Vitali Vougalter
email: vitali@math.toronto.edu

Vitaly Volpert
email: volpert@math.univ-lyon1.fr

Solvability Conditions for Some non Fredholm Operators

Keywords: solvability conditions, non Fredholm operators, elliptic problems AMS subject classification: 35J10, 35P10, 35P25

We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger type operators. One of the main results of the work concerns solvability conditions for the equation -∆u + V (x)u -au = f where a ≥ 0. They are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Introduction

Linear elliptic problems in bounded domains with a sufficiently smooth boundary satisfy the Fredholm property if and only if the ellipticity condition, proper ellipticity and the Lopatinskii conditions are satisfied. Fredholm property implies the solvability conditions: the nonhomogeneous operator equation Lu = f is solvable if and only if the right-hand side f is orthogonal to all solutions of the homogeneous adjoint problem L * v = 0. The orthogonality is understood in the sense of duality in the corresponding spaces.

In the case of unbounded domains, one more condition should be imposed in order to preserve the Fredholm property. This condition can be formulated in terms of limiting operators and requires that all limiting operators should be invertible or that the only bounded solution of limiting problems is trivial [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF]. Limiting operators are the operators with limiting values of the coefficients at infinity, if such limiting values exist. Otherwise, limiting coefficients are determined by means of sequences of shifted coefficients and locally convergent subsequences. If we consider for example the operator Lu = -∆u -au in R n , where a is a positive constant, then its only limiting operator is the same operator L. Since the limiting equation Lu = 0 has a nonzero bounded solution, then the operator L, considered in Sobolev or in Hölder spaces does not satisfy the Fredholm property. Therefore the solvability conditions are not applicable. However, the particular form of the equation -∆u -au = f in R n allows us to apply the Fourier transform and to find its solution. It can be easily verified that it has a solution u ∈ L 2 (R n ) if and only if f (ξ)/(ξ 2 -a) ∈ L 2 (R n ), where "hat" denotes the Fourier transform. In the other words, the solvability conditions are given by the equality R n e iξx f (x)dx = 0 for any ξ ∈ R n such that |ξ| 2 = a. This means that formally we obtain solvability conditions similar to those for Fredholm operators: the right-hand side is orthogonal to all solutions of the homogeneous formally adjoint problem.

In this example, we are able to obtain solvability conditions due to the fact that the operator has constant coefficients and we can apply the Fourier transform. In general, the question about solvability conditions for non Fredholm operators is open and represents one of the major challenges in the theory of elliptic problems. Some classes of reaction-diffusion operators without Fredholm property can be studied by the introduction of weighted spaces [START_REF] Volpert | Fredholm property of elliptic operators in unbounded domains[END_REF] or reducing them to integro-differential operators [START_REF] Ducrot | Systemes de réaction-diffusion sans propriété de Fredholm[END_REF], [START_REF] Ducrot | Reaction-diffusion problems with non Fredholm operators[END_REF]. Other types of solvability conditions, different from the usual orthogonality conditions, are obtained for some second order operators on the real axis or in cylinders [START_REF] Kryzhevich | Different types of solvability conditions for differential operators[END_REF]. Some elliptic problems in R 2 are studied in [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF] where the solvability conditions are obtained with the help of space decomposition of the operators.

A special class of elliptic operators in R n , A = A ∞ + A 0 , where A ∞ is a homogeneous operator with constant coefficients and A 0 is an operator with rapidly decaying coefficients is studied in specially chosen spaces with a polynomial weight. The finiteness of the kernel is proved in [W1], [NW], their Fredholm property in [W2], [L], [LM] in the case of weighted Sobolev spaces and in [B], [BP] for weighted Hölder spaces. The Fredholm property and the index of such operators are determined by their principal part A ∞ . The operator A 0 does not change them due to the rapid decay of the coefficients. Laplace operator in exterior domains is studied in [AB].

In this work we consider two classes of non Fredholm operators and establish the solvability conditions for the equations involving them. The methods cited above are not applicable here and we develop some new approaches. In the first case we study the operator

H a on L 2 (R 3 ), such that H a u = -∆u + V (x)u -au
where a ≥ 0 is a parameter, the potential V (x) decays to zero as x → ∞. We investigate the conditions on the function f ∈ L 2 (R 3 ) under which the equations

H a u = f (1.1)
and

H 0 u = f, (1.2)
the second one is the limiting case of the first one as a → 0, have the unique solution in L 2 (R 3 ). Since the potential equals zero at infinity, the operator H a has a unique limiting operator Lu = -∆u -au, which is the same as discussed above. The limiting problem Lu = 0 has nonzero bounded solutions. Therefore, the operator H a , a ≥ 0 does not satisfy the Fredholm property, and the solvability of equations (1.1) and (1.2) is not known. The coefficients of the operators are not constant any more and we cannot simply apply the Fourier transform as in the example above. We will use the spectral decomposition of selfadjoint operators. We note that in the case where a = 0 and the potential is rapidly decaying at infinity, the operator H 0 belong to the class of operators A ∞ + A 0 discussed above. The results of this work differ from the results in the cited papers. We do not work in the weighted spaces and obtain solvability conditions without proving the Fredholm property which may not hold. However, more important difference is that we consider also the case a > 0. It is essentially different and the previous methods are not applicable. To the best of our knowledge, solvability conditions for equation (1.1) with a > 0 and n ≥ 2 were not obtained before. The solvability conditions are formulated in terms of the orthogonality of the righthand side f to all solutions of the homogeneous adjoint equation H a v = 0 (the operator is self-adjoint).

For a function ψ(x) belonging to a L p (R d ) space with 1 ≤ p ≤ ∞, d ∈ N its norm is being denoted as ψ L p (R d ) . As technical tools for estimating the appropriate norms of functions we will be using, in particular the Young's inequality

f 1 * f 2 L ∞ (R 3 ) ≤ f 1 L 4 (R 3 ) f 2 L 4 3 (R 3 ) , f 1 ∈ L 4 (R 3 ), f 2 ∈ L 4 3 (R 3 ) ,
where * stands for the convolution and the Hardy-Littlewood-Sobolev inequality

R 3 R 3 f 1 (x)f 1 (y) |x -y| 2 dxdy ≤ c HLS f 1 2 L 3 2 (R 3 ) , f 1 ∈ L 3 2 (R 3 )
with the constant c HLS given on p.98 of [START_REF] Lieb | Loss: "Analysis[END_REF]. In our notations (f

1 (x), f 2 (x)) L 2 (R 3 ) := R 3 f 1 (x) f2 (x)
dx and for a vector function A(x) = (A 1 (x), A 2 (x), A 3 (x)) the inner product (f 1 (x), A(x)) L 2 (R 3 ) is the vector with the coordinates R 3 f 1 (x) Āi (x)dx, i = 1, 2, 3. Note that with a slight abuse the same notation will be used even if the functions above are not square integrable, like the so called perturbed plane waves ϕ k (x) which are normalized to a delta function (see the equation (2.1) in the Section 2). We make the following technical assumption.

Assumption 1.1. The potential function V (x) : R 3 → R is continuous and satisfies the bound |V (x)| ≤ C 1 + |x| 3.5+ε with some ε > 0 and x ∈ R 3 a.e. such that

4 1 9 9 8 (4π) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 ) < 1 and √ c HLS V L 3 2 (R 3 ) < 4π
The function f (x) ∈ L 2 (R 3 ) and |x|f (x) ∈ L 1 (R 3 ).

Here and further down C stands for a finite positive constant. Since under our assumptions on the potential the essential spectrum σ ess (H a ) of the Schrödinger type operator H a = H 0 -a fills the interval [-a, ∞) (see e.g. [JMST] ), the Fredholm alternative theorem fails to work in this case. The problem can be easily handled by the method of the Fourier transform in the absence of the potential term V (x). We show that this method can be generalized in the presence of a shallow, short-range V (x) by means of replacing the Fourier harmonics by the functions ϕ k (x), k ∈ R 3 of the continuous spectrum of the operator H 0 , which are the solutions of the Lippmann-Schwinger equation (see (2.1) in Section 2 and the explicit formula (2.2)).

While the wave vector k attains all the possible values in R 3 , the function ϕ 0 (x) corresponds to k = 0 in the formulas (2.1) and (2.2). The sphere of radius r in R d , d ∈ N centered at the origin is being designated as S d r , the unit one as S d and |S d | stands for its Lebesgue measure. Our first main result is as follows.

Theorem 1. Let the Assumption 1.1 hold. Then a) The problem (1.1) admits a unique solution u ∈ L 2 (R 3 ) if and only if

(f (x), ϕ k (x)) L 2 (R 3 ) = 0 f or k ∈ S 3 √ a a.e.
b) The problem (1.2) has a unique solution u ∈ L 2 (R 3 ) if and only if

(f (x), ϕ 0 (x)) L 2 (R 3 ) = 0
In the second part of the article we consider the operator L = -∆ x -∆ y + V(y) on L 2 (R n+m ) with the Laplacian operators ∆ x and ∆ y in x = (x 1 , x 2 , ..., x n ) ∈ R n , y = (y 1 , y 2 , ..., y m ) ∈ R m and prove the necessary and sufficient conditions for the solvability in L 2 (R n+m ) of the inhomogeneous problem Lu = g(x, y),

(1.3)

where g(x, y) ∈ L 2 (R n+m ). We assume the following.

Assumption 1.2. The function V(y) : R m → R is continuous and lim y→∞ V(y) = V + > 0.

Thus for the operator h := -∆ y + V(y) the essential spectrum σ ess (h) = [V + , ∞). Let us denote the eigenvalues of the operator h located below V + as e j , e j < e j+1 , j ≥ 1 and the corresponding elements of the orthonormal set of eigenfunctions as ϕ k j , such that

hϕ k j = e j ϕ k j , 1 ≤ k ≤ m j , (ϕ k i , ϕ l j ) L 2 (R m ) = δ i,j δ k,l
, where m j stands for the eigenvalue multiplicity, which is finite since the essential spectrum starts only at V + and δ i,j for the Kronecker symbol. We make the following key assumption on the discrete spectrum of the operator h relevant to the problem (1.3).

Assumption 1.3. The eigenvalues e j < 0 for all 1 ≤ j ≤ N -1 and e N = 0.

Thus under our assumptions the operator L is not Fredholm. Zero is the bottom of the essential spectrum of the operator -∆ x and h has the square integrable zero modes. Moreover, the operator h has the negative eigenvalues e j , j = 1, ..., N -1 and -∆ x the Fourier harmonics e ipx (2π)

n 2 , such that p ∈ S n √ -e j
. However, the equation (1.3) can be solved on the proper subspace and the orthogonality conditions will strongly depend on the dimensions of the problem.

Let us introduce the following subspace weighted in the first variable for the right side of the equation (1.3).

L 2 α, x = {g(x, y) : g(x, y) ∈ L 2 (R n+m ) and |x| α 2 g(x, y) ∈ L 2 (R n+m )}, α > 0 (1.4)
Our second main result is as follows.

Theorem 2. Let the Assumptions 1.2 and 1.3 hold. Then for the equation (1.3):

I) When n = 1 and g(x, y) ∈ L 2 α,
x for some α > 5 there exists a unique solution u ∈ L 2 (R 1+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), xϕ k N (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N and (g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j II) When n = 2 such that x = (x 1 , x 2 ) ∈ R 2 and g(x, y) ∈ L 2 α,
x for some α > 6 there exists a unique solution u ∈ L 2 (R 2+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x i ϕ k N (y)) L 2 (R 2+m ) = 0, i = 1, 2, 1 ≤ k ≤ m N and (g(x, y), e ipx 2π ϕ k j (y)) L 2 (R 2+m ) = 0, a.e. p ∈ S 2 √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j
III) When n = 3, 4 and g(x, y) ∈ L 2 α, x for some α > n + 2 there exists a unique solution u ∈ L 2 (R n+m ) if and only if:

(g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, 1 ≤ k ≤ m N and (g(x, y), e ipx (2π) n 2 ϕ k j (y)) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j IV) When n ≥ 5 and g(x, y) ∈ L 2 α,
x for some α > n + 2 there exists a unique solution u ∈ L 2 (R n+m ) if and only if:

(g(x, y), e ipx (2π) n 2 ϕ k j (y)) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ j ≤ N -1, 1 ≤ k ≤ m j
Proving solvability conditions for linear elliptic problems with non-Fredholm operators plays the crucial role in various applications including those to travelling wave solutions of reaction-diffusion systems (see [START_REF] Volpert | Solvability conditions for elliptic problems with non-Fredholm operators[END_REF]). Let us first establish several important properties for the functions of the spectrum of the Schrödinder operator in the left side of the equation (1.1) and for the related quantities.

2. Spectral properties of the operator H 0 and proof of Theorem 1

The functions of the continuous spectrum satisfy the Lippmann-Schwinger equation (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics, III. Scattering Theory[END_REF] p.98)

ϕ k (x) = e ikx (2π) 3 2 - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ k )(y)dy (2.1)
and the orthogonality relations

(ϕ k (x), ϕ q (x)) L 2 (R 3 ) = δ(k -q), k, q ∈ R 3 . We define the integral operator (Qϕ)(x) := - 1 4π R 3 e i|k||x-y| |x -y| (V ϕ)(y)dy, ϕ ∈ L ∞ (R 3 )
Let us show that the norm of the operator

Q : L ∞ (R 3 ) → L ∞ (R 3 ) denoted as Q ∞ is small
when the potential V (x) satisfies our assumptions. We prove the following lemma.

Lemma 2.1. Let the Assumption 1.1 hold. Then ||Q|| ∞ < 1.

Proof.

Clearly

Q ∞ ≤ sup x∈R 3 1 4π R 3 |V (y)| |x -y| dy
The expression involved in the right side of the inequality above can be written as

1 4π (χ {|x|≤R} 1 |x| ) * |V (x)| + 1 4π (χ {|x|>R} 1 |x| ) * |V (x)|
with some R > 0 and χ denoting the characteristic function of the correspondent set. This can be estimated above using the Young's inequality as

1 4π V L ∞ (R 3 ) R 0 4πrdr + 1 4π χ {|x|>R} 1 |x| L 4 (R 3 ) V L 4 3 (R 3 ) = = 1 2 V L ∞ (R 3 ) R 2 + 1 (4π) 3 4 V L 4 3 (R 3 ) R -1 4
We optimize the right side of the equality above over R. The minimum occurs when

R = V L ∞ (R 3 ) (4π) 3 4 4 V L 4 3 (R 3 ) -4 9 , such that Q ∞ ≤ 4 1 9 9 8 (4π) -2 3 V 1 9 L ∞ (R 3 ) V 8 9 L 4 3 (R 3 )
, which is k-independent. The Assumption 1.1 yields the statement of the Lemma. Note that V ∈ L 4 3 (R 3 ) which is guaranteed by its rate of decay given explicitly in the Assumption 1.1.

Corollary 2.2. Let the Assumption 1.1 hold. Then the functions of the continuous spectrum of the operator

H 0 are ϕ k (x) ∈ L ∞ (R 3 ) for all k ∈ R 3 , such that ϕ k (x) L ∞ (R 3 ) ≤ 1 1 -Q ∞ 1 (2π) 3 2 , k ∈ R 3
Proof. By means of the Lippmann-Schwinger equation (2.1) and the fact that Q ∞ < 1 the functions can be expressed as

ϕ k (x) = (I -Q) -1 e ikx (2π) 3 2 , k ∈ R 3 (2.2)
The Lemma 2.1 yields the bound on the operator norm (

I -Q) -1 ∞ ≤ 1 1 -Q ∞ .
The following elementary lemma shows that in our problem the operator H 0 possesses the spectrum analogous to the one of the minus Laplacian and therefore only the functions ϕ k (x), k ∈ R 3 are needed to be taken into consideration.

Lemma 2.3. Let the Assumption 1.1 be true. Then the operator H 0 is unitarily equivalent to -∆ on L 2 (R 3 ).

Proof. By means of the Hardy-Littlewood-Sobolev inequality (see e.g. p.98 [START_REF] Lieb | Loss: "Analysis[END_REF]) and the Assumption 1.1 we have

R 3 R 3 |V (x)||V (y)| |x -y| 2 dxdy ≤ c HLS V 2 L 3 2 (R 3 ) < (4π) 2
The left side of the inequality above is usually referred to as the Rollnik norm (see e.g. [S71]) and the upper bound we obtained on it is the sufficient condition for the operator

H 0 = -∆ + V (x) on L 2 (R 3
) to be self-adjoint and unitarily equivalent to -∆ via the wave operators (see e.g. [K65], also [START_REF] Rodnianski | Time decay for solutions of Schrödinger equations with rough and time-dependent potentials[END_REF]) given by Ω ± := s -lim t→∓∞ e it(-∆+V ) e it∆ where the limit is understood in the strong L 2 sense (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics, III. Scattering Theory[END_REF] p.34, [START_REF] Cycon | Schrödinger Operators with Application to Quantum Mechanics and Global Geometry[END_REF] p.90).

By means of the spectral theorem for the self-adjoint operator H 0 any function ψ(x) ∈ L 2 (R 3 ) can be expanded through the functions ϕ k (x), k ∈ R 3 forming the complete system in L 2 (R 3 ). The generalized Fourier transform with respect to these functions is being denoted as ψ(k

) := (ψ(x), ϕ k (x)) L 2 (R 3 ) , k ∈ R 3 (2.3)
We prove the following technical estimate concerning the above mentioned generalized Fourier transform for the right side of the equations (1.1) and (1.2).

Lemma 2.4. Let the Assumption 1.1 hold. Then

∇ k f (k) ∈ L ∞ (R 3 ) Proof. Obviously ∇ k f (k) = (f (x), ∇ k ϕ k (x)) L 2 (R 3
) . ¿From the Lippmann-Schwinger equation (2.1) we easily obtain

∇ k ϕ k = e ikx (2π) 3 2 ix + (I -Q) -1 Q e ikx (2π) 3 2 ix + (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2 , (2.4)
where

∇ k Q : L ∞ (R 3 ) → L ∞ (R 3 ; C 3
) stands for the operator with the integral kernel

∇ k Q(x, y, k) = - i 4π e i|k||x-y| k |k| V (y)
An elementary computation shows that its norm

∇ k Q ∞ ≤ 1 4π V L 1 (R 3 ) < ∞
due to the rate of decay of the potential V (x) given explicitly by the Assumption 1.1. It is clear from the identity (2.4) that we need to show the boundedness in the k-space of the three terms. The first one is

T 1 (k) := (f (x), e ikx (2π) 3 2 ix) L 2 (R 3 ) , such that |T 1 (k)| ≤ 1 (2π) 3 2
xf L 1 (R 3 ) < +∞ by the Assumption 1.1. The second term to be estimated is

T 2 (k) := (f (x), (I -Q) -1 Q e ikx (2π) 3 2 ix) L 2 (R 3 ) Thus |T 2 (k)| ≤ 1 (2π) 3 2 f L 1 (R 3 ) 1 1 -Q ∞ Qe ikx x L ∞ (R 3 ) . Note that f (x) ∈ L 1 (R 3
) by means of the Assumption 1.1 and Fact 1 of the Appendix. Using the definition of the operator Q along with the Young's inequality we have the upper bound

|Qe ikx x| ≤ 1 4π R 3 |V (y)||y| |x -y| dy = 1 4π {(χ {|x|≤1} 1 |x| ) * |V (x)||x| + (χ {|x|>1} 1 |x| ) * |V (x)||x|} ≤ ≤ 1 4π { V (y)y L ∞ (R 3 ) 1 0 4πrdr + χ {|x|>1} 1 |x| L 4 (R 3 ) V (x)x L 4 3 (R 3 ) } < +∞ and k-independent since V (x)x ∈ L ∞ (R 3 ) ∩ L 4 3 (R 3
) due to the explicit rate of decay of the potential V (x) stated in the Assumption 1.1. Therefore, T 2 (k) ∈ L ∞ (R 3 ). We complete the proof of the lemma with the estimate on the remaining term

T 3 (k) := (f (x), (I -Q) -1 (∇ k Q)(I -Q) -1 e ikx (2π) 3 2 ) L 2 (R 3 ) ,
such that we easily arrive at the k-independent upper bound

|T 3 (k)| ≤ 1 4π(2π) 3 2 f L 1 (R 3 ) 1 (1 -Q ∞ ) 2 V L 1 (R 3 ) < ∞
Armed with the auxiliary lemmas established above we proceed to prove the first theorem.

