AAM  >> Vol. 7 No. 7 (July 2018)

    污染环境下具有尺度结构的捕食种群模型解的存在唯一性
    Existence and Uniqueness of Solution for Predator-Prey Population Model with Size-Structured in Polluted Environment

  • 全文下载: PDF(500KB) HTML   XML   PP.758-765   DOI: 10.12677/AAM.2018.77091  
  • 下载量: 295  浏览量: 2,007   国家自然科学基金支持

作者:  

曹雪靓:兰州交通大学数学系,甘肃 兰州

关键词:
尺度结构存在唯一性环境污染Banach不动点定理Size-Structured Existence and Uniqueness Environmental Pollution Banach Fixed Point Theorem

摘要:

近年来,环境污染问题受到人们的广泛关注,同时研究污染环境下种群的发展规律也显得尤为重要。本文提出一个污染环境下具有尺度结构的捕食种群模型,通过建立相应的假设并运用特征线法得到了系统的形式解,然后通过不等式估计和Banach不动点定理证明了模型解的存在唯一性。

In recent years, people pay more and more attention to the problem of environmental pollution, and it is very important to study the law of population development in polluted environment. In this paper, a predator-prey population model with size-structured in polluted environment is proposed. The formal solution of the model is obtained by establishing the corresponding as-sumptions and using the characteristic line method. Then the existence and uniqueness of the so-lution are proved by inequality estimation and Banach fixed point theorem.

1. 建立模型

随着工业的发展,环境污染日趋严重,研究毒素对生物种群的影响显得尤重要。对这一问题的研究始于T.G.Hallam和他同事发表的论文 [1] [2] [3] ,随后国内外学者对这方面的问题进行了深入研究。2014年,雒志学首次将毒素种群模型和具有年龄结构的种群模型结合进行了研究 [4] [5] 。基于上述文献启发,本文提出一类具有尺度结构的捕食-被捕食者种群模型并研究其解的存在唯一性。模型如下:

{ p ( s , t ) t + [ g ( s ) p ( s , t ) ] s = [ μ 1 ( s , c 10 ( t ) ) + λ 1 ( t ) q ( t ) ] p ( s , t ) , ( s , t ) Q , d q ( t ) d t = [ μ 2 ( c 20 ( t ) ) + λ 2 ( t ) P ( t ) ] q ( t ) , t ( 0 , + ) , d c 10 ( t ) d t = k c e ( t ) l c 10 ( t ) n c 10 ( t ) , t ( 0 , + ) , d c 20 ( t ) d t = k c e ( t ) l c 20 ( t ) n c 20 ( t ) , t ( 0 , + ) , d c e ( t ) d t = k 1 c e ( t ) [ P ( t ) + q ( t ) ] + l 1 [ c 10 ( t ) P ( t ) + c 20 ( t ) q ( t ) ] h c e ( t ) + v ( t ) , g ( 0 ) p ( 0 , t ) = λ 3 ( t ) q ( t ) 0 m β 1 ( s , c 10 ( t ) ) p ( s , t ) d s , p ( s , 0 ) = p 0 ( s ) , P ( t ) = 0 m p ( s , t ) d s , q ( 0 ) = q 0 , s ( 0 , m ) , 0 c 0 ( 0 ) 1 , 0 c e ( 0 ) 1. (1)

其中 Q = ( 0 , m ) × ( 0 , + ) ,m为捕食者种群个体的最大尺度。模型中其他参数表示含义如下:

p 1 ( s , t ) :t时刻尺度为s的捕食者种群个体密度;

q ( t ) :t时刻尺度为s的食饵种群个体密度;

g ( s ) :捕食者种群个体的尺度增长函数;

c 10 ( t ) , c 20 ( t ) :分别表示t时刻捕食者种群个体、食饵种群个体体内的毒素浓度;

c e ( t ) :t时刻环境中的毒素浓度;

v ( t ) :t时刻外界向环境中输入的毒素浓度;

λ 1 ( t ) , λ 2 ( t ) :t时刻捕食者、食饵的相互作用因子;

β 1 ( s , c 10 ( t ) ) , μ 1 ( s , c 10 ( t ) ) :分别表示尺度为s,体内毒素浓度为 c 10 ( t ) 的捕食者种群的出生率和死亡率;

μ 2 ( c 20 ( t ) ) :体内毒素浓度为 c 20 ( t ) 的食饵种群的死亡率;

本文作如下假设:

(H1) μ 1 ( s , c 10 ( t ) ) L loc 1 ( Q ) 0 < μ 1 ( s , c 10 ( t ) ) μ 0 0 m μ 1 ( s , c 10 ( t ) ) d s = +

(H2) β 1 ( s , c 10 ( t ) ) L loc 1 ( Q ) 0 < β 1 ( s , c 10 ( t ) ) β 0

(H3) λ i ( t ) L ( 0 , T ) 0 < λ i ( t ) < λ i 0 i = 1 , 2 , 3

(H4) g C 1 ( 0 , m ) 0 < g ( s ) g * s ( 0 , m )

(H5) v ( ) L 2 ( 0 , T ) 0 < v ( t ) < v 0 < +

(H6) | β 1 ( s , x 1 ) β 1 ( s , x 2 ) | L β 1 | x 1 x 2 | | μ 1 ( s , x 1 ) μ 1 ( s , x 2 ) | L μ 1 | x 1 x 2 | | β 2 ( x 1 ) β 2 ( x 2 ) | L β 2 | x 1 x 2 | | μ 2 ( x 1 ) μ 2 ( x 2 ) | L μ 2 | x 1 x 2 |

(H7) 0 < p ( s , t ) < p * 0 < q ( t ) < q

(H8) l < k < l + n v 0 < h [6] 。

2. 模型解的存在唯一性

定义1.1:称向量 ( p ( s , t ) , q ( t ) , c 10 ( t ) , c 20 ( t ) , c e ( t ) ) 为模型(1)沿特征线 Γ ( s ) = 0 s 1 / g ( r ) d r 的解如果满足如下方程:

{ p ( s , t ) t + [ g ( s ) p ( s , t ) ] s = [ μ 1 ( s , c 10 ( t ) ) + λ 1 ( t ) q ( t ) ] p ( s , t ) , a .e . ( s , t ) Q , d q ( t ) d t = [ μ 2 ( c 20 ( t ) ) + λ 2 ( t ) P ( t ) + u 2 ( t ) ] q ( t ) , a .e . t ( 0 , + ) , d c 10 ( t ) d t = k c e ( t ) l c 10 ( t ) n c 10 ( t ) , a .e . t ( 0 , + ) , d c 20 ( t ) d t = k c e ( t ) l c 20 ( t ) n c 20 ( t ) , a .e . t ( 0 , + ) , d c e ( t ) d t = k 1 c e ( t ) [ P ( t ) + q ( t ) ] + l 1 [ c 10 ( t ) P ( t ) + c 20 ( t ) q ( t ) ] h c e ( t ) + v ( t ) , lim ε 0 + g ( 0 ) p ( Γ 1 ( ε ) , t + ε ) = λ 3 ( t ) q ( t ) 0 m β 1 ( s , c 10 ( t ) ) p ( s , t ) d s , lim ε 0 + p ( Γ 1 ( Γ ( s ) + ε ) , ε ) = p 0 ( s ) , P ( t ) = 0 m p ( s , t ) d s , q ( 0 ) = q 0 , a .e . s ( 0 , m ) , 0 c 10 ( 0 ) 1 , 0 c 20 ( 0 ) 1 , 0 c e ( 0 ) 1. (2)

定义1.2:模型(1)的解空间为:

X = { ( p , q , c 10 , c 20 , c e ) L ( 0 , T ; L 1 ( 0 , m ) ) × [ L ( 0 , T ) ] 4 | p ( s , t ) > 0 , q ( t ) > 0 , 0 0 m p ( s , t ) d s M , a .e . ( s , t ) Q } .

定理1.1:模型(1)解的形式为

p ( s , t ) = { exp { 0 t [ μ 1 ( Γ 1 ( Γ ( s ) + τ t ) , c 10 ( τ ) ) + g s ( Γ 1 ( Γ ( s ) + τ t ) ) + λ 1 ( τ ) q ( τ ) ] d τ } p 0 ( Γ 1 ( Γ ( s ) t ) ) , Γ ( s ) > t , exp { 0 Γ ( s ) [ μ 1 ( Γ 1 ( τ ) , c 10 ( t + τ Γ ( s ) ) ) + λ 2 ( t + τ Γ ( s ) ) P ( t + τ Γ ( s ) ) + g s ( Γ 1 ( τ ) ) ] d τ } B ( t Γ ( s ) ) , Γ ( s ) < t ,

