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**Harvesting in seasonal environments.**
*(English)*
Zbl 1066.92057

Summary: Most harvest theory is based on an assumption of a constant or stochastic environment, yet most populations experience some form of environmental seasonality. Assuming that a population follows logistic growth we investigate harvesting in seasonal environments, focusing on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. We show that the optimal strategy depends dramatically on the intrinsic growth rate of the population and the magnitude of seasonality. The ordered effectiveness of these alternative harvest strategies is given for different combinations of intrinsic growth rate and seasonality. Also, for piecewise continuous-time harvest strategies (i.e., open \(/\) closed harvest, and pulse harvest) harvest timing is of crucial importance to annual yield. Optimal timing for harvests coincides with maximal rate of decline in the seasonally fluctuating carrying capacity.

For large intrinsic growth rates and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to the optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.

For large intrinsic growth rates and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to the optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.

### MSC:

92D40 | Ecology |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |

34C25 | Periodic solutions to ordinary differential equations |

49N90 | Applications of optimal control and differential games |

### Keywords:

Differential equations; Population dynamics; Environmental seasonality; Maximum annual yield; Population persistence; Wildlife management
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\textit{C. Xu} et al., J. Math. Biol. 50, No. 6, 663--682 (2005; Zbl 1066.92057)

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### References:

[1] | Alvarez, L.H.R., Shepp, L.A.: Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177 (1998) · Zbl 0940.92029 |

[2] | Andrewartha, H.G., Birch, L.C.: The distribution and abundance of animals. Chicago: Chicago University Press, 1954 |

[3] | Annegars, J.F.: Seasonal food shortages in West Africa. Ecol. Food Nutr. 2, 251–257 (1973) |

[4] | Bartmann, R.M., White, G.C., Carpenter, L.H.: Compensatory mortality in a Colorado mule deer population. Wildl. Monogr. 121, 1–39 (1992) |

[5] | Batzli, G.O.: Can seasonal changes in density dependence drive population cycles? Trends Ecol. Evol. 14, 129–131 (1999) |

[6] | Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977) |

[7] | Bobek, B., Perzanowski, K., Sajdak, G., Szulakowska, G.: Seasonal changes in quality and quantity of deer browse in a deciduous forest. Proc. Int. Congr. Game Biol. 11, 545–552 (1974) |

[8] | Boyce, M.S.: Seasonality and patterns of natural selection for life histories. Am. Nat. 114, 569–583 (1979) |

[9] | Boyce, M.S., Daley, D.J.: Population tracking of fluctuating environments and natural selection for tracking ability. Am. Nat. 115, 480–491 (1980) |

[10] | Boyce, M.S., Sinclair, A.R.E., White, G.C.: Seasonal compensation of predation and harvesting. Oikos 87, 419–426 (1999) |

[11] | Caughley, G.: Analysis of vertebrate populations. New York: Wiley, 1977 |

[12] | Clark, C.W.: Mathematical bioeconomics. The optimal management of renewable resources. New York: Wiley, 1976 · Zbl 0364.90002 |

[13] | Clark, C.W.: Mathematical bioeconomics. The optimal management of renewable resources. 2d ed. New York: Wiley, 1990 · Zbl 0712.90018 |

[14] | Cooke, K.L., Witten, M.: One-dimensional linear and logistic harvesting models. Math. Model. 7, 301–340 (1986) · Zbl 0611.92025 |

[15] | Crombie, A.C.: On competition between different species of graminivorous insects. Proc. R. Soc. Lond., B. 132, 362–395 (1945) |

[16] | Cromer, T.L.: Harvesting in a seasonal environment. Mathl. Comput. Modelling 10, 445–450 (1988) · Zbl 0646.90022 |

[17] | Crowder, L.B., Murawski, S.A.: Fisheries by-catch: implications for management. Fisheries 23, 8–17 (1998) |

[18] | Davidson, J.: On the ecology of the growth of the sheep population in South Australia. Trans. R. Soc. S. Aust. 62, 141–148 (1938) |

[19] | De Roos, A.M., Persson, L., McCauley, E.: The influence of size-dependent life-history traits on the structure and dynamics of populations and communities. Ecol. Letters 6, 473–487 (2003) |

[20] | Emmel, T.C.: Population biology. New York: Harper& Row, 1976 |

[21] | Errington, P.L.: Predation and vertebrate populations. Q. Rev. Biol. 21, 144–177 (1946) |

[22] | Flaaten, O.: The optimal harvesting of a natural resource with seasonal growth. Can. J. Econ. 16, 447–462 (1983) |

[23] | Fretwell, S.D.: Populations in a seasonal environment. Princeton: Princeton University Press, 1972 |

[24] | Gaillard, J.M., Festa-Bianchet, M., Yoccoz, N.G.: Population dynamics of large herbivores: variable recruitment with constant adult survival. Trends Ecol. Evol. 13, 58–63 (1998) |

