Contents lists available at ScienceDirect Fisheries Research journal homepage: www.elsevier.com/locate/fishres Fisheries management in randomly varying environments: Comparison of constant, variable and penalized eforts policies for the Gompertz model Nuno M. Brites a, ⁎ ,CarlosA.Braumann a,b a Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora, Évora, 7000-671, Portugal b Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora, Évora, 7000-671, Portugal ARTICLEINFO Handled by A.E. Punt Keywords: Fisheries management Random environments Stochastic diferential equations Proft optimization Gompertz model ABSTRACT In a previous paper, we discussed the use of an optimal variable efort fshing policy versus an optimal sus- tainableconstantefort fshingpolicyintermsoftheexpectedaccumulateddiscountedproftduringafnitetime interval. We concluded that there is only a slight reduction in proft when choosing the applicable optimal sustainable constant efort fshing policy instead of the optimal variable efort fshing policy, which leads to major disadvantages in practice. In this paper, we confrm these conclusions by considering a diferent model, theGompertzmodel,andbyusinganotherrealisticdatasetofparametersandamoregeneralproft structure. Wealsoshowthatsomeofthedisadvantagesoftheoptimalvariableefort fshingpolicy,namelythoserelatedto socialobjectives,areeliminatedbyconsideringapenalizedproftwithanartifcialrunningenergycostonthe efort. However, the applicability problems remain. We also show that the proft advantage of this optimal penalized variable efort policy over the optimal sustainable constant efort policy is even smaller than the alreadyverysmalladvantageofthenon-penalizedpolicy.Thisfurtherreinforcestherobustnessofourprevious conclusions that the optimal sustainable constant efort policy, which does not have the shortcomings of the optimal variable efort fshing policy, is only slightly less proftable than it. 1. Introduction Stochastic diferential equations (SDE) have been applied to the growthdynamicsofharvestedpopulationslivinginarandomlyvarying environment, with the purpose of obtaining optimal fshing policies (Alvarez and Shepp, 1998; Braumann, 1985; Hanson and Ryan, 1998; Suri, 2008). In a random environment, the typical approach to obtain suchpoliciesusesstochasticoptimalcontrolmethods(thefshingefort being the control), to maximize the expected accumulated discounted proft (or the yield) over a time horizon T . However, contrary to the deterministic case (see, for instance, Clark, 1990), the population cannot be kept at an equilibrium size and will rather continue to ex- perience random fuctuations. Therefore, the optimal fshing efort mustbeadjustedateveryinstant,sothatthesizeofthepopulationis below (and close to) some threshold value. Consequently, the optimal fshing efort will have frequent transitions between maximum/high efortandlow/nullefort.Thesetransitionsarenotcompatiblewiththe operation of fsheries. In addition, the period of low/no harvesting poses undesirable socio-economic implications (intermittent un- employment is just one of them). In addition, these optimal policies requiretheknowledgeofthepopulationsizeateveryinstanttodefne theappropriatelevelofefort.Theestimationofthepopulationsizeis difcult, costly, time consuming and inaccurate. For all these reasons, such policies should be considered unacceptable and inapplicable. Braumann(2008, 1985, 1981) assumedconstantfshingefortand, foralargeclassofmodels,foundthatthereis,undermildconditions,a stochastic sustainable behaviour. Namely, the probability distribution of the population size at time t will converge, as + t ,toanequi- librium probability distribution (stationary or steady-state) having a probability density function (stationary density). The optimal sustain- ableyieldandefortwere foundbuttheissueofproftoptimizationwas not addressed. Brites (2017) considered, as an alternative to the optimal variable efort fshing policies, optimal sustainable constant efort fshing po- licies based on proft optimization, which are extremely easy to im- plement and lead to a stochastic steady-state. We determined the con- stant efort that maximizes the expected proft per unit time at the steady-state, considering general population SDE growth models and also (as in Brites and Braumann (2017)) specifc models such as the logistic. One might think that an optimal sustainable constant efort policywouldresultinasubstantialproftreductioncomparedwiththe optimal variable efort policy. Through Monte Carlo simulations, we https://doi.org/10.1016/j.fshres.2019.03.016 Received 20 September 2018; Received in revised form 11 March 2019; Accepted 12 March 2019 ⁎ Corresponding author. E-mail address: brites@uevora.pt (N.M. Brites). Fisheries Research 216 (2019) 196–203 0165-7836/ © 2019 Elsevier B.V. All rights reserved. T