Proof of Theorem 1. First of all if the equation (1.1) admits two solutions u 1 (x), u 2 (x) ∈ L 2 (R 3 ) their difference v(x) := u 1 (x) -u 2 (x) would satisfy the homogeneous problem H a v = 0. Since the operator H a possesses no nontrivial square integrable zero modes, v(x) will vanish a.e. The analogous argument holds for the solutions of the equation (1.2). ¿From the equation (1.1) by applying the transform (2.3) we obtain

ũ(k) = f (k) k 2 -a , k ∈ R 3 ,
which is convenient to write as the sum of the singular and the nonsingular parts

ũ(k) = f (k) k 2 -a χ Aσ + f (k) k 2 -a χ A c σ , (2.5)
where χ Aσ is the characteristic function of the spherical layer

A σ := {k ∈ R 3 : √ a -σ ≤ |k| ≤ √ a + σ} , 0 < σ < √ a
and χ A c σ of the layer's complement in the three-dimensional k-space. For the second term in the right side of the identity (2.5)

f (k) k 2 -a χ A c σ ≤ | f (k)| √ aσ ∈ L 2 (R 3 )
To estimate the remaining term we will make use of the identity

f (k) = f ( √ a, ω) + |k| √ a ∂ f (|s|, ω) ∂|s| d|s|
Here and further ω stands for the angle variables on the sphere and dω will denote integration with respect to these variables. Thus we can split the first term in the right side of (2.5) as ũ1 (k) + ũ2 (k), where

ũ1 (k) = |k| √ a ∂ f (|s|,ω) ∂|s| d|s| k 2 -a χ Aσ , ũ2 (k) = f ( √ a, ω) k 2 -a χ Aσ (2.6)
Clearly, we have the bound

|ũ 1 (k)| ≤ ∇ k f (k) L ∞ (R 3 ) |k| + √ a χ Aσ ∈ L 2 (R 3 )
by means of Lemma 2.4. We complete the proof of the Part a) of the theorem by estimating the norm

ũ2 (k) 2 L 2 (R 3 ) = √ a+σ √ a-σ d|k| |k| 2 (|k| - √ a) 2 (|k| + √ a) 2 S 3 dω| f( √ a, ω)| 2 < ∞ if and only if (f (x), ϕ k (x)) L 2 (R 3 ) = 0 for k a.
e. on the sphere S 3 √ a . Then we turn our attention to the equation (1.2) by applying to it the generalized Fourier transform with respect to the eigenfunctions of the continuous spectrum of the operator H 0 , which yields

ũ(k) = f (k) k 2 = f (k) k 2 χ {|k|≤1} + f (k) k 2 χ {|k|>1} Clearly f (k) k 2 χ {|k|>1} ≤ | f (k)| ∈ L 2 (R 3
). We use the formula

f (k) = f(0) + |k| 0 ∂ f (|s|, ω) ∂|s| d|s| with f (0) = (f (x), ϕ 0 (x)) L 2 (R 3
) and ϕ 0 (x) is given by (2.2) with k = 0. Hence

|k| 0 ∂ f (|s|,ω) ∂|s| d|s| k 2 χ {|k|≤1} ≤ ∇ k f (k) L ∞ (R 3 ) χ {|k|≤1} |k| ∈ L 2 (R 3 )
via Lemma 2.4. Therefore it remains to estimate the norm f (0)

k 2 χ {|k|≤1} 2 L 2 (R 3 ) = 4π 1 0 d|k| | f(0)| 2 |k| 2 < ∞
if and only if (f (x), ϕ 0 (x)) L 2 (R 3 ) = 0 which completes the proof of the theorem.

Note that if we let the potential function V (x) in the statement of the Theorem 1 vanish, we obtain precisely the usual orthogonality conditions in terms of the Fourier harmonics.

In the following chapter we prove the Theorem 2. As distinct from the first example the dimensions of the problem are not fixed anymore and we show how robust the dependence of the solvability conditions on these dimensions can be.

Spectral properties of the operator L and proof of Theorem 2

Let P ± and P 0 be the orthogonal projections onto the positive, negative and zero subspaces of the operator h. Applying these operators to both sides of the equation (1.3) via the spectral theorem we relate the problem to the equivalent system of the three following equations.

L + u + = g + , (3.1) L -u -= g -, (3.2)
and

L 0 u 0 = g 0 , (3.3)
where the operators L ± = P ± LP ± and L 0 = P 0 LP 0 act on the functions u ± = P ± u and u 0 = P 0 u respectively and the right sides of the equations above are g ± = P ± g and g 0 = P 0 g. Without loss of generality we can assume that

g 0 (x, y) = v 0 (x)ϕ 1 N (y) , (3.4)
where

v 0 (x) = (g 0 , ϕ 1 N ) L 2 (R m ) = (g, ϕ 1 N ) L 2 (R m ) .
Let us first turn our attention to the equation (3.1). We have the following lemma. Proof. By means of the orthogonal decomposition of the right side of the equation (1.3) g = g + + g 0 + g -we have the estimate

g + L 2 (R n+m ) ≤ g L 2 (R n+m )
The lower bound in the sense of the quadratic forms

L + ≥ P + hP + ≥ e N +1 > 0 ,
where e N +1 is either the bottom of the essential spectrum V + of the operator h or its lowest positive eigenvalue, whichever is smaller. Thus L + is the self-adjoint operator on the product of spaces L 2 (R n ) and the range Ran(P + ) such that the bottom of its spectrum is located above zero. Therefore it is invertible and the norm of the inverse L -1

+ : L 2 (R n ) ⊗ Ran(P + ) → L 2 (R n+m ) is bounded above by 1 e N +1
. Thus the equation (3.1) has the solution u + = L -1 + g + and its norm

u + L 2 (R n+m ) ≤ 1 e N +1 g L 2 (R n+m ) < ∞
Let us turn our attention to the analysis of the solvability conditions for the equation

(3.3) which is equivalent to (-∆ x )u 0 = g 0 (3.5)
The solution of this Poisson equation can be expressed as

u 0 = g 0 p 2 χ 1 + g 0 p 2 χ 1 c , (3.6)
where χ 1 stands for the characteristic function of the unit ball in the Fourier space centered at the origin and χ 1 c for the characteristic function of its complement. Here and below the hat symbol stands for the Fourier transform in the first variable, such that

ψ(p) := 1 (2π) n 2 R n ψ(x)e -ipx dx
The second term in the right side of (3.6) is square integrable for all dimensions n, m ∈ N since g 0 ∈ L 2 (R n+m ) and 1 p 2 is bounded away from the origin. Thus it remains to analyze the first term. We have the following lemma when the dimension n = 1. Lemma 3.2 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R 1+m ), m ∈ N if and only if

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), ϕ k N (y)x) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N
Proof. We will make use of the following representation

g 0 (p, y) = g 0 (0, y) + ∂ ∂p g 0 (0, y)p + p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds , where ∂ 2 ∂p 2 g 0 (p, y) = - 1 √ 2π +∞ -∞ g 0 (x, y)e -ipx x 2 dx
Hence the first term in the right side of (3.6) in our case equals to

g 0 (0, y) p 2 χ 1 + ∂ ∂p g 0 (0, y) χ 1 p + p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds χ 1 p 2 (3.7)
Clearly we have the upper bound

∂ 2 ∂q 2 g 0 (q, y) ≤ 1 √ 2π +∞ -∞ |g 0 (x, y)|x 2 dx
By means of the Schwarz inequality and (3.4) we have the estimate valid in a space of arbitrary dimensions

|g 0 (x, y)| ≤ R m |g(x, z)| 2 dz|ϕ 1 N (y)| , x ∈ R n , y ∈ R m , n, m ≥ 1 (3.8) which yields ∂ 2 ∂q 2 g 0 (q, y) ≤ 1 √ 2π +∞ -∞ dx x 2 1 + |x| α 1 + |x| α R m |g(x, s)| 2 ds|ϕ 1 N (y)|
with α > 5 such that g(x, y) ∈ L 2 α, x . The Schwarz inequality yields the upper bound

1 √ 2π +∞ -∞ dx x 4 1 + |x| α g 2 L 2 (R 1+m ) + |x| α 2 g 2 L 2 (R 1+m ) |ϕ 1 N (y)| = C|ϕ 1 N (y)|
Therefore, for the last term in (3.7) we obtain

p 0 s 0 ∂ 2 ∂q 2 g 0 (q, y)dq ds χ 1 p 2 ≤ C 2 |ϕ 1 N (y)|χ 1 ∈ L 2 (R 1+m )
Due to the behavior in the Fourier space of the first two terms, the expression (3.7) belongs to L 2 (R 1+m ) if and only if

g 0 (0, y) = 0, ∂ ∂p g 0 (0, y) = 0 a.e. ,
which is equivalent to

(g(x, y), ϕ k N (y)) L 2 (R 1+m ) = 0, (g(x, y), ϕ k N (y)x) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m N
When the dimension n = 2 we come up with the following analogous statement.

