# American Institute of Mathematical Sciences

March  2016, 21(2): 699-719. doi: 10.3934/dcdsb.2016.21.699

## Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  January 2015 Revised  June 2015 Published  November 2015

In this paper, we investigate a class of population model with seasonal constant-yield harvesting and discuss the effect of the seasonal harvesting on the survival of the population. It is shown that the population can be survival if and only if the model has at least a periodic solution and the initial amount of population is not lower than the minimum periodic solution. And if the population goes to extinction then it must be in the finite time. As an application of the conclusion, we systemically study the global dynamics of a logstic equation with seasonal constant-yield harvesting, and prove that there exists a threshold value $h_{MSY}$ of the intensity of harvesting, called the maximum sustainable yield, which is strictly greater than the maximum sustainable yield of logstic equation with constant-yield harvesting, such that the model has exactly two periodic solutions: one is attracting and the other is repelling if $0 < h < h_{MSY}$, a unique periodic solution which is semi-stable if $h=h_{MSY}$ and all solutions which go down to zero in the finite time if $h>h_{MSY}$. Hence, the logstic equation with seasonal constant-yield harvesting undergoes saddle-node bifurcation of the periodic solution as $h$ passes through $h_{MSY}$. Biologically, these theoretic results reveal that the seasonal constant-yield harvesting can increase the maximum sustainable yield such that the ecological system persists comparing to the constant-yield harvesting.
Citation: Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 699-719. doi: 10.3934/dcdsb.2016.21.699
##### References:
 [1] F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey systems, J.Math.Biol., 7 (1979), 319-337. doi: 10.1007/BF00275152. [2] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. [3] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1982), 101-114. doi: 10.1007/BF00275206. [4] F. Brauer and D. A. Sánchez, Periodic environments and periodic harvesting, Natural Resource Modeling, 16 (2003), 233-244. doi: 10.1111/j.1939-7445.2003.tb00113.x. [5] J. Chen, J. Huang, S. Ruan and J. Wang, Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J. Appl. Math., 73 (2013), 1876-1905. doi: 10.1137/120895858. [6] J. P. Cohen and J. S. Foale, Sustaining small-scale fisheries with periodically harvested marine reserves, Marine Policy, 37 (2013), 278-287. doi: 10.1016/j.marpol.2012.05.010. [7] C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, New York -Toronto, 1990. [8] G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210. doi: 10.1137/S0036139994275799. [9] R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalissed Gause model with prey harvesting and a generalized Holling response function of type III, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021. [10] M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Mathematical Biosciences, 152 (1998), 165-177. doi: 10.1016/S0025-5564(98)10024-X. [11] M. Hasanbulli , P. S. Rogovchenko and Y. V. Rogovchenko, Dynamics of a single species in a fluctuating environment under periodic yield harvesting, Journal of Applied Mathematics, Volume 2013, Article ID 167671, 12 pages. [12] L. S. Hill, J. E. Murphy, K. Reid, P. N. Trathan and J. A. Constable, Modelling Southern Ocean ecosystems: Krill, the food-web, and the impacts of harvesting, Biol. Rev., 81 (2006), 581-608. [13] M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and An Introduction to Chaos, Elsevier, New York, 2004. doi: 10.1007/978-1-4612-0873-0. [14] S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6. [15] S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage, Ecology, 76 (1995), 2278-2294. doi: 10.2307/1941702. [16] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-depedent predator-prey models with nonconstant harvesting, Discrete Contin. Dynam. Syst. Ser. S 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [17] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. doi: 10.1126/science.205.4403.267. [18] M. B. Schaefer, Some aspects of the dynamics of populations important to the management of commercial marine fisheries, Bull. Inter-Am. Trop. Tuna Comm., 1 (1954), 25-56. [19] R. J. Schmitt and S. J. Holbrook, Seasonally fluctuating resources and temporal variability of interspecific competition, Oecologia, 69 (1986), 1-11. doi: 10.1007/BF00399030. [20] Y. L. Tang and D. Xiao, Periodic solutions for a predator-prey model with periodic harvesting rate, International Journal of Bifurcation and Chaos, 24 (2014), 1450096(12 pages). doi: 10.1142/S0218127414500965. [21] C. J. Waiters and P. J. Bandy, Periodic harvest as a method of increasing big game yields, J. Wildl. Manage., 36 (1972), 128-134. [22] D. Xiao and L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. [23] D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048. [24] C. Xu, M. S. Boyce and D. J. Daley, Harvesting in seasonal environments, Journal of Mathematical Biology, 50 (2005), 663-682. doi: 10.1007/s00285-004-0303-5.

