American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[2]

V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments,, Springer-Verlag, (2007).   Google Scholar

[3]

D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[4]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).   Google Scholar

[5]

L. Arnold, Random Dynamical Systems,, Springer, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires,, Rev. Math. Pures. Appl., 4 (1959), 267.   Google Scholar

[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.  Google Scholar

[10]

J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).   Google Scholar

[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.  Google Scholar

[12]

C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources,, John Wiley and Sons Inc., (1990).   Google Scholar

[13]

H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999).  doi: 10.1007/b97846.  Google Scholar

[14]

T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357.  doi: 10.1007/BF02462011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Amsterdam, (1981).   Google Scholar

[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.  Google Scholar

[19]

F. C. Klebaner, Introduction to Stochastic Calculus With Applications,, Imperial College Press, (2005).  doi: 10.1142/p386.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.  doi: 10.1080/17442508008833146.  Google Scholar

[25]

A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).   Google Scholar

[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.  Google Scholar

[27]

X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).  doi: 10.1533/9780857099402.  Google Scholar

[28]

X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[32]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001).   Google Scholar

[33]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[35]

D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[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.  Google Scholar

[37]

R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering,, Springer, (2005).   Google Scholar

[38]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.   Google Scholar

[39]

K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).   Google Scholar

[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.  Google Scholar

[41]

C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. S. Anishchenko, Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments,, Springer-Verlag, (2007).   Google Scholar

[3]

D. Applebaum, Lévy Processes and Stochastics Calculus,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511809781.  Google Scholar

[4]

L. Arnold, Stochastic Differential Equations: Theory and Applications,, Wiley, (1974).   Google Scholar

[5]

L. Arnold, Random Dynamical Systems,, Springer, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

I. Barbalat, Systemes d'équations différentielles d'oscillations non linéaires,, Rev. Math. Pures. Appl., 4 (1959), 267.   Google Scholar

[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.  Google Scholar

[10]

J. X. Chen, C. H. Yu and L. Jin, Mathematical Analysis,, Higher Education Press, (2004).   Google Scholar

[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.  Google Scholar

[12]

C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewal Resources,, John Wiley and Sons Inc., (1990).   Google Scholar

[13]

H. Crauel and M. Gundlach, Stochastic Dynamics,, Springer-Verlag, (1999).  doi: 10.1007/b97846.  Google Scholar

[14]

T. C. Gard, Persistence in stochastic food web models,, Bull. Math. Biol., 46 (1984), 357.  doi: 10.1007/BF02462011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, Amsterdam, (1981).   Google Scholar

[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.  Google Scholar

[19]

F. C. Klebaner, Introduction to Stochastic Calculus With Applications,, Imperial College Press, (2005).  doi: 10.1142/p386.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. S. Liptser, A strong law of large numbers for local martingales,, Stochastics, 3 (1980), 217.  doi: 10.1080/17442508008833146.  Google Scholar

[25]

A. J. Lotka, Elements of Physical Biology,, William and Wilkins, (1925).   Google Scholar

[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.  Google Scholar

[27]

X. Mao, Stochastic Differential Equations and Applications,, Horwood, (1997).  doi: 10.1533/9780857099402.  Google Scholar

[28]

X. Mao, Stationary distribution of stochastic population systems,, Syst. Control Letters, 60 (2011), 398.  doi: 10.1016/j.sysconle.2011.02.013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, (2006).  doi: 10.1142/p473.  Google Scholar

[32]

R. M. May, Stability and Complexity in Model Ecosystems,, Princeton University Press, (2001).   Google Scholar

[33]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[35]

D. Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems,, Cambridge University Press, (1996).  doi: 10.1017/CBO9780511662829.  Google Scholar

[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.  Google Scholar

[37]

R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering,, Springer, (2005).   Google Scholar

[38]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie d'animali conviventi,, Mem. Acad. Lincei, 2 (1926), 31.   Google Scholar

[39]

K. Wang, Stochastic Biomathematics Models,, Science Press, (2010).   Google Scholar

[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.  Google Scholar

[41]

C. Zhu and G. Yin, On hybrid competitive lotka-volterra ecosystems,, Nonlinear Anal., 71 (2009).  doi: 10.1016/j.na.2009.01.166.  Google Scholar

[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.  Google Scholar

[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]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences