应用数学进展  >> Vol. 8 No. 5 (May 2019)

一类具有连续分布时滞的Nicholson飞蝇模型正周期解的全局指数稳定性
The Global Exponential Stability of the Positive Periodic Solution for a Class of Nicholson’s Blowflies Model with Continuously Distributed Delays

DOI: 10.12677/AAM.2019.85115, PDF, HTML, XML, 下载: 332  浏览: 442  国家自然科学基金支持

作者: 陈秋凤, 李建利:湖南师范大学,数学与统计学院,湖南 长沙

关键词: Nicholson飞蝇模型连续分布时滞周期解全局指数稳定性Nicholson’s Blowflies System Continuous Distributed Delays Periodic Solution Global Exponential Stability

摘要: 该文研究了具有连续分布时滞的Nicholson飞蝇模型,证明了具有连续分布时滞的Nicholson飞蝇模型存在唯一的正周期解以及该正周期解的全局指数稳定性,我们改变了相关文献的模型,改进了相关文献的条件。
Abstract: In this paper, we study the existence of positive periodic solution for Nicholson’s blowflies system with continuously distributed delay. Under appropriate conditions, we obtain that the system has unique positive periodic solution and its global exponential stability.

文章引用: 陈秋凤, 李建利. 一类具有连续分布时滞的Nicholson飞蝇模型正周期解的全局指数稳定性[J]. 应用数学进展, 2019, 8(5): 1007-1015. https://doi.org/10.12677/AAM.2019.85115

1. 引言

在生态动力系统中,环境对于人口模型的影响不可忽视,Nicholson飞蝇模型便是考虑了环境因素的模型。在文献 [1] [2] [3] 中,作者研究了具有线性死亡密度的Nicholson飞蝇模型。在文献 [4] [5] [6] 中,作者考虑了具有非线性死亡密度的Nicholson飞蝇模型,为了更精确地描述其发展规律,文献 [7] [8] [9] 研究了具有离散时滞的Nicholson飞蝇模型。随着对此类模型更深入的研究,发现分布时滞更符合真实环境下的模型。在该文中,我们研究一类具有非线性死亡密度连续分布时滞的Nicholson飞蝇模型正周期解的全局指数稳定性。

在文献 [7] 中,作者考虑了离散时滞的Nicholson飞蝇模型的概周期解及其指数稳定性,在文献 [4] 中,作者研究了非线性死亡密度函数为 a ( t ) b ( t ) e x 的Nicholson飞蝇模型。该文研究的是非线性死亡密度函数为 a ( t ) x b ( t ) + x 具有连续分布时滞的另一类Nicholson飞蝇模型。

该文考虑下面的具有连续分布时滞的Nicholson飞蝇模型

x ( t ) = a ( t ) x ( t ) b ( t ) + x ( t ) + j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) x ( t s ) e x ( t s ) d s (1.1)

其中a,b, β j : R ( 0 , + ) 是连续的T-周期函数, K j ( ) > 0 j = 1 , 2 , , m 为连续时滞核函数。

定义 g + = sup t R g ( t ) g = inf t R g ( t ) r = max 1 j m σ j + > 0

C = C ( [ r , 0 ] , R ) [ r , 0 ] 上全体连续函数的集合组成的Banach空间,赋予上确界范数 C + = C ( [ r , 0 ] , R + ) ,定义 x t ( θ ) = x ( t + θ ) θ [ r , 0 ] ,初始条件为 x t 0 = φ φ C + φ ( 0 ) > 0

结合 1 x e x 的单调性可知,存在 k ( 0 , 1 ) 使得

1 k e k = 1 e 2 (1.2)

显然

sup x k | 1 x e x | = 1 e 2 (1.3)

x e x [ 0 , 1 ] 上单调递增,在 [ 1 , + ) 上单调递减,则存在唯一的 k ˜ ( 1 , + ) ,使得

k e k = k ˜ e k ˜ (1.4)

