Skip to main content

Optimal harvesting control for an age-dependent competing population with diffusion

Abstract

This paper is mainly concerned with optimal harvesting policy for competing species with age dependence and diffusion. The existence and uniqueness of solutions for the system are proved by using the Banach fixed point theorem. The existence and uniqueness of optimal control are discussed by Ekeland’s variational principle. The maximum principle is obtained.

MSC: 35B10, 49K05, 65L12, 92B05.

1 Introduction

The optimal harvesting control of age-structured and size-structured single species have been widely studied in the literature [1]–[11]. The control problems of multi-species have been investigated in the literature [12]–[20]. The objective function represents the total harvesting, respectively, in [12]–[15]. The diffusion factor has not been considered in [14]–[20]. It is well known that the profit due to harvesting is a quite important problem for optimal harvesting. In this paper, our purpose is to consider the optimal control of the profit functional for diffusion population. Specifically, we deal with the following optimal harvesting problem:

sup i = 1 n Q [ u i ( a , t , x ) p i u ( a , t , x ) u i 2 ( a , t , x ) ] dadtdx
(1.1)

for all u=( u 1 (a,t,x), u 2 (a,t,x),, u n (a,t,x))U, where the corresponding state variable p u =( p 1 u , p 2 u ,, p n u ) satisfies the state system

{ p i a + p i t k i Δ p i = f i μ i ( a , t , x ) p i u i ( a , t , x ) p i p i a + p i t k i Δ p i = k = 1 , k i n λ i k ( a , t , x ) P k ( t , x ) p i , ( a , t , x ) Q , p i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , p i ( 0 , t , x ) = 0 A β i ( a , t , x ) p i ( a , t , x ) d a , ( t , x ) Ω T , p i ( a , 0 , x ) = p i 0 ( a , x ) , ( a , x ) Ω A , P i ( t , x ) = 0 A p i ( a , t , x ) d a , ( t , x ) Ω T , i = 1 , 2 , , n ,
(1.2)

where p i (a,t,x) represents the density of the ith population. k i is the diffusion rate of the ith population. We assume that the populations have the same life expectancy A, 0<A<+. μ i (a,t,x) is the average morality of the ith population, and β i (a,t,x) describes the average fertility of the ith population. λ i k (a,t,x) represents the interaction coefficients (i,k=1,2,,n, ki). Ω R N (N=1,2,3) is a bounded domain with a smooth enough boundary Ω, Q=(0,A)×(0,T)×Ω, Ω T =Ω×(0,T), Ω A =Ω×(0,A), =(0,A)×(0,T)×Ω, and ν is the outward unit normal. The function p i 0 (a,x) gives the initial density distribution of the population, and u i (a,t,x) represents the harvesting effort function, which is the control variable in the model and satisfies

u i U i = { v i L 2 ( Q ) | 0 γ i 1 ( a , t , x ) v i ( a , t , x ) γ i 2 ( a , t , x )  a.e. in  Q } ,U= i = 1 n U i ,

where γ i j L (Q), j=1,2, i=1,2,,n. The integral

J( u 1 , u 2 ,, u n )= i = 1 n Q [ u i ( a , t , x ) p i u ( a , t , x ) u i 2 ( a , t , x ) ] dadtdx
(1.3)

represents the profit due to harvesting.

Throughout this paper, we assume that:

(A1): β i (a,t,x) L (Q), 0 β i (a,t,x)M, where M is a constant.

(A2): μ i L loc ([0,A]×[0,T]× Ω ¯ ), μ i (a,t,x) μ 0 (a,t)0 a.e. in Q, i=1,2,,n, where μ 0 L loc ([0,A]×[0,T]), 0 A μ 0 (a,t+aA)da=+.

(A3): λ i k L (Q), 0 λ i k (a,t,x)B, where B is a positive constant (i,k=1,2,,n, ki).

(A4): f i L (Q), p i 0 (a,x) L ( Ω A ), p i 0 (a,x)0, 0 f i (a,t,x) B 0 , where B 0 is a constant, i=1,2,,n.

The rest of this paper is organized as follows. In Section 2, we prove that under the assumptions listed above, the system has a unique non-negative solution. In Section 3, the necessary conditions of optimality for the control problems is given. In the final section, we prove the existence and uniqueness of the optimal control.

2 The existence and uniqueness of solution for system (1.2)

For the sake of convenience, we introduce the following definitions of the solution.

Definition 2.1

Given the solution of system (1.2), the function p i L 2 (Q), i=1,2,,n, which belongs to C( S ¯ ; L 2 (Ω))AC(S; L 2 (Ω)) L 2 (S; H 1 (Ω)) L loc 2 (S; H 2 (Ω)) for almost any characteristic line S of the equation

at= a 0 t 0 ,(a,t)(0,A)×(0,T),( a 0 , t 0 ){0}×(0,T)(0,A)×{0},

satisfies

{ p i a + p i t k i Δ p i = f i μ i ( a , t , x ) p i u i ( a , t , x ) p i p i a + p i t k i Δ p i = k = 1 , k i n λ i k ( a , t , x ) P k ( t , x ) p i , a.e. in  Q , p i ν ( a , t , x ) = 0 , a.e. in  Σ , lim ε 0 + p ( ε , t + ε , ) = 0 A β i ( a , t , ) p i ( a , t , ) d a , in  L 2 ( Ω ) , a.e.  t ( 0 , T ) , lim ε 0 + p i ( a + ε , ε , ) = p i 0 ( a , ) , in  L 2 ( Ω ) , a.e.  a ( 0 , A ) ,
(2.1)

where P i (t,)= 0 A p i (a,t,)da, i=1,2,,n.

Then we rewrite the characteristic line S as

S= { ( a , t ) ( 0 , A ) × ( 0 , T ) ; a t = a 0 t 0 } = { ( a 0 + s , t 0 + s ) ; s ( 0 , α ) } ,

here ( a 0 +α, t 0 +α){A}×(0,T)(0,A)×{T}.

We introduce the following notations:

C ( S ¯ ; L 2 ( Ω ) ) = { h : S ¯ L 2 ( Ω ) ; h  continuous } , A C ( S ; L 2 ( Ω ) ) = { h : S L 2 ( Ω ) : h ( a 0 + , t 0 + ) : ( 0 , α ) L 2 ( Ω ) A C ( S ; L 2 ( Ω ) ) =  is absolutely continuous on any compact subinterval } .

Theorem 2.1

For any givenu=( u 1 , u 2 ,, u n )U, system (1.2) has a unique non-negative solution p u =( p 1 u , p 2 u ,, p n u ) L 2 (Q; R n )such that

  1. (i)

    0 p i u M 1 , (a,t,x)Q, i=1,2,,n, where M 1 is a positive constant;

  2. (ii)

    p u is continuous in u.

Proof

For any given h=( h 1 , h 2 ,, h n ) L 2 (Q; R n ), h0, we define

H i (t,x)= 0 A h i (a,t,x)da,i=1,2,,n.

The given system

{ p i a + p i t k i Δ p i = f i μ i ( a , t , x ) p i u i p i p i a + p i t k i Δ p i = k = 1 , k i n λ i k ( a , t , x ) H k ( t , x ) p i , ( a , t , x ) Q , p i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , p i ( 0 , t , x ) = 0 A β i ( a , t , x ) p i ( a , t , x ) d a , ( t , x ) Ω T , p i ( a , 0 , x ) = p i 0 ( a , x ) , ( a , x ) Ω A , P i ( t , x ) = 0 A p i ( a , t , x ) d a , ( t , x ) Ω T , i = 1 , 2 , , n ,
(2.2)

by Theorem 4.1.3 in [21], we know that the above system has a unique non-negative solution,

p h = ( p 1 h , p 2 h , , p n h ) L 2 ( Q ; R n ) , p i h (A,t,x)=0,(t,x) Ω T ,i=1,2,,n.

From the comparison principle of linear system [21], it follows that

p i h (a,t,x) p ¯ i (a,t,x),a.e. in Q,i=1,2,,n,

where p ¯ i (a,t,x) L (Q), and this is the solution of the following system:

{ p i a + p i t k i Δ p i = f i μ i ( a , t , x ) p i , ( a , t , x ) Q , p i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , p i ( 0 , t , x ) = 0 A β i ( a , t , x ) p i ( a , t , x ) d a , ( t , x ) Ω T , p i ( a , 0 , x ) = p i 0 ( a , x ) , ( a , x ) Ω A .

