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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  January 2020 Early access  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 and 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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

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

[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.

[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.

[30]

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

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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

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

[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.

[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.

[30]

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

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