Finding the optimal opening time of harvesting farmed fishery resources

Abstract

As an application of mathematics to engineering problems, this paper formulates a simple optimal stopping problem to decide the opening time of harvesting farmed fishery resources that maximizes an economic objective function. A sufficient condition for unique existence of the internal optimal opening time is provided and its concrete mathematical analysis is carried out. Comparative statics of the optimal opening time clearly reveals its dependence on the parameters of the farming environment. The problem is finally applied to analyzing management of a commercially important fishery resource in Japan.

Optimization model to start harvesting in stochastic aquaculture system

“…Several properties of the BE were analyzed from both analytical and numerical standpoints, and their implications to management of P. altivelis were discussed. The numerical simulation results confirmed some consistency on the profiles of the solution to the reduced BE with the mathematical analysis results in this and previous papers [45]. Although the presented mathematical analysis results are partly complemented by the numerical computation, they can effectively characterize the optimal management strategy of aquaculture systems subject to continuous harvesting.…”

“…This appendix shows a consistency result between the previous deterministic model [45] and the present stochastic process model. As conjectured in (55) based on the computational results, the function W * (t) is monotonically decreasing for 0 < t < T. Assume the initial condition W 0 = w > 0 and notice that W t can be seen as a bivariate function W t = W (t, w) with the couple of parameters (t, w), which yields W * = W ( * , w).…”

“…The officers of Hii River Fishery Cooperatives measured weight of individuals of P. altivelis during 2015 as plotted in Figure 2 for carrying out management strategies empirically considering their growth rate. Following the previous research [45], the classical Verhulst model [49] g with the parameter values W 0 = 9.9 (g), a = 76.7 (g), and b = 0.04 (1/day) is used, which accurately reproduces the observed growth curve as shown in Figure 2. In the figure, the day t = 0 is May 7, 2015.…”

“…Numerical simulation presented in the next section shows that this phenomenon would not be generic for variable g that satisfies Assumption 2.1. The optimal opening time * in (50) is the same with that in Proposition 3.1 of Yoshioka and Yaegashi [45], which focused on the deterministic problem ( = 0). This result shows that the present approach based on the BE and their direct approach is consistent with each other.…”

“…and where B t is the 1-D standard Brownian motion [46], R > 0 is the natural mortality rate, c > 0 is their harvesting rate after the opening time , ⩾ 0 is the noise intensity, g is a function such that W t with a sufficiently small W 0 > 0 and no stochasticity ( = 0) is increasing and bounded for t ∈ [0, T] (Proposition B.7 of Smith and Waltman [48]), and S is the usual indicator function for the set S. The noise intensity represents environmental stochasticity in the system that affects the growth of the aquatic organisms, which would involve unresolved factors, such as internal competitions among individuals and uncontrollable fluctuations of water quality. Following the previous research (Assumption 3.2 of Yoshioka and Yaegashi [45], Thieme [49]), in this paper, the growth rate g is assumed to satisfy the following condition to give a sigmoid-type growth of W t for t > 0 subject to W 0 > 0 when = 0:…”