American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries
  • DCDS-B Home
  • This Issue
  • Next Article
    Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks
January  2020, 25(1): 117-139. doi: 10.3934/dcdsb.2019175

Stochastic partial differential equation models for spatially dependent predator-prey equations

Nhu N. Nguyen and  George Yin

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Received  November 2018 Revised  March 2019 Published  July 2019

Fund Project: This research was supported in part by the National Science Foundation under grant DMS-1710827.

Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE models are more versatile. To incorporate more qualitative features of the ratio-dependent models, the Beddington-DeAngelis functional response is also used. To analyze the systems under consideration, first existence and uniqueness of solutions of the SPDEs are obtained using the notion of mild solutions. Then sufficient conditions for permanence and extinction are derived.

Keywords: Predator-prey model, SPDE, mild solution, positivity, extinction, permanence.
Mathematics Subject Classification: Primary: 60H15, 92D25, 92D40, 35Q92.
Citation: Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
References:
[1]

P. Acquistapace and B. Terreni, On the abstract nonautonomous parabolic Cauchy problem in the case of constant domains, Ann. Mat. Pura Appl., 140 (1985), 1-55.  doi: 10.1007/BF01776844.  Google Scholar

[2]

S. Ai, Y. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Eqs., 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021.  Google Scholar

[3]

W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, Evolutionary Equations, Handb. Differ. Equ., North-Holland, Amsterdam, 1 (2004), 1–85.  Google Scholar

[4]

R. Arditi and L. R. Ginzburg, Coupling in predatorprey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.   Google Scholar

[5]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[6]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001.  Google Scholar

[7]

S. Cerrai, Second Order PDEs in Finite and Infinite Dimension. A Probabilistic Approach, , Lecture Notes in Mathematics Series 1762, Springer Verlag, 2001. doi: 10.1007/b80743.  Google Scholar

[8]

S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Relat. Fields, 125 (2003), 271-304.  doi: 10.1007/s00440-002-0230-6.  Google Scholar

[9]

R. F. Curtain and P. L. Falez, Itȏ's Lemma in infinite dimensions, J. Math. Anal. Appl., 31 (1970), 434-448.  doi: 10.1016/0022-247X(70)90037-5.  Google Scholar

[10]

G. Da Prato and L. Tubaro, Some results on semilinear stochastic differential equations in Hilbert spaces, Stochastics, 15 (1985), 271-281.  doi: 10.1080/17442508508833360.  Google Scholar

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[12] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge University Press, London, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[13]

D. L. DeAngelis, R. A. Goldstein and R. V. ONeill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[14]

N.T. Dieu, N.H. Du, D.H. Nguyen and and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402.  doi: 10.1137/15M1032004.  Google Scholar

[15]

N. H. Du, N. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models., J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.  Google Scholar

[16]

M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response, Numer. Math., 107 (2007), 641-667.  doi: 10.1007/s00211-007-0106-x.  Google Scholar

[17]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomologist, 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.  Google Scholar

[18]

K.-Y. Lam, Y. Lou and F. Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662.  doi: 10.1137/15M1027887.  Google Scholar

[19]

S. Li and J. Wu, Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system, J. Differential Equations, 265 (2018), 3754-3791.  doi: 10.1016/j.jde.2018.05.017.  Google Scholar

[20]

H. Y. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 374 (2011), 644-654.  doi: 10.1016/j.jmaa.2010.08.029.  Google Scholar

[21]

K. Liu, R. Douglas, H. Brezis and A. Jeffrey, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall/CRC, New York, 2005. Google Scholar

[22]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. Google Scholar

[23]

Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

[24]

C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, Ann. Appl. Probab., 9 (1999), 1226-1259.  doi: 10.1214/aoap/1029962871.  Google Scholar

[25]

D. H. Nguyen, N. N. Nguyen and G. Yin, Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model, submitted. Google Scholar

[26] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ, 2005.   Google Scholar
[27]

G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Process. Appl., 77 (1998), 83-98.  doi: 10.1016/S0304-4149(98)00024-6.  Google Scholar

[28]

J. B. Walsh, An introduction to stochastic partial differential equations, École Dété de Probabilits de Saint-Flour, XIV-1984, volume 1180 of Lecture Notes in Math., pages 265–339. Springer, Berlin, 1986. doi: 10.1007/BFb0074920.  Google Scholar

[29]

M. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[30]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlga, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

show all references

References:
[1]

P. Acquistapace and B. Terreni, On the abstract nonautonomous parabolic Cauchy problem in the case of constant domains, Ann. Mat. Pura Appl., 140 (1985), 1-55.  doi: 10.1007/BF01776844.  Google Scholar

[2]

S. Ai, Y. Du and R. Peng, Traveling waves for a generalized Holling-Tanner predator-prey model, J. Differential Eqs., 263 (2017), 7782-7814.  doi: 10.1016/j.jde.2017.08.021.  Google Scholar

[3]

W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, Evolutionary Equations, Handb. Differ. Equ., North-Holland, Amsterdam, 1 (2004), 1–85.  Google Scholar

[4]

R. Arditi and L. R. Ginzburg, Coupling in predatorprey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.   Google Scholar

[5]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[6]

C. M. Carracedo and M. S. Alix, The Theory of Fractional Powers of Operators, Elsevier, Amsterdam, 2001.  Google Scholar

[7]

S. Cerrai, Second Order PDEs in Finite and Infinite Dimension. A Probabilistic Approach, , Lecture Notes in Mathematics Series 1762, Springer Verlag, 2001. doi: 10.1007/b80743.  Google Scholar

[8]

S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Relat. Fields, 125 (2003), 271-304.  doi: 10.1007/s00440-002-0230-6.  Google Scholar

[9]

R. F. Curtain and P. L. Falez, Itȏ's Lemma in infinite dimensions, J. Math. Anal. Appl., 31 (1970), 434-448.  doi: 10.1016/0022-247X(70)90037-5.  Google Scholar

[10]

G. Da Prato and L. Tubaro, Some results on semilinear stochastic differential equations in Hilbert spaces, Stochastics, 15 (1985), 271-281.  doi: 10.1080/17442508508833360.  Google Scholar

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[12] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math. 92, Cambridge University Press, London, 1989.  doi: 10.1017/CBO9780511566158.  Google Scholar
[13]

D. L. DeAngelis, R. A. Goldstein and R. V. ONeill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[14]

N.T. Dieu, N.H. Du, D.H. Nguyen and and G. Yin, Protection zones for survival of species in random environment, SIAM J. Appl. Math., 76 (2016), 1382-1402.  doi: 10.1137/15M1032004.  Google Scholar

[15]

N. H. Du, N. H. Dang and G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models., J. Appl. Probab., 53 (2016), 187-202.  doi: 10.1017/jpr.2015.18.  Google Scholar

[16]

M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response, Numer. Math., 107 (2007), 641-667.  doi: 10.1007/s00211-007-0106-x.  Google Scholar

[17]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomologist, 91 (1959), 293-320.  doi: 10.4039/Ent91293-5.  Google Scholar

[18]

K.-Y. Lam, Y. Lou and F. Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662.  doi: 10.1137/15M1027887.  Google Scholar

[19]

S. Li and J. Wu, Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system, J. Differential Equations, 265 (2018), 3754-3791.  doi: 10.1016/j.jde.2018.05.017.  Google Scholar

[20]

H. Y. Li and Y. Takeuchi, Dynamics of the density dependent predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 374 (2011), 644-654.  doi: 10.1016/j.jmaa.2010.08.029.  Google Scholar

[21]

K. Liu, R. Douglas, H. Brezis and A. Jeffrey, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall/CRC, New York, 2005. Google Scholar

[22]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. Google Scholar

[23]

Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.  Google Scholar

[24]

C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, Ann. Appl. Probab., 9 (1999), 1226-1259.  doi: 10.1214/aoap/1029962871.  Google Scholar

[25]

D. H. Nguyen, N. N. Nguyen and G. Yin, Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model, submitted. Google Scholar

[26] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, 31. Princeton University Press, Princeton, NJ, 2005.   Google Scholar
[27]

G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Process. Appl., 77 (1998), 83-98.  doi: 10.1016/S0304-4149(98)00024-6.  Google Scholar

[28]

J. B. Walsh, An introduction to stochastic partial differential equations, École Dété de Probabilits de Saint-Flour, XIV-1984, volume 1180 of Lecture Notes in Math., pages 265–339. Springer, Berlin, 1986. doi: 10.1007/BFb0074920.  Google Scholar

[29]

M. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558.  doi: 10.1016/j.jde.2017.11.027.  Google Scholar

[30]

A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlga, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[1]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263
[2]

Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148
[3]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371
[4]

Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162
[5]

Ching-Hui Wang, Sheng-Chen Fu. Traveling wave solutions to diffusive Holling-Tanner predator-prey models. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021007
[6]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[7]

Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
[8]

Telmo Peixe. Permanence in polymatrix replicators. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020032
[9]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[10]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[11]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[12]

Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006
[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[14]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[15]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[16]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[17]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[18]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[19]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[20]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (175)
  • HTML views (403)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences