1.Introduction PermanenceforNicholson-typeDelaySystemswithPatchStructureandNonlinearDensity-dependentMortalityTerms*

Electronic Journal of Qualitative Theory of Differential Equations 2012, No.73, 1-14;http://www.math.u-szeged.hu/ejqtde/

Permanence for Nicholson-type Delay Systems with Patch Structure and Nonlinear Density-dependent Mortality Terms*

Wei Chen^†

School of Mathematics and Information, Shanghai Lixin University of Commerce, Shanghai, 201620, People’s Republic of China

Abstract: In this paper, we study the Nicholson-type delay systems with patch structure and nonlinear density-dependent mortality terms. Under appropriate conditions, we establish some criteria to ensure the permanence of this model. Moreover, we give some examples to illustrate our main results.

Keywords: Nicholson-type delay system; permanence; patch structure; Nonlinear density- dependent mortality terms.

AMS(2000) Subject Classification: 34C25; 34K13.

1. Introduction

To reveal the rule of population of the Australian sheep blowfly that obtained in exper- imental data [1], Gurney et al [2] put forward the following Nicholson’s blowflies model

N^′(t) =−δN(t) +pN(t−τ)e^{−aN(t−τ)}. (1.1) Here, N(t) is the size of the population at time t, p is the maximum per capita daily egg production,¹_a is the size at which the population reproduces at its maximum rate,δis the per capita daily adult death rate, andτ is the generation time. As a class of biological systems,

*This work was supported by National Natural Science Foundation of China (grant nos. 11101283, 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (grant no. Z201122436).

†Corr. author. Tel.:+86 02136430738; fax: +86 02136430738. E-mail:chenweiwang2009@yahoo.com.cn

Nicholson’s blowflies model and its analogous equation have attracted much attention. There have been a large number of results about this model and its modifications. We refer the reader to [3-9] and the references cited therein. Moreover, the main focus of Nicholson’s blowflies model is on the scalar equation and results about patch structure of this model are gained rarely (see e.g.[10-13] and the reference therein). On the other hand, L. Berezansky et al [9] pointed out that a new study indicates that 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 fishery models with nonlinear density-dependent mortality rates. Consequently, B. Liu and S. Gong [14] and Liu [15] presented extensive results on the permanence of the following Nicholson’s blowflies model with a nonlinear density-dependent mortality term

N^′(t) =−D(N(t)) +P N(t−τ)e^{−aN(t−τ)} (1.2) whereP is a positive constant and functionDmight have one of the following forms: D(N) =

aN

N+b orD(N) =a−be^−N with positive constants a, b >0.

However, to the best of our knowledge, there have been few publications concerned with the permanence for Nicholson-type delay system with patch structure and nonlinear density- dependent mortality terms. Motivated by this, the main purpose of this paper is to give the conditions to guarantee the permanence for the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms:

N_i^′(t) =−D_ii(t, Ni(t)) +

n

X

j=1,j6=i

D_ij(t, Nj(t)) +

l

X

j=1

c_ij(t)Ni(t−τ_ij(t))e^{−γij(t)Ni(t−τij(t))}, (1.3) where

D_ij(t, N) = a_ij(t)N

b_ij(t) +N or D_ij(t, N) =a_ij(t)−b_ij(t)e^−N,

a_ij, b_ij, c_ik, γ_ik :R→(0,+∞) are all continuous functions bounded above and below by pos- itive constants, andτ_ik(t)≥0 are bounded continuous functions,r_i= max

1≤j≤l{sup_t∈Rτ_ij(t)}>

0, andi, j= 1,2· · ·, n,k= 1,2· · ·, l. Furthermore, in the caseD_ij(t, N) =a_ij(t)−b_ij(t)e^−N, we assume that aij(t) > bij(t) for t ∈ R and i, j = 1,2· · ·, n, which show the biological significance of the mortality terms.

For convenience, we introduce some notations. Throughout this paper, given a bounded continuous functiongdefined onR, let g⁺ and g⁻ be defined as

g⁻= inf

t∈Rg(t), g⁺= sup

t∈R

g(t).

LetRⁿ(R₊ⁿ) be the set of all (nonnegative) real vectors, we will usex= (x₁, . . . , x_n)^T ∈Rⁿ to denote a column vector, in which the symbol ()^T denotes the transpose of a vector. we let |x| denote the absolute-value vector given by |x| = (|x₁|, . . . ,|x_n|)^T and define ||x|| = max_1≤i≤n|xi|. Denote C = ^Qn

i=1

C([−ri,0], R¹) and C₊ = ^Qn

i=1

C([−ri,0], R¹₊) as Banach space equipped with the supremum norm defined by||ϕ||= sup

−ri≤t≤0

1max≤i≤n|ϕ_i(t)|for allϕ(t) = (ϕ₁(t), . . . , ϕ_n(t))^T ∈ C (or ∈ C₊). If x_i(t) is defined on [t₀ −r_i, ν) with t₀, ν ∈ R¹ and i = 1, . . . , n, then we define xt ∈ C as xt = (x¹_t, . . . xⁿ_t)^T where xⁱ_t(θ) = xi(t+θ) for all θ∈[−r_i,0] andi= 1, . . . , n.

The initial conditions associated with system (1.3) are of the form:

N_t0 =ϕ, ϕ= (ϕ1, . . . , ϕ_n)^T ∈C₊ and ϕ_i(0)>0, i= 1, . . . , n. (1.4) We write N_t(t0, ϕ)(N(t;t₀, ϕ)) for a solution of the initial value problem (1.3) and (1.4) . Also, let [t₀, η(ϕ)) be the maximal right-interval of existence ofNt(t₀, ϕ).

Definition 1.1. The system (1.3) with initial conditions (1.4) is said to be permanent, if there are positive constants ki and Ki such that

k_i ≤lim inf

t→+∞N_i(t;t₀, ϕ)≤lim sup

t→+∞

N_i(t;t₀, ϕ)≤K_i, i= 1,2· · ·, n.

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 (1.3). 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 D_ij(t, N) = _b^aij(t)N

ij(t)+N(i, j = 1,2,· · ·, n)

Theorem 2.1. Assume that the following conditions are satisfied

1min≤i≤n{a⁻_ii}>

n

X

i=1 n

X

j=1,j6=i

a⁺_ij+

n

X

i=1 l

X

j=1

c⁺_ij

eγ_ij⁻, (2.1)

sup

t∈R

a_ii(t) bii(t) ^Pn

j=1

cij(t)

<1, i= 1,2,· · ·, n. (2.2)

Then, the model (1.3) and (1.4) withD_ij(t, N) = _b^aij(t)N

ij(t)+N(i, j= 1,2,· · ·, n) is permanent.

Proof. Set N(t) = N(t;t₀, ϕ) for all t∈[t₀, η(ϕ)). In view of ϕ∈C₊, using Theorem 5.2.1 in [16, p.81], we have N_t(t₀, ϕ) ∈C₊ for all t∈[t₀, η(ϕ)). From(1.3) and the fact that

aii(t)N

bii(t)+N ≤ ^aiibii^(t)N(t) for all t∈R, N ≥0, we obtain N_i^′(t) = −D_ii(t, Ni(t)) +

n

X

j=1,j6=i

D_ij(t, Nj(t)) +

l

X

j=1

c_ij(t)Ni(t−τ_ij(t))e^{−γij(t)Ni(t−τij(t))}

≥ −a_ii(t)Ni(t) bii(t) +

l

X

j=1

cij(t)Ni(t−τij(t))e^{−γij(t)Ni(t−τij(t))}, i= 1,2,· · ·, n. (2.3)

In view ofNi(t₀) =ϕi(0)>0, integrating (2.3) fromt₀ to t, we get N_i(t) ≥ e⁻

Rt t0

aii(u) bii(u)du

N_i(t₀) + e⁻

Rt t0

aii(u) bii(u)duZ _t

t₀

e Rs

t0 aii(v) bii(v)dv l

X

j=1

cij(s)Ni(s−τij(s))e^{−γij(s)Ni(s−τij(s))}ds

> 0, for all t∈[t₀, η(ϕ)), i= 1,2,· · ·, n.

