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Optimal harvesting of diffusive models in a nonhomogeneous environment
Nonlinear Analysis 71 (2009) e2173–e2181
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Optimal harvesting of diffusive models in a nonhomogeneous environment Elena Braverman a,∗ , Leonid Braverman b,c a
Department of Mathematics & Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N1N4
b
Athabasca University, 1 University Drive, Athabasca, AB, Canada T9S3A3
c
St. Mary’s University College, 14500 Bannister Road SE, Calgary, AB, Canada T2X1Z4
article
info
Keywords: Reaction-diffusion equation Population dynamics Optimal harvesting Maximum sustainable yield Impulsive equations Logistic growth Gompertz growth Gilpin–Ayala model
abstract We study the optimal harvesting strategy for populations whose dynamics is described by reaction-diffusion equations. The production function can be of logistic, Gilpin–Ayala or Gompertz type. The diffusion structure is discussed; we suggest to consider ∆(u/K ), where K is the carrying capacity of the environment, rather than ∆u, and study optimal harvesting for models with this diffusion type. Maximum yield is investigated for both continuous and impulsive models. For continuous harvesting, the optimal policy is obtained; for the impulsive equation some limit cases are considered. The paper also outlines a variety of open problems. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Optimal resources management, including sustainable harvesting, is one of the most important problems of population ecology [1,2]. In the case when spatial distribution and interaction are neglected, optimal harvesting has recently become an area of active research, see, for example, [3–7]. This includes continuous [8–10] and impulsive harvesting strategies [6,7,11–13] ; some comparison of these two types of harvesting can be found in [5,14]. However, in many cases, populations are either age structured or spatially distributed. For example, the level of harvesting may depend on the age continuously, or be stage-structured (juvenile–adult population), see [15–17]. In fisheries, age structure of harvesting corresponds to different quotas for certain sizes of species. For spatially distributed species with diffusion, the optimal harvesting strategy was studied in [18,19]. Let u(t , x) be the population density at point x and time t. As a spatially distributed analogue of one-species onedimensional population models, the following reaction-diffusion equation was considered [19]
∂u = D∆u + r (t , x)f (u, K (t , x)) − E (t , x)u, ∂t
(1.1)
where E (t , x) > 0 is the combined harvesting effort and efficiency which will be in future referred to as the harvesting effort, E (t , x)u describes the level of harvesting, r (t , x) > 0 is the intrinsic growth rate, K (t , x) > 0 and the carrying capacity of the environment at point x and time t, D > 0 is the diffusion coefficient which describes the dispersal speed for a studied population. Function f (u, K ) is positive for 0 < u < K and is negative for u > K .
∗
Corresponding author. E-mail address: [email protected] (E. Braverman).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.04.025
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There are different possible growth functions f (u, K ). The simplest approach assumes that per capita growth rate is a linear decreasing function of population r (1 − u/K ), which leads to the logistic type of growth f (u, K ) = ru(1 − u/K ) which was first introduced by Verhulst in 1845 and later studied by Pearl and Reed in 1920 (see [17] for historical reference). This type of growth is widely applied in fisheries [1,2], for example, model (1.1) with logistic-type growth and without diffusion du dt
u = ru 1 − − Eu
(1.2)
K
is known as Schaefer equation [1,2]. However the growth of certain fish populations deviated from the logistic law, which motivated the consideration of more general growth functions
f1 (u, K ) = ru 1 −
u θ K
,
f2 (u, K ) = ru 1 −
u δ K
,
(1.3)
where θ > 0 describes the intensity of intraspecific interference, δ allows the accommodation of a non-symmetric (with respect to K /2) growth curve which occurs in fisheries. It is to be noted that f1 (which is called Gilpin–Ayala growth law) and f2 in (1.3) become logistic growth functions for θ = 1 and δ = 1, respectively. These modifications allow more a realistic description of various fish species. The Gilpin–Ayala growth law was suggested in [20] for Lotka–Volterra systems and was later referred to as a Gilpin–Ayala model and studied in [21]. However, both logistic and Gilpin–Ayala laws describe per capita growth rate of similar types: it is positive and finite at u = 0 and is decreasing for u > 0 as a polynomial function of u/K . The Gompertz growth function f (u, K ) = ru ln(u/K ) was designed to describe populations with unlimited per capita growth rate for low population levels. The rate of its decay is faster than for any logistic-type model. Originally, the Gompertz equation was formulated as a law of decreasing survivorship (in 1825 by Gompertz, see [17,22]). It appears in fish ecology in the Fox surplus yield model [23]. In addition to fisheries, the Gompertz equation was applied to model the growth of plants and tumors (see [17] for relevant references). In the case of the logistic reproduction function Eq. (1.1) becomes [19]
u ∂u = D∆u + r (t , x)u 1 − − E (t , x)u. ∂t K ( t , x)
(1.4)
Reaction-diffusion equations incorporate a certain growth law at each point with the spatial distribution of populations which can relocate (in (1.4), to less populated spots). Diffusion coefficient D > 0 can be high for some populations (for example, in fisheries) and lower for others (slugs, bacteria, plants). In future, we will also consider some other types of reproduction functions f : Gilpin–Ayala and Gompertz laws of growth. Optimal harvesting for a reaction-diffusion equation with the Gilpin–Ayala type of growth
" θ # ∂u u = D∆u + r (x)u 1 − − E (x)u ∂t K (x)
(1.5)
and time-independent parameters r, K and E, was studied in [18]. The purpose of the present paper is to study the following aspects of spatially distributed populations and optimal harvesting. (1) We are going to discuss the diffusion structure in (1.1). This equation involves a possibility of a massive migration from places with a high carrying capacity and still unused growth potential to locations with low resources where the migrating population is destined to die. This dispersal is just due to the fact that fertile places are more populated. In contrast to this ‘‘blind’’ diffusion scheme, here we suggest a ‘‘judicious’’ diffusion where the direction of the flow is defined by maximum per capita available resources. (2) For this new type of diffusion and time-independent carrying capacity, we describe optimal harvesting policies for the analogues of (1.4) and (1.5) with logistic and Gilpin–Ayala growth, respectively, and the reaction-diffusion equation which is subject to the Gompertz growth law. Unlike models with a usual diffusion, no additional conditions should be imposed on the carrying capacity and the growth rate to achieve this strategy. This is the first time when models with such type of diffusion are introduced, so all Theorems of the present paper are new results. (3) In the case when r (t , x) and K (t , x) are periodic in t and time-dependent, optimal harvesting policy is investigated. (4) We consider impulsive harvesting for models with diffusion and obtain some results on optimal harvesting. (5) Last but not the least, we state some open problems for both continuous and impulsive harvesting. In particular, for a ‘‘judicious’’ diffusion, can we claim that the level of optimal harvesting grows with the growth of the diffusion coefficient? For a usual diffusion and steeply changing K (x), can impulsive harvesting in space rather than in time (we set traps which serve as semi-conducting filters) be an efficient harvesting policy? These and many other open problems are presented in the last section of this paper.
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2. Optimal continuous harvesting for time-independent parameters Let us notice that if K (x) is variable, then models (1.4) and (1.5) may assume massive migration from resource rich fertile places with a high carrying capacity to scanty places. We argue that this scheme is not realistic for many populations and suggest that the direction of migration should correspond to the gradient of the ratio of the population density u(t , x) to the carrying capacity K (t , x) which means that emigration begins as far as per capita available resources are higher in the adjacent place than at a given point in space. This corresponds to the equation
∂u u = D∆ + r (t , x)f (u, K (t , x)) − E (t , x)u, x ∈ Ω . ∂t K (t , x)
(2.1)
Here Ω is a closed bounded domain in Rn (n = 1, 2 or 3). First, we consider Eq. (2.1) where the carrying capacity is time-independent (the environment enrichment distribution is the same at any time) and only space-dependent. Optimal harvesting strategy for time-dependent parameters will be considered in Section 3. 2.1. Logistic growth law Consider the logistic equation with time-independent carrying capacity and growth rate and with the diffusion defined in (2.1)
∂u u u = D∆ + r (x)u 1 − − E (x)u, x ∈ Ω , t ≥ 0. ∂t K (x) K (x)
(2.2)
Let us first formulate the following obvious statement. Lemma 1. The maximum of the function
f ( u, K ) = u 1 − for a fixed K equals u=
K 2
K 4
u
(2.3)
K
and is attained at
.
We proceed to the main result on the optimal continuous harvesting for (2.2). Since the term E (t , x)u describes the density of population harvested at point x and time t, then the optimal sustainable yield per unit time is
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
Z Ω
E (x)p(x) dx,
where p(x) is an optimal stationary solution, E (x) is the optimal harvesting effort. This is relevant in the case when both the optimal solution and optimal effort are time-independent. Generally (see Section 3), the maximal sustainable yield per unit time is T
Z 1
max
E (t ,x),p(t ,x)
Z dt
T
Ω
0
E (t , x)p(t , x) dx .
Theorem 1. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
1
Z
4
Ω
K (x)r (x) dx
(2.4)
is attained for the Eq. (2.2) have the form p(x) =
K (x) 2
,
E (x) =
r ( x) 2
.
(2.5)
Proof. Let us demonstrate that for a stationary (time-independent) solution the maximum sustainable yield is attained at every location and time stage. Then this is obviously an unbeatable strategy. K (x) u(t ,x) For u(t , x) ≡ p(x) = 2 the diffusion term vanishes since K (x) ≡ 21 is constant, the left hand side equals zero, since u(t , x) is time-independent. Furthermore, by Lemma 1 this population size provides the maximum of the production term in (2.2). Since
r (x)u 1 −
u K (x)
= E (x)u
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for any stationary (time-independent) solution then E (x)u also attains its maximum at any x and E (x) = r (x)/2 as in (2.5), which completes the proof. Remark 1. Formula (2.4) means that the maximal sustainable yield is proportional to the integrated product of the growth rate by the carrying capacity over the domain. It is attained when the optimal population level of K (x)/2 is maintained by persistent harvesting (otherwise, the population density will grow towards K (x), with the production level decreased), the optimal harvesting effort of r (x)/2 means that the population can be kept at the desired level. Remark 2. Comparing our results to [19] we observe that the maximum sustainable yield in Theorem 1 is the same as in the case of Eq. (1.4) with the usual diffusion ∆u. However there are no requirements on either r or K , while in [19] the condition r (x) 2
+D
∆K (x) ≥ 0, K (x)
x ∈ Ω,
is imposed. If the growth rate is not enough to compensate for the population outflow from fertile regions, the optimal harvesting strategy described in [19] is impossible. Remark 3. Let us mention that, when formulating the problem, we have not imposed any initial or boundary conditions. For the Neumann boundary conditions we will have for the optimal solution
∂(u/K (x)) = 0, ∂n
u ∈ ∂Ω.
(2.6)
If
∂ K =0 ∂ n ∂ Ω
(2.7)
(this assumption was also imposed in Example 1 of [19], p. 879) then the usual zero Neumann boundary conditions
∂u = 0, ∂n
u ∈ ∂Ω
(2.8)
apply to the solution. This means that there is no population inflow or outflow in the domain Ω . Conditions (2.6) and (2.7) are satisfied in the case when the carrying capacity vanishes at the boundary (for example, there is a natural bound to the population growth and dispersal outside the domain) together with its first derivative in the direction normal to the boundary ∂ Ω . Remark 4. Unlike other papers (see, for example, [18]), where an optimal harvesting strategy for a diffusive model was considered under the assumption that r , K and E are at least Hölder continuous functions, Theorem 1 does not involve any assumptions of this kind. We can choose the optimal strategy for any K (x), which may have a discontinuous patchy structure; then the optimal solution will not be continuous in x anymore; discontinuity of r in space will also cause jumps in harvesting effort E. We can consider discontinuous in x solutions, once the ratio u/K is smooth enough in x and t. 2.2. Gilpin–Ayala and Gompertz growth laws In this section we consider the Gilpin–Ayala reproduction function
" θ # ∂u u u = D∆ + r (x)u 1 − − E (x)u, ∂t K (x) K ( x)
x ∈ Ω , t ≥ 0,
(2.9)
and the Gompertz growth rate
∂u u K (x) = D∆ + r (x)u ln − E (x)u, ∂t K (x) u
x ∈ Ω , t ≥ 0.
(2.10)
The following two results are obvious. Lemma 2. The maximum of the function
f (u, K ) = u 1 − for a fixed K equals u=
u θ K
θK (1+θ)(1+θ)/θ
K
(1 + θ )1/θ
.
and is attained at
(2.11)
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Lemma 3. The maximum of the function f (u, K ) = u ln K e
for a fixed K equals u=
K e
K
(2.12)
u and is attained at
.
Lemmas 2 and 3 can be applied to describe the best possible strategy when the maximum production is attained at each time and location, similar to the proof of Theorem 1. They yield the following results on the optimal harvesting policy in the cases of the Gilpin–Ayala and the Gompertz growth laws. Theorem 2. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
θ (1 + θ )(1+θ)/θ
Z Ω
r (x)K (x) dx
(2.13)
is attained for the Eq. (2.9) have the form p(x) =
K (x) , (1 + θ )1/θ
r (x)θ
E (x) =
1+θ
.
(2.14)
Remark 5. By formula (2.13), the maximal sustainable yield is proportional to the integrated product of the growth rate by the carrying capacity over the domain. It is attained when the optimal population level of K (x)/(1 + θ )1/θ is attained at all points of the domain by persistent harvesting (otherwise, the population density will grow towards K (x), with the production level decreased), the optimal harvesting effort of r (x)θ /(1 +θ ) allows to keep the population at the desired level. Theorem 3. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
1
Z
e
Ω
K (x)r (x) dx
(2.15)
is attained for the Eq. (2.10) are as follows p(x) =
K (x) e
,
E (x) = r (x).
(2.16)
Remark 6. Formula (2.15) outlines that the maximal sustainable yield which is attained when the optimal population level of K (x)/e is maintained at all points of the domain by persistent harvesting. Without constant population deduction, the population density will grow towards K (x), with the production level decreased. The optimal harvesting effort of r (x) allows to keep the population at the desired level. 3. Optimal harvesting: time-dependent periodic parameters Now consider the problem of the continuous optimal harvesting when the parameters of the equation are not only spacedependent but also time-dependent:
∂u u u = D∆ + r (t , x)u 1 − − E (t , x)u, ∂t K (t , x) K (t , x)
x ∈ Ω , t ≥ 0.
(3.1)
Theorem 4. Let r (t , x) and K (t , x) be periodic functions in t of the same period T for each x ∈ Ω . If for any t , x, 0 ≤ t ≤ T , x ∈ Ω we have r (t , x) ≥
∂ K ( t , x) , K (t , x) ∂t 2
(3.2)
then the solution p(t , x) = u(t , x) and the harvesting effort E (t , x) such that the maximum yield per unit time T
Z max
E (t ,x),p(t ,x)
1
T
Z dt Ω
0
E (t , x)p(t , x) dx
=
1 4T
T
Z
Z dt
0
Ω
K (t , x)r (t , x) dx
(3.3)
is attained for the Eq. (3.1) have the form p(t , x) =
K ( t , x) 2
,
E (t , x) =
r ( t , x) 2
−
∂ K ( t , x) . K (t , s) ∂t 1
(3.4)
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Proof. The optimal production rate is attained at each point and each time stage if the population level is p(t , x) =
K (t , x) 2
.
Then
∆
p(t , x)
K ( t , x)
= 0 and E (t , x)
K (t , x) 2
=
r (t , x)K (t , x) 4
−
1 ∂ K (t , x) 2
∂t
,
which is nonnegative for any t , x if (3.2) holds. This immediately implies the harvesting effort E (t , x) in (3.4) and formula (3.3) for the maximum yield. Corollary 1. If the carrying capacity in (3.1) is time-independent
u u ∂u = D∆ + r (t , x)u 1 − − E (t , x)u, ∂t K (x) K (x)
x ∈ Ω , t ≥ 0,
(3.5)
then the optimal solution is also stationary (time-independent). It exists for any r (t , x); the maximum yield defined in (3.3) is attained for p(t , x) =
K (x) 2
,
E (t , x) =
r (t , x) 2
.
(3.6)
4. Impulsive harvesting Next, let us consider the model when a persistent harvesting effort cannot be applied but harvesting occurs once in a while during a relatively short period of time; here we assume that we deduct a part of the population every T time units (T > 0 is a constant). For time-independent parameters and the logistic growth, this corresponds to the impulsive partial differential equation
∂u u u = D∆ + r (x)u 1 − , ∂t K (x) K (x)
x ∈ Ω , t ≥ 0, t 6= nT ,
u(t + nT + , x) − u(t + nT , x) = −E (x)u(t + nT , x),
n = 1, 2, . . . .
(4.1) (4.2)
Here 0 < E < 1, f (t ) is the right side limit of the function f (t ) = lims→t ,s>t f (s), the solution u(t , x) is piecewise continuous in t. Let us refer to a known result in the theory of impulsive ordinary differential equations (see, for example, [6,13]). +
+
Lemma 4 ([13]). For the logistic equation
y0 (t ) = ry 1 −
y
(4.3)
K
with the impulsive conditions y(t + nT + ) − y(t + nT ) = Ey(t + nT ),
n = 1, 2, . . .
(4.4)
optimal sustainable harvesting is attained for x(0) = x(nT + ) = K e−rT /2 + 1
(4.5)
and the maximum sustainable yield (MSY) per unit time is MSY =
K (erT /2 − 1) T (erT /2 + 1)
.
(4.6)
We will consider only some aspects of optimal impulsive harvesting (4.2) for the Eq. (4.1). In particular, the limit cases when either D = 0 or D → ∞ will be investigated. Limit case D = 0. Without diffusion, we set optimal harvesting for each point x of domain Ω individually. Thus by Lemma 4 the total harvest over Ω and time period T between impulses will be
Z MSYT =
Ω
K (x)(eTr (x)/2 − 1) eTr (x)/2 + 1
dx
(4.7)
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or maximum sustainable yield per unit time is K (x)(eTr (x)/2 − 1) dx
Z MSY =
T (eTr (x)/2 + 1)
Ω
.
Limit case D → ∞. In this case we will have a homogeneous distribution of u(t , x)/K (x) in x over domain Ω at each time step. If we make a substitution u(t , x)
z (t , x) =
K (x)
= z (t )
(4.8)
in (4.1) then taking into account ∆z = 0 for any t , x, we obtain z 0 (t ) = rz (1 − z ),
(4.9)
where
R
ΩR
r =
r (x) dx
Ω
(4.10)
dx
is the average growth rate over domain Ω . Thus, Lemma 4 can be applied to the ordinary differential equation (4.9) which leads to MSYT =
erT /2 − 1
Z
erT /2 + 1
Ω
K (x) dx.
(4.11)
Let us compare yields (4.7) and (4.11). The following example demonstrates that, generally, they are not equal. This demonstrates the importance of diffusion effects in harvested populations. Below we choose K = T = 1 which can be achieved by scaling for any model with constant carrying capacity and time between impulses, and slowly changing in space time-independent intrinsic growth rate r (x). Example 1. Let r (x) = 2 ln(10x), x ∈ [1, 2], K ≡ 1, T = 1. Then in (4.7) 2
Z
K ( x) 1
eTr (x)/2 − 1
2
Z =
eTr (x)/2 + 1
1−
1
0.2 x + 0.1
dx = 1 − 0.2 ln
2.1
1.1
≈ 0.8707.
The average growth rate is
Z
2
ln(10x) = 2 ln 10 + 2(−x + x ln x)|21 ≈ 5.37776.
r =2 1
Then MSY in (4.11) exceeds (4.7) erT /2 − 1 erT /2 + 1
Z Ω
K (x) dx ≈ 0.8727 > 0.8707 ≈
2
Z
K (x) 1
eTr (x)/2 − 1 eTr (x)/2 + 1
.
Still, it remains an open problem as to whether higher levels of diffusion improve the maximum sustainable yield, see Section 5. In addition to the above limit cases, we will also consider (4.1) with a constant growth rate r (x) ≡ r. u(x,t ) Case r (x) ≡ r. For constant r, for each point x, the same ratio K (x) of the initial population level to the carrying capacity will correspond to the optimal impulsive harvesting policy. Thus, for the optimal solution at each point x we have ∆(u/K ) = 0. This harvesting strategy is optimal for the whole system. Hence the maximum sustainable yield over time T is MSYT =
erT /2 − 1 erT /2 + 1
Z Ω
K (x) dx,
(4.12)
which coincides with (4.7) and (4.11). 5. Discussion and open problems In the present paper, we considered the problem of optimal harvesting for various models described by reaction-diffusion equations. (1) We have argued that the regular diffusion with a tendency to a uniform density is not reasonable for models of population dynamics with a non-uniformly distributed carrying capacity. A new type of diffusion was proposed where the ratio of population density to a carrying capacity tends to be equal in the domain. All models in the present paper were considered for this type of diffusion. It is to be noted that models with such type of diffusion are introduced for the first time, so all Theorems presented in the paper are new results.
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(2) In the case when a growth rate and a carrying capacity depend only on location but are time-independent, optimal harvesting results were obtained for reaction-diffusion models with the following growth laws: • logistic • Gilpin–Ayala • Gompertz. (3) We considered models with time-dependent growth rates and carrying capacities. Theorem 4 extends Theorem 3.1 in [19] to the new type of diffusion. (4) To the best of our knowledge, there are very few publications on optimal harvesting incorporating both diffusion and impulsive behavior, see, for example, [24]. The present paper is the first publication to consider optimal impulsive harvesting for reaction-diffusion equations. Finally, we will state some open problems. (1) Prove that solutions p(t , x) outlined in Theorems 1–4 are asymptotically stable. Prove that the appropriate optimal solution of (4.1) and (4.2) is asymptotically stable. According to the common definition [25], sustainable yield is attained by an asymptotically stable solution. In the present paper, we have omitted the discussion on stability of optimal solutions. It would also be interesting to study the stability with respect to perturbations of harvesting effort E. (2) Extend the results of Theorem 4 to the equations with time-dependent growth rates and carrying capacities for Gilpin–Ayala and Gompertz laws of growth. (3) Find an optimal harvesting strategy for the impulsive model (4.1) and (4.2) in the general case (both the growth rate and the carrying capacity are location-dependent). (4) Prove or disprove that maximum sustainable yield MSY (D) is a nondecreasing function of diffusion coefficient D for all models considered in the paper. So far, for impulsive models we have obtained MSY in the cases when diffusion effects are either missing or are very strong. Is it possible to claim that MSY in the former case does not exceed MSY in the latter case? (5) Extend the results of the previous two items to models with the Gilpin–Ayala and Gompertz growth laws. (6) Everywhere above we assumed that there is no by-catch mortality or any other negative effect caused by harvesting (noise, pollution, disturbance of mate and foraging habits). Generally, this is not true. Consider Eq. (4.1) with the impulsive conditions u(t + nT + , x) − u(t + nT , x) = −E (x)u(t + nT , x) − d,
(5.1)
where deduction d can be interpreted as by-catch mortality or other harm to the population caused by the harvesting event. It is not involved in harvest computation, the yield over time T
Z Ω
E (x)u(t + nT , x) dx =
Z
u(t + nT , x) − u(t + (n − 1)T + ) dx
Ω
should be maximal. For logistic and Gompertz models which are not spatially structured, the impulsive harvesting of type (5.1) with constant deduction was considered in [12,14]. (7) For equations with a usual diffusion, a ‘‘space impulsive harvesting’’ can be considered rather than deduction which occurs once in a while over the whole open domain Ω . The model where the culling was continuous in time but occurred only at particular discrete points in space was considered in [26] in the context of population eradication, not sustainable harvesting. If there is a steep decrease in the carrying capacity, the traps at the boundary of the fertile domain Ω1 will harvest the outflow from this domain. This will keep the population inside Ω1 at a desired level and avoid dispersal to non-fertile areas. The strategy can incorporate sending back some of trapped species. We introduce a closed simple curve ∂ Ω1 ⊂ Ω and a culling function B(t , x). Following [26], we define the impulsive condition
∂ u( t , x ) ∂n
− +
∂ u( t , x ) ∂n
= −B(t , x)u(t , x),
x ∈ ∂ Ω1 ,
(5.2)
−
where normal to ∂ Ω1 direction is considered. It is possible to introduce several culling curves (surfaces) ∂ Ωi . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edition, Wiley, New York, 1990. S.J. Gao, L.S. Chen, L.H. Sun, Optimal pulse fishing policy in stage-structured models with birth pulses, Chaos Solitons Fractals 25 (5) (2005) 1209–1219. D. Ludwig, Harvesting strategies for a randomly fluctuating population, J. Cons. Int. Explor. Mer. 39 (1980) 168–174. C. Xu, M.S. Boyce, D.J. Daley, Harvesting in seasonal environments, J. Math. Biol. 50 (2005) 663–682. X. Zhang, Z. Shuai, K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. RWA 4 (2003) 639–651. Y. Zhang, Z. Xiu, L. Chen, Optimal impulsive harvesting of a single species with Gompertz law of growth, J. Biol. Systems 14 (2006) 303–314. K.L. Cooke, M. Witten, One-dimensional linear and logistic harvesting models, Math. Modelling 7 (1986) 301–340. L. Dong, L. Chen, L. Sun, Optimal harvesting policies for periodic Gompertz systems, Nonlinear Anal. Real World Appl. 8 (2007) 572–578. M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci. 152 (1998) 165–177. J. Angelova, A. Dishliev, Optimization problems for one-impulsive models from population dynamics, Nonlinear Anal. 39 (2000) 483–497.
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[12] E. Braverman, R. Mamdani, On optimal impulsive sustainable harvesting, in: Advances in Dynamical Systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14(Suppl. S2) (2007) 112–116. [13] Y. Xiao, D. Cheng, H. Qin, Optimal impulsive control in periodic ecosystems, Systems Control Lett. 55 (2006) 556–565. [14] E. Braverman, R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol. 57 (3) (2008) 413–434. [15] N. Hritonenko, Y. Yatsenko, Optimization of harvesting age in an integral age-dependent model of population dynamics, Math. Biosci. 195 (2) (2005) 154–167. [16] N. Hritonenko, Y. Yatsenko, The structure of optimal time- and age-dependent harvesting in the Lotka–McKendrik population model, Math. Biosci. 208 (1) (2007) 48–62. [17] M. Kot, Elements of Mathematical Ecology, Cambridge Univ. Press, 2001. [18] L. Bai, K. Wang, Gilpin–Ayala model with spatial diffusion and its optimal harvesting policy, Appl. Math. Comput. 171 (2005) 531–546. [19] L. Bai, K. Wang, A diffusive single-species model with periodic coefficients and its optimal harvesting policy, Appl. Math. Comput. 187 (2007) 873–882. [20] M.E. Gilpin, F.J. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci. USA 70 (1973) 3590–3593. [21] B.S. Goh, T.T. Agnew, Stability in Gilpin and Ayala’s models of competition, J. Math. Biol. 4 (1977) 275–279. [22] D.M. Easton, Gompertz survival kinetics: Fall in number of alive or growth in the number of dead? Theoret. Popul. Ecol. 48 (1995) 1–6. [23] W.W. Fox, An exponential surplus yield model for optimizing in exploited fish populations, Trans. Amer. Fisheries Soc. 99 (1970) 80–88. [24] L. Dong, L. Chen, L. Sun, Optimal harvesting policy for inshore-offshore fishery model with impulsive diffusion, Acta Math. Sci. Ser. B Engl. Ed. 27 (2) (2007) 405–412. [25] Z. Artstein, Chattering limit for a model of harvesting in a rapidly changing environment, Appl. Math. Optim. 28 (1993) 133–147. [26] R.R.L. Simons, S.A. Gourley, Extinction criteria in stage-structured population models with impulsive culling, SIAM J. Appl. Math. 66 (6) (2006) 1853–1870.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Optimal harvesting of diffusive models in a nonhomogeneous environment Elena Braverman a,∗ , Leonid Braverman b,c a
Department of Mathematics & Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N1N4
b
Athabasca University, 1 University Drive, Athabasca, AB, Canada T9S3A3
c
St. Mary’s University College, 14500 Bannister Road SE, Calgary, AB, Canada T2X1Z4
article
info
Keywords: Reaction-diffusion equation Population dynamics Optimal harvesting Maximum sustainable yield Impulsive equations Logistic growth Gompertz growth Gilpin–Ayala model
abstract We study the optimal harvesting strategy for populations whose dynamics is described by reaction-diffusion equations. The production function can be of logistic, Gilpin–Ayala or Gompertz type. The diffusion structure is discussed; we suggest to consider ∆(u/K ), where K is the carrying capacity of the environment, rather than ∆u, and study optimal harvesting for models with this diffusion type. Maximum yield is investigated for both continuous and impulsive models. For continuous harvesting, the optimal policy is obtained; for the impulsive equation some limit cases are considered. The paper also outlines a variety of open problems. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Optimal resources management, including sustainable harvesting, is one of the most important problems of population ecology [1,2]. In the case when spatial distribution and interaction are neglected, optimal harvesting has recently become an area of active research, see, for example, [3–7]. This includes continuous [8–10] and impulsive harvesting strategies [6,7,11–13] ; some comparison of these two types of harvesting can be found in [5,14]. However, in many cases, populations are either age structured or spatially distributed. For example, the level of harvesting may depend on the age continuously, or be stage-structured (juvenile–adult population), see [15–17]. In fisheries, age structure of harvesting corresponds to different quotas for certain sizes of species. For spatially distributed species with diffusion, the optimal harvesting strategy was studied in [18,19]. Let u(t , x) be the population density at point x and time t. As a spatially distributed analogue of one-species onedimensional population models, the following reaction-diffusion equation was considered [19]
∂u = D∆u + r (t , x)f (u, K (t , x)) − E (t , x)u, ∂t
(1.1)
where E (t , x) > 0 is the combined harvesting effort and efficiency which will be in future referred to as the harvesting effort, E (t , x)u describes the level of harvesting, r (t , x) > 0 is the intrinsic growth rate, K (t , x) > 0 and the carrying capacity of the environment at point x and time t, D > 0 is the diffusion coefficient which describes the dispersal speed for a studied population. Function f (u, K ) is positive for 0 < u < K and is negative for u > K .
∗
Corresponding author. E-mail address: [email protected] (E. Braverman).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.04.025
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There are different possible growth functions f (u, K ). The simplest approach assumes that per capita growth rate is a linear decreasing function of population r (1 − u/K ), which leads to the logistic type of growth f (u, K ) = ru(1 − u/K ) which was first introduced by Verhulst in 1845 and later studied by Pearl and Reed in 1920 (see [17] for historical reference). This type of growth is widely applied in fisheries [1,2], for example, model (1.1) with logistic-type growth and without diffusion du dt
u = ru 1 − − Eu
(1.2)
K
is known as Schaefer equation [1,2]. However the growth of certain fish populations deviated from the logistic law, which motivated the consideration of more general growth functions
f1 (u, K ) = ru 1 −
u θ K
,
f2 (u, K ) = ru 1 −
u δ K
,
(1.3)
where θ > 0 describes the intensity of intraspecific interference, δ allows the accommodation of a non-symmetric (with respect to K /2) growth curve which occurs in fisheries. It is to be noted that f1 (which is called Gilpin–Ayala growth law) and f2 in (1.3) become logistic growth functions for θ = 1 and δ = 1, respectively. These modifications allow more a realistic description of various fish species. The Gilpin–Ayala growth law was suggested in [20] for Lotka–Volterra systems and was later referred to as a Gilpin–Ayala model and studied in [21]. However, both logistic and Gilpin–Ayala laws describe per capita growth rate of similar types: it is positive and finite at u = 0 and is decreasing for u > 0 as a polynomial function of u/K . The Gompertz growth function f (u, K ) = ru ln(u/K ) was designed to describe populations with unlimited per capita growth rate for low population levels. The rate of its decay is faster than for any logistic-type model. Originally, the Gompertz equation was formulated as a law of decreasing survivorship (in 1825 by Gompertz, see [17,22]). It appears in fish ecology in the Fox surplus yield model [23]. In addition to fisheries, the Gompertz equation was applied to model the growth of plants and tumors (see [17] for relevant references). In the case of the logistic reproduction function Eq. (1.1) becomes [19]
u ∂u = D∆u + r (t , x)u 1 − − E (t , x)u. ∂t K ( t , x)
(1.4)
Reaction-diffusion equations incorporate a certain growth law at each point with the spatial distribution of populations which can relocate (in (1.4), to less populated spots). Diffusion coefficient D > 0 can be high for some populations (for example, in fisheries) and lower for others (slugs, bacteria, plants). In future, we will also consider some other types of reproduction functions f : Gilpin–Ayala and Gompertz laws of growth. Optimal harvesting for a reaction-diffusion equation with the Gilpin–Ayala type of growth
" θ # ∂u u = D∆u + r (x)u 1 − − E (x)u ∂t K (x)
(1.5)
and time-independent parameters r, K and E, was studied in [18]. The purpose of the present paper is to study the following aspects of spatially distributed populations and optimal harvesting. (1) We are going to discuss the diffusion structure in (1.1). This equation involves a possibility of a massive migration from places with a high carrying capacity and still unused growth potential to locations with low resources where the migrating population is destined to die. This dispersal is just due to the fact that fertile places are more populated. In contrast to this ‘‘blind’’ diffusion scheme, here we suggest a ‘‘judicious’’ diffusion where the direction of the flow is defined by maximum per capita available resources. (2) For this new type of diffusion and time-independent carrying capacity, we describe optimal harvesting policies for the analogues of (1.4) and (1.5) with logistic and Gilpin–Ayala growth, respectively, and the reaction-diffusion equation which is subject to the Gompertz growth law. Unlike models with a usual diffusion, no additional conditions should be imposed on the carrying capacity and the growth rate to achieve this strategy. This is the first time when models with such type of diffusion are introduced, so all Theorems of the present paper are new results. (3) In the case when r (t , x) and K (t , x) are periodic in t and time-dependent, optimal harvesting policy is investigated. (4) We consider impulsive harvesting for models with diffusion and obtain some results on optimal harvesting. (5) Last but not the least, we state some open problems for both continuous and impulsive harvesting. In particular, for a ‘‘judicious’’ diffusion, can we claim that the level of optimal harvesting grows with the growth of the diffusion coefficient? For a usual diffusion and steeply changing K (x), can impulsive harvesting in space rather than in time (we set traps which serve as semi-conducting filters) be an efficient harvesting policy? These and many other open problems are presented in the last section of this paper.
