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

Previous Article
Boundary layer separation of 2-D incompressible Dirichlet flows
DCDS-B Home
This Issue
Next Article

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

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
Related Papers

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 & 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. 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, (2007).
[3]	D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781.
[4]	L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).
[5]	L. Arnold, Random Dynamical Systems,, Springer, (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. 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. 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.
[9]	J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463.
[10]	J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).
[11]	C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets,, Physica A., 387* (2008), 3837. 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., (1990).
[13]	H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999). doi: 10.1007/b97846.
[14]	T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357. doi: 10.1007/BF02462011.
[15]	T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Anal., 10 (1986), 1411. 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. doi: 10.1016/j.apm.2010.12.025.
[17]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Amsterdam, (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. doi: 10.1016/j.jmaa.2011.12.049.*
[19]	F. C. Klebaner, Introduction to Stochastic Calculus With Applications,, Imperial College Press, (2005). doi: 10.1142/p386.
[20]	H. Kunita, Itô's stochastic calculus: Its surprising power for applications,, Stochastic Process. Appl., 120* (2010), 622. 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. 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. 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. doi: 10.3934/dcds.2009.24.523.
[24]	R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146.
[25]	A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).
[26]	E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Math. Biosci., 145* (1997), 47. doi: 10.1016/S0025-5564(97)00029-1.*
[27]	X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997). doi: 10.1533/9780857099402.
[28]	X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398. 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. 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. doi: 10.1016/S0022-247X(03)00539-0.*
[31]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (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, (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. doi: 10.1016/j.spa.2005.08.004.
[35]	D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (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. 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, (2005).
[38]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.
[39]	K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).
[40]	C. Zhu and G. Yin, On competitive lotka-volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066.
[41]	C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009). 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. 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. 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, (2007).
[3]	D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511809781.
[4]	L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).
[5]	L. Arnold, Random Dynamical Systems,, Springer, (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. 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. 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.
[9]	J. R. Beddington and R. M. May, Harvesting natural populations in a randomly fluctuating environment,, Science, 197 (1977), 463. doi: 10.1126/science.197.4302.463.
[10]	J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).
[11]	C. Chiarella, X. He, D. Wang and M. Zheng, The stochastic bifurcation behaviour of speculative financial markets,, Physica A., 387* (2008), 3837. 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., (1990).
[13]	H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999). doi: 10.1007/b97846.
[14]	T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357. doi: 10.1007/BF02462011.
[15]	T. C. Gard, Stability for multispecies population models in random environments,, Nonlinear Anal., 10 (1986), 1411. 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. doi: 10.1016/j.apm.2010.12.025.
[17]	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Amsterdam, (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. doi: 10.1016/j.jmaa.2011.12.049.*
[19]	F. C. Klebaner, Introduction to Stochastic Calculus With Applications,, Imperial College Press, (2005). doi: 10.1142/p386.
[20]	H. Kunita, Itô's stochastic calculus: Its surprising power for applications,, Stochastic Process. Appl., 120* (2010), 622. 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. 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. 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. doi: 10.3934/dcds.2009.24.523.
[24]	R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217. doi: 10.1080/17442508008833146.
[25]	A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).
[26]	E. M. Lungu and B. Øksendal, Optimal harvesting from a population in a stochastic crowded environment,, Math. Biosci., 145* (1997), 47. doi: 10.1016/S0025-5564(97)00029-1.*
[27]	X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997). doi: 10.1533/9780857099402.
[28]	X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398. 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. 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. doi: 10.1016/S0022-247X(03)00539-0.*
[31]	X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (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, (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. doi: 10.1016/j.spa.2005.08.004.
[35]	D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (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. 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, (2005).
[38]	V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.
[39]	K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).
[40]	C. Zhu and G. Yin, On competitive lotka-volterra model in random environments,, J. Math. Anal. Appl., 357 (2009), 154. doi: 10.1016/j.jmaa.2009.03.066.
[41]	C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009). 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. doi: 10.1016/j.cnsns.2013.09.010.

[1]	Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[2]	Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[3]	Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[4]	José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[5]	Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[6]	Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[7]	Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[8]	Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[9]	Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[10]	Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[11]	Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[12]	Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[13]	Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
[14]	Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[15]	Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044
[16]	Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[17]	John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[18]	Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124
[19]	Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519
[20]	Igor Nazarov, Bai-Lian Li. Maximal sustainable yield in a multipatch habitat. Conference Publications, 2005, 2005 (Special) : 682-691. doi: 10.3934/proc.2005.2005.682

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences