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Modelling the Impact of Marine Reserves on a Population with Depensatory Dynamics

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Abstract

In this study, we use a spatially implicit, stage-structured model to evaluate marine reserve effectiveness for a fish population exhibiting depensatory (strong Allee) effects in its dynamics. We examine the stability and sensitivity of the equilibria of the modelled system with regards to key system parameters and find that for a reasonable set of parameters, populations can be protected from a collapse if a small percentage of the total area is set aside in reserves. Furthermore, the overall abundance of the population is predicted to achieve a maximum at a certain ratio \(A\) of reserve area to fished area, which depends heavily on the other system parameters such as the net export rate of fish from the marine reserves to the fished areas. This finding runs contrary to the contested “equivalence at best” result when comparing fishery management through traditional catch or effort control and management through marine reserves. Lastly, we analyse the problem from a bioeconomics perspective by computing the optimal harvesting policy using Pontryagin’s Maximum Principle, which suggests that the value for \(A\) which maximizes the optimal equilibrium fishery yield also maximizes population abundance when the cost per unit harvest is constant, but can increase substantially when the cost per unit harvest increases with the area being harvested.

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Acknowledgments

The work of MHC was supported by the Australian Postgraduate Award. PSK was supported by the ARC Discovery Early Career Research Award (DE120101113).

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Matthew H. Chan & Peter S. Kim

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  1. Matthew H. Chan
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  2. Peter S. Kim
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Correspondence to Matthew H. Chan.

Appendix

Appendix

We list here two-parameter bifurcation diagrams of the system with respect to parameters \(n\), \(B\), \(\sigma _1\) and \(\sigma _2\).

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Chan, M.H., Kim, P.S. Modelling the Impact of Marine Reserves on a Population with Depensatory Dynamics. Bull Math Biol 76, 2122–2143 (2014). https://doi.org/10.1007/s11538-014-0003-2

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