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Optimal Control of a Fishery Utilizing Compensation and Critical Depensation Models

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Optimal Control of a Fishery Utilizing Compensation and Critical Depensation Models Appl. Math. Inf. Sci. 14, No. 3, 467-479 (2020) 467 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/140314 Optimal Control of a Fishery Utilizing Compensation and Critical Depensation Models Mahmud Ibrahim Department of Mathematics, University of Cape Coast, Cape Coast, Ghana Received: 12 Dec. 2019, Revised: 2 Jan. 2020, Accepted: 15 Feb. 2020 Published online: 1 May 2020 Abstract: This study proposes optimal control problems with two different biological dynamics: a compensation model and a critical depensation model. The static equilibrium reference points of the models are defined and discussed. Also, bifurcation analyses on the models show the existence of transcritical and saddle-node bifurcations for the compensation and critical depensation models respectively. Pontyagin’s maximum principle is employed to determine the necessary conditions of the model. In addition, sufficiency conditions that guarantee the existence and uniqueness of the optimality system are defined. The characterization of the optimal control gives rise to both the boundary and interior solutions, with the former indicating that the resource should be harvested if and only if the value of the net revenue per unit harvest (due to the application of up to the maximum fishing effort) is at least the value of the shadow price of fish stock. Numerical simulations with empirical data on the sardinella are carried out to validate the theoretical results. Keywords: Optimal control, Compensation, Critical depensation, Bifurcation, Shadow price, Ghana sardinella fishery 1 Introduction harvesting of either species. The study showed that, with harvesting effort serving as a control, the system could be steered towards a desirable state, and so breaking its Fish stocks across the world are increasingly under cyclic behavior. A two-predator one-prey system was pressure as a result of the excessive harvesting capacity analyzed and discussed by Pal et al. [6]. They focused on prevalent in the fishing industry. Thus, it is common that defining the sustainable yield of the resource while abundant stocks may suddenly collapse. A review by concurrently maintaining the ecological resilience of the Mullon et al. [1] of the FAO world fisheries data on catch system. Panja [7] developed and discussed a dynamical for the past five decades showed that as much as 25% of model involving the interactions of a prey, predator and fisheries’ collapses occurred during that period. Thus, the scavenger species. The study revealed that harvesting of prudent and effective management of fishery resources in the predator can serve as a control mechanism for the order to ensure their sustainability for future generations chaotic behavior of the system. The optimal exploitation cannot be overemphasized. of a fishery under critical depensation was analyzed by Various studies addressed the sustainable harvesting Kar and Misra [8]. They addressed the stability properties of renewable natural resources, especially fisheries [2–8]. and defined the optimal tax policy for the system. Dubey and Patra [2] proposed and analyzed a dynamic model with crowding effect of a renewable resource that However, the present study investigates the relative humans used for their nourishment. They obtained the performance of both compensation and critical optimal level of harvesting for the resource through the depensation models when subjected to harvesting. application of Pontryagin’s maximum principle. The Therefore, an optimal control problem with quadratic sustainable harvesting of prey species in a predator-prey revenues in the objective functional and a compensation model in the Sundarbans ecosystem was investigated by model as the biological dynamics is analyzed. Moreover, Hasan et al. [4]. They established the local and global another control problem with the biological dynamics stability of the model as well as obtaining the optimal representing a critical depensation model is tackled. It harvesting strategy. Kar [5] explored a predator-prey must be noted that, instead of the linear revenues in most system with a prey refuge that also included independent resource problems, quadratic revenues are employed in ∗ Corresponding author e-mail: mibrahim@ucc.edu.gh c 2020 NSP Natural Sciences Publishing Cor. 468 M. Ibrahim: Optimal control of a fishery... this study to account for diminishing returns whenever a The equilibrium point 0 is unstable while xeqm is bumper harvest occurs [9]. Another novelty of this study stable. Since these equilibrium points are hyperbolic, the is the exploration of the relationship between the shadow dynamical system is structurally stable. It