# Asymptotic harvesting of populations in random environments

- 34 Downloads

## Abstract

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction—instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang–bang property: there exists a threshold \(x^*>0\) such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is \(C^2\) and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang–bang type. This shows that one cannot always expect bang–bang type optimal controls.

## Keywords

Ergodic control Stochastic harvesting Ergodicity Stochastic logistic model Stochastic environment## Mathematics Subject Classification

92D25 60J70 60J60## Notes

### Acknowledgements

We thank two anonymous referees for very insightful comments and suggestions that led to major improvements.

## References

- Abakuks A (1979) An optimal hunting policy for a stochastic logistic model. J Appl Probab 16(2):319–331MathSciNetCrossRefMATHGoogle Scholar
- Arapostathis A, Borkar V S, Ghosh M K (2012) Ergodic control of diffusion processes. Cambridge University Press, CambridgeMATHGoogle Scholar
- Alvarez LHR (2000) Singular stochastic control in the presence of a state-dependent yield structure. Stoch Process Appl 86(2):323–343MathSciNetCrossRefMATHGoogle Scholar
- Abakuks A, Prajneshu (1981) An optimal harvesting policy for a logistic model in a randomly varying environment. Math Biosci 55(3–4):169–177MathSciNetCrossRefMATHGoogle Scholar
- Alvarez LHR, Shepp LA (1998) Optimal harvesting of stochastically fluctuating populations. J Math Biol 37(2):155–177MathSciNetCrossRefMATHGoogle Scholar
- Beddington JR, May RM (1977) Harvesting natural populations in a randomly fluctuating environment. Science 197(4302):463–465CrossRefGoogle Scholar
- Berg C, Pedersen HL (2006) The Chen–Rubin conjecture in a continuous setting. Methods Appl Anal 13(1):63–88MathSciNetMATHGoogle Scholar
- Berg C, Pedersen HL (2008) Convexity of the median in the gamma distribution. Ark Mat 46(1):1–6MathSciNetCrossRefMATHGoogle Scholar
- Braumann CA (2002) Variable effort harvesting models in random environments: generalization to density-dependent noise intensities. Math Biosci 177(178):229–245 (Deterministic and stochastic modeling of biointeraction (West Lafayette, IN, 2000))MathSciNetCrossRefMATHGoogle Scholar
- Benaïm M, Schreiber SJ (2009) Persistence of structured populations in random environments. Theor Popul Biol 76(1):19–34CrossRefMATHGoogle Scholar
- Borodin AN, Salminen P (2012) Handbook of Brownian motion-facts and formulae. Birkhäuser, BaselMATHGoogle Scholar
- Dennis B, Patil GP (1984) The gamma distribution and weighted multimodal gamma distributions as models of population abundance. Math Biosci 68(2):187–212MathSciNetCrossRefMATHGoogle Scholar
- Drèze J, Stern N (1987) The theory of cost-benefit analysis. Handb Pub Econ 2:909–989MathSciNetCrossRefGoogle Scholar
- Evans SN, Hening A, Schreiber SJ (2015) Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments. J Math Biol 71(2):325–359MathSciNetCrossRefMATHGoogle Scholar
- Ethier S N, Kurtz T G (2009) Markov processes: characterization and convergence. Wiley, New YorkMATHGoogle Scholar
- Evans SN, Ralph PL, Schreiber SJ, Sen A (2013) Stochastic population growth in spatially heterogeneous environments. J Math Biol 66(3):423–476MathSciNetCrossRefMATHGoogle Scholar
- Gard TC (1984) Persistence in stochastic food web models. Bull Math Biol 46(3):357–370MathSciNetCrossRefMATHGoogle Scholar
- Gard TC (1988) Introduction to stochastic differential equations. M. Dekker, New YorkMATHGoogle Scholar
- Gard TC, Hallam TG (1979) Persistence in food webs. I. Lotka-Volterra food chains. Bull Math Biol 41(6):877–891MathSciNetMATHGoogle Scholar
- Gulland JA (1971) The effect of exploitation on the numbers of marine animals. In: Proceedings of the advanced study institute on dynamics of numbers in populations, pp 450–468Google Scholar
- Hening A, Kolb M (2018) Quasistationary distributions for one-dimensional diffusions with singular boundary points (submitted). arXiv:1409.2387
- Hening A, Nguyen D (2018) Coexistence and extinction for stochastic Kolmogorov systems. Ann Appl Probab. arXiv:1704.06984
- Hening A, Nguyen D (2018) Persistence in stochastic Lotka-Volterra food chains with intraspecific competition (preprint). arXiv:1704.07501
- Hening A, Nguyen D (2018) Stochastic Lotka-Volterra food chains. J Math Biol. arXiv:1703.04809
- Hening A, Nguyen D, Yin G (2018) Stochastic population growth in spatially heterogeneous environments: the density-dependent case. J Math Biol 76(3):697–754MathSciNetCrossRefMATHGoogle Scholar
- Hutchings JA, Reynolds JD (2004) Marine fish population collapses: consequences for recovery and extinction risk. AIBS Bull 54(4):297–309Google Scholar
- Khasminskii R (2012) Stochastic stability of differential equations. In: Rozovskiĭ B, Glynn PW (eds) Stochastic modelling and applied probability, vol 66, 2nd edn. Springer, Heidelberg (With contributions by G. N. Milstein and M. B. Nevelson)Google Scholar
- Kokko H (2001) Optimal and suboptimal use of compensatory responses to harvesting: timing of hunting as an example. Wildl Biol 7(3):141–150CrossRefGoogle Scholar
- Leigh EG (1981) The average lifetime of a population in a varying environment. J Theor Biol 90(2):213–239MathSciNetCrossRefGoogle Scholar
- Lande R, Engen S, Saether B-E (1995) Optimal harvesting of fluctuating populations with a risk of extinction. Am Nat 145(5):728–745CrossRefGoogle Scholar
- Ludwig D, Hilborn R, Walters C (1993) Uncertainty, resource exploitation, and conservation: lessons from history. Ecol Appl 3:548–549Google Scholar
- Lungu EM, Øksendal B (1997) Optimal harvesting from a population in a stochastic crowded environment. Math Biosci 145(1):47–75MathSciNetCrossRefMATHGoogle Scholar
- May RM, Beddington JR, Horwood JW, Shepherd JG (1978) Exploiting natural populations in an uncertain world. Math Biosci 42(3–4):219–252MathSciNetCrossRefGoogle Scholar
- Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory, vol 1. Oxford University Press, New YorkMATHGoogle Scholar
- Merton RC (1969) Lifetime portfolio selection under uncertainty: the continuous-time case. Rev Econ Stat 51:247–257CrossRefGoogle Scholar
- Merton RC (1971) Optimum consumption and portfolio rules in a continuous-time model. J Econ Theory 3(4):373–413MathSciNetCrossRefMATHGoogle Scholar
- Primack RB (2006) Essentials of conservation biology. Sinauer Associates, SunderlandGoogle Scholar
- Reiter J, Panken KJ, Le Boeuf BJ (1981) Female competition and reproductive success in northern elephant seals. Anim Behav 29(3):670–687CrossRefGoogle Scholar
- Roth G, Schreiber SJ (2014) Persistence in fluctuating environments for interacting structured populations. J Math Biol 69(5):1267–1317MathSciNetCrossRefMATHGoogle Scholar
- Schreiber SJ, Benaïm M, Atchadé KAS (2011) Persistence in fluctuating environments. J Math Biol 62(5):655–683MathSciNetCrossRefMATHGoogle Scholar
- Shaffer ML (1981) Minimum population sizes for species conservation. Bioscience 31(2):131–134MathSciNetCrossRefGoogle Scholar
- Smith JB (1978) An analysis of optimal replenishable resource management under uncertainty. Digitized Theses, Paper 1074Google Scholar
- Schreiber SJ, Ryan ME (2011) Invasion speeds for structured populations in fluctuating environments. Theor Ecol 4(4):423–434CrossRefGoogle Scholar
- Traill LW, Bradshaw CJA, Brook BW (2007) Minimum viable population size: a meta-analysis of 30 years of published estimates. Biol Conserv 139(1):159–166CrossRefGoogle Scholar
- Tyson R, Lutscher F (2016) Seasonally varying predation behavior and climate shifts are predicted to affect predator-prey cycles. Am Nat 188(5):539–553CrossRefGoogle Scholar
- Turelli M (1977) Random environments and stochastic calculus. Theor Popul Biol 12(2):140–178MathSciNetCrossRefMATHGoogle Scholar