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

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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

Abstract
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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.

Keywords: Logstic population models, bifurcation., seasonal harvesting, global dynamics.

Mathematics Subject Classification: Primary: 34C25, 92D25; Secondary: 58F1.

Citation: Dongmei Xiao. Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting. Discrete & 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. doi: 10.1007/BF00275152. Google Scholar
[2]	F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting,, J. Math. Biol., 8 (1979), 55. doi: 10.1007/BF00280586. Google Scholar
[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. doi: 10.1007/BF00275206. Google Scholar
[4]	F. Brauer and D. A. Sánchez, Periodic environments and periodic harvesting,, Natural Resource Modeling, 16 (2003), 233. doi: 10.1111/j.1939-7445.2003.tb00113.x. Google Scholar
[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. doi: 10.1137/120895858. Google Scholar
[6]	J. P. Cohen and J. S. Foale, Sustaining small-scale fisheries with periodically harvested marine reserves,, Marine Policy, 37 (2013), 278. doi: 10.1016/j.marpol.2012.05.010. Google Scholar
[7]	C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990). Google Scholar
[8]	G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system,, SIAM J. Appl. Math., 58 (1998), 193. doi: 10.1137/S0036139994275799. Google Scholar
[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. doi: 10.1016/j.jde.2010.06.021. Google Scholar
[10]	M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients,, Mathematical Biosciences, 152 (1998), 165. doi: 10.1016/S0025-5564(98)10024-X. Google Scholar
[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, (2013). Google Scholar
[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. Google Scholar
[13]	M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and An Introduction to Chaos,, Elsevier, (2004). doi: 10.1007/978-1-4612-0873-0. Google Scholar
[14]	S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109. doi: 10.1007/s00285-011-0408-6. Google Scholar
[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. doi: 10.2307/1941702. Google Scholar
[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), 1 (2008), 303. doi: 10.3934/dcdss.2008.1.303. Google Scholar
[17]	R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries,, Science, 205 (1979), 267. doi: 10.1126/science.205.4403.267. Google Scholar
[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. Google Scholar
[19]	R. J. Schmitt and S. J. Holbrook, Seasonally fluctuating resources and temporal variability of interspecific competition,, Oecologia, 69 (1986), 1. doi: 10.1007/BF00399030. Google Scholar
[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). doi: 10.1142/S0218127414500965. Google Scholar
[21]	C. J. Waiters and P. J. Bandy, Periodic harvest as a method of increasing big game yields,, J. Wildl. Manage., 36 (1972), 128. Google Scholar
[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. doi: 10.1137/S0036139903428719. Google Scholar
[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. doi: 10.1016/j.jmaa.2005.11.048. Google Scholar
[24]	C. Xu, M. S. Boyce and D. J. Daley, Harvesting in seasonal environments,, Journal of Mathematical Biology, 50 (2005), 663. doi: 10.1007/s00285-004-0303-5. Google Scholar

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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. doi: 10.1007/BF00275152. Google Scholar
[2]	F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant rate prey harvesting,, J. Math. Biol., 8 (1979), 55. doi: 10.1007/BF00280586. Google Scholar
[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. doi: 10.1007/BF00275206. Google Scholar
[4]	F. Brauer and D. A. Sánchez, Periodic environments and periodic harvesting,, Natural Resource Modeling, 16 (2003), 233. doi: 10.1111/j.1939-7445.2003.tb00113.x. Google Scholar
[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. doi: 10.1137/120895858. Google Scholar
[6]	J. P. Cohen and J. S. Foale, Sustaining small-scale fisheries with periodically harvested marine reserves,, Marine Policy, 37 (2013), 278. doi: 10.1016/j.marpol.2012.05.010. Google Scholar
[7]	C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990). Google Scholar
[8]	G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system,, SIAM J. Appl. Math., 58 (1998), 193. doi: 10.1137/S0036139994275799. Google Scholar
[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. doi: 10.1016/j.jde.2010.06.021. Google Scholar
[10]	M. Fan and K. Wang, Optimal harvesting policy for single population with periodic coefficients,, Mathematical Biosciences, 152 (1998), 165. doi: 10.1016/S0025-5564(98)10024-X. Google Scholar
[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, (2013). Google Scholar
[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. Google Scholar
[13]	M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems and An Introduction to Chaos,, Elsevier, (2004). doi: 10.1007/978-1-4612-0873-0. Google Scholar
[14]	S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109. doi: 10.1007/s00285-011-0408-6. Google Scholar
[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. doi: 10.2307/1941702. Google Scholar
[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), 1 (2008), 303. doi: 10.3934/dcdss.2008.1.303. Google Scholar
[17]	R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries,, Science, 205 (1979), 267. doi: 10.1126/science.205.4403.267. Google Scholar
[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. Google Scholar
[19]	R. J. Schmitt and S. J. Holbrook, Seasonally fluctuating resources and temporal variability of interspecific competition,, Oecologia, 69 (1986), 1. doi: 10.1007/BF00399030. Google Scholar
[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). doi: 10.1142/S0218127414500965. Google Scholar
[21]	C. J. Waiters and P. J. Bandy, Periodic harvest as a method of increasing big game yields,, J. Wildl. Manage., 36 (1972), 128. Google Scholar
[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. doi: 10.1137/S0036139903428719. Google Scholar
[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. doi: 10.1016/j.jmaa.2005.11.048. Google Scholar
[24]	C. Xu, M. S. Boyce and D. J. Daley, Harvesting in seasonal environments,, Journal of Mathematical Biology, 50 (2005), 663. doi: 10.1007/s00285-004-0303-5. Google Scholar

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