Contents lists available at ScienceDirect Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs Renewable resource management in a seasonally fluctuating environment with restricted harvesting effort Srinivasu D.N. Pichika ⁎ , Simon D. Zawka Department of Mathematics, Andhra University, Visakhapatnam, 530003, India ARTICLE INFO Keywords: Bioeconomics Binding constraint Blocked interval Optimal periodic solution ABSTRACT This paper presents bio-economics of a renewable resources in a seasonally changing environment in which the resource exploitation is subjected to restrictions on harvesting effort. The dynamics of the resource is assumed to be governed by the logistic equation. Seasonality is incorporated into the system by choosing the coefficients in the growth equation to be periodic functions with the same period. A linear optimal control problem involving binding constraints on the control variable has been considered. As a result the concept of blocked interval plays a key role in the construction of optimal solution. In view of the periodicity associated with the considered problem, we first construct an optimal periodic solution. The optimal solution is established using the most rapid approach path to the said optimal periodic solution. The global asymptotic stability property of the optimal periodic solution enables construction of a suboptimal solution. The optimal solution is found to be periodic after some finite time and suboptimal solution approaches the optimal periodic solution asymptotically. Key results are illustrated through numerical simulation. 1. Introduction Optimal exploitation of a renewable resource has been a common concern for researchers, owners of the resource as well as open access users. The books authored by Clark [2,3] offer an excellent exposure to various aspects associated with bio-economics of resources such as optimal resource management and sustainable exploitation of re- sources. Some other works in this area can be found in [7,11,12,18,23,25,27,29,32]. Enormous contributions made on the optimal exploitation of resources pertaining to autonomous systems have driven the researchers to expand the domain to include non-au- tonomous systems. One of the vital concepts considered for inclusion in this regard is the influence of environmental fluctuations on the re- source, in particular, seasonal variations. Incorporating seasonality into the resource dynamics makes the mathematical model more reliable and takes the study closer to reality. Profound influence of the seasonal variations on the dynamics of re- newable resources such as fisheries, naturally guide the investigators to model the optimal exploitation problems in a periodic environment. This is to ensure that the study mimics the real world as closely as possible. We expect that outcomes of such studies will be in a position to offer solutions to some practical problems encountered by the ex- ploiters. One of the recent works which is appropriate in this framework is Hasanbulli et al. [5] and references therein. Some recent and relevant work on seasonally varying environment can be found in [8,10,14–17,20–22,24,26]. In this context, the work presented in Castilho and Srinivasu [9] becomes significant and it contains several references pertaining to the dynamics and management of renewable resources in periodically fluctuating environment. It presents a method to solve a linear optimal control problem wherein the dynamic constraints are periodic differ- ential equations. Solutions were constructed using Pontryagin’s Max- imum principle [34]. Thus an optimal periodic singular control and optimal periodic singular stock path are obtained for the considered optimal harvest problem. In practice we come across situations where it may not be possible to follow the singular stock path due to some restrictions on the fishing effort such as maximum/minimum number of vessels to be used for the harvesting activity. The major driving force for restricting the capacity of fishing vessels is to overcome the over exploitation of the resource and avoid resource depletion. On the other hand the minimum possible number of vessels or the minimum possible fleet size may also be considered in some fisheries management activities due to socio- economic conditions. The study presented in [13] can be considered as a good example for restriction on vessels to overcome over exploitation. The book by Barkin and DeSombre [4] highlights the need to reduce fishing capacity and to decrease the number of people and amount of capital employed in the industry so that the reduction of over https://doi.org/10.1016/j.mbs.2017.12.008 Received 16 December 2016; Received in revised form 16 December 2017; Accepted 26 December 2017 ⁎ Corresponding author. E-mail address: pdnsrini@gmail.com (S.D.N. Pichika). Mathematical Biosciences xxx (xxxx) xxx–xxx 0025-5564/ © 2018 Elsevier Inc. All rights reserved. Please cite this article as: Pichika, S., Mathematical Biosciences (2018), https://doi.org/10.1016/j.mbs.2017.12.008