Harvesting in a Random Varying Environment: Optimal, Stepwise and Sustainable Policies for the Gompertz Model
AbstractIn a random environment, the growth of a population subjected to harvesting can be described by a stochastic differential equation.We consider a population with natural growth following a Gompertz model with harvesting proportional to population size and to the exerted fishing effort. The main goal of this work is to compare the performance of three fishing policies: one with variable effort, named optimal policy, the second based on variable effort but with small periods with constant effort, named stepwise policy, and another with constant effort, denoted by optimal sustainable policy.We will show that the first policy is inapplicable from the practical point of view, whereas the second one, although being suboptimal, is applicable but has some shortcomings common to the optimal policy. On the contrary, the constant effort policy is easily implemented and predicts the sustainability of the population as well as the existence of a stationary density for its size. The performance of the policies will be assessed by the profit obtained over a finite time horizon. We also show how changes in the parameter values affect the profit differences between the variable effort policy and the constant effort policy.
ASMFC, Weakfish stock assessment report: a report of the ASMFC weakfish technical committee (SAW-SARC 48), NOAA, 2009.
L. H. R. Alvarez and L. A. Shepp, Optimal harvesting of stochastically fluctuating populations, Journal of Mathematical Biology, 37:155–177, 1998.
R. Arnason, L. K. Sandal, S. I. Steinshamn, and N. Vestergaard, Optimal feedback controls: comparative evaluation of the cod fisheries in Denmark, Iceland, and Norway, American Journal of Agricultural Economics, 86(2):531–542, 2004.
J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197(4302):463,1977.
R. Bellman, Dynamic Programming, Princeton University Press, New Jersey, 1957.
C. A. Braumann, Pescar num mundo aleat´orio: um modelo usando equacoes diferenciais estoc´asticas, In Actas VIII Jornadas Luso Espanholas Matem´atica, pages 301–308, Coimbra, 1981.
C. A. Braumann, Stochastic differential equation models of fisheries in an uncertain world: extinction probabilities, optimal fishing effort, and parameter estimation, In V Capasso, E Grosso, and S L Paveri Fontana, editors, Mathematics in Biology and Medicine,pages 201–206, Berlin, 1985. Springer.
C. A. Braumann, Growth and extinction of populations in randomly varying environments, Computers and Mathematics with Applications, 56(3):631–644, 2008.
N. M. Brites, Stochastic Differential Equation Harvesting Models: Sustainable Policies and Profit Optimization, PhD thesis, Universidade deEvora, 2017.
N. M. Brites and C. A. Braumann, Fisheries management in random environments: Comparison of harvesting policies for the logistic model, Fisheries Research, 195:238 – 246, 2017.
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources (2nd ed.), Wiley, New York, 1990.
F. B. Hanson, Applied Stochastic Processes and Control for Jump Diffusions: Modeling, Analysis, and Computation, Society for Industrial and Applied Mathematics, Philadelphia, 2007.
F. B. Hanson and D. Ryan, Optimal harvesting with both population and price dynamics, Mathematical Biosciences, 148(2):129–146, 1998.
T. K. Kar and K. Chakraborty, A bioeconomic assessment of the Bangladesh shrimp fishery, World Journal of Modelling and Simulation, 7(1):58 – 69, 2011.
S. Karlin and H. M. Taylor. A Second Course in Stochastic
Processes, Academic Press, New York, 1981.
B. Øksendal, Stochastic Differential Equations: An introduction with applications, 6th ed., Springer-Verlag, Berlin Heidelberg,2003.
R. Suri, Optimal Harvesting Strategies for Fisheries: A Differential Equations Approach, PhD thesis, Massey University, Albany,New Zealand, 2008.
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