )c1 (t )
=
3 4
< 1,
sup t ∈R
a22 (t ) b22 (t )c2 (t )
=
4 5
< 1.
< 7,
B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
1937
It follows that the Nicholson-type delay system (4.1) satisfies all the conditions in Theorem 2.1. Hence, from Theorem 2.1, the system (4.1) with initial conditions (1.6) is permanent. Example 4.2. Consider the following Nicholson-type delay system:
′ x1 (t ) = −(4 + sin t ) + (8 + | cos t |)e−x1 (t ) + (1 + cos t ) − (1 + | sin t |)e−x2 (t ) | arctan t | ) + (1 + cos2 t )x1 (t − 2e| arctan t | )e−(2+| arctan t |)x1 (t −2e , ′ −x1 (t ) x ( t ) = −( 6 + cos t ) + ( 10 + | sin t |) e + ( 2 + sin t ) − ( 1 + | cos t |)e−x1 (t ) 2 2 | arctan 2t | −(2+| arctan 2t |)x1 (t −2e| arctan 2t | ) + (1 + sin t )x2 (t − 2e )e .
(4.2)
− + − + − + − − − + It is easy to obtain that a+ 11 = 5, a11 = 3, a12 = 2, a12 = 0, a21 = 3, a21 = 1, a22 = 7, a22 = 5, b11 = 8, b22 = 10, b12 = π
π
+ + + + 2 2 2, b+ 21 = 2, c1 = 2, c2 = 2, γ1 = 2, γ2 = 2, r1 = supt ∈R τ1 (t ) = 2e , r2 = supt ∈R τ2 (t ) = 2e . So − − + a+ 11 − b11 < a12 − b12 ,
− − + a+ 22 − b22 < a21 − b21
and + a+ 12 + c1
1 eγ1−
< a− 11 ,
+ a+ 21 + c2
1 eγ2−
< a− 22 .
Hence, from Theorem 3.1, the model (4.2) is permanent. Remark 4.1. To the best of our knowledge, few authors have considered the problems of the permanence of Nicholson-type delay systems with nonlinear density-dependent mortality terms. It is clear that all the results in [4–6] and the references therein cannot be applicable to prove the permanence of (4.1) and (4.2). This implies that the results of this paper are new. Acknowledgements The authors would like to thank the referees very much for the helpful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8]
A. Nicholson, An outline of the dynamics of animal populations, Australian Journal of Zoology 2 (1954) 9–65. W. Gurney, S. Blythe, R. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. R. Nisbet, W. Gurney, Modelling Fluctuating Populations, John Wiley and Sons, NY, 1982. L. Berezansky, L. Idels, L. Troib, Global dynamics of Nicholson-type delay systems with applications, Nonlinear Analysis: Real World Applications 12 (1) (2011) 436–445. L. Berezansky, E. Braverman, L. Idels, Nicholson’s blowflies differential equations revisited: main results and open problems, Applied Mathematical Modelling 34 (2010) 1405–1417. B. Liu, Permanence for a delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term, Annales Polonici Mathematici, 2011 (APM 2204, in press). H.L. Smith, Monotone Dynamical Systems, in: Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1995. J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms✩ Bingwen Liu, Shuhua Gong ∗ College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China
article
info
Article history: Received 21 October 2010 Accepted 13 December 2010 Keywords: Nonlinear density-dependent mortality term Time-varying delays Permanence Nicholson-type delay differential system
abstract In this paper, we study the generalized Nicholson-type delay systems with nonlinear density-dependent mortality terms. Under proper conditions, we establish some criteria to guarantee the permanence of this model. Moreover, we give two examples to illustrate our main results. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction In a classic study of population dynamics, a delay differential equation model: x′ (t ) = −δ x(t ) + Px(t − τ )e−ax(t −τ ) ,
(1.1)
is frequently used, where x(t ) is the size of the population at time t , P is the maximum per capita daily egg production, 1a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time. This equation was introduced by Nicholson [1] to model laboratory fly population. Its dynamics was later studied in [2,3], where this model was referred to as the Nicholsons blowflies equation [2]. Recently, to describe the models of Marine Protected Areas and B-cell Chronic Lymphocytic Leukemia dynamics that belong to the Nicholson-type delay differential systems, Berezansky et al. [4] proposed the following Nicholson-type delay systems:
x′1 (t ) = −a1 x1 (t ) + b1 x2 (t ) + c1 x1 (t − τ )e−x1 (t −τ ) , x′2 (t ) = −a2 x2 (t ) + b2 x1 (t ) + c2 x2 (t − τ )e−x2 (t −τ ) ,
(1.2)
with initial conditions: xi (s) = ϕi (s),
s ∈ [−τ , 0], ϕi (0) > 0,
(1.3)
where ϕi ∈ C ([−τ , 0], [0, +∞)), ai , bi , ci and τ are nonnegative constants, i = 1, 2. According to the new studies in population dynamics (see [5,6]), a linear model of density-dependent mortality will be most accurate for populations at low densities, and marine ecologists are currently in the process of constructing new
✩ This work was supported by the Key Project of Chinese Ministry of Education (Grant No. 210 151), and the Scientific Research Fund of Zhejiang Provincial Education Department of PR China (Grant No. Y200907784). ∗ Corresponding author. Tel.: +86 057383643075. E-mail address: [email protected] (S. Gong).
1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.12.009
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fishery models with nonlinear density-dependent mortality rates. Therefore, Berezansky et al. [5] point out an open problem: How about the dynamic behaviors of the following scalar Nicholson’s blowflies model with a nonlinear density-dependent mortality term: x′ (t ) = −D(x(t )) + Px(t − τ )e−x(t −τ ) ,
(1.4)
where the nonlinear density-dependent mortality term D(x) might have one of the following forms: D(x) = bax or +x D(x) = a − be−x with constants a, b > 0. Now, a corresponding question arises: How about the dynamic behaviors of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. The main purpose of this paper is to give the conditions to ensure the permanence of Nicholson-type delay differential systems with nonlinear density-dependent mortality terms. Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, we will consider the following Nicholson-type delay systems with nonlinear density-dependent mortality terms: x′1 (t ) = −D11 (t , x1 (t )) + D12 (t , x2 (t )) + c1 (t ) x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) , x′2 (t ) = −D22 (t , x2 (t )) + D21 (t , x1 (t )) + c2 (t ) x2 (t − τ2 (t ))e−γ2 (t )x2 (t −τ2 (t )) ,
(1.5)
where Dij (t , x) =
aij (t )x bij (t ) + x
or Dij (t , x) = aij (t ) − bij (t )e−x ,
aij , bij , ci , γi : R → (0, +∞) are all continuous functions bounded above and below by positive constants, τi : R → [0, +∞) is a bounded continuous function, ri = supt ∈R τi (t ) > 0, and i, j = 1, 2. Throughout this paper, the set of all (nonnegative) real vectors will be denoted by R(R+ ). Let C = C ([−r1 , 0], R) × C ([−r2 , 0], R) be the continuous functions space equipped with the usual supremum norm ‖ · ‖, and let C+ = C ([−r1 , 0], R+ ) × C ([−r2 , 0], R+ ). For the sake of convenience, we set g + = sup g (t ), t ∈R
g − = inf g (t ), t ∈R
where g is a bounded continuous function defined on R. If xi (t ) is defined on [t0 − ri , σ ) with t0 , σ ∈ R and i = 1, 2, then we define xt ∈ C as xt = (x1t , x2t ) where xit (θ ) = xi (t + θ ) for all θ ∈ [−ri , 0] and i = 1, 2. For any ϕ, ψ ∈ C , we write ϕ ≤ ψ if ψ − ϕ ∈ C+ , ϕ < ψ if ϕ ≤ ψ and ϕ ̸= ψ . Due to the biological interpretation of model (1.5), only positive solutions are meaningful and therefore admissible. Thus, we just consider the admissible initial conditions xt0 = ϕ,
ϕ ∈ C+ and ϕi (0) > 0,
i = 1, 2.
(1.6)
We write xt (t0 , ϕ)(x(t ; t0 , ϕ)) for a solution of the initial value problems (1.5) and (1.6). Also, let [t0 , η(ϕ)) be the maximal right-interval of existence of xt (t0 , ϕ). The remaining part of this paper is organized as follows. In Sections 2 and 3, we shall derive new sufficient conditions for checking the permanence of model (1.5). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections. 2. Permanence of Nicholson-type delay systems with Dij (t , x) =
aij (t )x bij (t )+x
(i, j = 1, 2)
Theorem 2.1. Assume that the following conditions are satisfied. c1+ c2+ − + + min{a− , 11 , a22 } > a12 + a21 + − + eγ1 eγ2− sup
aii (t )
t ∈R bii (t )ci (t )
< 1,
(2.1)
i = 1, 2.
Then, the system (1.5) and (1.6) with Dij (t , x) =
(2.2) aij (t )x bij (t )+x
(i, j = 1, 2) is permanent.
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1933
Proof. Let x(t ) = x(t ; t0 , ϕ). First, under the conditions (2.1) and (2.2) we show that x(t ) for all t ∈ [t0 , η(ϕ)) is bounded, and η(ϕ) → +∞. In view of ϕ ∈ C+ , using Theorem 5.2.1 in [7, p. 81], we have xt (t0 , ϕ) ∈ C+ for all t ∈ [t0 , η(ϕ)). From (1.5) and the fact that a(t )x b(t ) + x
a(t )x
≤
for all t ∈ R, x ≥ 0,
b(t )
we get x′1 (t ) = −
≥−
a11 (t )x1 (t ) b11 (t ) + x1 (t ) a11 (t )x1 (t ) b11 (t )
+
a12 (t )x2 (t ) b12 (t ) + x2 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t ))
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) .
(2.3)
In view of x1 (t0 ) = ϕ1 (0) > 0, integrating (2.3) from t0 to t, we have x1 (t ) ≥ e
−
a11 (u) t0 b11 (u) du
t
x1 (t0 ) + e
−
a11 (u) t0 b11 (u) du
∫
t
t
a11 (v) t0 b11 (v) dv
s
e
c1 (s)x1 (s − τ1 (s))e−γ1 (s)x1 (s−τ1 (s)) ds
t0
> 0,
for all t ∈ [t0 , η(ϕ)).
x2 (t ) > 0,
for all t ∈ [t0 , η(ϕ)).
Similarly
It follows that xi (t ) > 0
for all t ∈ [t0 , η(ϕ)), i = 1, 2.
Set y(t ) = x1 (t ) + x2 (t ),
where t ∈ [t0 − r , η(ϕ)), r = min{r1 , r2 }.
Notice that max xe−x = x≥0
1 e
and
α1 + α2 α1 α2 ≤ + , where αi > 0, 1 ≤ i ≤ 6. α3 + α4 + α5 + α6 α3 + α4 α5 + α6
We obtain y′ (t ) = −
− ≤− ≤−
a11 (t )x1 (t ) b11 (t ) + x1 (t ) a22 (t )x2 (t )
+
a12 (t )x2 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t ))
b12 (t ) + x2 (t ) a21 (t )x1 (t )
b22 (t ) + x2 (t ) b21 (t ) + x1 (t ) a11 (t )x1 (t ) + a22 (t )x2 (t ) b11 (t ) + b22 (t ) + x1 (t ) + x2 (t ) − a− 11 x1 (t ) + a22 x2 (t )
b11 (t ) + b22 (t ) + x1 (t ) + x2 (t ) min{a11 , a22 }y(t ) −
≤−
+
+ c2 (t )x2 (t − τ2 (t ))e−γ2 (t )x2 (t −τ2 (t ))
+ a12 (t ) + a21 (t ) + + + a+ 12 + a21 +
−
+
b11 (t ) + b22 (t ) + y(t )
+ + a+ 12 + a21 +
c1
eγ1
−
c1+ eγ1−
c1 (t )
+
eγ1 (t )
+
c2 (t ) eγ2 (t )
c2+ eγ2−
+
+
c2
eγ2−
.
For each t ∈ [t0 − r , η(ϕ)), we define M (t ) = max{ξ : ξ ≤ t , y(ξ ) =
max y(s)}.
t0 −r ≤s≤t
Now we are in a position to prove that y(t ) is bounded on [t0 , η(ϕ)). In the contrary case, observe that M (t ) → η(ϕ) as t → η(ϕ), we have lim y(M (t )) = +∞.
t →η(ϕ)
But y(M (t )) = maxt0 −r ≤s≤t y(s), and so y′ (M (t )) ≥ 0 for all M (t ) > t0 . Thus, 0 ≤ y′ (M (t ))
≤−
− min{a− 11 , a22 }y(M (t ))
b11 (M (t )) + b22 (M (t )) + y(M (t ))
+ + a+ 12 + a21 +
c1+ eγ1−
+
c2+ eγ2−
,
for all M (t ) > t0 ,
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which yields − min{a− 11 , a22 }y(M (t ))
b11 (M (t )) + b22 (M (t )) + y(M (t ))
+ ≤ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
,
for all M (t ) > t0 .
(2.4)
Noting that the continuities and boundedness of the functions bij (t ), i, j = 1, 2, we can select a sequence {Tn }+∞ n=1 such that lim Tn = η(ϕ),
lim y(M (Tn )) = +∞,
n→+∞
n→+∞
lim bij (M (Tn )) = b∗ij .
n→+∞
(2.5)
In view of (2.4), we get − min{a− 11 , a22 }y(M (Tn ))
b11 (M (Tn )) + b22 (M (Tn )) + y(M (Tn ))
+ ≤ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
.
(2.6)
Letting n → +∞, (2.5) and (2.6) imply that c1+ c2+ − + + , min{a− 11 , a22 } ≤ a12 + a21 + − + eγ1 eγ2− which contradicts with (2.1). This implies that y(t ) is bounded on [t0 , η(ϕ)). From Theorem 2.3.1 in [8], we easily obtain η(ϕ) = +∞. Thus, every solution x(t ; t0 , ϕ) of (1.5) and (1.6) is positive and bounded on [t0 , +∞). So there exist two positive constants K1 and K2 such that 0 < x1 (t ) ≤ K1 ,
0 < x2 (t ) ≤ K2
for all t > t0 .
It follows that lim sup x1 (t ) ≤ K1 , t →+∞
lim sup x2 (t ) ≤ K2 .
(2.7)
t →+∞
We next prove that there exist two positive constants k1 and k2 such that lim inf x1 (t ) ≥ k1 , t →+∞
lim inf x2 (t ) ≥ k2 .
(2.8)
t →+∞
From the first equation of system (1.5), we have x′1 (t ) ≥ −
a11 (t )x1 (t ) b11 (t )
+ c1 (t )x1 (t − τ1 (t ))e−γ1 (t )x1 (t −τ1 (t )) ,
x1 (t ) > 0,
(2.9)
where t ∈ [t0 , +∞). Suppose the first expression of (2.8) does not hold, that is, lim inf x1 (t ) = 0. t →+∞
For each t ≥ t0 , we define
θ (t ) = max{ξ : ξ ≤ t , x1 (ξ ) = min x1 (s)}. t0 ≤s≤t
Observe that θ (t ) → +∞ as t → +∞ and that lim x1 (θ (t )) = 0.
(2.10)
t →+∞
However, x1 (θ (t )) = mint0 ≤s≤t x1 (s), and so x′1 (θ (t )) ≤ 0, where θ (t ) > t0 . According to (2.9), we have 0 ≥ x′1 (θ (t ))
≥−
a11 (θ (t ))x1 (θ (t )) b11 (θ (t ))
+ c1 (θ (t ))x1 (θ (t ) − τ1 (θ (t )))e−γ1 (θ (t ))x1 (θ (t )−τ1 (θ (t ))) ,
which is equivalent to a11 (θ (t )) b11 (θ (t ))
x1 (θ (t )) ≥ c1 (θ (t ))x1 (θ (t ) − τ1 (θ (t )))e−γ1 (θ (t ))x1 (θ (t )−τ1 (θ (t ))) ,
(2.11)
where θ (t ) > t0 . This, together with (2.10), implies that lim x1 (θ (t ) − τ1 (θ (t ))) = 0.
t →+∞
(2.12)
B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
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Now we select a sequence {tn }+∞ n=1 such that
lim x1 (θ (tn )) = 0,
lim tn = +∞,
n→+∞
n→+∞
lim b11 (θ (tn )) = b11 ,
n→+∞
lim a11 (θ (tn )) = a∗11 ,
n→+∞
lim c1 (θ (tn )) = c1 ,
∗
∗
n→+∞
lim γ1 (θ (tn )) = γ1∗ .
(2.13)
n→+∞
Thus, we get a11 (θ (tn ))
≥ c1 (θ (tn ))e−γ1 (θ(tn ))x1 (θ(tn )−τ1 (θ(tn ))) ,
b11 (θ (tn ))
where θ (tn ) > 0.
(2.14)
Letting n → +∞, from (2.12)–(2.14) we know that sup t ∈R
a11 (t ) b11 (t )c1 (t )
≥ lim
n→+∞
a11 (θ (tn )) b11 (θ (tn ))c1 (θ (tn ))
=
a∗11 b∗11 c1∗
≥ 1,
which contradicts with (2.2). Hence, the first inequality of (2.8) holds and similarly the second one does. Combining (2.7) and (2.8) the whole proof of Theorem 2.1 is complete. 3. Permanence of Nicholson-type delay systems with Dij (t , x) = aij (t ) − bij (t )e−x (i, j = 1, 2) Theorem 3.1. Suppose − − + a+ 11 − b11 < a12 − b12 ,
− − + a+ 22 − b22 < a21 − b21
(3.1)
and + a+ 12 + c1
1 eγ1
−
< a− 11 ,
+ a+ 21 + c2
1 eγ2−
< a− 22
(3.2)
are satisfied. Then, system (1.5) and (1.6) with Dij (t , x) = aij (t ) − bij (t )e−x (i, j = 1, 2) is permanent. Proof. Let x(t ) = x(t ; t0 , ϕ). We first claim that xi (t ) > 0,
for all t ∈ (t0 , η(ϕ)), i = 1, 2.
(3.3)
Contrarily, one of the following cases must occur. Case 1: There exists t1 ∈ (t0 , η(ϕ)) such that x1 (t1 ) = 0,
xi (t ) > 0
for all t ∈ (t0 , t1 ), and i = 1, 2.
Case 2: There exists t2 ∈ (t0 , η(ϕ)) such that x2 (t2 ) = 0,
xi (t ) > 0
for all t ∈ (t0 , t2 ), and i = 1, 2.
If Case 1 holds, we have 0 ≥ x′1 (t1 )
= −a11 (t1 ) + b11 (t1 )e−x1 (t1 ) + a12 (t1 ) − b12 (t1 )e−x2 (t1 ) + c1 (t1 )x1 (t1 − τ1 (t1 ))e−γ1 (t1 )x1 (t1 −τ1 (t1 )) ≥ −a11 (t1 ) + b11 (t1 ) + a12 (t1 ) − b12 (t1 ) − + − ≥ −a+ 11 + b11 − b12 + a12 . − − + It follows that a+ 11 − b11 ≥ a12 − b12 which contradicts with the first inequality of (3.1). Similarly we can prove that Case 2 do not occur. This implies that (3.3) holds. For all t ∈ [t0 , η(ϕ)), we define
m(t ) = max{ξ : ξ ≤ t , x1 (ξ ) =
max
t0 −r1 ≤s≤t
x1 (s)}.