For any h k =( h 1 k , h 2 k ,, h n k ) L 2 (Q; R n ), 0 h i k p ¯ i , let the corresponding state be p k =( p 1 k , p 2 k ,, p n k ) (k=1,2), w=( w 1 , w 2 ,, w n ):= p 1 p 2 . It follows from (1.2) that

{ w i a + w i t k i Δ w i = ( μ i + u i ) w i k = 1 , k i n λ i k ( a , t , x ) H k 1 ( t , x ) w i w i a + w i t k i Δ w i = k = 1 , k i n λ i k ( a , t , x ) [ H k 1 ( t , x ) H k 2 ( t , x ) ] p i 2 , ( a , t , x ) Q , w i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , w i ( 0 , t , x ) = 0 A β i ( a , t , x ) w i ( a , t , x ) d a , w i ( a , 0 , x ) = 0 , ( a , x ) Ω A , H i j ( t , x ) = 0 A h i j ( a , t , x ) d a , ( t , x ) Ω T , i = 1 , 2 , , n , j = 1 , 2 .
(2.3)

Multiplying the ith equation in system (2.3) by w i (i=1,2,,n), integrating on (0,A)×(0,t)×Ω, and according to the Hölder inequality, we have

w i ( , t , ) L 2 ( Ω A ) 2 ( A M 2 + B p ¯ i L ( Q ) ) 0 t w i ( , τ , ) L 2 ( Ω A ) 2 d τ + k = 1 n A 2 B p ¯ i L ( Q ) 0 t h k 1 h k 2 L 2 ( Ω A ) 2 d τ = C ˜ 0 t ( k = 1 n h k 1 ( , τ , ) h k 2 ( , τ , ) L 2 ( Ω A ) 2 + w i ( , τ , ) L 2 ( Ω A ) 2 ) d τ ,
(2.4)

where C ˜ = max 1 i n {A M 2 +B p ¯ i L ( Q ) , A 2 B p ¯ i L ( Q ) }. By using the Gronwall inequality, we have

w i ( , t , ) L 2 ( Ω A ) 2 C k = 1 n 0 t h k 1 ( , τ , ) h k 2 ( , τ , ) L 2 ( Ω A ) 2 dτ,
(2.5)

where C= C ˜ e C ˜ T . Set

I:= { h = ( h 1 , h 2 , , h n ) L 2 ( Q ; R n ) : 0 h i ( a , t , x ) p ¯ i ( a , t , x ) , ( a , t , x ) Q } .

Define the mapping G:II

(Gh)(a,t,x)= p h (a,t,x),(a,t,x)Q.

Denote the norm on L 2 (Q; R N ) by

h L 2 ( Q ) = ( i = 1 n h i L 2 ( Q ) 2 ) 1 2 .

Define the following norm for C>0:

h i = 0 T h i ( , t ) L 2 ( Ω A ) 2 e 4 n C t dt, h 2 = ( i = 1 n h i 2 ) 1 2 ,i=1,2,,n.

Obviously the norm L 2 ( Q ) is equivalent to the norm . Using (2.5), we get

G h 1 G h 2 = p 1 p 2 = ( i = 1 n w i 2 ) 1 2 = ( i = 1 n 0 T w i ( , t , ) L 2 ( Ω A ) 2 e 4 n C t d t ) 1 2 ( i = 1 n 0 T C k = 1 n 0 t h k 1 ( , τ , ) h k 2 ( , τ , ) L 2 ( Ω A ) 2 d τ e 4 n C t d t ) 1 2 = ( 0 T k = 1 n h k 1 ( , s , ) h k 2 ( , s , ) L 2 ( Ω A ) 2 s T n C e 4 n C t d t d s ) 1 2 1 2 ( 0 T k = 1 n h k 1 ( , s , ) h k 2 ( , s , ) L 2 ( Ω A ) 2 e 4 n C s d s ) 1 2 = 1 2 h 1 h 2 .

It is obvious that GII and G is a contraction on (I, ), so there is a unique fixed point, which is the solution of system (1.2).

Since p ¯ i L (Q), let M 1 =max{esssup| p ¯ i (a,t,x)|,1in}, we have 0 p i u M 1 , a.e. (a,t,x)Q. In the following, we study the continuity of the solution of (1.2) for the control variable u. Let u k =( u 1 k , u 2 k ,, u n k )U, k=1,2, y i (a,t,x)= p i u 1 (a,t,x) p i u 2 (a,t,x). From (1.2), it follows that

{ y i a + y i t k i Δ y i = ( μ i + u i 1 ) y i y i a + y i t k i Δ y i = k = 1 , k i n λ i k p i u 2 0 A y k ( a , t , x ) d a ( u i 1 u i 2 ) p i u 2 y i a + y i t k i Δ y i = k = 1 , k i n λ i k ( a , t , x ) P k u 1 ( t , x ) y i , ( a , t , x ) Q , y i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , y i ( 0 , t , x ) = 0 A β i ( a , t , x ) y i ( a , t , x ) d a , ( t , x ) Ω T , y i ( a , 0 , x ) = 0 , ( a , x ) Ω A , P i u j ( t , x ) = 0 A p i u j ( a , t , x ) d a , ( t , x ) Ω T , i = 1 , 2 , , n , j = 1 , 2 .
(2.6)

In a similar manner as that in (2.4), we deduce that

y i ( , τ , ) L 2 ( Ω A ) 2 ( A M 2 + p ¯ i L ( Q ) + B p ¯ i L ( Q ) ) 0 t y i ( , τ , ) L 2 ( Ω A ) 2 d τ + p ¯ i L ( Q ) 0 t u i 1 u i 2 L 2 ( Ω A ) 2 d τ + 2 k = 1 , k i n A B p ¯ k L ( Q ) 0 t Ω A y i 2 d x d a d τ .

Let C ¯ =A M 2 + p ¯ i L ( Q ) +B p ¯ i L ( Q ) +2AB k = 1 , k i n p ¯ k L ( Q ) , we have

y i ( , t , ) L 2 ( Ω A ) 2 C ¯ ( 0 t u i 1 ( , τ , ) u i 2 ( , τ , ) L 2 ( Ω A ) 2 d τ + 0 t y i ( , τ , ) L 2 ( Ω A ) 2 d τ ) .
(2.7)

Adding up (2.7) from i=1 to n and using the Gronwall inequality, we get

i = 1 n y i ( , t , ) L 2 ( Ω A ) 2 K i = 1 n 0 t u i 1 ( , τ , ) u i 2 ( , τ , ) L 2 ( Ω A ) 2 dτ,
(2.8)

where K= C ¯ e C ¯ T . Multiplying (2.8) by e n r t and integrating on (0,T), we have

p 1 p 2 K T u 1 u 2 .
(2.9)

Inequality (2.9) implies that p u is continuous with respect to u. This completes the proof. □

3 The necessary conditions

Before stating our main results, we prove the following lemma, which is useful in proving our results.

Lemma 3.1

Let( u , p )be an optimal pair for Problem (1.1), for any0<ε<1and for anyv=( v 1 , v 2 ,, v n ) L 2 (Q; R n ), v i >0, u +ε(v u )= u ε U, where u ε =( u 1 ε , u 2 ε ,, u n ε ), then the following limit holds:

p u ε p u ε z,in  L 2 ( Q ; R n ) ,as ε 0 + ,

where p =( p 1 u , p 2 u ,, p n u ), u =( u 1 , u 2 ,, u n ), andz=( z 1 , z 2 ,, z n )is the solution of

{ z i a + z i t k i Δ z i = ( μ i ( a , t , x ) + u i ) z i ( v i u i ) p i u z i a + z i t k i Δ z i = k = 1 , k i n λ i k ( a , t , x ) Z k ( t , x ) p i u z i a + z i t k i Δ z i = k = 1 , k i n λ i k P k u ( t , x ) z i , ( a , t , x ) Q , z i ν ( a , t , x ) = 0 , ( a , t , x ) Σ , z i ( 0 , t , x ) = 0 A β i ( a , t , x ) z i ( a , t , x ) d a , ( t , x ) Ω T , z i ( a , 0 , x ) = 0 , ( a , x ) Ω A , P i u ( t , x ) = 0 A p i u ( a , t , x ) d a , ( t , x ) Ω T , Z i ( t , x ) = 0 A z i ( a , t , x ) d a , ( t , x ) Ω T , i = 1 , 2 , , n .
(3.1)