引理1.1:假设存在 K ( k , k ˜ ] 使得

{ sup t R j = 1 m β j ( t ) ( b ( t ) + K ) a ( t ) K e 0 σ j ( t ) K j ( s ) d s < 1 , inf t R j = 1 m β j ( t ) b ( t ) a ( t ) e k 0 σ j ( t ) K j ( s ) d s > 1 , (1.5)

成立,则系统(1.1)的解 x ( t ; t 0 , φ ) [ t 0 , η ( φ ) ) 上是有界的,且系统(1.1)是持久的。

证明:令 x ( t ) = x ( t ; t 0 , φ ) ,因为 φ C + ,由引理5.2.1 [10] 有 x t ( t 0 , φ ) C + t [ t 0 , η ( φ ) ) 。由 a ( t ) x ( t ) b ( t ) + x ( t ) a ( t ) x ( t ) b ( t ) ,对 t R x 0 ,我们有

x ( t ) = a ( t ) x ( t ) b ( t ) + x ( t ) + j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) x ( t s ) e x ( t s ) d s a ( t ) x ( t ) b ( t ) + j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) x ( t s ) e x ( t s ) d s

已知 x ( t 0 ) = φ ( 0 ) > 0 ,对上式积分有

x ( t ) e t o t a ( s ) b ( s ) d s t 0 t e t o s a ( τ ) b ( τ ) d τ j = 1 m β j ( s ) 0 σ j ( s ) K j ( τ ) x ( s τ ) e x ( s τ ) d τ d s + x ( t 0 ) e t o t a ( s ) b ( s ) d s > 0 , t [ t 0 , η ( φ ) )

定义

N ( t ) = max { γ : γ t , x ( γ ) = max t 0 r s t x ( s ) } t [ t 0 , η ( φ ) )

下证 x ( t ) [ t 0 , η ( φ ) ) 上是有界的。

否则,当 t η ( φ ) 时, N ( t ) η ( φ ) ,有 lim t η ( φ ) x ( N ( t ) ) = + 。则存在序列 { t n } n = 1 + ,使得 lim n + t n = η ( φ ) lim n + x ( N ( t n ) ) = +

另一方面, x ( N ( t ) ) = max t 0 r s t x ( s ) ,则 x ( N ( t ) ) 0 , N ( t ) > t 0

因此,

0 x ( N ( t ) ) = a ( N ( t ) ) x ( N ( t ) ) b ( N ( t ) ) + x ( N ( t ) ) + j = 1 m β j ( N ( t ) ) 0 σ j ( N ( t ) ) K j ( s ) x ( N ( t ) s ) e x ( N ( t ) s ) d s

a ( N ( t ) ) x ( N ( t ) ) b ( N ( t ) ) + x ( N ( t ) ) j = 1 m β j ( N ( t ) ) 0 σ j ( N ( t ) ) K j ( s ) x ( N ( t ) s ) e x ( N ( t ) s ) d s

sup x 0 x e x = 1 e ,有

a ( N ( t n ) ) x ( N ( t n ) ) b ( N ( t n ) ) + x ( N ( t n ) ) j = 1 m β j ( N ( t n ) ) 0 σ j ( N ( t n ) ) K j ( s ) d s 1 e

x ( N ( t n ) ) b ( N ( t n ) ) + x ( N ( t n ) ) j = 1 m β j ( N ( t n ) ) a ( N ( t n ) ) e 0 σ j ( N ( t n ) ) K j ( s ) d s sup t R j = 1 m β j ( t ) a ( t ) e 0 σ j ( t ) K j ( s ) d s

n + ,有

1 sup t R j = 1 m β j ( t ) a ( t ) e 0 σ j ( t ) K j ( s ) d s

与(1.5)矛盾。因此, x ( t ) [ t 0 , η ( φ ) ) 上是有界的。由 [11] 中的定理2.3.1知, η ( φ ) = +

下证 x ( t ) < K t [ t 1 , + ) t 1 > t 0 。否则 t 2 ( t 1 , + ) ,使得 x ( t 2 ) = K x ( t ) < K t [ t 1 , t 2 )

0 x ( t 2 ) = a ( t 2 ) x ( t 2 ) b ( t 2 ) + x ( t 2 ) + j = 1 m β j ( t 2 ) 0 σ j ( t 2 ) K j ( s ) x ( t 2 s ) e x ( t 2 s ) d s a ( t 2 ) K b ( t 2 ) + K + j = 1 m β j ( t 2 ) 0 σ j ( t 2 ) K j ( s ) d s 1 e

所以,有

1 sup t R j = 1 m β j ( t 2 ) ( b ( t 2 ) + K ) a ( t 2 ) K e 0 σ j ( t 2 ) K j ( s ) d s sup t R j = 1 m β j ( t ) ( b ( t ) + K ) a ( t ) K e 0 σ j ( t ) K j ( s ) d s

这与(1.5)矛盾。所以有 lim t + sup x ( t ) K

下证 c = lim t + inf x ( t ) > k ,首先证明 c > 0 ,否则 c = 0

定义

h ( t ) = max { γ : γ t , x ( γ ) = min t 0 s t x ( s ) }

则有

lim t + h ( t ) = + lim t + x ( h ( t ) ) = 0 (1.6)

h ( t ) 的定义,有 x ( h ( t ) ) = min t 0 s t x ( s ) x ( h ( t ) ) 0 h ( t ) > t 0 。因此,

0 x ( h ( t ) ) = a ( h ( t ) ) x ( h ( t ) ) b ( h ( t ) ) + x ( h ( t ) ) + j = 1 m β j ( h ( t ) ) 0 σ j ( h ( t ) ) K j ( s ) x ( h ( t ) s ) e x ( h ( t ) s ) d s a ( h ( t ) ) x ( h ( t ) ) b ( h ( t ) ) + j = 1 m β j ( h ( t ) ) 0 σ j ( h ( t ) ) K j ( s ) x ( h ( t ) s ) e x ( h ( t ) s ) d s

由第一积分中值定理,有

a ( h ( t ) ) x ( h ( t ) ) b ( h ( t ) ) + x ( h ( t ) ) j = 1 m β j ( h ( t ) ) 0 σ j ( h ( t ) ) K j ( s ) x ( h ( t ) s ) e x ( h ( t ) s ) d s = j = 1 m β j ( h ( t ) ) 0 σ j ( h ( t ) ) K j ( s ) d s x ( h ( t ) τ j ( h ( t ) ) ) e x ( h ( t ) τ j ( h ( t ) ) )

结合(1.6)有 lim t + x ( h ( t ) τ j ( h ( t ) ) ) = 0 ,其中, τ j ( t ) [ 0 , σ j ( h ( t ) ) ] j = 1 , 2 , , m

由系数函数的连续性及周期性,选取序列 { t n } n = 1 + ,使得

{ t n + , x ( h ( t n ) ) 0 , a ( h ( t n ) ) a b ( h ( t n ) ) b , β j ( h ( t n ) ) β j , σ j ( h ( t n ) ) σ j , τ j ( h ( t n ) ) τ j

因此,

a ( h ( t n ) ) b ( h ( t n ) ) + x ( h ( t n ) ) j = 1 m β j ( h ( t n ) ) 0 σ j ( h ( t n ) ) K j ( s ) d s x ( h ( t n ) τ j ( h ( t n ) ) ) x ( h ( t n ) ) e x ( h ( t n ) τ j ( h ( t n ) ) ) j = 1 m β j ( h ( t ) ) 0 σ j ( h ( t ) ) K j ( s ) d s e x ( h ( t n ) τ j ( h ( t n ) ) )