其中 B ( t ) = p ( 0 , t )

q ( t ) = q 0 exp { 0 t [ μ 2 ( c 20 ( τ ) ) + λ 2 ( τ ) P ( τ ) ] d τ }

c i 0 ( t ) = c i 0 ( 0 ) exp { ( l + n ) t } + k 0 t c e ( τ ) exp { ( τ t ) ( l + n ) } d τ i = 1 , 2

c e ( t ) = 0 t [ l 1 c 10 ( τ ) P ( τ ) + l 1 c 20 ( τ ) q ( τ ) + v ( τ ) ] exp { t τ [ k 1 P ( r ) + k 1 q ( r ) + h ] d r } d τ + c e ( 0 ) exp { 0 t ( k 1 P ( r ) + k 1 q ( r ) + h ) d r } .

证明:模型(1)的解由特征线法和常数变易法即可导出 [7] [8] [9] 。

定理1.2:如果假设(H1)~(H8)成立,模型(1)存在唯一解

( p ( s , t ) , q ( t ) , c 10 ( t ) , c 20 ( t ) , c e ( t ) ) X

证明:首先,定义映射 F : X X

F ( p , q , c 10 , c 20 , c e ) = ( F 1 ( p , q , c 10 , c 20 , c e ) , F 2 ( p , q , c 10 , c 20 , c e ) , , F 5 ( p , q , c 10 , c 20 , c e ) ) ,

其中

F 1 ( p , q , c 10 , c 20 , c e ) = { exp { 0 t [ μ 1 ( Γ 1 ( Γ ( s ) + τ t ) , c 10 ( τ ) ) + g s ( Γ 1 ( Γ ( s ) + τ t ) ) + λ 1 ( τ ) q ( τ ) ] d τ } p 0 ( Γ 1 ( Γ ( s ) t ) ) , Γ ( s ) > t , exp { 0 Γ ( s ) [ μ 1 ( Γ 1 ( τ ) , c 10 ( t + τ Γ ( s ) ) ) + λ 2 ( t + τ Γ ( s ) ) P ( t + τ Γ ( s ) ) + g s ( Γ 1 ( τ ) ) ] d τ } B ( t Γ ( s ) ) , Γ ( s ) < t ,

F 2 ( p , q , c 10 , c 20 , c e ) = q 0 exp { 0 t [ μ 2 ( c 20 ( τ ) ) + λ 2 ( τ ) P ( τ ) ] d τ } ,

F 3 ( p , q , c 10 , c 20 , c e ) = c 10 ( 0 ) exp { ( l + n ) t } + k 0 t c e ( τ ) exp { ( τ t ) ( l + n ) } d τ ,

F 4 ( p , q , c 10 , c 20 , c e ) = c 20 ( 0 ) exp { ( l + n ) t } + k 0 t c e ( τ ) exp { ( τ t ) ( l + n ) } d τ ,

F 5 ( p , q , c 10 , c 20 , c e ) = 0 t [ l 1 c 10 ( τ ) P ( τ ) + l 1 c 20 ( τ ) q ( τ ) + v ( τ ) ] exp { t τ [ k 1 P ( r ) + k 1 q ( r ) + h ] d r } d τ + c e ( 0 ) exp { 0 t ( k 1 P ( r ) + k 1 q ( r ) + h ) d r } .

显然, F ( p , q , c 10 , c 20 , c e ) X

另外,

0 m p ( s , t ) d s = 0 Γ 1 ( t ) p ( s , t ) d s + Γ 1 ( t ) m p ( s , t ) d s = 0 Γ 1 ( t ) exp { 0 Γ ( s ) [ μ 1 ( Γ 1 ( τ ) , c 10 ( t + τ Γ ( s ) ) ) + λ 2 ( t + τ Γ ( s ) ) P ( t + τ Γ ( s ) ) + g s ( Γ 1 ( τ ) ) ] d τ } B ( t Γ ( s ) ) d s + Γ 1 ( t ) m exp { 0 t [ μ 1 ( Γ 1 ( Γ ( s ) + τ t ) , c 10 ( τ ) ) + g s ( Γ 1 ( Γ ( s ) + τ t ) ) + λ 1 ( τ ) q ( τ ) ] d τ } p 0 ( Γ 1 ( Γ ( s ) t ) ) d s 0 Γ 1 ( t ) B ( t Γ ( s ) ) d s + Γ 1 ( t ) m p 0 ( Γ 1 ( Γ ( s ) t ) ) d s g 1 ( 0 ) λ 3 0 q β 0 0 t 0 m p ( s , τ ) d s d τ + m p

则由Gronwall引理可得

0 m p ( s , t ) d s m p exp { 0 t g 1 ( 0 ) λ 3 0 q β 0 d t } = M .

接下来证明解对变量的连续依赖性。

x i = ( p i , q i , c 10 i , c 20 i , c e i ) i = 1 , 2 。当 Γ ( s ) < t 时,有如下不等式成立:

0 m | p 1 ( s , t ) p 2 ( s , t ) | d s = 0 Γ 1 ( t ) | p 1 ( s , t ) p 2 ( s , t ) | d s + Γ 1 ( t ) m | p 1 ( s , t ) p 2 ( s , t ) | d s = 0 Γ 1 ( t ) | B 1 exp { 0 Γ ( s ) [ μ 1 ( Γ 1 ( τ ) , c 10 1 ( t + τ Γ ( s ) ) ) + λ 2 ( t + τ Γ ( s ) ) P 1 ( t + τ Γ ( s ) ) + g s ( Γ 1 ( τ ) ) ] d τ } B 2 exp { 0 Γ ( s ) [ μ 1 ( Γ 1 ( τ ) , c 10 2 ( t + τ Γ ( s ) ) )

+ g s ( Γ 1 ( τ ) ) + λ 2 ( t + τ Γ ( s ) ) P 2 ( t + τ Γ ( s ) ) ] d τ } | d s + Γ 1 ( t ) m | exp { 0 t [ μ 1 ( Γ 1 ( Γ ( s ) + τ t ) , c 10 1 ( τ ) ) + g s ( Γ 1 ( Γ ( s ) + τ t ) ) + λ 1 ( τ ) q 1 ( τ ) ] d τ } exp { 0 t [ μ 1 ( Γ 1 ( Γ ( s ) + τ t ) , c 10 2 ( τ ) ) + g s ( Γ 1 ( Γ ( s ) + τ t ) ) + λ 1 ( τ ) q 2 ( τ ) ] d τ } | p 0 ( Γ 1 ( Γ ( s ) t ) ) d s

0 Γ 1 ( t ) | B 1 B 2 | d s + B 2 0 Γ 1 ( t ) 0 Γ ( s ) | μ 1 ( Γ 1 ( τ ) , c 10 1 ( t + τ Γ ( s ) ) ) μ 1 ( Γ 1 ( τ ) , c 10 2 ( t + τ Γ ( s ) ) ) | d τ d s + B 2 0 Γ 1 ( t ) 0 Γ ( s ) λ 2 ( t + τ Γ ( s ) ) | P 1 ( t + τ Γ ( s ) ) P 2 ( t + τ Γ ( s ) ) | d τ d s + Γ 1 ( t ) m p 0 ( Γ 1 ( Γ ( s ) t ) ) 0 t | μ 1 ( Γ 1 ( τ ) , c 10 1 ( t + τ Γ ( s ) ) ) μ 1 ( Γ 1 ( τ ) , c 10 2 ( t + τ Γ ( s ) ) ) | d τ d s

+ Γ 1 ( t ) m p 0 ( Γ 1 ( Γ ( s ) t ) ) 0 t λ 1 ( τ ) | q 1 ( τ ) q 2 ( τ ) | d τ d s ( 1 g * λ 3 0 β 0 q + λ 2 0 λ 3 0 β 0 q M T ) 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ + ( 1 g * λ 3 0 β 0 M + m λ 1 0 p * ) 0 t | q 1 ( τ ) q 2 ( τ ) | d τ + ( 1 g * λ 3 0 q M L β 1 + T λ 3 0 q M L μ 1 + m p * L μ 1 ) 0 t | c 10 1 ( τ ) c 10 2 ( τ ) | d τ .

因此,

0 m | F 1 ( x 1 ) F 1 ( x 2 ) | d s M 1 ( 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ + 0 t | q 1 ( τ ) q 2 ( τ ) | d τ + 0 t | c 10 1 ( τ ) c 10 2 ( τ ) | d τ ) , (3)

M 1 = max { 1 g * λ 3 0 β 0 q + λ 2 0 λ 3 0 β 0 q M T , 1 g * λ 3 0 β 0 M + m λ 1 0 p * , 1 g * λ 3 0 q M L β 1 + T λ 3 0 q M L μ 1 + m p * L μ 1 } .

Γ ( s ) > t 时同理可得上述不等式成立。

| q 1 q 2 | = q 0 | exp { 0 t [ μ 2 ( c 20 1 ( τ ) ) + λ 2 ( τ ) P 1 ( τ ) ] d τ } exp { 0 t [ μ 2 ( c 20 2 ( τ ) ) + λ 2 ( τ ) P 2 ( τ ) ] d τ } | q 0 0 t | μ 2 ( c 20 1 ( τ ) ) μ 2 ( c 20 2 ( τ ) ) | d τ + q 0 0 t λ 2 ( τ ) | P 1 ( τ ) P 2 ( τ ) | d τ q 0 L μ 2 0 t | c 20 1 ( τ ) c 20 2 ( τ ) | d τ + q 0 λ 2 0 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ M 2 ( 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ + 0 t | c 20 1 ( τ ) c 20 2 ( τ ) | d τ ) (4)

其中, M 2 = max { q 0 L μ 2 , q 0 λ 2 0 }

| c i 0 1 c i 0 2 | = | c i 0 ( 0 ) exp { ( l + n ) t } + k 0 t c e 1 ( τ ) exp { ( τ t ) ( l + n ) } d τ c i 0 ( 0 ) exp { ( l + n ) t } + k 0 t c e 2 ( τ ) exp { ( τ t ) ( l + n ) } d τ | = k 0 t | c e 1 ( τ ) c e 2 ( τ ) | exp { ( τ t ) ( l + n ) } d τ M 3 0 t | c e 1 ( τ ) c e 2 ( τ ) | d τ ( M 3 = k ) (5)

| c e 1 c e 2 | = | 0 t [ l 1 c 10 1 ( τ ) P 1 ( τ ) + l 1 c 20 1 ( τ ) q 1 ( τ ) + v ( τ ) ] exp { t τ [ k 1 P 1 ( r ) + k 1 q 1 ( r ) + h ] d r } d τ + c e ( 0 ) exp { 0 t ( k 1 P 1 ( r ) + k 1 q 1 ( r ) + h ) d r } 0 t [ l 1 c 10 2 ( τ ) P 2 ( τ ) + l 1 c 20 2 ( τ ) q 2 ( τ ) + v ( τ ) ] exp { t τ [ k 1 P 2 ( r ) + k 1 q 2 ( r ) + h ] d r } d τ + c e ( 0 ) exp { 0 t ( k 1 P 2 ( r ) + k 1 q 2 ( r ) + h ) d r } |

c e ( 0 ) 0 t | k 1 ( P 1 ( r ) P 2 ( r ) ) + k 1 ( q 1 ( r ) q 2 ( r ) ) | d r + 0 t | l 1 ( c 10 1 ( τ ) P 1 ( τ ) c 10 2 ( τ ) P 2 ( τ ) ) + l 1 ( c 20 1 ( τ ) q 1 ( τ ) c 20 2 ( τ ) q 2 ( τ ) ) | d τ + 0 t [ l 1 c 10 2 ( τ ) P 2 ( τ ) + l 1 c 20 2 ( τ ) q 2 ( τ ) + v ( τ ) ] | exp { t τ [ k 1 P 1 ( r ) + k 1 q 1 ( r ) + h ] d r } exp { t τ [ k 1 P 2 ( r ) + k 1 q 2 ( r ) + h ] d r } | d τ

( k 1 + l 1 + T M k 1 l 1 + T q k 1 l 1 + T k 1 v 0 ) 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ + ( k 1 + l 1 + T M k 1 l 1 + T q k 1 l 1 + T k 1 v 0 ) 0 t | q 1 ( τ ) q 2 ( τ ) | d τ + M l 1 0 t | c 10 1 ( τ ) c 10 2 ( τ ) | d τ + q l 1 0 t | c 20 1 ( τ ) c 20 2 ( τ ) | d τ M 4 ( 0 t 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s d τ + 0 t | q 1 ( τ ) q 2 ( τ ) | d τ + i = 1 2 0 t | c 20 1 ( τ ) c 20 2 ( τ ) | d τ ) (6)

M 4 = max { k 1 + l 1 + T M k 1 l 1 + T q k 1 l 1 + T k 1 v 0 , M l 1 , q l 1 }

最后,定义解空间X上的范数为

( p , q , c 10 , c 20 , c e ) X = E s s sup t ( 0 , T ) e λ t { 0 m | p ( s , t ) | d s + | q ( t ) | + i = 1 2 | c i 0 ( t ) | + | c e ( t ) | } ,

λ > 0 足够大。

则由不等式(3)-(6)可得

F ( x 1 ) F ( x 2 ) X = F 1 ( x 1 ) F 1 ( x 2 ) , F 2 ( x 1 ) F 2 ( x 2 ) , , F 5 ( x 1 ) F 5 ( x 2 ) X M 5 E s s sup t ( 0 , T ) e λ t 0 t { 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s + | q 1 ( τ ) q 2 ( τ ) | + i = 1 2 | c i 0 1 ( τ ) c i 0 2 ( τ ) | + | c e 1 ( τ ) c e 2 ( τ ) | } = M 5 E s s sup t ( 0 , T ) e λ t 0 t e λ τ { e λ τ [ 0 m | p 1 ( s , τ ) p 2 ( s , τ ) | d s + | q 1 ( τ ) q 2 ( τ ) | + i = 1 2 | c i 0 1 ( τ ) c i 0 2 ( τ ) | + | c e 1 ( τ ) c e 2 ( τ ) | ] } d τ M 5 x 1 x 2 X E s s sup t ( 0 , T ) { e λ t 0 t e λ τ d τ } M 5 λ x 1 x 2 X

成立,其中 M 5 = max { M 1 , M 2 , M 3 , M 4 } 是常数, λ > M 5 时F是压缩的,由Banach不动点定理可得不动点 ( p , q , c 10 , c 20 , c e ) 是F的唯一解同时也是模型(1)的解。证明完毕。

3. 结论

受毒素种群模型和具有尺度结构的捕食种群模型的启发,本文在第一节中建立了一个污染环境下具有尺度结构的捕食种群模型,并对模型中相应的参数进行了解释,给出基本假设。第二节中通过运用特征线法得到了模型(1)的形式解,然后通过不等式估计和Banach不动点定理证明了系统解的存在唯一性。

基金项目

本论文由国家自然基金项目(11561041)和甘肃省自然科学基金项目(1506RJZA071)资助。

文章引用:
曹雪靓. 污染环境下具有尺度结构的捕食种群模型解的存在唯一性[J]. 应用数学进展, 2018, 7(7): 758-765. https://doi.org/10.12677/AAM.2018.77091

参考文献

[1] Hallam, T.G., Clark, C.E. and Lassiter, R.R. (1983) Effects of Toxicants on Populations: A Qualitative Approach I. Equilibrium En-vironmental Exposure. Ecological Modelling, 18, 291-304.
https://doi.org/10.1016/0304-3800(83)90019-4
[2] Hallam, T.G., Clark, C.E. and Jordan, G.S. (1983) Effects of Toxicants on Populations: A Qualitative Approach II. First Order Kinetics. Journal of Mathematical Biology, 18, 25-37.
https://doi.org/10.1007/BF00275908
[3] Hallam, T.G. and De Luna, J.T. (1984) Effects of Toxicants on Populations: A Qual-itative Approach III. Environmental and Food Chain Pathways. Journal of Theoretical Biology, 109, 411-429.
https://doi.org/10.1016/S0022-5193(84)80090-9
[4] Luo, Z.X. and He, Z.R. (2014) Optimal Control for Age-Dependent Pop-ulation Hybrid System in a Polluted Environment. Applied Mathematics and Computation, 228, 68-76.
https://doi.org/10.1016/j.amc.2013.11.070
[5] Luo, Z.X. and Fan, X.L. (2014) Optimal Control for an Age-Dependent Com-petitive Species Model in a Polluted Environment. Applied Mathematics and Computation, 228, 91-101.
https://doi.org/10.1016/j.amc.2013.11.069
[6] 马知恩. 种群生态学的数学建模与研究[M]. 合肥: 安徽教育出版社, 1996: 168-175.
[7] Anita, S. (2000) Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic, Boston, 67-70.
https://doi.org/10.1007/978-94-015-9436-3
[8] 刘炎, 何泽荣. 具有size结构的捕食种群系统的最优收获策略[J]. 数学物理学报, 2012, 32(1): 90-102.
[9] He, Z.R. and Liu, Y. (2012) An Optimal Birth Control Problem for a Dynamical Population Model with Size-Structure. Nonlinear Analysis: Real World Applications, 13, 1369-1378.
https://doi.org/10.1016/j.nonrwa.2011.11.001