Proof

The existence and uniqueness of the solution of (3.1) follow in the same manner as by Theorem 2.1, the solution z=( z 1 , z 2 ,, z n ) L 2 (Q; R n ), and 0 z i L, where L is a constant. We introduce the following notations:

z ε ( a , t , x ) = 1 ε [ p u ε ( a , t , x ) p u ( a , t , x ) ] , ( a , t , x ) Q , x ε ( a , t , x ) = ε z ε ( a , t , x ) , ( a , t , x ) Q .

It is straightforward that z ε is the solution of

{ z i ε a + z i ε t k i Δ z i ε = ( μ i + u i ) z i ε ( v i u i ) p i u ε z i ε a + z i ε t k i Δ z i ε = k = 1 , k i n λ i k ( a , t , x ) Z k ε ( t , x ) p i u ε z i ε a + z i ε t k i Δ z i ε = k = 1 , k i n λ i k ( a , t , x ) P k u ( t , x ) z i ε , ( a , t , x ) Q , z i ε ν ( a , t , x ) = 0 , ( a , t , x ) Σ , z i ε ( 0 , t , x ) = 0 A β i ( a , t , x ) z i ε ( a , t , x ) d a , ( t , x ) Ω T , z i ( a , 0 , x ) = 0 , ( a , x ) Ω A ,
(3.2)

where P i u (t,x)= 0 A p i u (a,t,x)da, Z i ε (t,x)= 0 A z i ε (a,t,x)da, (t,x) Ω T .

By (2.9), we have

x ε = p u ε p u K T ε ( v u ) = K T ε v u .
(3.3)

So we can claim that x ε 0 in L 2 (Q; R n ), as ε 0 + . By (3.1) and (3.2), we see that z ε z=( z 1 ε z 1 , z 2 ε z 2 ,, z n ε z n ) L 2 (Q; R n ) is the solution of the following equation:

{ ( z i ε z i ) a + ( z i ε z i ) t k i Δ ( z i ε z i ) = μ i ( z i ε z i ) u i ( z i ε z i ) ( v i u i ) ( p i u ε p i u ) k = 1 , k i n λ i k ( 0 A ( z k ε z k ) d a ) p i u ε k = 1 , k i n λ i k ( a , t , x ) Z k ( t , x ) ( p i u ε p i u ) + k = 1 , k i n λ i k ( 0 A ( p k u ε p k u ) d a ) z i ε k = 1 , k i n λ i k ( 0 A p k u ( a , t , x ) d a ) ( z i ε z i ) , ( a , t , x ) Q , ( z i ε z i ) v ( a , t , x ) = 0 , ( a , t , x ) Σ , ( z i ε z i ) ( 0 , t , x ) = 0 A β i ( a , t , x ) ( z i ε z i ) d a , ( t , x ) Ω T , z i ( a , 0 , x ) = 0 , ( a , x ) Ω A , i = 1 , 2 , , n ,
(3.4)

where P i u (t,x)= 0 A p i u (a,t,x)da, Z i (t,x)= 0 A z i (a,t,x)da, (t,x) Ω T .

Multiplying the first equation in (3.4) by z i ε z i , integrating on (0,A)×(0,t)×Ω, using the Gronwall inequality, in the same manner as in (2.5), and taking x ε 0 in L 2 (Q; R n ), we deduce

z ε z,in  L 2 ( Q , R n ) ,as ε 0 + .

Consequently, the proof is completed. By the same argument as in (2.9), the following lemma holds. □

In order to give the necessary conditions, we introduce the adjoint equations of (3.1).

