- Email: [email protected]
Optimal harvesting policy of a stochastic food chain population model
Applied Mathematics and Computation 245 (2014) 265–270
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Optimal harvesting policy of a stochastic food chain population model Meng Liu ⇑, Chuanzhi Bai School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China
a r t i c l e
i n f o
Keywords: Food chain model Stochastic perturbations Optimal harvesting policy Fokker–Planck equation
a b s t r a c t This note studies the optimal harvesting policy of a stochastic three-species food chain model. The almost sufficient and necessary conditions for the existence of optimal harvesting policy are established. Meanwhile, the optimal harvesting effort and the maximum of expectation of sustainable yield are obtained. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Optimal policy in managing natural resources is one of the most important topics in mathematical biology. Clark [1], Fan and Wang [2], Dong et al. [3] and Shuai et al. [4] have considered the optimal harvesting policy of some deterministic singlespecies models. Goh et al. [5] and Kar [6] have investigated the optimal harvesting policy of species under a deterministic predator–prey framework. Braverman and Mamdani [7] have studied the optimal harvesting policy of both autonomous and nonautonomous population models with either impulsive or continuous harvesting. Braverman and Braverman [8] have established the optimal harvesting strategy of logistic, Gilpin-Ayala and Gompertz models with diffusion and harvesting. Korobenko et al. [9] have investigated the existence, positivity, persistence, extinction and stability of solutions of reaction–diffusion logistic equations with harvesting. On the other hand, in the natural world, the growth of population is inevitably subject to environmental perturbations. Thus many scholars have studied the optimal harvesting policy of stochastic population models. Beddington and May [10] have obtained the optimal harvesting policy of a classical stochastic logistic equation. The optimal harvesting of some single-species population models in stochastic environments were also examined in [11–19]. As species do not exist alone in nature, and food-chain interaction is a frequently observed phenomena among wildlife species, it is of great biological significance to consider the optimal harvesting policy of stochastic food-chain population model. However, to the best of our knowledge, there are few investigations of this kind. One reason is that it is quite difficult to solve the corresponding Fokker–Planck equation. In this note, using the results in Liu [20], we solve the corresponding Fokker–Planck equation of our food-chain model. Then we establish the almost sufficient and necessary conditions for the existence of optimal harvesting policy. Meanwhile, the optimal harvest effort and the maximum of expectation of sustainable yield (ESY) are given.
⇑ Corresponding author. E-mail address: [email protected] (M. Liu). http://dx.doi.org/10.1016/j.amc.2014.07.103 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
266
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
2. Model formulation and preparation works Our system is based on the following three-species food chain system (see e.g. [1]):
8 dx ðtÞ 1 > ¼ x1 ðtÞða1 b1 x1 ðtÞ c1 x2 ðtÞÞ; > < dt dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ; dt
ð1Þ
where xi is the size of the ith population, ai ; bi and c1 ; c2 are positive constants, i ¼ 1; 2; 3. Suppose that the population x1 is subject to exploitation with hx1 , where h is the harvesting effort, then system (1) becomes
8 dx ðtÞ 1 > ¼ x1 ðtÞða1 h b1 x1 ðtÞ c1 x2 ðtÞÞ; > < dt dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ: dt
ð2Þ
On the other hand, the harvesting effort h is inevitably affected by environmental noises, for example, the increasing or the decreasing of purchasing power. In this paper we assume that the noise in both nature and human society is multiplicative and generated by white noise, that is
_ h ! h þ rBðtÞ; where BðtÞ is a standard Brownian motion defined on a complete probability space ðX; fF t gt2Rþ ; PÞ with a filtration fF t gt2Rþ and r2 denotes the intensity of the noise, where Rþ ¼ ½0; þ1Þ. Then we obtain the following stochastic system
8 dx ðtÞ ¼ x1 ðtÞða1 h b1 x1 ðtÞ c1 x2 ðtÞÞdt þ rx1 ðtÞdBðtÞ; > < 1 dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ: dt
ð3Þ
Our aim is to find out the optimal harvesting effort h such that (i) ESY YðhÞ ¼ limt!þ1 Eðhx1 ðtÞÞ is maximum; (ii) x1 ðtÞ; x2 ðtÞ and x3 ðtÞ will not become extinct. Now let us do some preparation. Lemma 1. For any initial data ðx1 ð0Þ; x2 ð0Þ; x2 ð0ÞÞ 2 R3þ ¼: fa ¼ ða1 ; a2 ;3 Þjai > 0; i ¼ 1; 2; 3g, there is a unique global positive solution to model (3) almost surely (a.s.). Proof. The proof is standard and hence is omitted (see e.g. [21,22]).
h
If x1 ðtÞ; x2 ðtÞ; x3 ðtÞ is the solution of system (3), then there is a transition probability density of this solution. Let
U½x1 ðt þ DtÞ; x2 ðt þ DtÞ; x3 ðt þ DtÞjx1 ðtÞ ¼ x1 ; x1 ðtÞ ¼ x2 ; x3 ðtÞ ¼ x3 be the transition probability density function of x1 ; x2 and x3 at t þ Dt given that x1 ðtÞ ¼ x1 ; x2 ðtÞ ¼ x2 and x3 ðtÞ ¼ x3 . Then U must obey the following Fokker–Planck equation
@U @ @ @ @2 ¼ ½x1 ða1 h b1 x1 c1 x2 ÞU ½x2 ða2 þ b2 x1 c2 x3 ÞU ½x3 ða3 þ b3 x2 ÞU þ 0:5 2 r2 x21 U : @x1 @x2 @x3 @t @x1
Lemma 2. For system (3), (i) If
a1 0:5r2
c1 a3 b1 a2 < h; b3 b2
ð4Þ
then x3 ðtÞ goes to extinction a.s., i.e., limt!þ1 x3 ðtÞ ¼ 0; a:s. (ii) If
a1 0:5r2
c1 a3 b1 a2 > h; b3 b2
ð5Þ
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
267
then the density of stationary distribution of solution of system (3) is
Uðx1 ; x2 ; x3 Þ ¼ m exp
2b1
r2
2z1 2b1 z2 1 2b1 c1 z3 1 1 2b1 c1 2b1 c1 c2 r2 b r2 b b ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 exp 2 ðx3 Þ 1 x3 2 3 ; r b2 r b2 b3
ð6Þ
where
z1 ¼ a1 0:5r2 h ( m¼
exp Z
2b1
r2
þ
2b1 c1 2b1 c1 c2 þ r2 b2 r2 b2 b3
þ1
exp
0
c1 a3 c1 a3 b2 b2 c1 a3 a2 ; ; z2 ¼ ; z3 ¼ z1 a 2 ¼ a1 0:5r2 h b3 b3 b1 b1 b3
Z
þ1
2b1 c1 c2 x x r2 b 2 b 3 3 3
r
0
2b1 c1 z3 1 r2 b2 b3
2z1 Z 1 2b1 exp 2 x1 x1r2 dx1
þ1 0
2b1 z2 1 2b1 c1 r2 b exp 2 x2 x2 2 dx2 r b2
)1 dx3
ð7Þ
Proof. Denote y1 ¼ ln x1 ; y2 ¼ ln x2 ; y3 ¼ ln x3 , then by Itô’s formula,
8 2 y1 y2 > < dy1 ¼ ða1 0:5r h b1 e c1 e Þdt þ rdBðtÞ; y1 y3 dy2 ¼ ða2 þ b2 e c2 e Þdt; > : dy3 ¼ ða3 þ b3 ey2 Þdt;
ð8Þ
Let W½y1 ðt þ DtÞ; y2 ðt þ DtÞ; y3 ðt þ DtÞjy1 ðtÞ ¼ y1 ; y1 ðtÞ ¼ y2 ; y3 ðtÞ ¼ y3 stand for the transition probability density function of the solution process of (8). Then W must obey the following Fokker–Planck equation in steady state:
@ @ @ @2 ða1 0:5r2 h b1 ey1 c1 ey2 ÞW þ ½ða2 þ b2 ey1 c2 ey3 ÞW þ ½ða3 þ b3 ey2 ÞW 0:5 2 r2 W @y1 @y2 @y3 @y1 @ b2 @ r2 @ W @ b3 @ r2 b2 @ W ðz1 b1 ey1 ÞW þ ðz2 c1 ey2 ÞW ¼ @y1 b1 @y2 @y1 c1 @y3 2 @y1 2 b1 @y2 2 @ r b b @ W 2 3 : ðz3 c2 ey3 ÞW þ @y2 2 b1 c1 @y3
0¼
ð9Þ
According to Liu’s analysis [20], if an equation can be written in the form
@W @W þ L2 ðy1 ; y2 ; y3 Þ H2 ðy2 ÞW þ K 2 ðy2 Þ 0 ¼ L1 ðy1 ; y2 ; y3 Þ H1 ðy1 ÞW þ K 1 ðy1 Þ @y1 @y2 @W þ L3 ðy1 ; y2 ; y3 Þ H3 ðy3 ÞW þ K 3 ðy3 Þ ; @y3
ð10Þ
where L1 ðy1 ; y2 ; y3 Þ; L2 ðy1 ; y2 ; y3 Þ and L3 ðy1 ; y2 ; y3 Þ are partial differential operators, then a unique solution of W can be gotten by solving
H1 ðy1 ÞW þ K 1 ðy1 Þ
@W @W @W ¼ 0; H2 ðy2 ÞW þ K 2 ðy2 Þ ¼ 0; H3 ðy3 ÞW þ K 3 ðy3 Þ ¼ 0: @y1 @y2 @y3
That is to say,
Z
Wðy1 ; y2 ; y3 Þ ¼ m exp
y1 0
Z y2 Z y3 H1 ðsÞ H2 ðsÞ H3 ðsÞ ds exp ds exp ds ; K 1 ðsÞ K 2 ðsÞ K 3 ðsÞ 0 0
where m is the normalization factor satisfying
Z
þ1
1
Z
þ1
1
Z
þ1
Wðy1 ; y2 ; y3 Þdy1 dy2 dy3 ¼ 1:
1
Clearly, Eq. (9) has the form of (10) with
L1 ðy1 ; y2 ; y3 Þ ¼
@ b2 @ ; @y1 b1 @y2
L2 ðy1 ; y2 ; y3 Þ ¼
@ b3 @ ; @y1 c1 @y3
H1 ðy1 Þ ¼ z1 b1 ey1 ; H2 ðy2 Þ ¼ z2 c1 ey2 ; H3 ðy3 Þ ¼ z3 c2 ey3 ; K 1 ðy1 Þ ¼
r2 2
; K 2 ðy2 Þ ¼
r2 b 2 2 b1
; K 3 ðy3 Þ ¼
r2 b 2 b 3 2 b1 c 1
:
L3 ðy1 ; y2 ; y3 Þ ¼
@ ; @y2
268
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Therefore the solution of Eq. (9) is
"Z # "Z # y2 y3 z1 b 1 e s z2 c1 es z3 c 2 e s exp exp ds ds ds r2 b2 r 2 b2 b3 r2 =2 0 0 0 2 b1 2 b1 c 1 2 2b1 2b1 c1 y1 y2 ðz3 y3 c2 ey3 þ c2 Þ ¼ m exp 2 ðz1 y1 b1 e þ b1 Þ exp 2 ðz2 y2 c1 e þ c1 Þ exp 2 r r b2 r b2 b3 2b1 z2 2b1 c1 z3 2z1 2b1 y1 2b c 2b1 c1 c2 y3 1 1 ðe 1Þ ðey3 Þ r2 b2 b3 : ¼ m exp 2 ðe 1Þ ðey1 Þ r2 exp 2 ðey2 1Þ ðey2 Þ r2 b2 exp 2 r r b2 r b2 b3
Wðy1 ; y2 ; y3 Þ ¼ m exp
Z
y1
Note that x1 ¼ ey1 ; x2 ¼ ey2 ; x3 ¼ ey3 , hence the joint steady state probability function of ðx1 ; x2 ; x3 Þ is
Uðx1 ; x2 ; x3 Þ ¼ m exp
2b1 z2 2b1 c1 z3 2z1 2b1 c1 2b1 c1 c2 y3 ðey1 1Þ ðey1 Þ r2 exp 2 ðey2 1Þ ðey2 Þ r2 b2 exp 2 ðe 1Þ ðey3 Þ r2 b2 b3 r b2 r b2 b3
2b1
r2
1 x1 x2 x3 2z1 2b1 z2 1 2b1 c1 z3 1 1 2b1 2b1 c1 2b1 c1 c2 r2 b r2 b b ¼ m exp 2 ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 exp 2 ðx3 1Þ x3 2 3 ; r r b2 r b2 b3
R þ1 R þ1 R þ1 where m is determined by the following integral 0 Uðx1 ; x2 ; x3 Þdx1 dx2 dx3 ¼ 1. By the theory of improper integral, 0 0 R þ1 R þ1 R þ1 if (4) holds, that is z3 < 0, then the density of x3 will tend to 0. Else if (5) holds, then 0 Wðx1 ; x2 ; x3 Þdx1 dx2 dx3 is 0 0 convergent and (Z )1 2z1 2b1 z2 1 2b1 c1 z3 1 Z þ1 Z þ1 þ1 1 2b1 2b1 c1 2b1 c1 c2 r2 b r2 b b m¼ exp 2 ðx1 1Þ x1r2 dx1 exp 2 ðx2 1Þ x2 2 dx2 exp 2 ðx3 1Þ x3 2 3 dx3 r r b2 r b2 b3 0 0 0 ( Z þ1 2z1 2b1 z2 1 Z þ1 1 2b1 2b1 c1 2b1 c1 c2 2b1 2b1 c1 r2 b ¼ exp þ þ exp 2 x1 x1r2 dx1 exp 2 x2 x2 2 dx2 r 2 r 2 b2 r 2 b2 b3 0 r r b2 0 )1 2b1 c1 z3 1 Z þ1 2b1 c1 c2 r2 b b exp 2 x3 x3 2 3 dx3 : r b2 b3 0 This completes the proof.
h
3. Optimal harvesting policy Now we are in the position to give our main result. Theorem 1. For model (3),
(I) If a1 0:5r2 c1ba3 3 bb1 a2 2 < h, then limt!þ1 x3 ðtÞ ¼ 0; (II) Suppose that a1 0:5r (a) If 2
c1 a3 b3
b1 a2 b2
a:s.
> h.
c1 a3 b1 a2 0:5 a1 0:5r2 < 0; b3 b2 then the optimal harvesting policy does not exist. (b) If
c1 a3 b1 a2 0:5 a1 0:5r2 > 0; b3 b2
then the optimal harvesting effort h is
c1 a3 : h ¼ 0:5 a1 0:5r2 b3 Meanwhile, the maximum of ESY is 2
Y ¼
ða1 0:5r2 c1 a3 =b3 Þ : 4b1
ð11Þ
269
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Proof. By Lemma 2, we need only to prove (II). The expectation of sustainable yield is
YðhÞ ¼ lim Eðhx1 ðtÞÞ ¼ t!þ1
Z
þ1
hx1 U1 ðx1 Þdx1 ;
0
where
U1 ðx1 Þ ¼
Z
þ1
Z
0
þ1
Uðx1 ; x2 ; x3 Þdx2 dx3 :
0
It follows from (6) and (7) that
2b1 z2 1 2b1 c1 z3 1 Z þ1 2b1 c1 2b1 c1 c2 r2 b2 r2 b b x x3 2 3 dx3 exp ðx 1Þ dx exp ðx 1Þ 2 2 3 2 r2 r2 b2 r2 b 2 b 3 0 0 ( Z þ1 2z1 2b1 z2 1 Z þ1 1 2b1 2b1 c1 2b1 c1 c2 2b1 2b1 c1 r2 b ¼ exp þ 2 þ 2 exp 2 x1 x1r2 dx1 exp 2 x2 x2 2 dx2 2 r r b2 r b2 b3 0 r r b2 0 )1 2b1 c1 z3 1 2z1 2b1 z2 1 Z þ1 Z þ1 1 2b1 c1 c2 2b1 2b1 c1 r2 b b r2 b exp 2 x3 x3 2 3 dx3 exp 2 ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 dx2 r b2 b3 r r b2 0 0 h i 2z1 1 2b1 2b1 c1 z3 1 Z þ1 exp x x1r2 1 2 2b1 c1 c2 r r2 b b exp 2 ðx3 1Þ x3 2 3 dx3 ¼ : h i 2z1 1 R þ1 r b2 b3 0 2b1 r2 exp x dx x 1 1 1 0 r2
U1 ðx1 Þ ¼ mexp
2b1
2z1 Z 1 ðx1 1Þ x1r2
þ1
Consequently
h i 2z1 2z1 R þ1 r2 1 2z exp 2b hr2 0 ev v r2 dv hr2 Cð r21 þ 1Þ hr2 2z1 hz1 r2 x1 x1 dx1 ¼ ¼ ¼ ¼ 2z1 h i 2z1 1 R 2b1 þ1 v r2 1 2b1 Cð2z21 Þ 2b1 r2 b1 r2 1 r e v dv exp 2b dx1 0 r2 x1 x1
R þ1 YðhÞ ¼ h
0
R þ1 0
¼
hða1 0:5r2 h c1 a3 =b3 Þ : b1
where CðÞ is the Gamma function. Note that YðhÞ has a unique extreme value at h ¼ 0:5ða1 0:5r2 c1 a3 =b3 Þ. Thus if h obeys
z3 jh¼h ¼ b2 ða1 0:5r2 h c1 a3 =b3 Þ=b1 a2 > 0;
i.e. 0:5b2 ða1 0:5r2 c1 a3 =b3 Þ=b1 a2 > 0, then the optimal harvesting effort is h and the maximum of expectation of sustainable yield is
Y ¼
2
h ða1 0:5r2 h c1 a3 =b3 Þ ða1 0:5r2 c1 a3 =b3 Þ ¼ : b1 4b1
If 0:5b2 ða1 0:5r2 c1 a3 =b3 Þ=b1 a2 < 0, i.e. z3 jh¼h < 0, then the optimal harvesting effort h will make x3 extinct and thus the optimal harvesting policy does not exist. h Remark 1. Consider the following stochastic logistic population model:
dxðtÞ ¼ xðtÞða h bxðtÞÞdt þ rxdBðtÞ:
By Theorem 1, the optimal harvesting policy exists if and only if h < a 0:5r2 , the harvest effort is h ¼ 0:5ða 0:5r2 Þ and 2 Þ2
r the maximum of ESY is Y ¼ ða0:5 . These results coincide with the classical results obtained by Beddington and May [10] 4b in Science.
Remark 2. In this note, we suppose that only the top prey is subjected to exploitation. It is interesting to investigate the optimal harvesting of the following model:
8 > < dx1 ðtÞ ¼ x1 ðtÞða1 h1 b1 x1 ðtÞ c1 x2 ðtÞÞdt þ r1 x1 dBðtÞ; dx2 ðtÞ ¼ x2 ðtÞða2 h2 þ b2 x1 ðtÞ c2 x3 ðtÞÞdt þ r2 x2 dBðtÞ; > : dx3 ðtÞ ¼ x3 ðtÞða3 h3 þ b3 x2 ðtÞÞdt þ r3 x3 dBðtÞ;
ð12Þ
where hi is the harvesting effort of xi ; i ¼ 1; 2; 3. In fact, we have also attempted to investigate this question. Unfortunately, we cannot obtain the explicit solution of the Fokker–Planck equation of (12) now. It is also interesting to investigate the optimal harvesting of other stochastic models.
270
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Acknowledgments The authors thank the editor and reviewers for their valuable comments. The authors also thank the National Natural Science Foundation of China (Nos. 11301207, 11171081, 11271364, 11301112), Natural Science Foundation of Jiangsu Province (Nos. BK2011407 and BK20130411), Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 13KJB110002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, Wiley, New York, 1976. M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci. 152 (1998) 165–177. L. Dong, L. Chen, L. Sun, Optimal harvesting policies for periodic Gompertz systems, Nonlinear Anal. Real World Appl. 8 (2007) 572–578. Z. Shuai, L. Bai, Optimization problems for general simple population with n-impulsive harvest, J. Math. Anal. Appl. 329 (2007) 634–646. B.S. Goh, G. Leitman, T.L. Vincent, Optimal control of a prey–predator system, Math. Biosci. 19 (1974) 263–286. T.K. Kar, Modelling and analysis of a harvested prey–predator system incorporating a prey refuge, J. Comput. Appl. Math. 185 (2006) 19–33. E. Braverman, R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol. 57 (2008) 413–434. E. Braverman, L. Braverman, Optimal harvesting of diffusive models in a nonhomogeneous environment, Nonlinear Anal. 71 (2009) e2173–e2181. L. Korobenko, M. Kamrujjaman, E. Braverman, Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting, J. Math. Anal. Appl. 399 (2013) 352–368. J.R. Beddington, R.M. May, Harvesting natural populations in a randomly fluctuating environment, Science 197 (1977) 463–465. D. Ludwig, J.M. Varah, Optimal harvesting of a randomly fluctuating resource II: numerical methods and results, SIAM J. Appl. Math. 37 (1979) 185– 205. R. Lande, S. Engen, B.E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat. 145 (1995) 728–745. E.M. Lungu, B. ;ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci. 145 (1997) 47–75. L.H.R. Alvarez, L.A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol. 37 (1998) 155–177. C.A. Braumann, Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. Biosci. 177&178 (2002) 229–245. Q.S. Song, R. Stockbridge, C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim. 49 (2011) 859–889. W. Li, K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl. 368 (2010) 420–428. X. Zou, K. Wang, Optimal harvesting for a logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl. 161 (2014) 969–979. M. Liu, C. Bai, Optimal harvesting policy for a stochastic predator–prey model, Appl. Math. Lett. 34 (2014) 22–26. S.C. Liu, Solutions of Fokker–Planck equation with applications in nonlinear random vibration, Bell Syst. Tech. J. 48 (1969) (1988) 2031–2051. C. Ji, D. Jiang, N. Shi, Analysis of a predator–prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl. 359 (2009) 482–498. M. Liu, K. Wang, Stochastic Lotka–Volterra systems with Lévy noise, J. Math. Anal. Appl. 410 (2014) 750–763.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Optimal harvesting policy of a stochastic food chain population model Meng Liu ⇑, Chuanzhi Bai School of Mathematical Science, Huaiyin Normal University, Huaian 223300, PR China
a r t i c l e
i n f o
Keywords: Food chain model Stochastic perturbations Optimal harvesting policy Fokker–Planck equation
a b s t r a c t This note studies the optimal harvesting policy of a stochastic three-species food chain model. The almost sufficient and necessary conditions for the existence of optimal harvesting policy are established. Meanwhile, the optimal harvesting effort and the maximum of expectation of sustainable yield are obtained. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Optimal policy in managing natural resources is one of the most important topics in mathematical biology. Clark [1], Fan and Wang [2], Dong et al. [3] and Shuai et al. [4] have considered the optimal harvesting policy of some deterministic singlespecies models. Goh et al. [5] and Kar [6] have investigated the optimal harvesting policy of species under a deterministic predator–prey framework. Braverman and Mamdani [7] have studied the optimal harvesting policy of both autonomous and nonautonomous population models with either impulsive or continuous harvesting. Braverman and Braverman [8] have established the optimal harvesting strategy of logistic, Gilpin-Ayala and Gompertz models with diffusion and harvesting. Korobenko et al. [9] have investigated the existence, positivity, persistence, extinction and stability of solutions of reaction–diffusion logistic equations with harvesting. On the other hand, in the natural world, the growth of population is inevitably subject to environmental perturbations. Thus many scholars have studied the optimal harvesting policy of stochastic population models. Beddington and May [10] have obtained the optimal harvesting policy of a classical stochastic logistic equation. The optimal harvesting of some single-species population models in stochastic environments were also examined in [11–19]. As species do not exist alone in nature, and food-chain interaction is a frequently observed phenomena among wildlife species, it is of great biological significance to consider the optimal harvesting policy of stochastic food-chain population model. However, to the best of our knowledge, there are few investigations of this kind. One reason is that it is quite difficult to solve the corresponding Fokker–Planck equation. In this note, using the results in Liu [20], we solve the corresponding Fokker–Planck equation of our food-chain model. Then we establish the almost sufficient and necessary conditions for the existence of optimal harvesting policy. Meanwhile, the optimal harvest effort and the maximum of expectation of sustainable yield (ESY) are given.
⇑ Corresponding author. E-mail address: [email protected] (M. Liu). http://dx.doi.org/10.1016/j.amc.2014.07.103 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
266
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
2. Model formulation and preparation works Our system is based on the following three-species food chain system (see e.g. [1]):
8 dx ðtÞ 1 > ¼ x1 ðtÞða1 b1 x1 ðtÞ c1 x2 ðtÞÞ; > < dt dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ; dt
ð1Þ
where xi is the size of the ith population, ai ; bi and c1 ; c2 are positive constants, i ¼ 1; 2; 3. Suppose that the population x1 is subject to exploitation with hx1 , where h is the harvesting effort, then system (1) becomes
8 dx ðtÞ 1 > ¼ x1 ðtÞða1 h b1 x1 ðtÞ c1 x2 ðtÞÞ; > < dt dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ: dt
ð2Þ
On the other hand, the harvesting effort h is inevitably affected by environmental noises, for example, the increasing or the decreasing of purchasing power. In this paper we assume that the noise in both nature and human society is multiplicative and generated by white noise, that is
_ h ! h þ rBðtÞ; where BðtÞ is a standard Brownian motion defined on a complete probability space ðX; fF t gt2Rþ ; PÞ with a filtration fF t gt2Rþ and r2 denotes the intensity of the noise, where Rþ ¼ ½0; þ1Þ. Then we obtain the following stochastic system
8 dx ðtÞ ¼ x1 ðtÞða1 h b1 x1 ðtÞ c1 x2 ðtÞÞdt þ rx1 ðtÞdBðtÞ; > < 1 dx2 ðtÞ ¼ x2 ðtÞða2 þ b2 x1 ðtÞ c2 x3 ðtÞÞ; dt > : dx3 ðtÞ ¼ x3 ðtÞða3 þ b3 x2 ðtÞÞ: dt
ð3Þ
Our aim is to find out the optimal harvesting effort h such that (i) ESY YðhÞ ¼ limt!þ1 Eðhx1 ðtÞÞ is maximum; (ii) x1 ðtÞ; x2 ðtÞ and x3 ðtÞ will not become extinct. Now let us do some preparation. Lemma 1. For any initial data ðx1 ð0Þ; x2 ð0Þ; x2 ð0ÞÞ 2 R3þ ¼: fa ¼ ða1 ; a2 ;3 Þjai > 0; i ¼ 1; 2; 3g, there is a unique global positive solution to model (3) almost surely (a.s.). Proof. The proof is standard and hence is omitted (see e.g. [21,22]).
h
If x1 ðtÞ; x2 ðtÞ; x3 ðtÞ is the solution of system (3), then there is a transition probability density of this solution. Let
U½x1 ðt þ DtÞ; x2 ðt þ DtÞ; x3 ðt þ DtÞjx1 ðtÞ ¼ x1 ; x1 ðtÞ ¼ x2 ; x3 ðtÞ ¼ x3 be the transition probability density function of x1 ; x2 and x3 at t þ Dt given that x1 ðtÞ ¼ x1 ; x2 ðtÞ ¼ x2 and x3 ðtÞ ¼ x3 . Then U must obey the following Fokker–Planck equation
@U @ @ @ @2 ¼ ½x1 ða1 h b1 x1 c1 x2 ÞU ½x2 ða2 þ b2 x1 c2 x3 ÞU ½x3 ða3 þ b3 x2 ÞU þ 0:5 2 r2 x21 U : @x1 @x2 @x3 @t @x1
Lemma 2. For system (3), (i) If
a1 0:5r2
c1 a3 b1 a2 < h; b3 b2
ð4Þ
then x3 ðtÞ goes to extinction a.s., i.e., limt!þ1 x3 ðtÞ ¼ 0; a:s. (ii) If
a1 0:5r2
c1 a3 b1 a2 > h; b3 b2
ð5Þ
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
267
then the density of stationary distribution of solution of system (3) is
Uðx1 ; x2 ; x3 Þ ¼ m exp
2b1
r2
2z1 2b1 z2 1 2b1 c1 z3 1 1 2b1 c1 2b1 c1 c2 r2 b r2 b b ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 exp 2 ðx3 Þ 1 x3 2 3 ; r b2 r b2 b3
ð6Þ
where
z1 ¼ a1 0:5r2 h ( m¼
exp Z
2b1
r2
þ
2b1 c1 2b1 c1 c2 þ r2 b2 r2 b2 b3
þ1
exp
0
c1 a3 c1 a3 b2 b2 c1 a3 a2 ; ; z2 ¼ ; z3 ¼ z1 a 2 ¼ a1 0:5r2 h b3 b3 b1 b1 b3
Z
þ1
2b1 c1 c2 x x r2 b 2 b 3 3 3
r
0
2b1 c1 z3 1 r2 b2 b3
2z1 Z 1 2b1 exp 2 x1 x1r2 dx1
þ1 0
2b1 z2 1 2b1 c1 r2 b exp 2 x2 x2 2 dx2 r b2
)1 dx3
ð7Þ
Proof. Denote y1 ¼ ln x1 ; y2 ¼ ln x2 ; y3 ¼ ln x3 , then by Itô’s formula,
8 2 y1 y2 > < dy1 ¼ ða1 0:5r h b1 e c1 e Þdt þ rdBðtÞ; y1 y3 dy2 ¼ ða2 þ b2 e c2 e Þdt; > : dy3 ¼ ða3 þ b3 ey2 Þdt;
ð8Þ
Let W½y1 ðt þ DtÞ; y2 ðt þ DtÞ; y3 ðt þ DtÞjy1 ðtÞ ¼ y1 ; y1 ðtÞ ¼ y2 ; y3 ðtÞ ¼ y3 stand for the transition probability density function of the solution process of (8). Then W must obey the following Fokker–Planck equation in steady state:
@ @ @ @2 ða1 0:5r2 h b1 ey1 c1 ey2 ÞW þ ½ða2 þ b2 ey1 c2 ey3 ÞW þ ½ða3 þ b3 ey2 ÞW 0:5 2 r2 W @y1 @y2 @y3 @y1 @ b2 @ r2 @ W @ b3 @ r2 b2 @ W ðz1 b1 ey1 ÞW þ ðz2 c1 ey2 ÞW ¼ @y1 b1 @y2 @y1 c1 @y3 2 @y1 2 b1 @y2 2 @ r b b @ W 2 3 : ðz3 c2 ey3 ÞW þ @y2 2 b1 c1 @y3
0¼
ð9Þ
According to Liu’s analysis [20], if an equation can be written in the form
@W @W þ L2 ðy1 ; y2 ; y3 Þ H2 ðy2 ÞW þ K 2 ðy2 Þ 0 ¼ L1 ðy1 ; y2 ; y3 Þ H1 ðy1 ÞW þ K 1 ðy1 Þ @y1 @y2 @W þ L3 ðy1 ; y2 ; y3 Þ H3 ðy3 ÞW þ K 3 ðy3 Þ ; @y3
ð10Þ
where L1 ðy1 ; y2 ; y3 Þ; L2 ðy1 ; y2 ; y3 Þ and L3 ðy1 ; y2 ; y3 Þ are partial differential operators, then a unique solution of W can be gotten by solving
H1 ðy1 ÞW þ K 1 ðy1 Þ
@W @W @W ¼ 0; H2 ðy2 ÞW þ K 2 ðy2 Þ ¼ 0; H3 ðy3 ÞW þ K 3 ðy3 Þ ¼ 0: @y1 @y2 @y3
That is to say,
Z
Wðy1 ; y2 ; y3 Þ ¼ m exp
y1 0
Z y2 Z y3 H1 ðsÞ H2 ðsÞ H3 ðsÞ ds exp ds exp ds ; K 1 ðsÞ K 2 ðsÞ K 3 ðsÞ 0 0
where m is the normalization factor satisfying
Z
þ1
1
Z
þ1
1
Z
þ1
Wðy1 ; y2 ; y3 Þdy1 dy2 dy3 ¼ 1:
1
Clearly, Eq. (9) has the form of (10) with
L1 ðy1 ; y2 ; y3 Þ ¼
@ b2 @ ; @y1 b1 @y2
L2 ðy1 ; y2 ; y3 Þ ¼
@ b3 @ ; @y1 c1 @y3
H1 ðy1 Þ ¼ z1 b1 ey1 ; H2 ðy2 Þ ¼ z2 c1 ey2 ; H3 ðy3 Þ ¼ z3 c2 ey3 ; K 1 ðy1 Þ ¼
r2 2
; K 2 ðy2 Þ ¼
r2 b 2 2 b1
; K 3 ðy3 Þ ¼
r2 b 2 b 3 2 b1 c 1
:
L3 ðy1 ; y2 ; y3 Þ ¼
@ ; @y2
268
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Therefore the solution of Eq. (9) is
"Z # "Z # y2 y3 z1 b 1 e s z2 c1 es z3 c 2 e s exp exp ds ds ds r2 b2 r 2 b2 b3 r2 =2 0 0 0 2 b1 2 b1 c 1 2 2b1 2b1 c1 y1 y2 ðz3 y3 c2 ey3 þ c2 Þ ¼ m exp 2 ðz1 y1 b1 e þ b1 Þ exp 2 ðz2 y2 c1 e þ c1 Þ exp 2 r r b2 r b2 b3 2b1 z2 2b1 c1 z3 2z1 2b1 y1 2b c 2b1 c1 c2 y3 1 1 ðe 1Þ ðey3 Þ r2 b2 b3 : ¼ m exp 2 ðe 1Þ ðey1 Þ r2 exp 2 ðey2 1Þ ðey2 Þ r2 b2 exp 2 r r b2 r b2 b3
Wðy1 ; y2 ; y3 Þ ¼ m exp
Z
y1
Note that x1 ¼ ey1 ; x2 ¼ ey2 ; x3 ¼ ey3 , hence the joint steady state probability function of ðx1 ; x2 ; x3 Þ is
Uðx1 ; x2 ; x3 Þ ¼ m exp
2b1 z2 2b1 c1 z3 2z1 2b1 c1 2b1 c1 c2 y3 ðey1 1Þ ðey1 Þ r2 exp 2 ðey2 1Þ ðey2 Þ r2 b2 exp 2 ðe 1Þ ðey3 Þ r2 b2 b3 r b2 r b2 b3
2b1
r2
1 x1 x2 x3 2z1 2b1 z2 1 2b1 c1 z3 1 1 2b1 2b1 c1 2b1 c1 c2 r2 b r2 b b ¼ m exp 2 ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 exp 2 ðx3 1Þ x3 2 3 ; r r b2 r b2 b3
R þ1 R þ1 R þ1 where m is determined by the following integral 0 Uðx1 ; x2 ; x3 Þdx1 dx2 dx3 ¼ 1. By the theory of improper integral, 0 0 R þ1 R þ1 R þ1 if (4) holds, that is z3 < 0, then the density of x3 will tend to 0. Else if (5) holds, then 0 Wðx1 ; x2 ; x3 Þdx1 dx2 dx3 is 0 0 convergent and (Z )1 2z1 2b1 z2 1 2b1 c1 z3 1 Z þ1 Z þ1 þ1 1 2b1 2b1 c1 2b1 c1 c2 r2 b r2 b b m¼ exp 2 ðx1 1Þ x1r2 dx1 exp 2 ðx2 1Þ x2 2 dx2 exp 2 ðx3 1Þ x3 2 3 dx3 r r b2 r b2 b3 0 0 0 ( Z þ1 2z1 2b1 z2 1 Z þ1 1 2b1 2b1 c1 2b1 c1 c2 2b1 2b1 c1 r2 b ¼ exp þ þ exp 2 x1 x1r2 dx1 exp 2 x2 x2 2 dx2 r 2 r 2 b2 r 2 b2 b3 0 r r b2 0 )1 2b1 c1 z3 1 Z þ1 2b1 c1 c2 r2 b b exp 2 x3 x3 2 3 dx3 : r b2 b3 0 This completes the proof.
h
3. Optimal harvesting policy Now we are in the position to give our main result. Theorem 1. For model (3),
(I) If a1 0:5r2 c1ba3 3 bb1 a2 2 < h, then limt!þ1 x3 ðtÞ ¼ 0; (II) Suppose that a1 0:5r (a) If 2
c1 a3 b3
b1 a2 b2
a:s.
> h.
c1 a3 b1 a2 0:5 a1 0:5r2 < 0; b3 b2 then the optimal harvesting policy does not exist. (b) If
c1 a3 b1 a2 0:5 a1 0:5r2 > 0; b3 b2
then the optimal harvesting effort h is
c1 a3 : h ¼ 0:5 a1 0:5r2 b3 Meanwhile, the maximum of ESY is 2
Y ¼
ða1 0:5r2 c1 a3 =b3 Þ : 4b1
ð11Þ
269
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Proof. By Lemma 2, we need only to prove (II). The expectation of sustainable yield is
YðhÞ ¼ lim Eðhx1 ðtÞÞ ¼ t!þ1
Z
þ1
hx1 U1 ðx1 Þdx1 ;
0
where
U1 ðx1 Þ ¼
Z
þ1
Z
0
þ1
Uðx1 ; x2 ; x3 Þdx2 dx3 :
0
It follows from (6) and (7) that
2b1 z2 1 2b1 c1 z3 1 Z þ1 2b1 c1 2b1 c1 c2 r2 b2 r2 b b x x3 2 3 dx3 exp ðx 1Þ dx exp ðx 1Þ 2 2 3 2 r2 r2 b2 r2 b 2 b 3 0 0 ( Z þ1 2z1 2b1 z2 1 Z þ1 1 2b1 2b1 c1 2b1 c1 c2 2b1 2b1 c1 r2 b ¼ exp þ 2 þ 2 exp 2 x1 x1r2 dx1 exp 2 x2 x2 2 dx2 2 r r b2 r b2 b3 0 r r b2 0 )1 2b1 c1 z3 1 2z1 2b1 z2 1 Z þ1 Z þ1 1 2b1 c1 c2 2b1 2b1 c1 r2 b b r2 b exp 2 x3 x3 2 3 dx3 exp 2 ðx1 1Þ x1r2 exp 2 ðx2 1Þ x2 2 dx2 r b2 b3 r r b2 0 0 h i 2z1 1 2b1 2b1 c1 z3 1 Z þ1 exp x x1r2 1 2 2b1 c1 c2 r r2 b b exp 2 ðx3 1Þ x3 2 3 dx3 ¼ : h i 2z1 1 R þ1 r b2 b3 0 2b1 r2 exp x dx x 1 1 1 0 r2
U1 ðx1 Þ ¼ mexp
2b1
2z1 Z 1 ðx1 1Þ x1r2
þ1
Consequently
h i 2z1 2z1 R þ1 r2 1 2z exp 2b hr2 0 ev v r2 dv hr2 Cð r21 þ 1Þ hr2 2z1 hz1 r2 x1 x1 dx1 ¼ ¼ ¼ ¼ 2z1 h i 2z1 1 R 2b1 þ1 v r2 1 2b1 Cð2z21 Þ 2b1 r2 b1 r2 1 r e v dv exp 2b dx1 0 r2 x1 x1
R þ1 YðhÞ ¼ h
0
R þ1 0
¼
hða1 0:5r2 h c1 a3 =b3 Þ : b1
where CðÞ is the Gamma function. Note that YðhÞ has a unique extreme value at h ¼ 0:5ða1 0:5r2 c1 a3 =b3 Þ. Thus if h obeys
z3 jh¼h ¼ b2 ða1 0:5r2 h c1 a3 =b3 Þ=b1 a2 > 0;
i.e. 0:5b2 ða1 0:5r2 c1 a3 =b3 Þ=b1 a2 > 0, then the optimal harvesting effort is h and the maximum of expectation of sustainable yield is
Y ¼
2
h ða1 0:5r2 h c1 a3 =b3 Þ ða1 0:5r2 c1 a3 =b3 Þ ¼ : b1 4b1
If 0:5b2 ða1 0:5r2 c1 a3 =b3 Þ=b1 a2 < 0, i.e. z3 jh¼h < 0, then the optimal harvesting effort h will make x3 extinct and thus the optimal harvesting policy does not exist. h Remark 1. Consider the following stochastic logistic population model:
dxðtÞ ¼ xðtÞða h bxðtÞÞdt þ rxdBðtÞ:
By Theorem 1, the optimal harvesting policy exists if and only if h < a 0:5r2 , the harvest effort is h ¼ 0:5ða 0:5r2 Þ and 2 Þ2
r the maximum of ESY is Y ¼ ða0:5 . These results coincide with the classical results obtained by Beddington and May [10] 4b in Science.
Remark 2. In this note, we suppose that only the top prey is subjected to exploitation. It is interesting to investigate the optimal harvesting of the following model:
8 > < dx1 ðtÞ ¼ x1 ðtÞða1 h1 b1 x1 ðtÞ c1 x2 ðtÞÞdt þ r1 x1 dBðtÞ; dx2 ðtÞ ¼ x2 ðtÞða2 h2 þ b2 x1 ðtÞ c2 x3 ðtÞÞdt þ r2 x2 dBðtÞ; > : dx3 ðtÞ ¼ x3 ðtÞða3 h3 þ b3 x2 ðtÞÞdt þ r3 x3 dBðtÞ;
ð12Þ
where hi is the harvesting effort of xi ; i ¼ 1; 2; 3. In fact, we have also attempted to investigate this question. Unfortunately, we cannot obtain the explicit solution of the Fokker–Planck equation of (12) now. It is also interesting to investigate the optimal harvesting of other stochastic models.
270
M. Liu, C. Bai / Applied Mathematics and Computation 245 (2014) 265–270
Acknowledgments The authors thank the editor and reviewers for their valuable comments. The authors also thank the National Natural Science Foundation of China (Nos. 11301207, 11171081, 11271364, 11301112), Natural Science Foundation of Jiangsu Province (Nos. BK2011407 and BK20130411), Natural Science Research Project of Ordinary Universities in Jiangsu Province (No. 13KJB110002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, Wiley, New York, 1976. M. Fan, K. Wang, Optimal harvesting policy for single population with periodic coefficients, Math. Biosci. 152 (1998) 165–177. L. Dong, L. Chen, L. Sun, Optimal harvesting policies for periodic Gompertz systems, Nonlinear Anal. Real World Appl. 8 (2007) 572–578. Z. Shuai, L. Bai, Optimization problems for general simple population with n-impulsive harvest, J. Math. Anal. Appl. 329 (2007) 634–646. B.S. Goh, G. Leitman, T.L. Vincent, Optimal control of a prey–predator system, Math. Biosci. 19 (1974) 263–286. T.K. Kar, Modelling and analysis of a harvested prey–predator system incorporating a prey refuge, J. Comput. Appl. Math. 185 (2006) 19–33. E. Braverman, R. Mamdani, Continuous versus pulse harvesting for population models in constant and variable environment, J. Math. Biol. 57 (2008) 413–434. E. Braverman, L. Braverman, Optimal harvesting of diffusive models in a nonhomogeneous environment, Nonlinear Anal. 71 (2009) e2173–e2181. L. Korobenko, M. Kamrujjaman, E. Braverman, Persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting, J. Math. Anal. Appl. 399 (2013) 352–368. J.R. Beddington, R.M. May, Harvesting natural populations in a randomly fluctuating environment, Science 197 (1977) 463–465. D. Ludwig, J.M. Varah, Optimal harvesting of a randomly fluctuating resource II: numerical methods and results, SIAM J. Appl. Math. 37 (1979) 185– 205. R. Lande, S. Engen, B.E. Saeher, Optimal harvesting of fluctuating populations with a risk of extinction, Am. Nat. 145 (1995) 728–745. E.M. Lungu, B. ;ksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci. 145 (1997) 47–75. L.H.R. Alvarez, L.A. Shepp, Optimal harvesting of stochastically fluctuating populations, J. Math. Biol. 37 (1998) 155–177. C.A. Braumann, Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math. Biosci. 177&178 (2002) 229–245. Q.S. Song, R. Stockbridge, C. Zhu, On optimal harvesting problems in random environments, SIAM J. Control Optim. 49 (2011) 859–889. W. Li, K. Wang, Optimal harvesting policy for general stochastic Logistic population model, J. Math. Anal. Appl. 368 (2010) 420–428. X. Zou, K. Wang, Optimal harvesting for a logistic population dynamics driven by a Lévy process, J. Optim. Theory Appl. 161 (2014) 969–979. M. Liu, C. Bai, Optimal harvesting policy for a stochastic predator–prey model, Appl. Math. Lett. 34 (2014) 22–26. S.C. Liu, Solutions of Fokker–Planck equation with applications in nonlinear random vibration, Bell Syst. Tech. J. 48 (1969) (1988) 2031–2051. C. Ji, D. Jiang, N. Shi, Analysis of a predator–prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl. 359 (2009) 482–498. M. Liu, K. Wang, Stochastic Lotka–Volterra systems with Lévy noise, J. Math. Anal. Appl. 410 (2014) 750–763.