Lemma 3.3 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R 2+m ), m ∈ N if and only if

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), ϕ k N (y)x i ) L 2 (R 2+m ) = 0, i = 1, 2, 1 ≤ k ≤ m N Proof.
Let us use the expansion analogous to the one we had for proving the previous lemma.

g 0 (p, y) = g 0 (0, y) + ∂ ∂|p| g 0 (0, θ p , y)|p| + |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds , with g 0 (|p|, θ p , y) = 1 2π R 2 g 0 (x, y)e -i|p||x|cosθ dx
and the angle between the p = (|p|, θ p ) and x = (|x|, θ x ) vectors on the plane is θ = θ p -θ x . Therefore the first term in the right side of (3.6) when n = 2 equals to

g 0 (0, y) p 2 χ 1 + ∂ ∂|p| g 0 (0, θ p , y) χ 1 |p| + |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds χ 1 p 2 (3.9) Obviously ∂ 2 ∂|q| 2 g 0 ≤ 1 2π R 2 |g 0 (x, y)||x| 2 dx
Using the estimate (3.8) we arrive at the upper bound for the right side of this inequality

1 2π |ϕ 1 N (y)| R 2 dx |x| 2 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz with α > 6 such that g(x, y) ∈ L 2 α, x
. By means of the Schwarz inequality we estimate this from above as

1 √ 2π |ϕ 1 N (y)| ∞ 0 d|x| |x| 5 1 + |x| α g 2 L 2 (R 2+m ) + |x| α 2 g 2 L 2 (R 2+m ) = C|ϕ 1 N (y)|
Therefore, for the last term in (3.9) we arrive at

χ 1 p 2 |p| 0 s 0 ∂ 2 ∂|q| 2 g 0 (|q|, θ p , y)d|q| ds ≤ C 2 χ 1 |ϕ 1 N (y)| ∈ L 2 (R 2+m )
A simple computation using the Fourier transform yields

∂ ∂|p| g 0 (0, θ p , y) = - i 2π R 2 g 0 (x, y)|x|cosθdx = Q 1 (y)cosθ p + Q 2 (y)sinθ p , where Q 1 (y) := - i 2π R 2 g 0 (x, y)x 1 dx , Q 2 (y) := - i 2π R 2 g 0 (x, y)x 2 dx and x = (x 1 , x 2 ) ∈ R 2 .
Computing the square of the L 2 (R 2+m ) norm of the first two terms of (3.9) we arrive at 2π

1 0 d|p| |p| 3 R m dy| g 0 (0, y)| 2 + π 1 0 d|p| |p| R m (|Q 1 (y)| 2 + |Q 2 (y)| 2 )dy ,
which is finite if and only if the quantities g 0 (0, y), Q 1 (y) and Q 2 (y) vanish a.e. This is equivalent to the orthogonality conditions

(g(x, y), ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x 1 ϕ k N (y)) L 2 (R 2+m ) = 0, (g(x, y), x 2 ϕ k N (y)) L 2 (R 2+m ) = 0, with 1 ≤ k ≤ m N .
Let us investigate how the situation with solvability conditions differs in dimensions n = 3, 4.

Lemma 3.4 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution

u 0 ∈ L 2 (R n+m ), n = 3, 4, m ∈ N if and only if (g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, n = 3, 4, 1 ≤ k ≤ m N
Proof. Let us expand as

g 0 (p, y) = g 0 (0, y) + |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s|
Thus by means of (3.6) we need to estimate

χ 1 p 2 [ g 0 (0, y) + |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s|] (3.10)
By means of the Fourier transform

∂ ∂|p| g 0 (|p|, ω, y) = -i (2π) n 2 R n g 0 (x, y)e -i|p||x|cosθ |x|cosθdx ,
where θ is the angle between p and x in R n . Using (3.8) along with the Schwarz inequality and α > n + 2 such that g(x, y) ∈ L 2 α, x we easily obtain

∂ ∂|s| g 0 ≤ 1 (2π) n 2 R n dx|x| R m |g(x, z)| 2 dz|ϕ 1 N (y)| ≤ ≤ 1 (2π) n 2 ∞ 0 |S n | |x| n+1 1 + |x| α d|x| g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) |ϕ 1 N (y)| = C|ϕ 1 N (y)| ,
which implies the bound

χ 1 p 2 |p| 0 ∂ ∂|s| g 0 (|s|, ω, y)d|s| ≤ C χ 1 |p| |ϕ 1 N (y)| ∈ L 2 (R n+m ), n = 3, 4
We finalize the proof of the lemma by estimating the square of the L 2 norm of the first term in (3.10).

|S n | R m dy| g 0 (0, y)| 2 1 0 d|p||p| n-5 < ∞, n = 3, 4
if and only if g 0 (0, y) = 0 a.e., which is equivalent to

(g(x, y), ϕ k N (y)) L 2 (R n+m ) = 0, n = 3, 4, 1 ≤ k ≤ m N
Thus it remains only to establish the orthogonality conditions in dimensions five and higher in the x-variable under which the equation (3.5) admits a square integrable solution.

Lemma 3.5 Let the assumptions of the Theorem 2 hold. Then the equation (3.5) possesses a solution u 0 ∈ L 2 (R n+m ), n ≥ 5, m ∈ N.

Proof. We estimate the Fourier transform using the bound (3.8) along with the Schwarz inequality and α > n + 2 such that g(x, y) ∈ L 2 α, x .

| g 0 (p, y)| ≤ 1 (2π) n 2 R n |g 0 (x, y)|dx ≤ |ϕ 1 N (y)| (2π) n 2 R n dx R m |g(x, z)| 2 dz ≤ ≤ |ϕ 1 N (y)| (2π) n 2 R n dx 1 + |x| α R n dx(1 + |x| α ) R m |g(x, z)| 2 dz = = |ϕ 1 N (y)| (2π) n 2 ∞ 0 d|x| |x| n-1 1 + |x| α |S n | g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) = = C|ϕ 1 N (y)|, n ≥ 5, m
∈ N This enables us to obtain the bound on the square of the L 2 norm of the first term in the right side of (3.6).

R n dp R m dy | g 0 | 2 |p| 4 χ 1 ≤ C 1 0 d|p||p| n-5 |S n | R m |ϕ 1 N (y)| 2 dy < ∞
which completes the proof of the lemma.

We proceed with establishing the conditions under which the equation (3.2) admits a square integrable solution. Let {P -,j } N -1 j=1 be the orthogonal projections onto the subspaces correspondent to {e j } N -1 j=1 , the negative eigenvalues of the operator h, such that

P -= N -1 j=1 P -,j , P -,j P -,k = P -,j δ j,k , 1 ≤ j, k ≤ N -1
Applying these projection operators to both sides of the equation (3.2) and using the orthogonal decompositions u -=

N -1 j=1 u -,j and g -=

N -1 j=1 g -,j with P -,j u -= u -,j and P -,j g -= g -,j we easily obtain the system of equations equivalent to (3.2).

[-∆ x -∆ y + V(y)]u -,j = g -,j , 1 ≤ j ≤ N -1 (3.11)

Without loss of generality we can assume that g -,j (x, y) = v j (x)ϕ 1 j (y), 1 ≤ j ≤ N -1 , (3.12) where v j (x) := (g -,j , ϕ

1 j ) L 2 (R m ) = (g, ϕ 1 j ) L 2 (R m )
. By means of the Schwarz inequality

|v j (x)| ≤ R m |g(x, z)| 2 dz, x ∈ R n (3.13)
Hence the goal is to establish the conditions under which such an equation as (3.11) possesses a square integrable solution. We make the Fourier transform in the x-variable and using the fact that the operator -∆ x does not have positive eigenvalues on L 2 (R n ) obtain the expression for a solution of (3.11) as

u -,j (p, y) = v j (p) p 2 + e j ϕ 1 j (y), 1 ≤ j ≤ N -1
We distinguish the two cases dependent upon the dimension of the problem in the first variable.

Lemma 3.6 Let the assumptions of the Theorem 2 hold. Then the equation (3.11) possesses a solution u -,j (x, y) ∈ L 2 (R 1+m ), m ∈ N if and only if

(g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1
Proof. We express a solution of (3.11) as the sum of its regular and singular components

u -,j (p, y) = v j (p)χ Ω c δ p 2 + e j ϕ 1 j (y) + v j (p)χ Ω δ p 2 + e j ϕ 1 j (y) , (3.14)
where the set in the Fourier space Ω

δ := [ √ -e j -δ, √ -e j + δ] ∪ [- √ -e j -δ, - √ -e j + δ] = Ω + δ ∪ Ω - δ with 0 < δ <
√ -e j and Ω c δ is its complement, χ Ω δ and χ Ω c δ are their characteristic functions. It is trivial to estimate the first term in the right side of (3.14) since we are away from the positive and negative singularities ± √ -e j . Thus

v j (p)χ Ω c δ p 2 + e j ϕ 1 j (y) ≤ C|ϕ 1 j (y)|| v j (p)|χ Ω c δ ,
which along with (3.13) enables us to estimate the square of its L 2 norm.

+∞

-∞ dp R m dy|ϕ 1 j (y)| 2 | v j (p)| 2 χ Ω c δ ≤ v j 2 L 2 (R) ≤ g 2 L 2 (R 1+m ) < ∞
To obtain the conditions under which the remaining term in (3.14) is square integrable we first study its behavior near its negative singularity using the formula

v j (p) = p - √ -e j d v j (s) ds ds + v j (--e j )
Thus one needs to estimate

v j (- √ -e j ) + p - √ -e j db v j (s) ds ds p 2 + e j χ Ω - δ ϕ 1 j (y) (3.15)
We derive the upper bound for the derivative using (3.13) along with the Schwarz inequality with α > 5 such that g(x, y) ∈ L 2 α, x .

d v j (p) dp ≤ 1 √ 2π ∞ -∞ dx|x||v j (x)| ≤ 1 √ 2π ∞ -∞ dx |x| 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz ≤ ≤ 1 √ 2π +∞ -∞ dx x 2 1 + |x| α ∞ -∞ dx(1 + |x| α ) R m |g(x, z)| 2 dz = = 1 √ 2π +∞ -∞ dx x 2 1 + |x| α g 2 L 2 (R 1+m ) + |x| α 2 g 2 L 2 (R 1+m ) = C < ∞
This enables us to prove the square integrability for the second term in (3.15).