Lemma 3.2

If q u 1 , q u 2 are the solutions of the following system corresponding to the control u 1 , u 2 , respectively:

{ q j a + q j t + k j Δ q j = ( μ j + u j ) q j + u j q j ( 0 , t , x ) β j ( a , t , x ) q j a + q j t + k j Δ q j = + k = 1 , k j n λ j k ( a , t , x ) q j P k u ( t , x ) q j a + q j t + k j Δ q j = + k = 1 , k j n 0 A λ k j ( a , t , x ) p k u q j ( a , t , x ) d a , ( a , t , x ) Q , q j ν ( a , t , x ) = 0 , ( a , t , x ) Σ , q j ( a , T , x ) = q j ( A , t , x ) = 0 , ( a , x ) Ω A , P j u ( t , x ) = 0 A p j u ( a , t , x ) d a , ( t , x ) Ω T , j = 1 , 2 , , n .
(3.5)

Then

q u 1 q u 2 K T u 1 u 2 , q u M 2 ,

where q u 1 =( q 1 u 1 , q 2 u 1 ,, q n u 1 ), q u 2 =( q 1 u 2 , q 2 u 2 ,, q n u 2 ), u 1 =( u 1 1 , u 2 1 ,, u n 1 ), u 2 =( u 1 2 , u 2 2 ,, u n 2 ), K ¯ >0, M 2 are constant and independent of q u 1 , q u 2 , u 1 , u 2 .

We are now able to state the main result in this section.

Theorem 3.1

Suppose that (A1)-(A5) hold, if( u , p )is an optimal pair for Problem (1.1), andq=( q 1 , q 2 ,, q n )is the solution of (3.5), which is corresponding to the control u =( u 1 , u 2 ,, u n ), then we have the following necessary conditions:

i = 1 n 0 A 0 T Ω ( v i u i ) [ ( 1 + q i ) p i u 2 u i ] dxdtda0.

Proof

The existence and uniqueness of the solution q=( q 1 , q 2 ,, q n ) to (3.5) can be proved by Theorem 2.1. For any v i L 2 (Q) and any 0<ε<1, u i +ε( v i u i ) U i . Since u =( u 1 , u 2 ,, u n ) is an optimal control for (1.1), we get

i = 1 n Q [ u i p i u ( u i ) 2 ] dadtdx i = 1 n Q [ u i ε p i u ε u i ε 2 ] dadtdx,

which implies

i = 1 n 0 A 0 T Ω [ u i p i u ε p i u ε + ( v i u i ) ( p i u ε 2 u i ε ( v i u i ) ) ] dxdtda0.
(3.6)

By Lemma 3.1 and inequality (3.3), passing to the limit as ε 0 + in (3.6), we conclude

i = 1 n 0 A 0 T Ω [ u i z i + ( v i u i ) ( p i u 2 u i ) ] dxdtda0.
(3.7)

Multiplying (3.5) by z i , then integrating over Q, we deduce that

i = 1 n 0 A 0 T Ω ( z i u i ) ( a , t , x ) d x d t d a = i = 1 n 0 A 0 T Ω q i ( v i u i ) p i u ( a , t , x ) d x d t d a .
(3.8)

Combining (3.7) with (3.8), we have

i = 1 n 0 A 0 T Ω ( v i u i ) [ ( 1 + q i ) p i u 2 u i ] dxdtda0.
(3.9)

This completes the proof. □

4 The existence and uniqueness of optimal control

In this section, we prove the existence and uniqueness of optimal control. From [21], we give the following lemma.

Lemma 4.1

LetXbe a reflexive Banach space and letφ:X(,+]be a lower semi-continuous convex function. If X 0 is a closed, convex, and bounded subset ofX, thenφattains its infimum on X 0 . In other words, then there is x 0 X 0 such that

φ( x 0 )=inf { φ ( x ) , x X 0 } .

The main conclusion is presented as follows:

Theorem 4.1

IfTis small enough, then Problem (1.1) has a unique optimal control inU.

Proof

For any u=( u 1 , u 2 ,, u n ), v=( v 1 , v 2 ,, v n ), u i U i , v i U i , i=1,2,,n, we define

H(ε)=J ( ε u + ( 1 ε ) v ) ,

we shall prove that H (ε) is strictly monotone. Indeed, denote by ( p 1 ε , p 2 ε ,, p n ε ) and ( p 1 ε + δ , p 2 ε + δ ,, p n ε + δ ) the state corresponding to controls ε( u 1 , u 2 ,, u n )+(1ε)( v 1 , v 2 ,, v n ) and (ε+δ)( u 1 , u 2 ,, u n )+(1(ε+δ))( v 1 , v 2 ,, v n ), respectively.

By (1.3), we have

H ( ε ) = i = 1 n lim δ 0 1 δ [ J ( ( ε + δ ) u i + ( 1 ε δ ) v i ) J ( ε u i + ( 1 ε ) v i ) ] = lim δ 0 i = 1 n 1 δ Q [ ( p i ε + δ p i ε ) ( v i + ( ε + δ ) ( u i v i ) ) + p i ε δ ( u i v i ) ] d a d t d x lim δ 0 i = 1 n 1 δ Q [ δ 2 ( u i v i ) 2 + 2 ( v i + ε ( u i v i ) ) δ ( u i v i ) ] d a d t d x = i = 1 n Q [ z i ε ( v i + ε ( u i v i ) ) + p i ε ( u i v i ) ] d a d t d x i = 1 n Q 2 ( v i + ε ( u i v i ) ( u i v i ) ) d a d t d x ,
(4.1)

where z ε =( z 1 ε , z 2 ε ,, z n ε ) is the solution of Problem (3.1) corresponding to the control v i +ε( u i v i ). By the same argument as given in (3.8), we have

Q z i ε ( v i + ε ( u i v i ) ) dadtdx= Q p i ε q i ε ( u i v i )dadtdx.
(4.2)

Combining (4.1) and (4.2), we obtain

H (ε)= i = 1 n Q ( u i v i ) [ p i ε ( 1 + q i ε ) 2 ( v i + ε ( u i v i ) ) ] dtdtdx,
(4.3)

where ( q 1 ε , q 2 ε ) is the solution of Problem (3.5) corresponding to the control v 1 +ε( u 1 v 1 ) ( v 2 +ε( u 2 v 2 )).

Given ε,ρ(0,1), and ερ, and the norm L 2 ( Q ) being equivalent to the norm , combining Theorem 2.1 and Lemma 3.1, we have

[ H ( ε ) H ( ρ ) ] ( ε ρ ) = ( ε ρ ) i = 1 n Q ( u i v i ) [ p i ε ( 1 + q i ε ) p i ρ ( 1 + q i ρ ) ] d a d t d x 2 ( ε ρ ) 2 i = 1 n Q ( u i v i ) 2 d a d t d x = ( ε ρ ) i = 1 n Q [ ( u i v i ) ( p i ε p i ρ ) + ( u i v i ) ( p i ε q i ε p i ρ q i ρ ) ] d a d t d x 2 ( ε ρ ) 2 i = 1 n Q ( u i v i ) 2 d a d t d x ( ε ρ ) 2 K T i = 1 n u i v i L 2 ( Q ) 2 + ( ε ρ ) 2 M 2 K T i = 1 n u i v i L 2 ( Q ) 2 + ( ε ρ ) 2 M 1 K ¯ T i = 1 n u i v i L 2 ( Q ) 2 2 ( ε p ) 2 i = 1 n u i v i L 2 ( Q ) 2 ,

where T is small enough. If u i v i L 2 ( Q ) 0, then we have

[ H ( ε ) H ( ρ ) ] (ερ) ( ε ρ ) 2 u i v i L 2 ( Q ) 2 <0.

Hence, H (ε) is strictly monotone. Consequently, J(u) is strictly concave in U.

Define the function Φ: L 2 (Q; R N )(,+):

Φ(u)={ J ( u 1 , u 2 , , u n ) if  u = ( u 1 , u 2 , , u n ) U , if  u = ( u 1 , u 2 , , u n ) U .

It is clear that Φ is concave in L 2 (Q; R N ). It follows that the nonlinear function Φ: L 2 (Q; R N )(,+) is upper weakly semi-continuous. Since U is a closed, convex, and bounded subset in L 2 (Q; R N ) and J(u) is strictly convex in U, by Lemma 4.1, J(u) attains the unique maximum in U, which implies Problem (1.1) has a unique optimal control in U. □

References

  1. 1.