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##### References:
 [1] F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey systems, J.Math.Biol., 7 (1979), 319-337. doi: 10.1007/BF00275152. [2] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. [3] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1982), 101-114. doi: 10.1007/BF00275206. [4] F. Brauer and D. A. Sánchez, Periodic environments and periodic harvesting, Natural Resource Modeling, 16 (2003), 233-244. doi: 10.1111/j.1939-7445.2003.tb00113.x. [5] J. Chen, J. Huang, S. Ruan and J. Wang, Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J. Appl. Math., 73 (2013), 1876-1905. doi: 10.1137/120895858. [6] J. P. Cohen and J. S. Foale, Sustaining small-scale fisheries with periodically harvested marine reserves, Marine Policy, 37 (2013), 278-287. doi: 10.1016/j.marpol.2012.05.010. [7] C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, $2^{nd}$ edition, John Wiley & Sons, New York -Toronto, 1990. [8] G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210. doi: 10.1137/S0036139994275799. [9] R. M. Etoua and C. Rousseau, Bifurcation analysis of a generalissed Gause model with prey harvesting and a generalized Holling response function of type III, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021. [10] M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients, Mathematical Biosciences, 152 (1998), 165-177. doi: 10.1016/S0025-5564(98)10024-X. [11] M. Hasanbulli , P. S. Rogovchenko and Y. V. Rogovchenko, Dynamics of a single species in a fluctuating environment under periodic yield harvesting, Journal of Applied Mathematics, Volume 2013, Article ID 167671, 12 pages. [12] L. S. Hill, J. E. Murphy, K. Reid, P. N. Trathan and J. A. Constable, Modelling Southern Ocean ecosystems: Krill, the food-web, and the impacts of harvesting, Biol. Rev., 81 (2006), 581-608. [13] M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and An Introduction to Chaos, Elsevier, New York, 2004. doi: 10.1007/978-1-4612-0873-0. [14] S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6. [15] S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage, Ecology, 76 (1995), 2278-2294. doi: 10.2307/1941702. [16] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-depedent predator-prey models with nonconstant harvesting, Discrete Contin. Dynam. Syst. Ser. S 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [17] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. doi: 10.1126/science.205.4403.267. [18] M. B. Schaefer, Some aspects of the dynamics of populations important to the management of commercial marine fisheries, Bull. Inter-Am. Trop. Tuna Comm., 1 (1954), 25-56. [19] R. J. Schmitt and S. J. Holbrook, Seasonally fluctuating resources and temporal variability of interspecific competition, Oecologia, 69 (1986), 1-11. doi: 10.1007/BF00399030. [20] Y. L. Tang and D. Xiao, Periodic solutions for a predator-prey model with periodic harvesting rate, International Journal of Bifurcation and Chaos, 24 (2014), 1450096(12 pages). doi: 10.1142/S0218127414500965. [21] C. J. Waiters and P. J. Bandy, Periodic harvest as a method of increasing big game yields, J. Wildl. Manage., 36 (1972), 128-134. [22] D. Xiao and L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. [23] D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29. doi: 10.1016/j.jmaa.2005.11.048. [24] C. Xu, M. S. Boyce and D. J. Daley, Harvesting in seasonal environments, Journal of Mathematical Biology, 50 (2005), 663-682. doi: 10.1007/s00285-004-0303-5.
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