Let y(t) = ^Pn

i=1

x_i(t), where t ∈ [t₀−r, η(ϕ)), r = min

1≤i≤n{r_i}. Notice that max

x≥0 xe^−x = ¹_e, we have

y^′(t) = −

n

X

i=1

aii(t)Ni(t) b_ii(t) +N_i(t) +

n

X

i=1 n

X

j=1,j6=i

aij(t)Nj(t) b_ij(t) +N_j(t) +

n

X

i=1 l

X

j=1

c_ij(t)N_i(t−τ_ij(t))e^{−γij(t)Ni(t−τij(t))}

≤ −

n

P

i=1

a_ii(t)Ni(t)

n

P

i=1

b_ii(t) + ^Pn

i=1

N_i(t) +

n

X

i=1 n

X

j=1,j6=i

a_ij(t) +

n

X

i=1 l

X

j=1

cij(t) eγ_ij(t)

≤ −

n

P

i=1

a⁻_iiN_i(t)

n

P

i=1

b_ii(t) + ^Pn

i=1

N_i(t) +

n

X

i=1 n

X

j=1,j6=i

a⁺_ij +

n

X

i=1 l

X

j=1

c⁺_ij eγ_ij⁻

≤ −

1min≤i≤n{a⁻_ii}y(t)

n

P

i=1

b_ii(t) +y(t) +

n

X

i=1 n

X

j=1,j6=i

a⁺_ij +

n

X

i=1 l

X

j=1

c⁺_ij eγ_ij⁻.

For each t∈[t0−r, η(ϕ)), we define

M(t) = max{ξ:ξ ≤t, y(ξ) = max

t0−r≤s≤ty(s)}.

We now claim thaty(t) is bounded on [t₀, η(ϕ)). In the contrary case, observe thatM(t)→ η(ϕ) ast→η(ϕ), we get

t→limη(ϕ)y(M(t)) = +∞. Buty(M(t)) = max

t0−r≤s≤ty(s), and soy^′(M(t))≥0 for allM(t)> t₀. Thus, 0 ≤ y^′(M(t))

≤ −

1min≤i≤n{a⁻_ii}y(M(t))

n

P

i=1

b_ii(M(t)) +y(M(t)) +

n

X

i=1 n

X

j=1,j6=i

a⁺_ij +

n

X

i=1 l

X

j=1

c⁺_ij

eγ_ij⁻, for all M(t)> t₀,

which yields

1min≤i≤n{a⁻_ii}y(M(t))

n

P

i=1

b_ii(M(t)) +y(M(t))

≤

n

X

i=1 n

X

j=1,j6=i

a⁺_ij+

n

X

i=1 l

X

j=1

c⁺_ij

eγ_ij⁻, for all M(t)> t₀. (2.4) Therefore, from the continuities and boundedness of the functionsbij(t),i, j= 1,2,· · ·, n, we can select a sequence{T_n}⁺n=1^∞ such that

n→lim+∞T_n=η(ϕ), lim

n→+∞y(M(T_n)) = +∞, lim

n→+∞b_ij(M(T_n)) =b^∗_ij, (2.5) and

1min≤i≤n{a⁻_ii}y(M(Tn))

n

P

i=1

b_ii(M(T_n)) +y(M(T_n)) ≤

n

X

i=1 n

X

j=1,j6=i

a⁺_ij+

n

X

i=1 l

X

j=1

c⁺_ij

eγ⁻_ij. (2.6) Lettingn→+∞, (2.5) and (2.6) imply that

1min≤i≤n{a⁻_ii} ≤

n

X

i=1 n

X

j=1,j6=i

a⁺_ij+

n

X

i=1 l

X

j=1

c⁺_ij eγ_ij⁻.

which contradicts with (2.1). This implies thaty(t) is bounded on [t0, η(ϕ)) . From Theorem 2.3.1 in [17], we easily obtain η(ϕ) = +∞.Thus, every solution N(t;t₀, ϕ) of (1.3) and (1.4) is positive and bounded on [t0,+∞). So there exist positive constantsK_i, such that

0< N_i(t)≤K_i, for all t > t₀ , i= 1,2,· · ·, n It follows that

t→lim+∞supN_i(t)≤K_i, i= 1,2,· · ·, n. (2.7)

We next prove that there exist positive constants k_i, such that

t→lim+∞infN_i(t)≥k_i, i= 1,2,· · ·, n. (2.8) Fori= 1,2,· · ·, n, from (1.3) we have

N_i^′(t)≥ −aii(t)Ni(t) b_ii(t) +

l

X

j=1

cij(t)Ni(t−τij(t))e^{−γij(t)Ni(t−τij(t))}, (2.9) wheret∈[t₀,+∞). Suppose that (2.8) does not hold, that is,

t→lim+∞infN_i(t) = 0, i= 1,2,· · ·, n.

For each t≥t₀, we define

θi(t) = max{ξ:ξ ≤t, Ni(ξ) = min

t₀≤s≤tNi(s),}, i= 1,2,· · ·, n.

Observe thatθ_i(t)→+∞ ast→+∞,i= 1,2,· · ·, n, and

t→lim+∞N_i(θi(t)) = 0, i= 1,2,· · ·, n. (2.10) However, Ni(θi(t)) = min

t₀≤s≤tNi(s), and so N_i^′(θi(t)) ≤ 0, where θi(t) > t₀, i = 1,2,· · ·, n.

According to (2.9), we have 0 ≥ N_i^′(θi(t))

≥ −a_ii(θi(t))Ni(θi(t)) bii(θi(t)) +

l

X

j=1

c_ij(θi(t))Ni(θi(t)−τ_ij(θi(t)))e^{−γij(θi(t))Ni(θi(t)−τij(θi(t)))},

which is equivalent to a_ii(θi(t))

bii(θi(t))N_i(θi(t))≥

l

X

j=1

c_ij(θi(t))Ni(θi(t)−τ_ij(θi(t)))e^{−γij(θi(t))Ni(θi(t)−τij(θi(t)))}, (2.11) whereθ_i(t)> t₀, i= 1,2,· · ·, n. This, together with (2.10), implies that

t→lim+∞N_i(θi(t)−τ_ij(θi(t))) = 0, i= 1,2,· · ·, n (2.12) Now we select a sequence{t_n}⁺n=1^∞ such that







θi(tn)> t₀, lim

n→+∞tn= +∞, lim

n→+∞Ni(θi(tn)) = 0, lim

n→+∞aii(θi(tn)) =a^∗_ii

n→lim+∞b_ii(θ_i(t_n)) =b^∗_ii, lim

n→+∞c_ij(θ_i(t_n)) =c^∗_ij, lim

n→+∞γ_ij(θ_i(t_n)) =γ_ij^∗,

(2.13)

wherei= 1,2,· · ·, n, j = 1,2,· · ·, l.Thus, we obtain a_ii(θi(tn))

bii(θi(tn)) ≥

l

X

j=1

c_ij(θi(tn))e^{−γij(θi(tn))Ni(θi(tn)−τij(θi(tn)))}, (2.14) wherei= 1,2,· · ·, n. Lettingn→+∞, from (2.12)-(2.14) we know that

sup

t∈R

aii(t) b_ii(t) ^Pl

j=1

c_ij(t)

≥ lim

n→+∞

aii(θi(tn)) b_ii(θ_i(t_n))^Pl

j=1

c_ij(θ_i(t_n))

= a^∗_ii b^∗_ii ^Pl

j=1

c^∗_ij

≥1,

which contradicts with (2.2). Hence, inequality of (2.8) holds. Combining (2.7) and (2.8) the whole proof of Theorem 2.1 is complete.

3. Permanence of Nicholson-type delay systems with D_ij(t, N) =a_ij(t)−b_ij(t)e^−N(i, j = 1,2,· · ·, n)

Theorem 3.1. Assume that a⁺_ii−b⁻_ii <

n

X

j=1,j6=i

(a⁻_ij −b⁺_ij), i= 1,2,· · ·, n, (3.1)

n

X

j=1,j6=i

a⁺_ij +

l

X

j=1

c⁺_ij

eγ⁻_ij < a⁻_ii, i= 1,2,· · ·, n. (3.2) Then, the model (1.3) and (1.4) with Dij(t, N) = aij(t) −bij(t)e^−N(i, j = 1,2,· · ·, n) is permanent.