[25] | Grenfell, B.T., Finkenstädt, B.F.: Seasonality, stochasticity and population cycles. Res. Popul. Ecol. 40, 141–143 (1998) |

[26] | Hansson, L., Henttonen, H.: Rodent fluctuations in relation to seasonality in Fennoscandia and Hokkaido. Res. Popul. Ecol. 40, 127–129 (1998) |

[27] | Haramis, G.M., Thompson, D.Q.: Density-production characteristics of box-nesting wood ducks in a northern greenhouse impoundment. J. Wildl. Manage. 49, 429–436 (1985) |

[28] | Jenkins, D., Watson, A., Miller, G.R.: Predation and red grouse populations. J. Appl. Ecol. 1, 183–195 (1964) |

[29] | Jonzén, N., Lundberg, P.: Temporally structured density dependence and population management. Ann. Zool. Fenn. 36, 39–44 (1999) |

[30] | Klerk, P.D., Gatto, M.: Some remarks on periodic harvesting of a fish population. Math. Biosci. 56, 47–69 (1981) · Zbl 0476.92016 |

[31] | Kokko, H., Lindström, J.: Seasonal density dependence, timing of mortality, and sustainable harvesting. Ecol. Model. 110, 293–304 (1998) |

[32] | Kokko, H., Pöysä, H., Lindström, J., Ranta, E.: Assessing the impact of spring hunting on waterfowl populations. Ann. Zool. Fenn. 35, 195–204 (1998) |

[33] | Kot, M.: Elements of mathematical ecology. Cambridge: Cambridge University Press, 2001 · Zbl 1060.92058 |

[34] | Lande, R., Engen, S., Saether, B.E.: Optimal harvesting of fluctuating populations with a risk of extinction. Am. Nat. 145, 728–745 (1995) |

[35] | Ludwig, D.: Harvesting strategies for a randomly fluctuating population. J. Cons. Int. Explor. Mer. 39, 168–174 (1980) |

[36] | Lungu, E.M., Øksendal, B.: Optimal harvesting from a population in a stochastic crowded environment. Preprint Series, vol 10, Department of Mathematics, University of Oslo, 1996 · Zbl 0885.60052 |

[37] | May, R.M., Beddington, J.R., Horwood, J.W., Shepherd, J.G.: Exploiting natural populations in an uncertain world. Math. Biosci. 42, 219–252 (1978) |

[38] | Murton, R.K., Westwood, N.J., Isaacson, A.J.: A study of wood-pigeon shooting: the exploitation of a natural animal population. J. Appl. Ecol. 11, 61–81 (1974) |

[39] | Nichols, J.D., Conroy, M.J., Anderson, D.R., Burnham, K.P.: Compensatory mortality in waterfowl populations: a review of the evidence and implication for research and management. Trans. N. Am. Wildl. Nat. Resour. Conf. 49, 535–554 (1984) |

[40] | Odum, E.P.: Fundamental of ecology. 3d ed. Philadelphia: Saunders, 1971 |

[41] | Paine, R.T.: Phycology for the mammalogist: Marine rocky shores and mammal-dominated communities - How different are the structuring processes?J. Mammal. 81, 637–648 (2000) |

[42] | Reed, W.J.: The steady state of a stochastic harvesting model. Math. Biosci. 41, 273–307 (1978) · Zbl 0385.92016 |

[43] | Reed, W.J.: Optimal escapement levels in stochastic and deterministic harvesting models. J. Envir. Econ. Manage. 6, 350–363 (1979) · Zbl 0439.90020 |

[44] | Ribenboim, P.: Functions, limits, and continuity. New York: John Wiley& Sons, Inc., 1964 · Zbl 0118.28601 |

[45] | Saitoh, T., Stenseth, N.C., Bjørnstad, O.N.: The population dynamics of the vole Clethrionomys rufocanus in Hokkaido, Japan. Res. Popul. Ecol. 40, 61–76 (1998) |

[46] | Sanchez, D.A.: Periodic environments, harvesting, and a Ricatti equation. In: Nonlinear Phenomena in Mathematical Sciences. New York: Academic Press, 1982 |

[47] | Shepherd, J.G., Horwood, J.W.: The sensitivity of exploited populations to environment ”noise”, and the implications for management. J. Cons. Int. Explor. Mer 38, 318–323 (1979) |

[48] | Sullivan, M.: Active management of walleye fisheries in Alberta: dilemmas of managing recovering fisheries. N. Am. J. Fish. Manage. 23, 1341–1356 (2003) |

[49] | Swenson, J.E.: Compensatory reproduction in an introduced mountain goat population in the Absaroka Mountains, Montana. J. Wildl. Manage. 49, 837–843 (1985) |

[50] | Tang, S.Y., Chen, L.S.: The effect of seasonal harvesting on stage-structured population models. J. Math. Biol. 48, 357–374 (2004) · Zbl 1058.92051 |

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