We now show that x1 (t ) is bounded for all t ∈ [t0 , η(ϕ)). In the contrary case, observe that m(t ) → η(ϕ) as t → η(ϕ), we have lim x1 (m(t )) = +∞.
(3.4)
t →η(ϕ)
But x1 (m(t )) = maxt0 −r1 ≤s≤t x1 (s), and so x′1 (m(t )) ≥ 0 for all m(t ) > t0 . Thus 0 ≤ x′1 (m(t ))
≤ −a11 (m(t )) + b11 (m(t ))e−x1 (m(t )) + a12 (m(t )) + c1+
1 eγ1−
for all m(t ) > t0 .
(3.5)
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B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
Letting t → η(ϕ), (3.5) implies that + a+ 12 + c1
1 eγ1−
≥ a− 11 ,
which contradicts with the first inequality of (3.2). This shows that x1 (t ) > 0 and similarly x2 (t ) is bounded for all t ∈ [t0 , η(ϕ)). From Theorem 2.3.1 in [8], we easily obtain η(ϕ) = +∞. So there exist two positive constants L1 and L2 such that 0 < x1 (t ) ≤ L1 ,
0 < x2 (t ) ≤ L2
for all t > t0 .
It follows that lim sup x1 (t ) ≤ L1 , t →+∞
lim sup x2 (t ) ≤ L2 .
(3.6)
t →+∞
In what follows, we prove that there exist two positive constants l1 and l2 such that lim inf x1 (t ) ≥ l1 ,
lim inf x2 (t ) ≥ l2 .
t →+∞
(3.7)
t →+∞
Suppose the first expression of (3.7) does not hold, that is, lim inf x1 (t ) = 0. t →+∞
For each t ≥ t0 , we define
ω(t ) = max{ξ : ξ ≤ t , x1 (ξ ) = min x1 (s)}. t0 ≤s≤t
Observe that ω(t ) → +∞ as t → +∞ and that lim x1 (ω(t )) = 0.
(3.8)
t →+∞
However, x1 (ω(t )) = mint0 ≤s≤t x1 (s), and so x′1 (ω(t )) ≤ 0, where ω(t ) > t0 . Then 0 ≥ x′1 (ω(t ))
≥ −a11 (ω(t )) + b11 (ω(t ))e−x1 (ω(t )) + a12 (ω(t )) − b12 (ω(t ))e−x2 (ω(t )) + ≥ −a11 (ω(t )) + b11 (ω(t ))e−x1 (ω(t )) + a− 12 − b12 ,
where ω(t ) > t0 .
(3.9)
− − + Letting t → +∞, (3.9) implies that a+ 11 − b11 ≥ a12 − b12 , which contradicts with the first inequality of (3.1). Similarly we can prove that the second inequality of (3.2) holds. This ends the proof of Theorem 3.1.
4. Two examples In this section we present two examples to illustrate our results. Example 4.1. Consider the following Nicholson-type delay system:
√ √ (8 + | sin 2t |)x1 (t ) (1 + | cos 2t |)x2 (t ) ′ + x1 (t ) = − t t 12 + t 2 + + x1 (t ) 2 + t2+ + x2 ( t ) 1 1 2 | arctan t | −(4+| arctan t |)x1 (t −2e| arctan t | ) + (1 + cos t )x1 (t − 2e )e , √ √ ′ (7 + | sin 2t |)x2 (t ) (2 + | cos 2t |)x1 (t ) x2 ( t ) = − + t t + x2 (t ) + x1 ( t ) 10 + t 2 + 3 + t2+ 1 1 2 | arctan 2t | −(2+| arctan t |)x1 (t −2e| arctan 2t | ) + (1 + sin t )x1 (t − 2e )e .