p -√ -e j db v j (s)

ds ds p 2 + e j χ Ω - δ ϕ 1 j (y) ≤ C |p - √ -e j | χ Ω - δ |ϕ 1 j (y)| ≤ C 2 √ -e j -δ χ Ω - δ |ϕ 1 j (y)| ∈ L 2 (R 1+m )
Near the positive singularity we use the identity

v j (p) = p √ -e j d v j (s) ds ds + v j ( -e j )
to study the conditions of the square integrability of the term

v j ( √ -e j ) + p √ -e j db v j (s) ds ds p 2 + e j χ Ω + δ ϕ 1 j (y) (3.16)
Analogously to the situation at the negative singularity we prove the square integrability of the second term in (3.16) using the bound on the derivative involved in it. Hence p √ -e j db v j (s)

ds ds p 2 + e j χ Ω + δ ϕ 1 j (y) ≤ C |p + √ -e j | χ Ω + δ |ϕ 1 j (y)| ≤ ≤ C 2 √ -e j -δ χ Ω + δ |ϕ 1 j (y)| ∈ L 2 (R 1+m )
Thus it remains to derive the conditions under which the first term in (3.15) and the first term in (3.16) are square integrable. Estimating the square of the This bound implies that the necessary and sufficient conditions for the existence of u -,j (x, y) ∈ L 2 (R 1+m ) solving the equation (3.11) are v j ( -e j ) = 0, v j (--e j ) = 0 which by means of the definition of the functions v j (x) is equivalent to (g(x, y), e ±i √ -e j x √ 2π ϕ k j (y)) L 2 (R 1+m ) = 0, 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1

L 2 (R 1+m ) norm of v j (- √ -e j ) p 2 + e j χ Ω - δ ϕ 1 j (y) + v j ( √ -e j ) p 2 + e j χ Ω + δ ϕ 1 j (
After establishing the solvability conditions for the equation (3.11) when the situation is one dimensional in the first variable we turn our attention to the cases of dimensions two and higher. Proof. It is convenient to represent a solution of (3.11) as the sum of the singular and the regular parts u -,j (p, y) = v j (p)χ A δ p 2 + e j ϕ 1 j (y) + v j (p)χ A c δ p 2 + e j ϕ 1 j (y) , (3.17)

where the spherical layer in Fourier space A δ := {p ∈ R n : √ -e j -δ ≤ |p| ≤ √ -e j + δ}, its complement in R n is A c δ , their characteristic functions are χ A δ and χ A c δ respectively and 0 < δ < √ -e j . Clearly for the second term in the right side of (3.17

R n | v j (p)| 2 dp R m |ϕ 1 j (y)| 2 dy = v j 2 L 2 (R n ) ≤ g 2 L 2 (R n+m ) < ∞.
Hence the first term in the right side of (3.17) will play the crucial role for establishing the solvability conditions for the equation (3.11). We will make use of the formula Let us derive the upper bound for the derivative of the Fourier transform involved in it using (3.13) along with the Schwarz inequality, α > 6 for n = 2 and α > n + 2 for n ≥ 3 such that g(x, y) ∈ L 2 α, x . Fact 1 Let f (x) ∈ L 2 (R 3 ) and |x|f (x) ∈ L 1 (R 3 ). Then f (x) ∈ L 1 (R 3 ).

∂ v j ∂|p| ≤ 1 (2π) n 2 R n |v j (x)||x|dx ≤ 1 (2π) n 2 R n dx|x| R m |g(x, z)| 2 dz = = 1 (2π) n 2 R n dx |x| 1 + |x| α 1 + |x| α R m |g(x, z)| 2 dz ≤ ≤ 1 (2π) n 2 R n dx |x| 2 1 + |x| α R n dx(1 + |x| α ) R m |g(x, z)| 2 dz = = 1 (2π) n 2 ∞ 0 d|x||S n | |x| n+1 1 + |x| α g 2 L 2 (R n+m ) + |x| α 2 g 2 L 2 (R n+m ) = C < ∞
Proof. The norm f L 1 (R 3 ) is being estimated from above by means of the Schwarz inequality as

|x|≤1 |f (x)| 2 dx |x|≤1 dx + |x|>1 |x||f (x)|dx ≤ f L 2 (R 3 ) 4π 3 + |x|f L 1 (R 3 ) < ∞

Lemma 3. 1

 1 The equation (3.1) possesses a solution u + ∈ L 2 (R n+m ), n ∈ N, m ∈ N.

  j )| 2 (p 2 + e j ) 2 + √ -e j +δ √ -e j -δ dp | v j ( √ -e j )| 2 (p 2 + e j ) 2 ,which can be bounded below by| v j (-√ -e j )| 2

Lemma 3. 7

 7 Let the assumptions of the Theorem 2 hold. Then the equation (3.11) possesses a solution u -,j (x, y)∈ L 2 (R n+m ), n ≥ 2, m ∈ N if and only if ) L 2 (R n+m ) = 0, a.e. p ∈ S n √ -e j , 1 ≤ k ≤ m j , 1 ≤ j ≤ N -1

  ∂|s| (|s|, ω)d|s| + v j ( -e j , ω) to get the estimate for |p| √ -e j ∂b v j ∂|s| (|s|, ω)d|s| + v j ( √ -e j , ω) p 2 + e j χ A δ ϕ 1 j (y)

  ∂|s| (|s|, ω)d|s| p 2 + e j χ A δ ϕ 1 j (y) ≤ C √ -e j χ A δ |ϕ 1 j (y)| ∈ L 2 (R n+m )

Acknowledgement The first author thanks I.M.Sigal for partial support via NSERC grant No. 7901.

and it remains to estimate from below the square of the L 2 norm of the term v j ( √ -e j , ω)

which yields the necessary and sufficient conditions of solvability of the equation

. Using the definition of the functions v j (x) we easily arrive at

Having established the orthogonality conditions in the lemmas above which guarantee the existence of square integrable solutions for our equations we conclude the proof of Theorem 2.

Proof of Theorem 2. We construct the solution of the equation (1.3) as u := u + + u 0 + N -1 j=1 u -,j , where the existence of u + ∈ L 2 (R n+m ) is guaranteed by Lemma 3.1, of u 0 ∈ L 2 (R n+m ) by Lemmas 3.2-3.5, of {u -,j } N -1 j=1 ∈ L 2 (R n+m ) by Lemmas 3.6 and 3.7. Suppose the equation (1.3) admits two solutions u 1 , u 2 ∈ L 2 (R n+m ). Then their difference w := u 1 -u 2 ∈ L 2 (R n+m ) solves the homogeneous problem with separation of variables Lw = 0 which admits two types of solutions. The first ones are of the form γ(x)ϕ k N (y), 1 ≤ k ≤ m N with γ(x) harmonic. The second ones are of the kind e ipx (2π)

In both cases to vanish is the only possibility for them to belong to the space L 2 (R n+m ).

Appendix


Stephen Duncan
email: .stephen.duncan@eng.ox.ac.uk

Cameron Hepburn

Antonis Papachristodoulou

Optimal harvesting of fish stocks under a timevarying discount rate

Keywords: Resource management, Bio-economics, Optimal control, Dynamical systems

Optimal control theory has been extensively used to determine the optimal harvesting policy for renewable resources such as fish stocks. In such optimizations, it is common to maximise the discounted utility of harvesting over time, employing a constant time discount rate. However, evidence from human and animal behaviour suggests that we have evolved to employ discount rates which fall over time, often referred to as "hyperbolic discounting". This increases the weight on benefits in the distant future, which may appear to provide greater protection of resources for future generations, but also creates challenges of time-inconsistent plans. This paper examines harvesting plans when the discount rate declines over time. With a declining discount rate, the planner reduces stock levels in the early stages (when the discount rate is high) and intends to compensate by allowing the stock level to recover later (when the discount rate will be lower). Such a plan may be feasible and optimal, provided that the planner remains committed throughout. However, in practice there is a danger that such plans will be re-optimized and adjusted in the future. It is shown that repeatedly restarting the optimization can drive the stock level down to the point where the optimal policy is to harvest the stock to extinction. In short, a key contribution of this paper is to identify the surprising severity of the consequences flowing from incorporating a rather trivial, and widely prevalent, non-rational aspect of human behaviour into renewable resource management models. These ideas are related to the collapse of the Peruvian anchovy fishery in the 1970's.

Introduction

In this paper, we consider the optimal harvesting of a renewable resource when the discounting of the utility of harvesting varies over time. The analysis is based upon a standard model of the harvesting of a sole owner fishery [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF][START_REF] Sethi | Optimal Control Theory: Applications to Management Science and Economics[END_REF], but the results might also be applied to other renewable resources such as forestry, water resources and agricultural land use. We consider a system where the rate of change of the fish stock level is the difference between the natural growth rate of the population and the harvesting rate. In the absence of any discounting, if the utility is a monotonically increasing function of the harvesting rate, then the optimal integrated utility over an infinite time horizon is obtained by following a harvesting rate that drives the stock to the saddle point equilibrium point corresponding to the maximum sustainable yield, which will coincide with the peak of the natural growth rate curve [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. As shown in Figure 1, when a constant discount factor is included in the objective function, so that the optimal harvesting rate maximizes the integrated discounted utility over an infinite time horizon, the optimal sustainable policy converges to an equilibrium at a stock level that is below the maximum sustainable yield associated with the undiscounted objective function. The harvesting rate follows the stable manifold of the saddle point equilibrium and for an initial stock level below the optimal sustainable yield, the stock level will increase monotonically with time. However, for very low initial stock levels, it can take a long time for the stock to recover to the optimal sustainable yield and because the discount factor falls during this period, the optimal solution in this case is to collapse the stock by following a trajectory that converges to a stable equilibrium where there is no remaining stock, resulting in the extinction of the species [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. There is a critical initial stock level above which the optimal policy converges to the sustainable equilibrium, but when the initial stock is below this critical level, the optimal policy harvests all of the existing stock. This can be regarded as a consequence of the fact that the discount factor reduces the benefits of harvesting in the future.

[Figure 1

about here.]

It is usual to assume that the discount factor is described by an exponentially decaying function with a constant discount rate. However, evidence from behavioral economics discussed in Section 2 suggests that time-declining discount rates are commonly observed in human behaviour [START_REF] Thaler | Some empirical evidence on dynamic inconsistency[END_REF][START_REF] Kirby | Bidding on the future: evidence against normative discounting of delayed rewards[END_REF][START_REF] Harris | Dynamic choices of hyperbolic consumers[END_REF][START_REF] Ainslie | Picoeconomics[END_REF][START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF][START_REF] Harris | Dynamic choices of hyperbolic consumers[END_REF] and animal behaviour [START_REF] Mazur | The Effect of Delay and of Intervening Events on Reinforcement Value of Quantitative Analyses of Behavior[END_REF][START_REF] Green | Exponential versus hyperbolic discounting of delayed outcomes: risk and waiting times[END_REF]. Such behaviour can have some surprising and unappealing results, discussed in Section 2 and exemplified by the analysis in this paper. However, using time-declining discount rates also puts greater weight on distant future events. For instance, [START_REF] Li | Renewable resources and economic sustainability: a dynamic analysis with heterogeneous time preferences[END_REF] have examined the effect of using a "hyperbolic discount" model 1 when determining the optimal policy for achieving economic sustainability and they showed that it is necessary for the discount rate to decline with time in order to avoid the current generation having 'tyranny' over future generations.

