Theoretical Ecology (2019) 12:131–144 https://doi.org/10.1007/s12080-018-0399-7 ORIGINAL PAPER Marine reserves and optimal dynamic harvesting when fishing damages habitat Michael R. Kelly Jr. 1 · Michael G. Neubert 2 · Suzanne Lenhart 3 Received: 3 September 2018 / Accepted: 11 November 2018 / Published online: 29 November 2018 © Springer Nature B.V. 2018 Abstract Marine fisheries are a significant source of protein for many human populations. In some locations, however, destructive fishing practices have negatively impacted the quality of fish habitat and reduced the habitat’s ability to sustain fish stocks. Improving the management of stocks that can be potentially damaged by harvesting requires improved understanding of the spatiotemporal dynamics of the stocks, their habitats, and the behavior of the harvesters. We develop a mathematical model for both a fish stock as well as its habitat quality. Both are modeled using nonlinear, parabolic partial differential equations, and density dependence in the growth rate of the fish stock depends upon habitat quality. The objective is to find the dynamic distribution of harvest effort that maximizes the discounted net present value of the coupled fishery-habitat system. The value derives both from extraction (and sale) of the stock and the provisioning of ecosystem services by the habitat. Optimal harvesting strategies are found numerically. The results suggest that no-take marine reserves can be an important part of the optimal strategy and that their spatiotemporal configuration depends both on the vulnerability of habitat to fishing damage and on the timescale of habitat recovery when fishing ceases. Keywords Fisheries bioeconomics · Marine protected areas · Optimal control · Destructive fishing · Ecosystem-based management Introduction It is not uncommon for modern fishing gear to impact the habitats of the fish being harvested. Trawls and dredges— fishing gear that is dragged along the ocean’s floor—can be particularly harmful to the benthic habitat upon which many commercially valuable species rely. Indeed the effects of these gears in some locations have been compared to forest clearcutting (Watling and Norse 1998). These effects are Electronic supplementary material The online version of this article (https://doi.org/10.1007/s12080-018-0399-7) contains supplementary material, which is available to authorized users.  Michael G. Neubert mneubert@whoi.edu 1 Division of Natural Sciences and Mathematics, Transylvania University, 300 North Broadway, Lexington, KY 40508, USA 2 Biology Department and Marine Policy Center, Woods Hole Oceanographic Institution, Wood’s Hole, MA 02543-1049, USA 3 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA well known; reviews of fishing gear impacts can be found in papers by Dayton et al. (1995) and Chuenpagdee et al. (2003) and Grabowski et al. (2014). Recent analyses suggest that the extent of trawling, particularly in the deep sea, has been underestimated (Victorero et al. 2018). Given the collateral damage that fishing may impose on essential habitat, it is reasonable to ask whether marine re- serves—areas that are closed to fishing—might be a useful part of management. After all, reserves would protect both stocks and their habitats. Of course, closures might also negatively affect yields or rents, by closing off a portion of the stock to harvest as well as by changing the distribution of harvesters (Kaiser et al. 2002; Kellner et al. 2007). The analysis of bioeconomic models is useful for understanding this tradeoff. Spatially explicit models are necessary to fully address the question of marine reserve utility, since reserves are a form of spatial management (Herrera and Lenhart 2010). Previous analyses of spatial models (that ignore the potential for habitat damage) have found that reserves may be necessary as part of a strategy designed to maximize yield or rent depending on the ecological and economic circumstances (e.g., Neubert 2003; Joshi et al.