(4.1)
π − + + + − + − 2 Obviously, a− 11 = 8, a22 = 7, a12 = 2, a21 = 3, c1 = 2, γ1 = 4, c2 = 2, γ2 = 2, r1 = supt ∈R τ1 (t ) = 2e , r2 = π supt ∈R τ2 (t ) = 2e 2 . So
− min{a− 11 , a22 } = 7,
+ a+ 12 + a21 +
c1+ eγ1
−
+
c2+ eγ2−
=5+
3 2e
and sup t ∈R
a11 (t ) b11 (t )c1 (t )
=
3 4
< 1,
sup t ∈R
a22 (t ) b22 (t )c2 (t )
=
4 5
< 1.
< 7,
B. Liu, S. Gong / Nonlinear Analysis: Real World Applications 12 (2011) 1931–1937
1937
It follows that the Nicholson-type delay system (4.1) satisfies all the conditions in Theorem 2.1. Hence, from Theorem 2.1, the system (4.1) with initial conditions (1.6) is permanent. Example 4.2. Consider the following Nicholson-type delay system:
′ x1 (t ) = −(4 + sin t ) + (8 + | cos t |)e−x1 (t ) + (1 + cos t ) − (1 + | sin t |)e−x2 (t ) | arctan t | ) + (1 + cos2 t )x1 (t − 2e| arctan t | )e−(2+| arctan t |)x1 (t −2e , ′ −x1 (t ) x ( t ) = −( 6 + cos t ) + ( 10 + | sin t |) e + ( 2 + sin t ) − ( 1 + | cos t |)e−x1 (t ) 2 2 | arctan 2t | −(2+| arctan 2t |)x1 (t −2e| arctan 2t | ) + (1 + sin t )x2 (t − 2e )e .
(4.2)
− + − + − + − − − + It is easy to obtain that a+ 11 = 5, a11 = 3, a12 = 2, a12 = 0, a21 = 3, a21 = 1, a22 = 7, a22 = 5, b11 = 8, b22 = 10, b12 = π
π
+ + + + 2 2 2, b+ 21 = 2, c1 = 2, c2 = 2, γ1 = 2, γ2 = 2, r1 = supt ∈R τ1 (t ) = 2e , r2 = supt ∈R τ2 (t ) = 2e . So − − + a+ 11 − b11 < a12 − b12 ,
− − + a+ 22 − b22 < a21 − b21
and + a+ 12 + c1
1 eγ1−
< a− 11 ,
+ a+ 21 + c2
1 eγ2−
< a− 22 .
Hence, from Theorem 3.1, the model (4.2) is permanent. Remark 4.1. To the best of our knowledge, few authors have considered the problems of the permanence of Nicholson-type delay systems with nonlinear density-dependent mortality terms. It is clear that all the results in [4–6] and the references therein cannot be applicable to prove the permanence of (4.1) and (4.2). This implies that the results of this paper are new. Acknowledgements The authors would like to thank the referees very much for the helpful comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8]
A. Nicholson, An outline of the dynamics of animal populations, Australian Journal of Zoology 2 (1954) 9–65. W. Gurney, S. Blythe, R. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. R. Nisbet, W. Gurney, Modelling Fluctuating Populations, John Wiley and Sons, NY, 1982. L. Berezansky, L. Idels, L. Troib, Global dynamics of Nicholson-type delay systems with applications, Nonlinear Analysis: Real World Applications 12 (1) (2011) 436–445. L. Berezansky, E. Braverman, L. Idels, Nicholson’s blowflies differential equations revisited: main results and open problems, Applied Mathematical Modelling 34 (2010) 1405–1417. B. Liu, Permanence for a delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term, Annales Polonici Mathematici, 2011 (APM 2204, in press). H.L. Smith, Monotone Dynamical Systems, in: Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1995. J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.