Stochastic optimization model of aquacultured fish for sale and ecological education
- Hidekazu Yoshioka^{1}Email author and
- Yuta Yaegashi^{2}
https://doi.org/10.1186/s13362-017-0038-8
© The Author(s) 2017
Received: 13 March 2017
Accepted: 11 May 2017
Published: 19 May 2017
Abstract
A stochastic optimization model for management of aquacultured fish for sale and ecological education is established. Population dynamics of the fish is described with a system of stochastic differential equations assuming that they are stochastically harvested after an opening time: the variable to be optimized. A remarkable difference between the present and conventional models for the aquaculture is that the former considers the harvesting for ecological education, which is a key for current inland fishery especially in Japan. Finding the cost-effective optimal opening time effectively reduces to solving an algebraic equation. Mathematical analysis on the optimal opening time clearly reveals its unique existence and qualitative behavior, such as its dependence on the model parameters, which have practical implications. A demonstrative application example of the model is also presented focusing on an aquaculture of Plecoglossus altivelis in Japan.
Keywords
1 Introduction
Aquaculture is one of the most profitable fisheries sectors that conventionally farms, harvests, and sells fishery resources [1]. Cost-effective and sustainable management strategies for aquaculture have been studied from both scientific and engineering viewpoints. Clarke et al. [2] reviewed fish vaccine development using plant genetic engineering techniques for sustainable aquaculture. Mazid et al. [3] investigated cost-effective feeding strategy for aquacultured fish using locally available ingredients.
Population dynamics models can efficiently simulate behavior and dynamics of aquatic organisms [4–6]. For modeling the dynamics related to aquaculture, stochastic models turned out to be able to compute cost-effective, environmentally- and/or ecologically-sound management strategies of fishery resources under a variety of conditions [7–11]. Stochastic differential equations (SDEs) [12] have served as one of the most important tools for optimization problems in aquaculture because of their power to effectively handle population dynamics subject to uncertainties. Nøstbakken [13] numerically computed optimal management rules of aquaculture systems subject to regime-switching stochastic stocks and prices of farmed fishes. Reed and Clarke [14] derived optimal harvesting and pricing rules for aquacultured fishes with the stochastic and size-dependent growth rate. León-Santana et al. [15] analytically derived formulae to achieve environmentally-sound aquaculture based on linear SDEs.
The purpose of this paper is to develop a new, tractable stochastic process model for aquaculture considering the harvesting not for sale, which can be utilized for decision-making of the cost-effective and ecologically conscious aquaculture. Stochastic population dynamics in an artificial aquaculture system is described with a system of SDEs that govern the total number of the population and its representative weight. The opening time of harvesting is set as the variable to be optimized. The decision-maker in our optimization problem is the manager of an FC. The opening time is optimized by the manager so that the performance index, which is a net profit by the aquaculture, is maximized. The main difference between the present and conventional models [18, 25–27] is that fishes are harvested not only for sale but also for ecological education to local residents. This effect is incorporated into both the population dynamics model and the performance index. Mathematical analysis on the optimal opening time is carried out and its results are verified with numerical computation focusing on Plecoglossus altivelis, which is one of the most important inland fishery resources in Japan, managed by Hii River Fisheries Cooperative (HRFC). This paper is the first attempt to model aquaculture considering harvesting for ecological education.
The rest of the paper is organized as follows. Section 2 presents the mathematical model for cost-effective management of aquacultured fishes without reproduction. Section 3 presents mathematical analysis results on the model. Section 4 performs numerical computation to verify the mathematical analysis results with an application to Plecoglossus altivelis (Ayu), which is an important inland fishery resource in Japan. Section 5 concludes this paper and presents future perspectives of our research.
2 Methods: mathematical model
The population dynamics model in this paper is not significantly different from the conventional ones [18] except for that it is driven by Poisson noises. On the other hand, the performance index to be maximized, which has a term on profit by harvesting the fish for ecological education, distinguishes the present model from the conventional ones. Hereafter, \(a \wedge b\) for \(a, b \in \mathbb{R}\) represents the smaller value between a and b.
2.1 Governing equations
The time is denoted as \(t \in [ 0, T ]\) where \(t = 0, T\) are the initial and terminal times of an aquaculture, respectively. The fish is introduced into the system only at \(t = 0\). The population dynamics in an artificial aquaculture system is reasonably considered to be non-renewable [18]. The dynamics is described with the representative (average) weight of individuals \(W_{t} \ge 0\) and the total number of individuals \(N_{t} \ge 0\). The governing equations of \(W_{t}\) and \(N_{t}\) are assumed to be independent with each other. This is a reasonable assumption for realistic aquaculture system operated by well-experienced managers with a controlled, sufficiently large pool. The representative weight \(W_{t}\) monotonically increases throughout the period. On the other hand, the total number of individuals \(N_{t}\) monotonically decreases at the same time. All the stochastic integrals in what follow are defined in the Itô’s sense [28].
There is an opening time \(\tau \in [ 0, T ]\) of harvesting after which the population is harvested for sale and for exchange meetings for ecological education between FCs and local residents. The exchange meetings are hosted by FCs for ecological education, so that the residents can learn about their fishes and the contents of their works. At each exchange meeting, several hundred individuals of the aquacultured fishes are used for the catching competition by children of local residents: the main event of the meeting (Figure 1). In addition, several lectures on ecology and biology of aquatic organisms including fishes living in and around rivers, lakes, the seas, and in the aquaculture system are held at each meeting. Selling events during \([ \tau, T )\) is modelled with a Poisson process \(P_{t}^{ ( 1 )}\) with the intensity \(\lambda^{ ( 1 )} > 0\). The total number of potentially harvested individuals for sale, the demand, at the time t follows a non-negative continuous-time stationary Markov process \(c_{t}^{ ( 1 )}\). The exchange meetings are held during \([ \tau, T )\), which is also modelled as a Poisson process \(P_{t}^{ ( 2 )}\) with the intensity \(\lambda^{ ( 2 )} > 0\). The total number of potentially harvested individuals for ecological education at the time t follows a non-negative continuous-time stationary Markov process \(c_{t}^{ ( 2 )}\). Scheduled date and time of each exchange meeting are usually determined before \(t = 0\), but they are possibly held on different date and time due mainly to climate stochasticity. The mean values of \(c_{t}^{ ( 1 )}\) and \(c_{t}^{ ( 2 )}\) are assumed to exist and are denoted as \(C^{ ( 1 )} > 0\) and \(C^{ ( 2 )} > 0\), respectively. The stochastic processes \(P_{t}^{ ( 1 )}\), \(P_{t}^{ ( 2 )}\), \(c_{t}^{ ( 1 )}\), and \(c_{t}^{ ( 2 )}\) are assumed to be independent with each other. The stochastic population dynamics is considered under usual filtration generated by the processes \(P_{t}^{ ( 1 )}\), \(P_{t}^{ ( 2 )}\), \(c_{t}^{ ( 1 )}\), and \(c_{t}^{ ( 2 )}\) as in the conventional stochastic process models. Mathematical properties and applications of Poisson processes are presented in Ross [29].
2.2 Performance index
Remark 2.1
In practice, the orders of \(\lambda^{ ( 1 )}\) and \(\lambda^{ ( 2 )}\) are significantly different. As an example of P. altivelis farmed by HRFC during 2016, the exchange meetings for ecological education were held at most several times in each month. During the same year, the aquacultured fishes were sold almost every day. Therefore, we infer that \(\lambda^{ ( 1 )} = O ( 10^{0} )\) (1/day) to \(O ( 10^{1} )\) (1/day) and \(\lambda^{ ( 2 )} = O ( 10^{ - 2} )\) (1/day) to \(O ( 10^{ - 1} )\) (1/day). We also infer \(C^{ ( 1 )} = O ( 10^{0} )\) to \(O ( 10^{1} )\) and \(C^{ ( 2 )} = O ( 10^{2} )\) to \(O ( 10^{3} )\) based on interviews from HRFC during 2016.
3 Mathematical analysis
Mathematical analysis on existence and uniqueness of the optimal opening time \(\tau^{*}\) is carried out. A primitive stochastic process model is also analyzed in this section. Practical implications of the mathematical analysis results are presented as well, which are verified numerically in the next section.
3.1 Primitive model
Remark 3.1
3.2 Existence and uniqueness of optimal opening time
A sufficient condition for \(\tau^{*} > 0\), which is consistent with the actual strategies for managing aquaculture systems by FCs, is presented below. Practical implications of the condition are presented as well.
Proposition 3.1
Proof
\(\tau^{*} > 0\) if \(\frac{\mathrm{d}J_{\tau}}{\mathrm{d}\tau}|_{\tau = 0} > 0\) since \(J_{\tau}\) is continuously differentiable with respect to \(\tau \ge 0\). Substituting (25) and (26) into \(\frac{\mathrm{d}J_{\tau}}{\mathrm{d}\tau} |_{\tau = 0} > 0\) leads to (29) since \(\tilde{A} > 0\). □
Based on Proposition 3.1, the conditions to guarantee \(\tau^{*} > 0\) for several growth models are derived.
Proposition 3.2
For \(g = r\ ( = \mathrm{const} ) > 0\), the condition (29) is satisfied for sufficiently large r and \(\eta^{ - 1}\).
Proof
The condition (29) is satisfied if its left-hand side is positive and η is sufficiently small. The latter is always possible because the parameters p, β, \(\alpha^{ ( 1 )}\), and \(\alpha^{ ( 2 )}\) involved in η do not appear in the left-hand side of (29). Actually, choosing sufficiently small pβ or sufficiently large \(\tilde{A} = \lambda^{ ( 1 )}\alpha^{ ( 1 )}C^{ ( 1 )} + \lambda^{ ( 2 )}\alpha^{ ( 2 )}C^{ ( 2 )}\) can achieve arbitrary small and positive η.
Theorem 3.1
For the generalized Verhulst model, the condition (29) is satisfied for sufficiently large r and sufficiently small \(W_{0}K^{ - 1}\) and η.
Proof
Remark 3.2
Proposition 3.2 and Theorem 3.1 show that the optimal opening time \(\tau^{*}\) is positive, which is in accordance with the reality, if the aquacultured fishes grow sufficiently fast and large and the demand of harvesting them is sufficiently large. A practical implication of the analysis results on management of aquacultured fish is that the decision-maker should carefully observe their growth. Similar results have been theoretically derived in Yoshioka and Yaegashi [18] using a different population dynamics model based on an optimization approach. This fact implies that there exists an underlying universal principle for cost-effective management of aquacultured fish harvested at a certain rate after an opening time.
Uniqueness results of \(\tau^{*}\) for small p are finally presented in this sub-section. Key propositions for the case \(p = 0\), which seem to be not reasonable from a practical viewpoint but actually serve as the basis for dealing with the case with sufficiently small \(p > 0\), are presented below.
Proposition 3.3
For \(p = 0\), \(g = r\ ( = \mathrm{const} ) > 0\) and \(r \ne R\), \(\tau^{*} \in [ 0, \tau_{\mathrm{ex}} ]\) exists uniquely.
Proof
Proposition 3.4
Assume \(p = 0\), g is given by the generalized Verhulst model (2) with \(r > R\) and \(\theta \ge 1\). Then, \(\tau^{*} \in [ 0, \tau_{\mathrm{ex}} ]\) exists uniquely.
Proof
Now, an application of the classical implicit function theorem with the smoothness of \(F_{\tau} - \eta G_{\tau}\) with respect to τ and p immediately shows the following unique existence theorem of \(\tau^{*} \in ( 0, \tau_{\mathrm{ex}} )\) such that \(\frac{\mathrm{d}J_{\tau}}{ \mathrm{d}\tau}\) for small \(p > 0\).
Theorem 3.2
Assume (29). Then, under the assumption of Proposition 3.4 where ‘ \(p = 0\) ’ is replaced by ‘sufficiently small \(p > 0\) ’, \(\tau^{*} \in ( 0, \tau_{\mathrm{ex}} )\) exists uniquely.
Theorem 3.2, although it can effectively characterize \(\tau^{*} \in ( 0, \tau_{\mathrm{ex}} )\), is possibly not sharp as the numerical computation results demonstrate later.
3.3 Comparative statics of the critical and optimal opening times
Comparative statics of the critical opening time \(\tau_{\mathrm{ex}}\) and the optimal opening time \(\tau^{*}\) such that \(0 < \tau^{*} < \tau_{\mathrm{ex}}\), namely their dependences on model parameters, is carried out. The comparative statics of \(\tau_{\mathrm{ex}}\) is firstly carried out. A straightforward calculation shows the following proposition.
Proposition 3.5
Proof
Proposition 3.5 indicates that the critical opening time \(\tau_{\mathrm{ex}}\) increases as the natural mortality rate of the fish R, the mean harvesting rates \(C^{ ( n )}\), or the intensities \(\lambda^{ ( n )}\). On the other hand, \(\tau_{\mathrm{ex}}\) decreases as the initial total number of individuals \(N_{0}\) increases. The results thus indicate that the critical opening time \(\tau_{\mathrm{ex}}\), which is interpreted as an upper bound of the optimal opening time \(\tau^{*}\), is larger (smaller) for relatively sparse (abundant) population in the aquaculture system.
Proposition 3.6
Proof
Proposition 3.6 leads to the following theorem.
Theorem 3.3
Proof
Substituting \(\eta = \frac{\beta pBR}{\tilde{A}}\) and (52) into (53) with the positivity of \(G_{\tau}\) leads to (55). □
Theorem 3.3 has significant practical implications to management of aquacultured fish. Larger p or β gives smaller optimal opening time \(\tau^{*}\), meaning that increasing the cost of farming the fishes makes the manager harvest them earlier. This is a consequence of the fact that the cost by farming, which is mainly for feeding the fishes and for cleaning up their excrements decreases as the total number of individuals decreases. On the other hand, larger \(\alpha^{ ( 1 )}\) or \(\alpha^{ ( 2 )}\) gives larger optimal opening time \(\tau^{*}\). This implies that increasing the profit by harvesting the fishes, for sale and for exchange meetings for ecological education, makes the manager harvest them later. The aquacultured fishes are used for catching competition by children of local residents, and utility of the children can be larger with better-grown fishes since they can eat the caught fishes after the competition. Therefore, the dependence of \(\tau^{*}\) on the parameters \(\alpha^{ ( 1 )}\) or \(\alpha^{ ( 2 )}\) is considered to be consistent with the reality. It should be noted that similar comparative statics results have been obtained in the deterministic optimization model for aquacultured fish [18].
3.4 Analogy with a model for managing non-renewable released fish
This sub-section shows an analogy between the present model for managing aquacultured fish and that for released fish, P. altivelis in particular. The fish has been the major inland fishery resource in Japan for recreational and fishery purposes. P. altivelis is an endemic fish in Japan and has an annual life history, which is reviewed in detail in the literature and the references therein [36–39]. In Japan, in each spring, FCs release juvenile P. altivelis into the rivers that they authorize. Harvesting the fish starts at the coming summer. The harvesting period is closed at the end of the autumn, on which the fishes spawn. The important point is that the released fish is often non-renewable possibly due to genetic and environmental reasons [40].
As a simple model we can hypothesize that the population dynamics of released P. altivelis in a river follows the SDEs (1) and (3) where the parameters and variables have different meanings. Under this setting, R is the mortality rate of the fish by natural death and predation from waterfowls such as Phalacrocorax carbo [41]. The stochastic processes \(P_{t}^{ ( 1 )}\) and \(P_{t}^{ ( 2 )}\) represent the events of harvesting the fish for recreational and fishery purposes, respectively. We assume that the FC wants to choose the opening time of harvesting so that a performance index with the form (7) is maximized. The parameter p in this case represents the cost of exterminating the predators; the parameters R and p may be dependent with each other. The weight parameters β, \(\alpha^{ ( 1 )}\), and \(\alpha^{ ( 2 )}\) depend on the attitude of the decision-maker as in the presented model for aquaculture. The above discussion implies that the present framework of mathematical modeling can be used not only for aquacultured fishes but also for some of the non-renewable released fishes.
4 Results and discussion
The present optimization model is applied to a demonstrative numerical computation of management of aquacultured P. altivelis.
4.1 P. altivelis and HRFC
In each year, farming juveniles of P. altivelis in an aquaculture system in Japan starts in spring (early to middle May) and they mature in summer around which harvesting opens (beginning of July). The harvesting ends in the coming autumn (late August to early October). HRFC in Shimane Prefecture, Japan farms P. altivelis from May to October in each year. Recently, officers of HRFC measured mean body weight of the individuals in the aquaculture system to track their growth. Both feeding the fish and cleaning up the pool are constantly carried out by the officers. According to the officers of HRFC, aquaculture of P. altivelis is one of the most indispensable sources of its income. They are trying to find cost-effective and ecologically sound management strategy of harvesting aquacultured P. altivelis.
4.2 Parameter identification
The initial time \(\pmb{t = 0}\) , the identified \(\pmb{W_{0}}\) , K , and r , and \(\pmb{\mathrm{R}^{2}}\) value between the measured and identified \(\pmb{W_{t}}\) in each year
Year | t = 0 | \(\boldsymbol{W_{0}}\) (g) | K (g) | r (1/day) | \(\mathbf{R}^{\mathbf{2}}\) |
---|---|---|---|---|---|
2012 | May 9 | 8.6 | 72.8 | 0.047 | 0.980 |
2013 | May 9 | 8.7 | 83.8 | 0.040 | 0.995 |
2014 | May 9 | 7.8 | 71.0 | 0.046 | 0.991 |
2015 | May 7 | 10.1 | 78.0 | 0.042 | 0.997 |
2016 | May 10 | 8.4 | 106.7 | 0.036 | 0.976 |
The terminal time is set as \(T = 150\) (day) considering current management strategy by HRFC. The optimal opening time \(\tau^{*}\) is computed directly from (51) using the classical Simpson’s rule. The time increment for integration Δτ is set as 0.01 (day). The total number of individuals at the time \(t = 0\) is specified as \(N_{0} = 20\text{,}000\) based on an actual management strategy by HRFC. This \(N_{0}\) has been empirically determined by HRFC considering the size of the pool used for their aquaculture.
4.3 Computational results
Conjecture 4.1
\(\tau^{*} \in [ 0, T ]\) exists uniquely.
5 Conclusion
The present mathematical model is the first model that can evaluate the influences of the harvesting for ecological education on the aquaculture management. Mathematical and numerical analyses on the optimal opening time were carried out to comprehend its existence, uniqueness, and behavior. The analysis results focusing on P. altivelis in Japan demonstrated that the harvesting for ecological education is less robust than that for sale, but should not be abandoned since it has served as one of the most indispensable works of the FC. From the analysis results, we recommend FCs to create a working environment where the harvesting for ecological education can be more efficiently carried out with wider spillover effects. Financial and social supports from local governments would be helpful for FCs to achieve this attempt.
Future research will develop an optimization theory for long-term, multi-year management of aquacultured P. altivelis, so that FCs in Japan can find the way to achieve cost-effective management strategy of the fish. In addition, local economic dynamics around the FCs will also be mathematically described, so that the demand of the fishes is more reasonably tracked. Investigations both from theoretical and practical viewpoints are necessary for further advance of fishery science and engineering.
Declarations
Acknowledgements
The River Fund No. 285311020 in charges of The River Foundation, JSPS Research Grant Nos. 15H06417, 17K15345, and WEC Applied Ecology Research Grant No. 2016-02 support this research. The authors thank to Hii River Fishery Cooperatives for providing valuable data.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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