Proof. Let N(t) =N(t;t₀, ϕ), we first claim that

Ni(t)>0 for all t∈[t₀, η(ϕ)), i= 1,2,· · ·, n. (3.3) Contrarily, it must occur that there existt^∗ ∈[t₀, η(ϕ)) and k∈ {1,2,· · ·, n}such that

N_k(t^∗) = 0, Ni(t)>0 for all t∈[t₀, t^∗), i= 1,2,· · ·, n.

Then, we have 0 ≥ N_k^′(t^∗)

= −D_kk(t^∗, N_i(t^∗)) +

n

X

j=1,j6=k

D_kj(t^∗, N_j(t^∗)) +

l

X

j=1

c_kj(t^∗)N_k(t^∗−τ_kj(t^∗))e^{−γkj(t∗)Nk(t∗−τkj(t∗))}

≥ −a_kk(t^∗) +b_kk(t^∗) +

n

X

j=1,j6=k

a_kj(t^∗)−

n

X

j=1,j6=k

b_kj(t^∗)

≥ −a⁺_kk+b⁻_kk+

n

X

j=1,j6=k

a⁻_kj −

n

X

j=1,j6=k

b⁺_kj,

It follows thata⁺_kk−b⁻_kk≥ ^Pn

j=1,j6=k

(a⁻_kj−b⁺_kj) which contradicts with inequality of (3.1). This implies that (3.3) holds. For allt∈[t₀−r_i, η(ϕ)), we define

m_i(t) = max{ξ :ξ≤t, N_i(ξ) = max

t0−ri≤s≤tN_i(s)}, i= 1,2,· · ·, n.

We now show that N_i(t) are bounded on [t₀, η(ϕ)),i = 1,2,· · ·, n. In the contrary case, it existsk∈ {1,2,· · ·, n}and observe that m_k(t)→η(ϕ) ast→η(ϕ), we get

lim

t→η(ϕ))N_k(m_k(t)) = +∞. (3.4)

ButN_k(m_k(t)) = max

t0−rk≤s≤tN_k(s), and soN_k^′(m_k(t))≥0 for allm_k(t)> t₀. Thus, 0 ≤ N_k^′(m_k(t))

≤ −a_kk(m_k(t)) +b_kk(m_k(t))e^−Nk(mk(t))+

n

X

j=1,j6=k

a_kj(m_k(t)) +

l

X

j=1

c⁺_kj 1

eγ⁻_kj. (3.5)

Lettingt→η(ϕ), (3.5) implies that

n

X

j=1,j6=k

a⁺_kj+

l

X

j=1

c⁺_kj

eγ_kj⁻ ≥a⁻_kk.

which contradicts with the inequality of (3.2). This shows thatN_i(t) are positive and bounded for allt∈[t₀, η(ϕ)),i= 1,2,· · ·, n. From Theorem 2.3.1 in [17], we easily obtainη(ϕ) = +∞. So there exist positive constantsL_i such that

0< N_i(t)≤L_i, i= 1,2,· · ·, n.

It follows that

t→lim+∞supN_i(t)≤L_i, i= 1,2,· · ·, n. (3.6) In what follows, we prove that there exists a positive constantl_i such that

t→lim+∞infNi(t)≥li, i= 1,2,· · ·, n (3.7) Assume that (3.7) does not hold, then it existsk∈ {1,2,· · ·, n}, such that

t→lim+∞infN_k(t) = 0.

For each t≥t₀, we define

ω(t) = max{ξ :ξ≤t, Nk(ξ) = min

t0≤s≤tNk(s)}.

Observe thatω(t)→+∞ast→+∞and

t→lim+∞N_k(ω(t)) = 0. (3.8)

However,N_k(ω(t)) = min

t0≤s≤tN_k(s), and so N_k^′(ω(t))≤0, whereω(t)> t₀. Then 0 ≥ N_k^′(ω(t))

≥ −a_kk(ω(t)) +b_kk(ω(t))e^−Nk(ω(t))+

n

X

j=1,j6=k

(a_kj(ω(t))−b_kj(ω(t))e^−Nj(ω(t)))

≥ −a_kk(ω(t)) +b_kk(ω(t))e^−Nk(ω(t))+

n

X

j=1,j6=k

(a⁻_kj−b⁺_kj). (3.9)

Lettingt→+∞, (3.9) implies that

a⁺_kk−b⁻_kk≥

n

X

j=1,j6=k

(a⁻_kj −b⁺_kj),

which contradicts with the inequality of (3.1). This ends the proof of Theorem 3.1.

4. Some examples

In this section we present some examples to illustrate our results.

Example 4.1. Consider the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms:



















































N₁^′(t) = −^(13+|cos

√3t|)N1(t) 5+|sin√

2t|+N1(t) +₃₊⁽¹⁺_|^|_{cos 3t}^{sin 2t}_|^|_+N^)N2(t)

2(t) +⁽¹⁺_4+|^|_{sin 3t}^{cos 2t}_|+N^|)N3(t)

3(t)

+(1 + cos²t)N₁(t−2|sint|)e^{−4N1(t−2|sint|)} +(1 + sin²t)N₁(t−2|cost|)e^{−4N1(t−2|cost|)} N₂^′(t) = −^(14+|sin

√3t|)N2(t) 6+|cos√

2t|+N2(t) +⁽¹⁺_3+|^|_{sin 3t}^{cos 2t}_|+N^|)N1(t)

1(t) + ⁽¹⁺_4+|^|_{cos 3t}^{sin 2t}_|+N^|)N3(t)

3(t)

+(1 + sin²t)N₂(t−2|cost|)e^{−5N2(t−2|cost|)} +(1 + cos²t)N₂(t−2|sint|)e^{−5N2(t−2|sint|)} N₃^′(t) = −⁽¹⁵⁺_6+|cos^|sin√√_6t^5t_|+N^|)N₃³_(t)^(t) +⁽¹⁺_3+|^|_{sin 2t}^{cos 3t}_|+N^|)N1(t)

1(t) + ⁽¹⁺_4+|^|_{cos 2t}^{sin 3t}_|+N^|)N2(t)

2(t)

+(1 + sin²√

2t)N₃(t−2|cos 2t|)e^{−6N3(t−2|cos 2t|)} +(1 + cos²√

2t)N3(t−2|sin 3t|)e^{−6N3(t−2|sin 3t|)},

(4.1)

Obviously,a⁻₁₁= 13, a⁻₂₂ = 14, a⁻₃₃ = 15, a⁺_ij = 2,(i, j = 1,2,3, i 6=j), c⁺_ij = 2,(i= 1,2,3, j = 1,2), γ_1j⁻ = 4,γ_2j⁻ = 5,γ_3j⁻ = 6,(j = 1,2),ri= 2,(i= 1,2,3). So

13 = min

1≤i≤3{a⁻_ii}>

3

X

i=1 3

X

j=1,j6=i

a⁺_ij +

3

X

i=1 2

X

j=1

c⁺_ij

eγ_ij⁻ = 12 + 32 15e, and

1max≤i≤3{sup

t∈R

a_ii(t) bii(t) ^P2

j=1

cij(t)

}= max{14 15,5

6,8 9}= 14

15 <1.

It follows that the Nicholson’s blowflies model with patch structure and nonlinear density- dependent mortality terms (4.1) satisfies all the conditions in Theorem 2.1. Hence, from Theorem 2.1, the system (4.1) with initial conditions (1.4) is permanent.