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2. Optimal continuous harvesting for time-independent parameters Let us notice that if K (x) is variable, then models (1.4) and (1.5) may assume massive migration from resource rich fertile places with a high carrying capacity to scanty places. We argue that this scheme is not realistic for many populations and suggest that the direction of migration should correspond to the gradient of the ratio of the population density u(t , x) to the carrying capacity K (t , x) which means that emigration begins as far as per capita available resources are higher in the adjacent place than at a given point in space. This corresponds to the equation
∂u u = D∆ + r (t , x)f (u, K (t , x)) − E (t , x)u, x ∈ Ω . ∂t K (t , x)
(2.1)
Here Ω is a closed bounded domain in Rn (n = 1, 2 or 3). First, we consider Eq. (2.1) where the carrying capacity is time-independent (the environment enrichment distribution is the same at any time) and only space-dependent. Optimal harvesting strategy for time-dependent parameters will be considered in Section 3. 2.1. Logistic growth law Consider the logistic equation with time-independent carrying capacity and growth rate and with the diffusion defined in (2.1)
∂u u u = D∆ + r (x)u 1 − − E (x)u, x ∈ Ω , t ≥ 0. ∂t K (x) K (x)
(2.2)
Let us first formulate the following obvious statement. Lemma 1. The maximum of the function
f ( u, K ) = u 1 − for a fixed K equals u=
K 2
K 4
u
(2.3)
K
and is attained at
.
We proceed to the main result on the optimal continuous harvesting for (2.2). Since the term E (t , x)u describes the density of population harvested at point x and time t, then the optimal sustainable yield per unit time is
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
Z Ω
E (x)p(x) dx,
where p(x) is an optimal stationary solution, E (x) is the optimal harvesting effort. This is relevant in the case when both the optimal solution and optimal effort are time-independent. Generally (see Section 3), the maximal sustainable yield per unit time is T
Z 1
max
E (t ,x),p(t ,x)
Z dt
T
Ω
0
E (t , x)p(t , x) dx .
Theorem 1. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
1
Z
4
Ω
K (x)r (x) dx
(2.4)
is attained for the Eq. (2.2) have the form p(x) =
K (x) 2
,
E (x) =
r ( x) 2
.
(2.5)
Proof. Let us demonstrate that for a stationary (time-independent) solution the maximum sustainable yield is attained at every location and time stage. Then this is obviously an unbeatable strategy. K (x) u(t ,x) For u(t , x) ≡ p(x) = 2 the diffusion term vanishes since K (x) ≡ 21 is constant, the left hand side equals zero, since u(t , x) is time-independent. Furthermore, by Lemma 1 this population size provides the maximum of the production term in (2.2). Since
r (x)u 1 −
u K (x)
= E (x)u
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for any stationary (time-independent) solution then E (x)u also attains its maximum at any x and E (x) = r (x)/2 as in (2.5), which completes the proof. Remark 1. Formula (2.4) means that the maximal sustainable yield is proportional to the integrated product of the growth rate by the carrying capacity over the domain. It is attained when the optimal population level of K (x)/2 is maintained by persistent harvesting (otherwise, the population density will grow towards K (x), with the production level decreased), the optimal harvesting effort of r (x)/2 means that the population can be kept at the desired level. Remark 2. Comparing our results to [19] we observe that the maximum sustainable yield in Theorem 1 is the same as in the case of Eq. (1.4) with the usual diffusion ∆u. However there are no requirements on either r or K , while in [19] the condition r (x) 2
+D
∆K (x) ≥ 0, K (x)
x ∈ Ω,
is imposed. If the growth rate is not enough to compensate for the population outflow from fertile regions, the optimal harvesting strategy described in [19] is impossible. Remark 3. Let us mention that, when formulating the problem, we have not imposed any initial or boundary conditions. For the Neumann boundary conditions we will have for the optimal solution
∂(u/K (x)) = 0, ∂n
u ∈ ∂Ω.
(2.6)
If
∂ K =0 ∂ n ∂ Ω
(2.7)
(this assumption was also imposed in Example 1 of [19], p. 879) then the usual zero Neumann boundary conditions
∂u = 0, ∂n
u ∈ ∂Ω
(2.8)
apply to the solution. This means that there is no population inflow or outflow in the domain Ω . Conditions (2.6) and (2.7) are satisfied in the case when the carrying capacity vanishes at the boundary (for example, there is a natural bound to the population growth and dispersal outside the domain) together with its first derivative in the direction normal to the boundary ∂ Ω . Remark 4. Unlike other papers (see, for example, [18]), where an optimal harvesting strategy for a diffusive model was considered under the assumption that r , K and E are at least Hölder continuous functions, Theorem 1 does not involve any assumptions of this kind. We can choose the optimal strategy for any K (x), which may have a discontinuous patchy structure; then the optimal solution will not be continuous in x anymore; discontinuity of r in space will also cause jumps in harvesting effort E. We can consider discontinuous in x solutions, once the ratio u/K is smooth enough in x and t. 2.2. Gilpin–Ayala and Gompertz growth laws In this section we consider the Gilpin–Ayala reproduction function
" θ # ∂u u u = D∆ + r (x)u 1 − − E (x)u, ∂t K (x) K ( x)
x ∈ Ω , t ≥ 0,
(2.9)
and the Gompertz growth rate
∂u u K (x) = D∆ + r (x)u ln − E (x)u, ∂t K (x) u
x ∈ Ω , t ≥ 0.
(2.10)
The following two results are obvious. Lemma 2. The maximum of the function
f (u, K ) = u 1 − for a fixed K equals u=
u θ K
θK (1+θ)(1+θ)/θ
K
(1 + θ )1/θ
.
and is attained at
(2.11)
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Lemma 3. The maximum of the function f (u, K ) = u ln K e
for a fixed K equals u=
K e
K
(2.12)
u and is attained at
.
Lemmas 2 and 3 can be applied to describe the best possible strategy when the maximum production is attained at each time and location, similar to the proof of Theorem 1. They yield the following results on the optimal harvesting policy in the cases of the Gilpin–Ayala and the Gompertz growth laws. Theorem 2. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
θ (1 + θ )(1+θ)/θ
Z Ω
r (x)K (x) dx
(2.13)
is attained for the Eq. (2.9) have the form p(x) =
K (x) , (1 + θ )1/θ
r (x)θ
E (x) =
1+θ
.
(2.14)
Remark 5. By formula (2.13), the maximal sustainable yield is proportional to the integrated product of the growth rate by the carrying capacity over the domain. It is attained when the optimal population level of K (x)/(1 + θ )1/θ is attained at all points of the domain by persistent harvesting (otherwise, the population density will grow towards K (x), with the production level decreased), the optimal harvesting effort of r (x)θ /(1 +θ ) allows to keep the population at the desired level. Theorem 3. The solution p(x) = u(t , x) and the harvesting effort E (x) such that the optimal sustainable yield per unit time
Z max
E (x),p(x)
Ω
E (x)p(x) dx =
1
Z
e
Ω
K (x)r (x) dx
(2.15)
is attained for the Eq. (2.10) are as follows p(x) =
K (x) e
,
E (x) = r (x).
(2.16)
Remark 6. Formula (2.15) outlines that the maximal sustainable yield which is attained when the optimal population level of K (x)/e is maintained at all points of the domain by persistent harvesting. Without constant population deduction, the population density will grow towards K (x), with the production level decreased. The optimal harvesting effort of r (x) allows to keep the population at the desired level. 3. Optimal harvesting: time-dependent periodic parameters Now consider the problem of the continuous optimal harvesting when the parameters of the equation are not only spacedependent but also time-dependent:
∂u u u = D∆ + r (t , x)u 1 − − E (t , x)u, ∂t K (t , x) K (t , x)
x ∈ Ω , t ≥ 0.
(3.1)
Theorem 4. Let r (t , x) and K (t , x) be periodic functions in t of the same period T for each x ∈ Ω . If for any t , x, 0 ≤ t ≤ T , x ∈ Ω we have r (t , x) ≥
∂ K ( t , x) , K (t , x) ∂t 2
(3.2)
then the solution p(t , x) = u(t , x) and the harvesting effort E (t , x) such that the maximum yield per unit time T
Z max
E (t ,x),p(t ,x)
1
T
Z dt Ω
0
E (t , x)p(t , x) dx
=
1 4T
T
Z
Z dt
0
Ω
K (t , x)r (t , x) dx
(3.3)
is attained for the Eq. (3.1) have the form p(t , x) =
K ( t , x) 2
,
E (t , x) =
r ( t , x) 2
−
∂ K ( t , x) . K (t , s) ∂t 1
(3.4)
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Proof. The optimal production rate is attained at each point and each time stage if the population level is p(t , x) =
K (t , x) 2
.
Then
∆
p(t , x)
K ( t , x)
= 0 and E (t , x)
K (t , x) 2
=
r (t , x)K (t , x) 4
−
1 ∂ K (t , x) 2
∂t
,
which is nonnegative for any t , x if (3.2) holds. This immediately implies the harvesting effort E (t , x) in (3.4) and formula (3.3) for the maximum yield. Corollary 1. If the carrying capacity in (3.1) is time-independent
u u ∂u = D∆ + r (t , x)u 1 − − E (t , x)u, ∂t K (x) K (x)
x ∈ Ω , t ≥ 0,
(3.5)
then the optimal solution is also stationary (time-independent). It exists for any r (t , x); the maximum yield defined in (3.3) is attained for p(t , x) =
K (x) 2
,
E (t , x) =
r (t , x) 2
.
(3.6)
4. Impulsive harvesting Next, let us consider the model when a persistent harvesting effort cannot be applied but harvesting occurs once in a while during a relatively short period of time; here we assume that we deduct a part of the population every T time units (T > 0 is a constant). For time-independent parameters and the logistic growth, this corresponds to the impulsive partial differential equation
∂u u u = D∆ + r (x)u 1 − , ∂t K (x) K (x)
x ∈ Ω , t ≥ 0, t 6= nT ,
u(t + nT + , x) − u(t + nT , x) = −E (x)u(t + nT , x),
n = 1, 2, . . . .
(4.1) (4.2)
Here 0 < E < 1, f (t ) is the right side limit of the function f (t ) = lims→t ,s>t f (s), the solution u(t , x) is piecewise continuous in t. Let us refer to a known result in the theory of impulsive ordinary differential equations (see, for example, [6,13]). +
+
Lemma 4 ([13]). For the logistic equation
y0 (t ) = ry 1 −
y
(4.3)
K
with the impulsive conditions y(t + nT + ) − y(t + nT ) = Ey(t + nT ),
n = 1, 2, . . .
(4.4)
optimal sustainable harvesting is attained for x(0) = x(nT + ) = K e−rT /2 + 1
(4.5)
and the maximum sustainable yield (MSY) per unit time is MSY =
K (erT /2 − 1) T (erT /2 + 1)
.
(4.6)
We will consider only some aspects of optimal impulsive harvesting (4.2) for the Eq. (4.1). In particular, the limit cases when either D = 0 or D → ∞ will be investigated. Limit case D = 0. Without diffusion, we set optimal harvesting for each point x of domain Ω individually. Thus by Lemma 4 the total harvest over Ω and time period T between impulses will be
Z MSYT =
Ω
K (x)(eTr (x)/2 − 1) eTr (x)/2 + 1
dx
(4.7)
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or maximum sustainable yield per unit time is K (x)(eTr (x)/2 − 1) dx
Z MSY =
T (eTr (x)/2 + 1)
Ω
.