    Brokate M: Pontryagin’s principle for control problem in age-dependent population dynamics. J. Math. Biol. 1985, 23: 75-101. 10.1007/BF00276559

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Murphy LF, Smith SJ: Maximum sustainable yield of a nonlinear population model with continuous age structure. Math. Biosci. 1991, 104: 259-270. 10.1016/0025-5564(91)90064-P

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Murphy LF, Smith SJ: Optimal harvesting of age-structured population. J. Math. Biol. 1990, 29: 77-90. 10.1007/BF00173910

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Anita S, Iannelli M, Kim MY, Park EJ: Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 1998, 58: 1648-1666. 10.1137/S0036139996301180

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Anita S: Optimal harvesting for a nonlinear age-dependent population dynamics. J. Math. Anal. Appl. 1998, 226: 6-22. 10.1006/jmaa.1998.6064

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Ainseba B, Anita S, Langlais M: On the optimal harvesting of a nonlinear age-structured population dynamics. Abstr. Appl. Anal. 2001, 6: 357-368. 10.1155/S108533750100063X

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Busoni G, Matucci S: A problem of optimal harvesting policy in two-stage age-dependent populations. Math. Biosci. 1997, 143: 1-33. 10.1016/S0025-5564(97)00011-4

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Luo Z, Wang M: Optimal harvesting control problem for linear age-dependent population dynamics. Appl. Math. J. Chin. Univ. Ser. B 2003, 18: 412-420. 10.1007/s11766-003-0068-4

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Gurtin ME, Murphy LF: On the optimal harvesting of age-structured populations: some simple models. Math. Biosci. 1981, 55: 115-136. 10.1016/0025-5564(81)90015-8

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Gurtin ME, Murphy LF: On the optimal harvesting of persistent age-structured populations. J. Math. Biol. 1981, 13: 131-148. 10.1007/BF00275209

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Liu Y, Cheng XL, He ZR: On the optimal harvesting of size-structured population dynamics. Appl. Math. J. Chin. Univ. 2013, 28(2):173-186. 10.1007/s11766-013-2965-5

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Zhao C, Wang M, Zhao P: Optimal harvesting problems for age-dependent interacting species with diffusion. Appl. Math. Comput. 2005, 163: 117-129. 10.1016/j.amc.2004.01.030

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Luo Z:Optimal harvesting problem for an age-dependent n-dimensional food chain diffusion model. Appl. Math. Comput. 2007, 186: 1742-1752. 10.1016/j.amc.2006.08.168

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Luo Z:Optimal harvesting control problem for an age-dependent competing system of n species. Appl. Math. Comput. 2006, 183: 119-127. 10.1016/j.amc.2006.05.180

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    He ZR: Optimal harvesting of two competing species with age dependence. Nonlinear Anal., Real World Appl. 2006, 7: 769-788. 10.1016/j.nonrwa.2005.04.005

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    He ZR, Liu R, Liu LL: Optimal harvesting for populations system with age-dependent predator. Adv. Math. 2012, 42(5):691-700.

    MATH  Google Scholar 

  17. 17.

    Luo Z, Xing T, Li X: Optimal birth control of free horizon problems for predator-prey system with age-structure. J. Appl. Math. Comput. 2010, 34: 19-37. 10.1007/s12190-009-0302-1

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    He ZR, Liu LL, Luo ZX: Global positive periodic solutions of age-dependent competing systems. Appl. Math. J. Chin. Univ. 2011, 26(1):38-46. 10.1007/s11766-011-2503-2

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Luo Z, Fan X: Optimal control for an age-dependent competitive species model in a polluted environment. J. Appl. Math. Comput. 2014, 228: 91-101. 10.1016/j.amc.2013.11.069

    MathSciNet  Article  Google Scholar 

  20. 20.

    Luo ZX, Yu XD, Ba ZG:Overtaking optimal control problem for an age-dependent competition system of n species. Appl. Math. Comput. 2013, 218: 8561-8569. 10.1016/j.amc.2012.02.019

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Anita S: Analysis and Control of Age-Dependent Population Dynamics. Kluwer Academic, Dordrecht; 2000.

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. The project is supported by Graduate Student Innovation of Jilin Normal University (201114) and the Department of Education of Jilin Province (2013445).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jun Fu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and approved the final version.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fu, J., Wu, X. & Zhu, H. Optimal harvesting control for an age-dependent competing population with diffusion. Bound Value Probl 2014, 145 (2014). https://doi.org/10.1186/s13661-014-0145-z

Download citation

Keywords

  • competing
  • diffusion
  • optimal control
  • Ekeland’s principle
  • maximum principle