Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps

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March 2015, 20(2): 683-701. doi: 10.3934/dcdsb.2015.20.683

Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps

Xiaoling Zou ^1, and Ke Wang ^2,

1.	Department of Mathematics, Harbin Institute of Technology, Weihai 264209
2.	Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209

Received January 2014 Revised August 2014 Published January 2015

Abstract
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Optimization problem for a stochastic N-dimensional competitive Lotka-Volterra system is studied in this paper. The considered system is driven by both white noise and jumping noise, and the jumping noise is modeled by a stochastic integral with respect to a Poisson counting measure generated by a Poisson point process. For two types of objective functions, namely, time-averaged yield and sustained yield, the optimal harvesting efforts as well as the corresponding maximum yields are given respectively. Moreover, almost sure equivalence between these two objective functions is proved by ergodic method. This paper provides us a new idea to study the stochastic optimal harvesting problem with sustained yield, and this idea can be popularized to other stochastic systems.

Keywords: Brownian motion, sustained yield., Fokker-Planck equation, time-averaged yield, jumping noise, stationary probability density.

Mathematics Subject Classification: Primary: 60H10, 93E20; Secondary: 92B05, 37N2.

Citation: Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683

References:

[1]	L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math Biosci., 152 (1998), 63-85. doi: 10.1016/S0025-5564(98)10018-4.
[2]	V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, New York, 2007.
[3]	D. Applebaum, Lévy Processes and Stochastics Calculus, Cambridge University Press, 2 edition, 2009. doi: 10.1017/CBO9780511809781.
[4]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.
[5]	L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.
[6]	J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.
[7]	J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. doi: 10.1016/j.jmaa.2012.02.043.
[8]	I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures. Appl., 4 (1959), 267-270.
[9]	J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. doi: 10.1126/science.197.4302.463.
[10]	J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis, Higher Education Press, Beijing, 2 edition, 2004.
[11]	C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets, Physica A., 387 (2008), 3837-3846. doi: 10.1016/j.physa.2008.01.078.
[12]	C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, John Wiley and Sons Inc., New York, 2 edition, 1990.
[13]	H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846.
[14]	T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370. doi: 10.1007/BF02462011.
[15]	T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2.
[16]	G. Hu and K. Wang, Stability in distribution of competitive lotka-volterra system with markovian switching, Appl. Math. Model., 35 (2011), 3189-3200. doi: 10.1016/j.apm.2010.12.025.
[17]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam, North-Holland, 1981.
[18]	D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595. doi: 10.1016/j.jmaa.2011.12.049.
[19]	F. C. Klebaner, Introduction to Stochastic Calculus With Applications, Imperial College Press, London, 2005. doi: 10.1142/p386.
[20]	H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622-652. doi: 10.1016/j.spa.2010.01.013.
[21]	W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162. doi: 10.1016/j.amc.2011.05.079.
[22]	X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021.
[23]	X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation, Discret. Contin. Dyn. S., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.
[24]	R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146.
[25]	A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.
[26]	E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75. doi: 10.1016/S0025-5564(97)00029-1.
[27]	X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997. doi: 10.1533/9780857099402.
[28]	X. Mao, Stationary distribution of stochastic population systems, Syst. Control Letters, 60 (2011), 398-405. doi: 10.1016/j.sysconle.2011.02.013.
[29]	X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.
[30]	X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.
[31]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473.
[32]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.
[33]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 6 edition, 2003. doi: 10.1007/978-3-642-14394-6.
[34]	S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380. doi: 10.1016/j.spa.2005.08.004.
[35]	D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
[36]	M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment, Appl. Stochastic Models Bus. Ind., 19 (2003), 43-49. doi: 10.1002/asmb.490.
[37]	R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005.
[38]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113.
[39]	K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010.
[40]	C. Zhu and G. Yin, On competitive lotka-volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.
[41]	C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166.
[42]	X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557-1568. doi: 10.1016/j.cnsns.2013.09.010.

show all references

References:

[1]	L. H. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Math Biosci., 152 (1998), 63-85. doi: 10.1016/S0025-5564(98)10018-4.
[2]	V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments, Springer-Verlag, New York, 2007.
[3]	D. Applebaum, Lévy Processes and Stochastics Calculus, Cambridge University Press, 2 edition, 2009. doi: 10.1017/CBO9780511809781.
[4]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.
[5]	L. Arnold, Random Dynamical Systems, Springer, New York, 1998. doi: 10.1007/978-3-662-12878-7.
[6]	J. Bao, X. Mao,G. Yin and C. Yuan, Competitive lotka-volterra population dynamics with jumps, Nonlinear Anal., 74 (2011), 6601-6616. doi: 10.1016/j.na.2011.06.043.
[7]	J. Bao and C. Yuan, Stochastic population dtnamics driven by lévy noise, J. Math. Anal. Appl., 391 (2012), 363-375. doi: 10.1016/j.jmaa.2012.02.043.
[8]	I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires, Rev. Math. Pures. Appl., 4 (1959), 267-270.
[9]	J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465. doi: 10.1126/science.197.4302.463.
[10]	J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis, Higher Education Press, Beijing, 2 edition, 2004.
[11]	C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets, Physica A., 387 (2008), 3837-3846. doi: 10.1016/j.physa.2008.01.078.
[12]	C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources, John Wiley and Sons Inc., New York, 2 edition, 1990.
[13]	H. Crauel and M. Gundlach, Stochastic Dynamics, Springer-Verlag, New York, 1999. doi: 10.1007/b97846.
[14]	T. C. Gard, Persistence in stochastic food web models, Bull. Math. Biol., 46 (1984), 357-370. doi: 10.1007/BF02462011.
[15]	T. C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal., 10 (1986), 1411-1419. doi: 10.1016/0362-546X(86)90111-2.
[16]	G. Hu and K. Wang, Stability in distribution of competitive lotka-volterra system with markovian switching, Appl. Math. Model., 35 (2011), 3189-3200. doi: 10.1016/j.apm.2010.12.025.
[17]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Amsterdam, North-Holland, 1981.
[18]	D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous lotka-volterra competition system with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595. doi: 10.1016/j.jmaa.2011.12.049.
[19]	F. C. Klebaner, Introduction to Stochastic Calculus With Applications, Imperial College Press, London, 2005. doi: 10.1142/p386.
[20]	H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stochastic Process. Appl., 120 (2010), 622-652. doi: 10.1016/j.spa.2010.01.013.
[21]	W. Li, K. Wang and H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157-162. doi: 10.1016/j.amc.2011.05.079.
[22]	X. Li, D. Jiang and X. Mao, Population dynamical behavior of lotka-volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021.
[23]	X. Li and X. Mao, Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation, Discret. Contin. Dyn. S., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523.
[24]	R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146.
[25]	A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.
[26]	E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment, Math. Biosci., 145 (1997), 47-75. doi: 10.1016/S0025-5564(97)00029-1.
[27]	X. Mao, Stochastic Differential Equations and Applications, Horwood, New York, 1997. doi: 10.1533/9780857099402.
[28]	X. Mao, Stationary distribution of stochastic population systems, Syst. Control Letters, 60 (2011), 398-405. doi: 10.1016/j.sysconle.2011.02.013.
[29]	X. Mao, G. Marion and E. Renshaw, Environmental brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl., 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0.
[30]	X. Mao, S. Sabanis and E. Renshaw, Asymptotic behavior of the stochastic lotka-volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0.
[31]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473.
[32]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.
[33]	B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, 6 edition, 2003. doi: 10.1007/978-3-642-14394-6.
[34]	S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380. doi: 10.1016/j.spa.2005.08.004.
[35]	D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662829.
[36]	M. A. Shah and U. Sharma, Optimal harvesting policies for a generalized gordon-schaefer model in randomly varying environment, Appl. Stochastic Models Bus. Ind., 19 (2003), 43-49. doi: 10.1002/asmb.490.
[37]	R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005.
[38]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi, Mem. Acad. Lincei, 2 (1926), 31-113.
[39]	K. Wang, Stochastic Biomathematics Models, Science Press, Beijing, 2010.
[40]	C. Zhu and G. Yin, On competitive lotka-volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. doi: 10.1016/j.jmaa.2009.03.066.
[41]	C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems, Nonlinear Anal., 71 (2009), e1370-e1379. doi: 10.1016/j.na.2009.01.166.
[42]	X. Zou and K. Wang, Numerical simulations and modeling for stochastic biological systems with jumps, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1557-1568. doi: 10.1016/j.cnsns.2013.09.010.

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