is instructive to price and the marginal net revenue as it relates to the note that the model undergoes a transcritical bifurcation r optimal fishing effort. In Section Two, the optimal control at the point E = . That is, there exists a single problem is formulated comprising the biomass dynamics q – compensation model – as well as the complete nonhyperbolic equilibrium point, 0, which makes the bioeconomic model. In addition, it covers the optimality system structurally unstable. In other words, for any initial population x0, the resource will eventually go into of the model, which consists of the characterization of the r optimal control, as well as the existence and uniqueness extinction. When E > , there is a single non-negative of the optimality system. An optimal control problem q hyperbolic equilibrium point, 0. Of course, it is stable, with a critical depensation model as biological dynamics and the system is structurally stable. Thus, for any initial is addressed in Section Three. Furthermore, bifurcation population level, the resource will die out in finite time. analysis of the model is explored and numerical For the sake of completeness, the other equilibrium point simulations are conducted. Section Four is devoted to is x , which is hyperbolic and unstable. For further summary and conclusion. − eqm details, see Ibrahim [12]. Incorporating economic parameters [9,13] intoEq,(2), 2 Compensation Model the optimal control problem can be formulated as ∞ −δt p2 2 Let the generic biological model be given by max J(E)= e p1qEx − (qEx) − cE dt E 2 Z0 dx(t) h dx x i = g(x) − h(t), (1) subject to = rx 1 − − qEx (5) dt dt K x(0)= x where x(t) represents the size of fish stock (or biomass) at 0 time t, g(x) is the net natural growth rate and h(t) is the 0 ≤ E ≤ Emax , rate of harvest of stock at time t. g(x) where δ is the annual discount rate, p is the price of Models in which the per capita growth rate is a 1 x landed fish, p2 represents the price component of the term decreasing function of the stock size are known as pure involving diminishing returns when there is a huge compensation models. They imply that, irrespective of quantity of fish on sale, c represents the cost per unit of how low the population is, it will always recover when fishing effort and Emax is the maximum allowable fishing fishing ceases [10]. A typical example of a compensation effort. model is the logistic growth curve (see Figure 1 (a)). The biological and inflation-adjusted economic Thus, the Schaefer model [11] can be expressed as parameter values used for the study are presented in Table 1. dx(t) x(t) = rx(t) 1 − − qE(t)x(t), x(0)= x , (2) The linear and quadratic revenues are depicted in dt K 0 Figure 2. The diminishing returns on the linear revenues are such that when the fishing effort and biomass are at where x is the initial biomass level, r is the intrinsic 0 the maximum sustainable yield (MSY) reference point, growth rate of fish stock, q is the catchability coefficient, the quadratic gross revenues are 15% less than the linear E(t) is the rate of fishing effort at time t, and K denotes revenues. Therefore, p is computed as the carrying capacity of the ecosystem in the absence of 2 harvesting. It is easily noted that the net growth rate of the 0.30p1 logistic (without harvesting) is strictly concave and p2 = positive for 0 < x < K (see Figure 1 (b)). Moreover, the qEMSY xMSY per capita growth rate decreases for 0 ≤ x ≤ K. = $5.07 × 10−4 year/tonne2 . Note that the harvest (or yield) is given by Data on the static reference points, maximum economic h(t)= qE(t)x(t). (3) yield (MEY), bionomic equilibrium (BE) and the MSY are presented in Table 2. It shows that, at the MEY levels, the There are two hyperbolicequilibrium points associated maximum revenue and the highest stock size are attained. with Eq. (2), namely 0 and a positive equilibrium point Thus, apart from providing the most revenue, it is also the qE most conservationist among the three. Clearly, the level xeqm = K 1 − , (4) at MSY provides the highest yield, and at the BE level, r it provides the most effort, lowest stock size, lowest yield, r and of course, zero revenue. Thus, fisheries managers must provided E < . q guard against this scenario. c 2020 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 14, No. 3, 467-479 (2020) / www.naturalspublishing.com/Journals.asp 469 3 1800 (a) (b) 2.5 1600 1400 2 1200 1.5 1000 1 800 0.5 600 Per Capita Growth Rate 0 400 Net Growth Rate (thousand tonnes/year) −0.5 200 −1 0 Compensation Model Compensation Model Critical Depensation Model Critical Depensation Model −1.5 −200 0 200 400 600 800 1000 0 200 400 600 800 1000 Biomass (thousand tonnes) Biomass (thousand tonnes) Fig. 1: Per capita growth rate and net growth rate for compensation and critical depensation models Table 1: Parameter values for model Parameter Description
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