Example 4.2. Consider the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms:























































N₁^′(t) = −(9 +|cost|) + (8 +|sint|)e^−N1(t)+ (3 +|sint|)−(0.5 +|cost|)e^−N2(t)

+(3 +|cost|)−(0.5 +|sint|)e^−N3(t)+ (1 + cos²t)N1(t−2|sint|)e^{−4N1(t−2|sint|)} +(1 + sin²t)N₁(t−2|cost|)e^{−4N1(t−2|cost|)}

N₂^′(t) = −(9 +|sint|) + (8 +|cost|)e^−N2(t)+ (3 +|cost|)−(0.5 +|sint|)e^−N1(t)

+(3 +|sint|)−(0.5 +|cost|)e^−N3(t)+ (1 + sin²t)N₂(t−2|cost|)e^{−4N2(t−2|cost|)} +(1 + cos²t)N2(t−2|sint|)e^{−4N2(t−2|sint|)}

N₃^′(t) = −(9 +|sin 2t|) + (8 +|cos 2t|)e^−N3(t)+ (3 +|cos 2t|)−(0.5 +|sin 2t|)e^−N1(t) +(3 +|sin 2t|)−(0.5 +|cos 2t|)e^−N2(t)

+(1 + sin²2t)N₂(t−2|cos 2t|)e^{−4N2(t−2|cos 2t|)} +(1 + cos²2t)N₂(t−2|sin 2t|)e^{−4N2(t−2|sin 2t|)},

(4.2) Obviously, a⁺_ii = 10, a⁻_ii = 9, b⁺_ii = 9, b⁻_ii = 8,(i = 1,2,3), a⁺_ij == 4, b⁺_ij = 1.5, a⁻_ij = 3, b⁻_ij = 0.5,(i, j= 1,2,3, i6=j),c⁺_ij = 2, γ_ij⁻= 4,(i= 1,2,3, j = 1,2), r_i= 2,(i= 1,2,3). So

2 =a⁺_ii −b⁻_ii <

n

X

j=1,j6=i

(a⁻_ij −b⁺_ij) = 3, i= 1,2,3 , and

8 + 1 e =

n

X

j=1,j6=i

a⁺_ij +

2

X

j=1

c⁺_ij

eγ_ij⁻ < a⁻_ii = 9, i= 1,2,3.

Hence, from Theorem 3.1, the model (4.2) is permanent.

The above two examples that satisfy the conditions of Theorem 2.1 and Theorem 3.1 re- spectively are permanent. Next we shall give the example that does not satisfy the conditions of Theorem 2.1 is not permanent.

Example 4.3. Consider the following Nicholson-type delay system with patch structure and nonlinear density-dependent mortality terms:

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



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







N₁^′(t) = −⁽²⁵⁺_1+|sin^|cos_t|+N^t|)N₁¹_(t)^(t)+₁₃₊⁽¹²⁺_|^|_cos^sin_t^t_|^|_+N^)N2(t)

2(t) + (1 + cos²t)N₁(t− |sint|)e^{−4N1(t−|sint|)} +(1 + sin²t)N₁(t− |cost|)e^{−4N1(t−|cost|)}

N₂^′(t) = −⁽²⁵⁺_1+|cos^|sin_t|^t_+N^|)N₂²_(t)^(t) +₁₃₊^{(12+|cost|)N1(t)}

|sint|+(1+sin²t)N1(t) +N₂(t− |cost|)e^{−4N2(t−|cost|)} +(1 + cos²t)N₂(t− |sint|)e^{−4N2(t−|sint|)}

(4.3) Obviously,a⁻₁₁=a⁻₂₂= 25,a⁺_ij = 12,(i, j = 1,2, i6=j),c⁺_ij = 2,γ_ij⁻= 4,r_i = 1,(i, j= 1,2). So

25 = min

1≤i≤2{a⁻_ii}<

2

X

i=1 2

X

j=1,j6=i

a⁺_ij +

2

X

i=1 2

X

j=1

c⁺_ij

eγ_ij⁻ = 26 +2 e, and

1max≤i≤2{sup

t∈R

a_ii(t) b_ii(t) ^P2

j=1

c_ij(t) }= 26

3 >1.