Limit case D → ∞. In this case we will have a homogeneous distribution of u(t , x)/K (x) in x over domain Ω at each time step. If we make a substitution u(t , x)
z (t , x) =
K (x)
= z (t )
(4.8)
in (4.1) then taking into account ∆z = 0 for any t , x, we obtain z 0 (t ) = rz (1 − z ),
(4.9)
where
R
ΩR
r =
r (x) dx
Ω
(4.10)
dx
is the average growth rate over domain Ω . Thus, Lemma 4 can be applied to the ordinary differential equation (4.9) which leads to MSYT =
erT /2 − 1
Z
erT /2 + 1
Ω
K (x) dx.
(4.11)
Let us compare yields (4.7) and (4.11). The following example demonstrates that, generally, they are not equal. This demonstrates the importance of diffusion effects in harvested populations. Below we choose K = T = 1 which can be achieved by scaling for any model with constant carrying capacity and time between impulses, and slowly changing in space time-independent intrinsic growth rate r (x). Example 1. Let r (x) = 2 ln(10x), x ∈ [1, 2], K ≡ 1, T = 1. Then in (4.7) 2
Z
K ( x) 1
eTr (x)/2 − 1
2
Z =
eTr (x)/2 + 1
1−
1
0.2 x + 0.1
dx = 1 − 0.2 ln
2.1
1.1
≈ 0.8707.
The average growth rate is
Z
2
ln(10x) = 2 ln 10 + 2(−x + x ln x)|21 ≈ 5.37776.
r =2 1
Then MSY in (4.11) exceeds (4.7) erT /2 − 1 erT /2 + 1
Z Ω
K (x) dx ≈ 0.8727 > 0.8707 ≈
2
Z
K (x) 1
eTr (x)/2 − 1 eTr (x)/2 + 1
.
Still, it remains an open problem as to whether higher levels of diffusion improve the maximum sustainable yield, see Section 5. In addition to the above limit cases, we will also consider (4.1) with a constant growth rate r (x) ≡ r. u(x,t ) Case r (x) ≡ r. For constant r, for each point x, the same ratio K (x) of the initial population level to the carrying capacity will correspond to the optimal impulsive harvesting policy. Thus, for the optimal solution at each point x we have ∆(u/K ) = 0. This harvesting strategy is optimal for the whole system. Hence the maximum sustainable yield over time T is MSYT =
erT /2 − 1 erT /2 + 1
Z Ω
K (x) dx,
(4.12)
which coincides with (4.7) and (4.11). 5. Discussion and open problems In the present paper, we considered the problem of optimal harvesting for various models described by reaction-diffusion equations. (1) We have argued that the regular diffusion with a tendency to a uniform density is not reasonable for models of population dynamics with a non-uniformly distributed carrying capacity. A new type of diffusion was proposed where the ratio of population density to a carrying capacity tends to be equal in the domain. All models in the present paper were considered for this type of diffusion. It is to be noted that models with such type of diffusion are introduced for the first time, so all Theorems presented in the paper are new results.
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(2) In the case when a growth rate and a carrying capacity depend only on location but are time-independent, optimal harvesting results were obtained for reaction-diffusion models with the following growth laws: • logistic • Gilpin–Ayala • Gompertz. (3) We considered models with time-dependent growth rates and carrying capacities. Theorem 4 extends Theorem 3.1 in [19] to the new type of diffusion. (4) To the best of our knowledge, there are very few publications on optimal harvesting incorporating both diffusion and impulsive behavior, see, for example, [24]. The present paper is the first publication to consider optimal impulsive harvesting for reaction-diffusion equations. Finally, we will state some open problems. (1) Prove that solutions p(t , x) outlined in Theorems 1–4 are asymptotically stable. Prove that the appropriate optimal solution of (4.1) and (4.2) is asymptotically stable. According to the common definition [25], sustainable yield is attained by an asymptotically stable solution. In the present paper, we have omitted the discussion on stability of optimal solutions. It would also be interesting to study the stability with respect to perturbations of harvesting effort E. (2) Extend the results of Theorem 4 to the equations with time-dependent growth rates and carrying capacities for Gilpin–Ayala and Gompertz laws of growth. (3) Find an optimal harvesting strategy for the impulsive model (4.1) and (4.2) in the general case (both the growth rate and the carrying capacity are location-dependent). (4) Prove or disprove that maximum sustainable yield MSY (D) is a nondecreasing function of diffusion coefficient D for all models considered in the paper. So far, for impulsive models we have obtained MSY in the cases when diffusion effects are either missing or are very strong. Is it possible to claim that MSY in the former case does not exceed MSY in the latter case? (5) Extend the results of the previous two items to models with the Gilpin–Ayala and Gompertz growth laws. (6) Everywhere above we assumed that there is no by-catch mortality or any other negative effect caused by harvesting (noise, pollution, disturbance of mate and foraging habits). Generally, this is not true. Consider Eq. (4.1) with the impulsive conditions u(t + nT + , x) − u(t + nT , x) = −E (x)u(t + nT , x) − d,
(5.1)
where deduction d can be interpreted as by-catch mortality or other harm to the population caused by the harvesting event. It is not involved in harvest computation, the yield over time T
Z Ω
E (x)u(t + nT , x) dx =
Z
u(t + nT , x) − u(t + (n − 1)T + ) dx
Ω
should be maximal. For logistic and Gompertz models which are not spatially structured, the impulsive harvesting of type (5.1) with constant deduction was considered in [12,14]. (7) For equations with a usual diffusion, a ‘‘space impulsive harvesting’’ can be considered rather than deduction which occurs once in a while over the whole open domain Ω . The model where the culling was continuous in time but occurred only at particular discrete points in space was considered in [26] in the context of population eradication, not sustainable harvesting. If there is a steep decrease in the carrying capacity, the traps at the boundary of the fertile domain Ω1 will harvest the outflow from this domain. This will keep the population inside Ω1 at a desired level and avoid dispersal to non-fertile areas. The strategy can incorporate sending back some of trapped species. We introduce a closed simple curve ∂ Ω1 ⊂ Ω and a culling function B(t , x). Following [26], we define the impulsive condition
∂ u( t , x ) ∂n
− +
∂ u( t , x ) ∂n
= −B(t , x)u(t , x),
x ∈ ∂ Ω1 ,
(5.2)
−
where normal to ∂ Ω1 direction is considered. It is possible to introduce several culling curves (surfaces) ∂ Ωi . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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