An example of a failure to use a discounting policy that takes account of the benefits to future generations is the collapse of the stock of Peruvian anchovy, which was once the world's largest fishery [START_REF] Idyll | The anchovy crisis[END_REF]. The Peruvian fishing fleet expanded rapidly during the 1960's to such an extent that the authorities feared a collapse of anchovy stocks and they responded by restricting the length of the fishing season [START_REF] Aguero | A bioeonomic model of the Peruvian pelagic fishery[END_REF]. Experts such as the Instituto del Mar del Peru [START_REF] Glantz | Science, politics and economics of the Peruvian anchoveta fishery[END_REF] warned that the fishery should be closed to allow the stocks to recover so that it could survive the effect of the next El Niño. When the El Niño arrived in 1972-73, rather than closing the fishery, the Peruvian authorities continued to allow fishing, resulting in a collapse of the anchovy population, which had a worldwide effect on food prices [START_REF] Idyll | The anchovy crisis[END_REF]. [START_REF] Barrett | Population dynamics of the Peruvian anchovy[END_REF] highlighted the difference between a discount rate based purely on economic factors and a discount rate that takes into account the long term benefit to society of the anchovy fishery [START_REF] Cropper | The implications of hyperbolic discounting for project evaluation[END_REF]. They compared the net present value of the anchovy harvest associated with a policy that allowed the fish stocks to recover over a period of ten years, to the short-term policy based on a market discount rate and showed that the decision not to impose a conservation policy that would allow fish stocks to recover, was based on a short-term policy.

When a decreasing discount rate is applied to the problem of determining the optimal harvesting of fish stocks, the effect is to reduce the benefit of the collapse solution, because unlike the constant discount rate case, the discount rate decays asymptotically to zero as time becomes large, increasing the benefit of allowing the stock to recover to the point corresponding to the optimal sustainable yield. This paper shows how the optimal trajectories that converge to both the sustainable and the collapse solutions can be calculated for the case where the discount rate decays exponentially over time. The optimal policy can lead to a reduction in stocks during the early stages of the policy when the discount rate is high, which is then offset by a relatively low harvesting rate in the later stages of the policy when the discount rate is reduced, allowing the fish stocks to recover to the sustainable equilibrium. However, declining discount rates generally give rise to time-inconsistent plans [START_REF] Ramsey | A mathematical theory of saving[END_REF][START_REF] Strotz | Myopia and inconsistency in dynamic utility maximisation[END_REF], which implies that policies made today will not be carried out tomorrow unless a mechanism for ensuring commitment is in place. This paper shows that problems can occur if the planner does not remain committed to the optimal trajectory, but instead follows a policy leading to the sustainable equilibrium for only a finite time before recalculating the policy starting from the current stock level. If the stock level at which the re-optimization occurs is lower than the initial stock level and if the re-optimization is repeated, and even though the planner had been attempting to follow a policy that led to the sustainable solution, at some point in the future the collapse solution becomes optimal [START_REF] Hepburn | Behavioural economics, hyperbolic discounting and environmental policy[END_REF]. Although the factors leading to the collapse of the stocks of North Atlantic cod due to over-fishing are primarily the result of errors in estimating the existing stock level [START_REF] Walters | Lessons for stock assessment from the northern cod collapse[END_REF], it is not impossible that repeated re-optimization of policies based on an initial high discount rate contributed to the collapse.

The paper is organized as follows. Section 2 provides a review of the literature on discounting and hyperbolic discounting and its relationship to renewable resource management. In Section 3, the solution to the problem of determining the optimal policy for a time-varying discount rate is presented (the details of the derivation are given in the Appendix). Although this closely follows the solution for the constant discount rate case [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF], the version presented here explicitly includes the effects of the time-varying discounting and the requirement that the cost of harvesting must remain finite as the stock level approaches zero, which is relevant when determining the collapse solution associated with the extinction of the stock. Section 4 discusses the results and Section 5 concludes the paper.

Literature Review

A discount factor, D(t), is used in management of renewable resources to reflect the fact that the utility derived from a harvest may be worth more at one point in time than another. Normally, humans prefer utility now rather than later, so D(t) falls over time. An important distinction is drawn between discounting utility on the on hand, and goods on the other. In a realistic economy, where the rates of return on assets in the economy are not identical, the appropriate discount factor will be specific to the particular good being evaluated, and will also be specific to the individual or organisation performing the evaluation. The discount rate, δ(t), is defined as the rate of decline in the discount factor, D(t), and it is commonly assumed for ease of analysis that the discount rate δ(t) is constant, such that the discount factor D(t) falls exponentially.

Over the last couple of decades, debates over the theory and practice of discounting have occurred within the economic literature and even the mainstream media. One of the most significant areas of debate is whether the discounting behaviours implicit in real (but imperfect) human choices should be employed in economic analysis, or whether more "prescriptive" approaches should be employed, which might for instance explicitly take greater account of the long-term interests of future generations. We do not enter into these debates in this paper. 2 Instead, we examine the implications of incorporating imperfect but realistic human discounting behaviour into the management of a renewable resource. In other words, this paper provides a "descriptive" rather than a "normative" analysis, although we offer some reflections on the normative and policy recommendations flowing from our analysis in Section 5.

Given our objectives, we start from various experiments from behavioural economists and psychologists into human and animal "time preference". This research shows robust evidence of "preference reversals", where subjects choose a today rather than b tomorrow, but will choose b in one year and a day ahead of a in one year from now. These preference reversals are not consistent with rational choice under a constant discount rate δ(t) . They are consistent with δ(t) < 0 or "diminishing impatience". Indeed, many experiments suggest that impatience is higher in the present than with respect to trade-offs in the future [START_REF] Ainslie | Picoeconomics[END_REF][START_REF] Frederick | Time discounting and time preference: a critical review[END_REF][START_REF] Dellavigna | Psychology and economics: Evidence from the field[END_REF]. This has come to be referred to as "hyperbolic discounting", in reference to the fact that one relevant functional form was hyperbolic. Interestingly, diminishing impatience (or "hyperbolic discounting") appears prevalent in both humans and animals, and while they may at first seem irrational, there are plausible ways in which they could confer evolutionary advantage. For instance, [START_REF] Dasgupta | Uncertainty and hyperbolic discounting[END_REF] note that in the presence of given uncertainty and waiting costs, evolutionary pressure may have generated such preferences.

While there may be other explanations for some of the observed behaviour,3 careful reviews of the field [START_REF] Frederick | Time discounting and time preference: a critical review[END_REF][START_REF] Dellavigna | Psychology and economics: Evidence from the field[END_REF] conclude that the evidence for hyperbolic discounting is robust. Furthermore, recent evidence from functional imaging of the brain also supports a quasi-hyperbolic discounting model (McClure et al. 2007). Additionally, the hyperbolic discounting model provides explanations for otherwise baffling phenomena (Laibson et al. 2010) and there is robust evidence that people do have self-control problems of the sort implicated by hyperbolic discounting, that commitment mechanisms can be beneficial (Ariely and Wertenbroch 2002;Gine et al. 2009) and that some people are willing to effectively pay for commitment mechanisms to prevent time inconsistencies (Ashraf et al. 2006;[START_REF] Dellavigna | Psychology and economics: Evidence from the field[END_REF]Malmendier 2004, 2006).

If hyperbolic discounting does provide an accurate description of the time preferences of humans and animals in some contexts, the policy implications are challenging. In particular, although hyperbolic discounting may place greater weight on the interests of future generations, there are problems with incorporating hyperbolic discounting of utility into social cost-benefit analysis [START_REF] Pearce | Valuing the future: Recent advances in social discounting[END_REF][START_REF] Groom | Discounting the future: the long and the short of it[END_REF]. Because hyperbolic discounting can explain a range of perplexing human behaviours, including drug addiction [START_REF] Gruber | Is addiction rational? theory and evidence[END_REF], sub-optimally low savings rates [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF][START_REF] Laibson | Self-control and saving for retirement[END_REF][START_REF] Harris | Dynamic choices of hyperbolic consumers[END_REF], procrastination (ODonoghue & Rabin, 1999a,b;[START_REF] Benabou | Willpower and personal rules[END_REF]) and various others [START_REF] Akerlof | Procrastination and obedienc[END_REF], this paper does not make the normative recommendation that hyperbolic preferences be incorporated into optimal policy making. 4On the other hand, it is important to understand the descriptive implications of what might happen if a planner does use hyperbolic discounting in making policy, if only so that we can be aware of potential problems before they arise. The contribution of this paper is to provide such an analysis in the context of renewable resource management.

Hyperbolic discounting has been considered, particularly by economists, in renewable resource management and more specifically climate change over the last decade. 5 In the context of climate change, [START_REF] Karp | Global warming and hyperbolic discounting[END_REF], [START_REF] Karp | Time perspective and climate change policy[END_REF] and [START_REF] Fujii | Numerical analysis of non-constant pure rate of time preference: A model of climate policy[END_REF] develop models of hyperbolic discounting in which there are a succession of regulators developing plans which account for how future regulators will deviate from their plans. The paper thus finds a time-consistent Markov Perfect equilibrium. In contrast, in our model we consider the implications if the planner/regulator is even more imperfect (or realistic) such that, following the evidence outlined above, the planner determines a harvesting policy based on the erroneous assumption that they can commit future planners. The contribution of this paper is to show that where an (imperfect) planner is managing a renewable resource, such as a fishery, and they do not solve a sophisticated inter-temporal game but simply assume that future decisions will follow their plan, then the results can be dire. Of course, it is obvious that if a planner errs in anticipating their future behaviour, the outcome may be poor. The non-obvious aspect of the paper is that if the planner behaves much like as a normal human being, rather than as a hypothetical perfectly rational maximiser, or as a planner who can find Markov Perfect equilibria in intertemporal games, then there are conditions in which the erroneous assumption of future commitment can be catastrophic for the economic and biological survival of the fishery. In short, the surprising aspect of this paper is the potential severity of the consequences flowing from incorporating a rather trivial, and widely prevalent, non-rational aspect of human behaviour.