可得,

1 j = 1 m β j ( h ( t n ) ) ( b ( h ( t n ) ) + x ( h ( t n ) ) ) a ( h ( t n ) ) 0 σ j ( h ( t n ) ) K j ( s ) d s e x ( h ( t n ) τ j ( h ( t n ) ) )

n + ,有

1 lim n + j = 1 m β j ( h ( t n ) ) ( b ( h ( t n ) ) + x ( h ( t n ) ) ) a ( h ( t n ) ) 0 σ j ( h ( t n ) ) K j ( s ) d s e x ( h ( t n ) τ j ( h ( t n ) ) ) inf t R j = 1 m β j ( t ) b ( t ) a ( t ) 0 σ j ( t ) K j ( s ) d s

这与(1.5)矛盾,因此, c > 0

下证 c > k ,否则, c k

由波动引理( [10] Lemma A.1.)可知,存在数列 { t i } i 1 ,使得 t i + x ( t i ) lim t + inf x ( t ) x ( t i ) 0 ,当 i + 时。

因为 { x t i } 是一致有界且等度连续的,由Ascoli-Arzela定理可知,存在 { t i } i 1 的子列,不妨仍记为它本身,使得 x t i φ ,其中 φ C + 。此外, φ ( 0 ) = c φ ( θ ) K θ [ r , 0 )

由第一积分中值定理,有

x ( t ) = a ( t ) x ( t ) b ( t ) + x ( t ) + j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) d s x ( t τ j ( t ) ) e x ( t τ j ( t ) )

其中, τ j ( t ) [ 0 , σ j ( t ) ) j = 1 , 2 , , m

由上可推导出, c φ ( τ j ) K j = 1 , 2 , , m

x ( t i ) = a ( t i ) x ( t i ) b ( t i ) + x ( t i ) + j = 1 m β j ( t i ) 0 σ j ( t i ) K j ( s ) d s x ( t i τ j ( t i ) ) e x ( t i τ j ( t i ) )

t i + 时,有

0 = a c b + c + j = 1 m β j 0 σ j K j ( s ) d s φ ( τ j ) e φ ( τ j ) a c b + j = 1 m β j 0 σ j K j ( s ) d s c e c

与(1.5)矛盾。所以 c = lim t + inf x ( t ) > k 。综上,引理1.1得证。

引理1.2:假设引理1.1的条件成立,且

sup t R { j = 1 m β j ( t ) ( b ( t ) + K ) 2 a ( t ) b ( t ) e 2 0 σ j ( t ) K j ( s ) e s d s } < 1 (1.7)

x ( t ) = x ( t ; t 0 , φ ) x ( t ) = x ( t ; t 0 , φ ) ,则存在正常数 λ t 使得

| x ( t ; t 0 , φ ) x ( t ; t 0 , φ ) | L φ , φ e λ t t t

其中,

L φ , φ = e λ t ( max t [ t 0 r , t ] | x ( t ) x ( t ) | + 1 )

证明:令 y ( t ) = x ( t ) x ( t ) ,其中 t [ t 0 r , + ) ,则有

y ( t ) = [ a ( t ) x ( t ) b ( t ) + x ( t ) a ( t ) x ( t ) b ( t ) + x ( t ) ] + j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) [ x ( t s ) e x ( t s ) x ( t s ) e x ( t s ) ] d s

由引理1.1可知,存在 t φ , φ > t 0 k x ( t ) , x ( t ) K t [ t φ , φ r , + )

考虑Lyapunov函数 V ( t ) = | y ( t ) | e λ t

D ( V ( t ) ) = λ | y ( t ) | e λ t e λ t a ( t ) x ( t ) ( b ( t ) + x ( t ) ) ( b ( t ) + x ( t ) ) | x ( t ) x ( t ) | + sgn ( x ( t ) x ( t ) ) e λ t j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) [ x ( t s ) e x ( t s ) x ( t s ) e x ( t s ) ] d s

断言 V ( t ) = | y ( t ) | e λ t < L φ , φ

否则,存在 t ,使得 V ( t ) = L φ , φ V ( t ) < L φ , φ t [ t 0 r , t ) ,由条件知,存在 λ ( 0 , 1 ] 0 < μ < 1 ,且 λ < a ( t ) b ( t ) ( b ( t ) + K ) 2 ,使得

sup t R { j = 1 m β j ( t ) ( b ( t ) + K ) 2 ( a ( t ) b ( t ) λ ( b ( t ) + K ) 2 ) e 2 0 σ j ( t ) K j ( s ) e s d s } < μ < 1

0 D ( V ( t ) ) = λ V ( t ) a ( t ) b ( t ) ( b ( t ) + x ( t ) ) ( b ( t ) + x ( t ) ) V ( t ) + sgn ( x ( t ) x ( t ) ) e λ t j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) [ x ( t s ) e x ( t s ) x ( t s ) e x ( t s ) ] d s

所以,

( a ( t ) b ( t ) ( b ( t ) + x ( t ) ) ( b ( t ) + x ( t ) ) λ ) L φ , φ j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) e λ s V ( t s ) d s 1 e 2 j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) e s d s 1 e 2 L φ , φ

因此,有

1 sup t R { j = 1 m β j ( t ) ( b ( t ) + K ) 2 ( a ( t ) b ( t ) λ ( b ( t ) + K ) 2 ) e 2 0 σ j ( t ) K j ( s ) e s d s }

这与(1.7)矛盾。所以, | y ( t ) | < L φ , φ e λ t t > t

2. 周期解及全局指数稳定性

定理2.1:在引理1.2的条件假设下,系统(1.1)存在唯一的正T-周期解,且该正周期解是全局指数稳定的。

证明:令 x ( t ) = x ( t ; t 0 , φ ) ,由引理1.1可知,存在 t φ > t 0 ,使得 k x ( t ; t 0 , φ ) K t t φ 。由(1.1)的系数和时滞的周期性,对任意的自然数q,有

[ x ( t + ( q + 1 ) T ) ] = j = 1 m β j ( t ) 0 σ j ( t ) K j ( s ) x ( t + ( q + 1 ) T s ) e x ( t + ( q + 1 ) T s ) d s a ( t ) x ( t + ( q + 1 ) T ) b ( t ) + x ( t + ( q + 1 ) T ) , t + ( q + 1 ) T [ t 0 , + )

这表明 x ( t + ( q + 1 ) T ) t t 0 r ( q + 1 ) T 时为系统(1.1)的解。 x ( t + T ) ( t [ t 0 r , + ) ) 是系统(1.1)在初值 ψ ( s ) = x ( s + t 0 + T ) s [ r , 0 ] 下的解。

由引理1.2知,对任意的非负整数h,存在正常数 Q > t 0 ,使得当 t + q T > Q 时,有

| x ( t + ( q + 1 ) T ; t 0 , φ ) x ( t + q T ; t 0 , φ ) | = | x ( t + q T ; t 0 , ψ ) x ( t + q T ; t 0 , φ ) | L φ , ψ e λ ( t + q T )

其中

L φ , ψ = e λ Q ( max t [ t 0 r , Q ] | x ( t ) x ( t ) | + 1 )

[ a , b ] R 是R中的任意区间,选择非负整数 p 0 ,使得当 t [ a , b ] 时,有 t + p 0 T Q ,则对 t [ a , b ] p > p 0 ,有

x ( t + p T ) = x ( t + p 0 T ) + q = p 0 p 1 [ x ( t + ( q + 1 ) T ) x ( t + q T ) ]

[ a , b ] 的任意性可知, { x ( t + p T ) } p 在R上内闭一致收敛到 x ( t ) ,且 k x ( t ) K t R 。取极限有

x ( t + T ) = lim p + x ( ( t + T ) + p T ) = lim p + 1 + x ( t + ( p + 1 ) T ) = x (t)

所以, x ( t ) 是T-周期解。

x ( t + p T ) x ( t 0 + p T ) = t 0 t [ a ( s ) x ( s + p T ) b ( s ) + x ( s + p T ) + j = 1 m β j ( s ) 0 σ j ( s ) K j ( u ) x ( s + p T u ) e x ( s + p T u ) d u ] d s

p + ,有

x ( t ) x ( t 0 ) = t 0 t [ a ( s ) x ( s ) b ( s ) + x ( s ) + j = 1 m β j ( s ) 0 σ j ( s ) K j ( u ) x ( s u ) e x ( s u ) d u ] d s

所以, x ( t ) 是系统(1.1)在 [ t 0 r , + ) 上的解。类似于引理2.1的证明,可证正周期解 x ( t ) 是全局指数稳定的。

3. 举例应用

例3.1:考虑下面的系统

x ( t ) = ( 16 + 1 100 sin 2 ( t ) ) x ( t ) 12 + 1 100 sin 2 ( t ) + x ( t ) + 80 + cos 2 ( t ) 100 0 1 e s x ( t s ) e x ( t s ) d s + 80 + cos 2 ( 2 t ) 100 0 1 e s x ( t s ) e x ( t s ) d s

由上式可知,

a ( t ) = 16 + 1 100 sin 2 ( t ) b ( t ) = 12 + 1 100 sin 2 ( t ) β 1 ( t ) = 80 + cos 2 ( t ) 100 β 2 ( t ) = 80 + cos 2 ( 2 t ) 100 σ 1 ( t ) = σ 2 ( t ) = 1 K 1 ( s ) = K 2 ( s ) = e s

r = 1 a + = 16.01 a = 16 b + = 12.01 b = 12 β 1 + = β 2 + = 4 5 β 1 = β 2 = 81 100

K = 0.9 λ = 0.4 μ = 0.94 ,由计算可知, k 0.72154 k ˜ 1.34228

sup t R j = 1 m β j ( t ) ( b ( t ) + K ) a ( t ) K e 0 σ j ( t ) K j ( s ) d s = sup t R { 80 + cos 2 ( t ) 100 ( 12 + 1 100 sin 2 ( t ) + 9 10 ) 16 + 1 100 sin 2 ( t ) 9 10 e 0 1 e s d s + 80 + cos 2 ( 2 t ) 100 ( 12 + 1 100 sin 2 ( t ) + 9 10 ) 16 + 1 100 sin 2 ( t ) 9 10 e 0 1 e s d s } < 0.54 < 1

inf t R j = 1 m β j ( t ) b ( t ) a ( t ) e k 0 σ j ( t ) K j ( s ) d s = inf t R { 80 + cos 2 ( t ) 100 ( 12 + 1 100 sin 2 ( t ) ) ( 16 + 1 100 sin 2 ( t ) ) e k 0 1 e s d s + 80 + cos 2 ( 2 t ) 100 ( 12 + 1 100 sin 2 ( t ) ) ( 16 + 1 100 sin 2 ( t ) ) e k 0 1 e s d s } > 1.07 > 1

sup t R { j = 1 m β j ( t ) ( b ( t ) + K ) 2 a ( t ) b ( t ) e 2 0 σ j ( t ) K j ( s ) e s d s } = sup t R { ( 80 + cos 2 ( t ) 100 + 80 + cos 2 ( 2 t ) 100 ) ( 12 + 1 100 sin 2 ( t ) + 9 10 ) 2 ( ( 16 + 1 100 sin 2 ( t ) ) ( 12 + 1 100 sin 2 ( t ) ) 2 5 ( 12 + 1 100 sin 2 ( t ) + 9 10 ) 2 ) e 2 0 1 e 2 s d s } < 0.94 < 1

综上,满足定理2.1的全部条件,所以上述系统存在唯一的全局指数稳定的正2π-周期解。

基金项目

国家自然科学基金资助项目(No. 11571088,No. 11471109,No.11526111),浙江省自然科学项目(No. LY14A010024),湖南省教育厅项目(No. 14A098)。

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