It follows that the Nicholson’s blowflies model with patch structure and nonlinear density- dependent mortality terms (4.3) dose not satisfy the conditions of Theorem 2.1. Moreover, we shall prove the model (4.3) is not permanent with the initial condition ϕ^∗ satisfying ϕ^∗ ∈C₊, ϕ^∗_i(0)>0 and||ϕ^∗||< e,i= 1,2 . We write (4.3) as the following systems of delay differential equation:

N_i^′(t) =f_i(t, N_t), i= 1,2 where







































f₁(t, ϕ) = −⁽²⁵⁺_1+|sin^|cost|^t+ϕ^|)ϕ1¹(0)⁽⁰⁾ +⁽¹²⁺_13+|^|_cos^sint_t|+ϕ^|)ϕ2(0)

2(0)+ (1 + cos²t)ϕ1(−|sint|)e^{−4ϕ1(−|sint|)} +(1 + sin²t)ϕ₁(−|cost|)e^{−4ϕ1(−|cost|)}

= −b^a₁₁¹¹(t)+ϕ^(t)ϕ11⁽⁰⁾(0) +_b^a12(t)ϕ2(0)

12(t)+ϕ2(0) + ^P2

j=1

c_1j(t)ϕ1(−τ_1j(t))e^{−γ1j(t)ϕ1(−τ1j(t))} f₂(t, ϕ) = −⁽²⁵⁺_1+|cos^|sin_t|^t_+ϕ^|)ϕ₂²₍₀₎⁽⁰⁾+⁽¹²⁺_13+|^|_sin^cos_t|^t_+ϕ^|)ϕ1(0)

1(0) + (1 + sin²t)ϕ₂(−|cost|)e^{−4ϕ2(−|cost|)} +(1 + cos²t)ϕ₂(−|sint|)e^{−4ϕ2(−|sint|)}

= −_b^a₂₂²²_(t)+ϕ^(t)ϕ2₂⁽⁰⁾₍₀₎ +_b^a21(t)ϕ1(0)

21(t)+ϕ1(0) + ^P2

j=1

c_2j(t)ϕ₂(−τ_2j(t))e^{−γ2j(t)ϕ2(−τ2j(t))}

LetN(t) =N(t;t₀, ϕ^∗) be the solution of system (4.3) with the initial conditionϕ^∗ for all t∈[t₀, η(ϕ^∗)). In view of ϕ^∗ ∈C₊, using Theorem 5.2.1 in [16,p.81], we haveN_t(t, ϕ^∗)∈C₊ for allt∈[t₀, η(ϕ^∗)). Now we prove that

||N(t)||< e for all t∈[t₀, η(ϕ^∗)) and η(ϕ^∗) = +∞. (4.4) In the contrary case, there arei∈ {1,2} and t₁> t₀ such that

N_i(t₁) =e, 0≤N_j(t)< e for all t₀ ≤t < t₁, j= 1,2.

We have

0 ≤ N_i^′(t₁) =− a_ii(t₁)N_i(t₁) b_ii(t₁) +N_i(t₁) +

2

X

j=1,j6=i

a_ij(t₁)N_j(t₁) b_ij(t₁) +N_j(t₁) +

2

X

j=1

c_ij(t₁)Ni(t₁−τ_ij(t₁))e^{−γij(t1)Ni(t1−τij(t1))}

≤ − aii(t₁)Ni(t₁) b_ii(t₁) +N_i(t₁)+

2

X

j=1,j6=i

aij(t₁)Nj(t₁) b_ij(t₁) +

2

X

j=1

cij(t₁) γ_ij(t₁)

1 e

≤ − a⁻_iie b⁺_ii+e +

2

X

j=1,j6=i

a⁺_ije b⁻_ij +

2

X

j=1

c⁺_ij γ_ij⁻

1 e

= − 25e

2 +e+e+1 e <0,

which is a contradiction. This implies that (4.4) holds. Lety(t) =N(t)e^λt, whereλ >0 and satisfyingλ−²³_2+e^−e + 4e^λ <0. We claim that

||y(t)||< e for all t∈[t₀,+∞). (4.5) If this is not valid, there arei∈ {1,2} andt₂ > t₀ such that

yi(t₂) =e, 0≤yj(t)< e for all t₀ ≤t < t₂, j= 1,2.

We have

0 ≤ y^′_i(t2) =λN_i(t2)e^λt2 +e^λt2N_i^′(t2)

= λNi(t₂)e^λt2 −aii(t₂)Ni(t₂)e^λt2 b_ii(t₂) +N_i(t₂) +

2

X

j=1,j6=i

aij(t₂)Nj(t₂)e^λt2 b_ij(t₂) +N_j(t₂)