Solving the Optimization Problem

The analysis presented in this paper is based upon the mathematical model for the exploitation of a renewable resource that is presented in, for example, [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. Let x(t) ≥ 0 denote the fish stock at time t, and h(t) ≥ 0 denote the harvesting rate. If U (h) denotes the social utility of harvesting and c(x) denotes the unit cost of harvesting, the aim is to find the harvesting rate h(t) that optimizes the than a declining utility discount rate over time, relating to uncertainty. For instance, [START_REF] Sozou | On hyperbolic discounting and uncertain hazard rates[END_REF] and [START_REF] Weitzman | Why the far-distant future should be discounted at its lowest possible rate[END_REF] have shown that when the discount rate is constant but uncertain, the 'certainty-equivalent discount rate' will decline over time to the lowest possible rate. The UK Government Treasury guidelines on the appraisal of investments and policies requires the use of declining discount rates for these reasons [START_REF] Treasury | The Green Book: Appraisal and Evaluation in Central Government[END_REF]. [START_REF] Hepburn | Use of discount rates in the estimation of the costs of inaction with respect to selected environmental concerns[END_REF] provides a review of approaches to discounting within OECD countries. integrated discounted net benefit of harvesting (utility less costs) over an infinite time horizon max

h(t) ∞ 0 D(t) [U (h) -c(x)h(t)] dt (1) subject to ẋ(t) = F (x) -h(t) with x(0) = x 0 (2)
where x 0 is the initial stock level at t = 0 and F (x) describes the natural growth rate of the fish stock in the absence of any harvesting (i.e. the reproductive rate minus the death rate) [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. The discount factor is defined as

D(t) = exp - t 0 δ(τ )dτ (3)
with δ(t) > 0 being the (time-varying) discount rate.

Appendix A shows that the trajectories of x(t) and h(t) that optimize (1) for a time-varying discount rate satisfy

ẋ(t) = F (x) -h(t) ( 4 )
ḣ(t) = 1 U (h) {[δ(t) -F (x)] [U (h) -c(x)] + c (x)F (x)} (5)
with x(0) = x 0 . Note that when δ(t) is constant, (5) agrees with the expression given in [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF].

We consider the case where the discount rate decays over time according to

δ(t) = δ + (δ -δ)e -χt (6)
which has the effect of reducing the discount rate from δ at t = 0 to δ as t → ∞. From (3), the discount factor D(t) associated with this discount rate is given by

D(t) = exp -δt + (δ -δ) χ 1 -e -χt (7)
and Figure 2 plots D(t) for the parameters in Table 1. Because δ(t) is time-varying, the evolution equations in (4) and ( 5) are non-autonomous, but because

δ(t) = -χ(δ -δ)e -χt = -χ(δ(t) -δ) ( 8 )
combining ( 4), ( 5) and ( 8) gives a three state autonomous system

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ẋ(t) ḣ(t) δ(t) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ F (x) -h(t) 1 U (h) {[δ(t) -F (x)] [U (h) -c(x)] + c (x)F (x)} -χ(δ(t) -δ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (9)
It should be emphasized that this autonomous system is not used for analyzing the stability of the underlying non-autonomous system, but instead is introduced for the purpose of obtaining the optimal solution by numerical integration.

The evolution of δ(t) is independent of x(t) and h(t), so the autonomous system will only converge to an equilibrium in the limit δ(t) → δ, which occurs as t → ∞, and equilibrium points will occur at values of x and h satisfying

F (x) -h = 0 (10) 1 U (h) {[δ -F (x)] [U (h) -c(x)] + c (x)F (x)} = 0 (11)
This means that the equilibria associated with optimal harvesting in the presence of the time-varying discount rate in (7) are the same as the equilibria associated with optimal harvesting using a constant discount factor of δ [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF].

[Table 1 about here.]

[Figure 2 about here.]

Sustainable Solution

Assume that the natural growth rate of the fish population in the absence of any harvesting is described by a model with critical depensation that ignores the age structure of the population, so that

F (x) = rx x K 0 -1 1 - x K ( 12
)
where r is the net proportional growth rate of the population, K 0 is the minimum viable population and K is the environmental carrying capacity, with 0 < K 0 < K [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF][START_REF] Hanley | Environmental Economics in Theory and Practice[END_REF]. The instantaneous utility of consumption is assumed to be isoelastic and is taken to have the form

U (h) = h (1-γ) 1 -γ (13)
where γ > 0 is the Arrow-Pratt coefficient of relative risk-aversion [START_REF] Pratt | Risk aversion in the large and the small[END_REF][START_REF] Arrow | Aspects of the Theory of Risk Bearing[END_REF]. The unit cost of harvesting is taken to be

c(x) = C x α (14)
where C and α are constants and choosing α > 0 ensures that the cost of harvesting increases as the stock level reduces. For simplicity, we assume that γ ≤ 1 and α ≤ 1, although the analysis can be extended to handle larger values of these parameters.

Using these expressions for U (h) and c(x), the evolution of the optimal harvesting rate is

ḣ(t) = h γ [F (x) -δ(t)] 1 -C h γ x α + αC h γ x α F (x) x (15)
For this model, there exists an equilibrium point of the system in (9) denoted by [x s , h s , δ] T , where

x s > 0 and h s > 0 satisfy

F (x s ) -h s = 0 (16) h s γ [F (x s ) -δ] 1 -C h γ s x α s + αC h γ s x α s F (x s ) x s = 0 (17)
The Jacobian associated with this equilibrium is

J s = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ F (x s ) -1 0 J 21 J 22 -hs γ 1 -C h γ s x α s 0 0 -χ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (18)
where

J 21 = h s γ F (x s ) 1 -C h γ s x α s + 2 F (x s ) x s -δ αC h γ s x α s -α(α + 1)C h γ s x α s F (x s ) x 2 s (19) J 22 = C h γ s x α s α F (x s ) x s -(F (x s ) -δ) (20)
By ignoring the higher order terms in ( 9), in the region close to the equilibrium point

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x(t) -x s h(t) -h s δ(t) -δ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ≈ α 1 v 1 e λ1t + α 2 v 2 e λ2t + α 3 v 3 e λ3t ( 21
)
where α 1 , α 2 , α 3 ∈ R are constants and v 1 , v 2 , v 3 ∈ R 3 are the eigenvectors of the Jacobian associated with the eigenvalues λ 1 , λ 2 and λ 3 . One of the eigenvalues occurs at λ 1 = -χ, which is associated with the evolution of δ(t) to δ. For a wide range of model parameters, including the values given in Table 1, one of the remaining two eigenvalues is positive, while the other is negative, indicating the presence of a saddle-point equilibrium. Given the structure of J s , it can be shown that the third element of both v 2 and v 3 are zero, and that the first two elements of both of these eigenvectors are equal to the eigenvectors associated with the saddle-point equilibrium for the case with a constant discount rate of δ.

For values of t 0, so that e -χt ≈ 0, then x(t) and h(t) approach equilibrium along the eigenvector associated with the stable eigenvalue, which coincides with the stable trajectory of a system with constant discount rate, δ. This suggests that the following procedure can be used to find the trajectory of the time-varying system in ( 9) that approaches the sustainable equilibrium.

1. Substitute h s = F (x s ) into (17) and use a numerical root find routine to find x s . The expression in ( 16) can then be used to find h s .

2. Use numerical integration to determine the trajectory associated with the stable eigenvalue of the saddle point equilibrium for the system with constant discount rate δ [START_REF] Conrad | Natural Resource Economics[END_REF][START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF].

3. Numerically integrate (9) backwards from t = t 1 0 to t = 0, where t 1 is chosen such that e -χt ≈ 0, starting from δ(t 1 ) = δ and from [x(t 1 ) u(t 1 )] T that is a point on the trajectory obtained in step 2. In practice, δ(t) can be obtained directly from (6) rather than via numerical integration.

Figure 3 shows h(t) plotted against x(t) for a series of trajectories using a range of initial values [x(t 1 ) u(t 1 )] T along the constant discount rate trajectory for the parameters in Table 1. The simulations were carried out in MATLAB using the fzero routine to find x s and the ode113 routine for numerically integrating (9). From the Figure, it can be seen that when the initial harvesting rate satisfies h(0) > F (x(0)), then ẋ(t) is negative and the optimal policy to achieve a sustainable equilibrium is to harvest stock until ẋ(t) = 0, which occurs when the trajectory crosses the locus F (x) = h. After this point, the harvesting rate is sufficiently low to allow the stock to recover and increase towards the equilibrium level, x s . This means that unlike the constant discount rate case, the optimal policy reduces stock in the initial period where the discount rate is high and then reduces the harvesting rate in the later stages when the discount rate is lower.

The objective function associated with a given trajectory can be expressed in the form

∞ 0 D(t) h (1-γ) 1 -γ - C x α h(t) dt = t1 0 D(t) h (1-γ) 1 -γ - C x α h(t) dt+ ∞ t1 D(t) h (1-γ) 1 -γ - C x α h(t) dt (22)
where D(t) is given in ( 7). The first integral on the right hand side is obtained by combining the numerical integration of the system in (9) backwards from t = t 1 to t = 0, with the integration of

V = -D(t) h (1-γ) 1 -γ - C x α h(t) with V (t 1 ) = 0 (23)
where the negative sign is included to allow for the integration backwards in time. The second integral is obtained by integrating forwards along the trajectory associated with the constant discount rate δ, from t 1 to t 2 t 1 , where t 2 is chosen to be sufficiently large such that D(t 2 ) ≈ 0.

[Figure 3 about here.]

Collapse Solution

As [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF] points out, there appears to be no general results for determining the optimal approach to extinction, where [x(t) h(t)] T approaches [0 0] T in the limit t → ∞. However, for the system considered here, when the stock level is below the minimum viable stock, so that x(t) < K 0 , then from (12), F (x) < 0 and ẋ(t) < 0, so from (4) the stock will become extinct, although because F (x) → 0 as

x → 0, extinction will occur in the limit t → ∞. The difficulty is to determine the optimal harvesting as x(t) → ∞. Using the expressions for U (h) and c(x) in ( 13) and ( 14), the given that F (x) and F (x)/x are finite for finite x, the expression for ḣ(t) in ( 15) can become infinite as x(t) → 0. However, along any optimal trajectory that converges to the collapse solution, we require that h(t) → 0 as t → ∞, which will occur when lim t→∞ ḣ(t) ≤ 0. Given that lim t→∞ δ(t) = δ and that

lim x→0 F (x) = lim x→0 F (x) x = -r (24)
this condition will be met when

lim t→∞ h γ (t) x α (t) ≤ 1 C r + δ (1 -α)r + δ (25)
Provided that this condition holds in the limit h(t) → 0, then the cost of harvesting satisfies

c(x)h = h (1-γ) C h γ x α ≤ h (1-γ) 1 C r + δ (1 -α)r + δ (26)
which remains finite since it is assumed that γ ≤ 1 and α ≤ 1.

[Figure 4 about here.]

The optimal trajectory will have the largest allowable value of h, which will occur when equality is achieved in (25) as both h and x approach zero. This can be used to find the optimal trajectory for a constant discount rate δ, by numerically integrating

dh dx = h γ 1 F (x) -δ [F (x) -δ] 1 -C h γ x α + γC h γ x α F (x) x (27)
from x = ε with ε close to zero, starting from a value of h(ε) that achieves equality in (25). Once the trajectory for the constant discount rate has been found, the trajectories for the time-varying discount rate in (6) and the corresponding costs can be obtained using the same approach as in the sustainable case, where ( 9) is integrated backwards from t = t 1 0 to t = 0, starting from a point on the constant discount rate trajectory obtained from ( 27). Figure 4 shows the optimal trajectories for a range of initial stock levels, using the parameters in Table 1.

[Figure 5 about here.]

[Figure 6 about here.]

Discussion

Figure 5 shows the value of the objective function associated with the optimal sustainable and collapse trajectories plotted against initial stock level, x(0). The two curves intersect at a critical initial stock level, x c , and for any x(0) < x c , the collapse solution is optimal, while for x(0) > x c , the sustainable solution is optimal [START_REF] Clark | Mathematical Bioeconomics: Optimal Management of Renewable Resources[END_REF]. As can be seen in Figure 3, for some initial stock levels that satisfy

x(0) > x c , the optimal sustainable trajectory takes the stock level below the initial critical stock level,

x c , before reducing the harvesting rate to allow the stock level to recover and converge to the sustainable equilibrium. This is shown in Figure 6, which plots the time evolution of a sustainable trajectory that starts with x(0) > x c and for a period of time, the stock level will be below x c . This is acceptable provided that the policy is followed through to t → ∞. However, if the harvesting policy is stopped during this period and then re-optimized, the optimal policy is to follow the collapse solution, illustrating the potential danger associated with periodically restarting an optimal policy. Even if the trajectory does not cross the critical stock level x c , if the policy is restarted at time t r where

x(t r ) < x(0), then the optimization is started from a lower initial stock level. Repeatedly restarting the optimization will drive the initial stock level down to the point where x(0) < x c , making the collapse solution optimal. If the time between restarts becomes small, then the stock trajectory follows the "naive" policy shown by the dotted line in Figure 3, which drives down the stock level until it falls below x c . Note that continually restarting the policy does not produce the same trajectory as discounting with a constant discount rate at level δ, because the (different) anticipated future harvesting trajectories change the optimal harvest rates at the present decision point.

Conclusion

This paper has examined optimal harvesting policies for fish stocks when hyperbolic discounting described by an exponentially decaying discount rate is applied to the utility of the harvesting. There are two possible solutions to the optimization problem; one solution converges to a sustainable equilibrium, while the other converges to the collapse solution where the stock is reduced to zero.

Which of the two solutions is optimal depends upon whether the initial stock level x(0) exceeds an initial critical stock level x c . When x(0) is greater than x c , the optimal harvesting policy converges to the sustainable equilibrium, but when x(0) is less than x c , the optimal policy collapses the stock. With time-declining discount rates, the optimal policy leading to a sustainable solution may initially reduce the stock below the initial stock level during the period when the discount rate is high, which is then compensated by a low harvesting rate during the later stages when the discount rate is low. The problem is that a declining discount rate leads to time inconsistent policies, so that at some point in the future, it may appear that it is beneficial to stop the policy and recalculate the optimal harvesting rate, starting from the current stock level. If the policy is repeatedly restarted at times where the stock is below the initial stock, then this can result in the stock level being driven down to the point where the policy is restarted with an initial stock x(0) < x c and in this case, the optimal policy would be to drive the stock level to zero. The key conclusion is that care is required if declining utility discount rates, reflecting apparent human behaviour, are proposed to be used in renewable resource management.6

While proposals to use hyperbolic discounting in this way might be motivated by a desire to preserve renewable resources for future generations, this paper has made clear that perverse results are possible without putting safeguards in place. These might include placing a side-constraint on the objective function to rule out paths which risk resource collapse. Alternatively, as noted in this paper, to prevent the stock level being inadvertently reduced by repeated re-optimizations, a commitment mechanism might be found to ensure that the optimal policy is followed at least to the point where the stock exceeds the initial level before any changes to the policy are considered. parameters in Table 1. The plot also shows the discount factor for the constant discount rates δ = 0.15 (dashed) and δ = 0.025 (dash-dotted).

Figure Captions Figure 1 .

 Captions1 Figure Captions

Figure 2 .

 2 Figure 2. Plot of discount factor D(t) (solid) for exponentially decaying discount rate in (6) with

Figure 3 .

 3 Figure3. Plot of trajectories (solid lines) of {x(t), h(t)} that converge to the sustainable equilibrium, where each trajectory starts from a different initial stock level, x(0), marked by a circle. The trajectories are obtained by numerically integrating (9). The dashed curve shows the locus of the natural growth function F (x) = h and the location of the sustainable equilibrium, (x s , h s ), is marked by an asterisk. The trajectory of the naive policy is shown by the dotted curve.

Figure 4 .Figure 5 .

 45 Figure 4. Plot of trajectories (solid lines) of {x(t), h(t)} that converge to the collapse solution where each trajectory starts from a different initial stock level, x(0), marked by a circle. The trajectories are obtained by numerically integrating (9),

Figure 6 .

 6 Figure6. Plot of a single optimal sustainable trajectory against time for an initial stock level that satisfies x(0) > x c . The critical initial stock level x c is shown by the dashed line.

Table Parameter

 Parameter

		Value
	K 0	1.0
	K	3.0
	r	0.06
	γ	0.7
	δ	0.025
	δ	0.15
	χ	0.1
	C	1.0
	α	0.5

Table 1 :

 1 Values of parameters used in simulations

Although hyperbolic discounting was originally introduced to describe a discount rate that varied inversely with time, the term is now commonly used to describe any discounting with a declining discount rate(Dasgupta & Maskin,

2005).

The interested reader is referred to the discussion and analysis on discounting following the Stern Review on the Economics of Climate Change[START_REF] Stern | The Economics of Climate Change: the Stern Review[END_REF][START_REF] Beckerman | Ethics of the discount rate in the stern review on the economics of climate change[END_REF] Weitzman, 2007;[START_REF] Nordhaus | A review of the stern. review on the economics of climate change[END_REF][START_REF] Dasgupta | Discounting climate change[END_REF][START_REF] Heal | Climate economics: A meta-review and some suggestions for future research[END_REF].

For instance, there are potential confounding factors in some studies(Chabris et al. 2008), and other theories also appear consistent with the evidence for preference reversals[START_REF] Read | Is time-discounting hyperbolic or subadditive[END_REF][START_REF] Rubinstein | Is it 'economics and psychology' ?: The case of hyperbolic discounting[END_REF].

There are other reasons government policy may wish to employ a declining consumption discount rate over time, rather

See for instance[START_REF] Cropper | The implications of hyperbolic discounting for project evaluation[END_REF],[START_REF] Shogren | Hyperbolic discounting and tine inconsistency in a native-exotic conflict[END_REF],[START_REF] Voinov | Reconciling sustainability, systems theory and discounting[END_REF] and the 2010 Special Issue 46:2 of Environmental and Resource Economics on behavioural economics and the environment.

Note the distinction between hyperbolic utility discounting and the application of declining consumption discount rates which reflect inherent economic uncertainty, as employed by H M[START_REF] Treasury | The Green Book: Appraisal and Evaluation in Central Government[END_REF].

Acknowledgements

The authors gratefully acknowledge financial support for this work from the UK Engineering and Physical Sciences Research Council under grant EP/H03062X/1.

A Derivation of Optimal Harvesting Policies

For the discount factor in (3) dD dt = -δ(t)D(t) ( A . 1 ) so that the state evolution in (2) can be written as

where D ∈ [1, 0] and the initial condition is x(1) = x 0 . Using (A.1), the optimization in (1) becomes max

and the aim is to minimise this objective function subject to the state evolution

with x(1) = x 0 , together with h(D) ≥ 0 and x(D) ≥ 0.

The Hamiltonian associated with the minimization problem is

where

and λ(D) is the costate. The co-state satisfies

with λ(0) = 0, since there is no terminal cost at D = 0. The optimal h(D) satisfies .11) which can be rearranged as

Instead of expressing the problem in terms of (x, λ, D), it is more convenient to consider the system in the space defined by (x, h, D). Differentiating (A.12) gives

and after equating to (A.9) and simplifying, the system can be described by two coupled differential equations

with x(1) = x 0 . Although the terminal condition for (A.9) is λ(0) = 0, from (A.12), the value of h(D) is undefined when D = 0. However, as will be discussed below, we are interested in steady state solutions, which restricts the possible values of lim D→0 h(D). Transforming these equations back to the (h, x, t) domain gives the system described by (4) and (5).