# American Institute of Mathematical Sciences

2019, 27: 89-99. doi: 10.3934/era.2019011

## Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle

 School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received  September 2019 Revised  December 2019 Published  December 2019

This paper deals with the initial value problem of a predator-prey system with dispersal and delay, which does not admit the classical comparison principle. When the initial value has nonempty compact support, the initial value problem formulates that two species synchronously invade the same habitat in population dynamics. By constructing proper auxiliary equations and functions, we confirm the faster invasion speed of two species, which equals to the minimal wave speed of traveling wave solutions in earlier works.

Citation: Shuxia Pan. Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle. Electronic Research Archive, 2019, 27: 89-99. doi: 10.3934/era.2019011
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, J.A. Goldstein Eds., Lecture Notes in Mathematics, Vol. 446. Springer, Berlin, German, (1975), 5–49.  Google Scholar [2] X. Bao and W.-T. Li, Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats, Nonlinear Anal. Real World Appl., 51 (2020), 102975, 26 pp.  doi: 10.1016/j.nonrwa.2019.102975.  Google Scholar [3] X. Bao, W.-T. Li, W. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048-3091.  doi: 10.1016/j.jde.2018.05.003.  Google Scholar [4] P. W. Bates, On some nonlocal evolution equations arising in materials science, In: Nonlinear Dynamics and Evolution Equations (Ed. by H. Brunner, X.Q. Zhao, X. Zou), Fields Inst. Commun., 48 (2006), 13–52, AMS, Providence.  Google Scholar [5] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 725-755.  doi: 10.1017/S0308210504000721.  Google Scholar [6] W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar [7] A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system,, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.  Google Scholar [8] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differ. Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar [9] A. Ducrot, J. S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Art. 146, 25 pp. doi: 10.1007/s00033-019-1188-x.  Google Scholar [10] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar [11] W. F. Fagan and J. G. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens, Amer. Nat., 155 (2000), 238-251.  doi: 10.1086/303320.  Google Scholar [12] J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar [13] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In: Trends in Nonlinear Analysis (Ed. by M. Kirkilionis, S. Kr$\ddot{o}$mker, R. Rannacher, F. Tomi), 153–191, Springer: Berlin, 2003.  Google Scholar [14] L. Hopf, Introduction to Differential Equations of Physics, Dover: New York, 1948.  Google Scholar [15] Y. Jin and X. Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189.  doi: 10.1088/0951-7715/22/5/011.  Google Scholar [16] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269.  Google Scholar [17] X. Li, S. Pan and H. B. Shi, Minimal wave speed in a dispersal predator-prey system with delays, Boundary Value Problems, 2018 (2018), Paper No. 49, 26 pp. doi: 10.1186/s13661-018-0966-2.  Google Scholar [18] X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar [19] G. 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Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dyn. Differ. Equ., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar [25] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019), 291.  doi: 10.3390/math7030291.  Google Scholar [26] R. Lui, Biological growth and spread modeled by systems of recursions. Ⅰ. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar [27] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar [28] J. D. Murray, Mathematical Biology, II. Spatial Models and Biomedical Applications., Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag: New York, 2003.  Google Scholar [29] M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bull. Math. Biol., 63 (2001), 655-684.  doi: 10.1006/bulm.2001.0239.  Google Scholar [30] S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar [31] S. Pan, Convergence and traveling wave solutions for a predator-prey system with distributed delays, Mediterr. J. Math., 14 (2017), Art. 103, 15 pp. doi: 10.1007/s00009-017-0905-y.  Google Scholar [32] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar [33] S. Pan, G. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.  Google Scholar [34] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press: Oxford, UK, 1997. Google Scholar [35] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS: Providence, RI, USA, 1995.  Google Scholar [36] 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 [37] H. F. Weinberger, Long-time behavior of a class of biological model, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [38] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar [39] P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar [40] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011.   Google Scholar [41] Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems, IMA J. Appl. Math., 76 (2011), 493-513.  doi: 10.1093/imamat/hxq048.  Google Scholar [42] G. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar [43] X. Q. Zhao, Spatial dynamics of some evolution systems in biology, In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, Y. Du, H. Ishii, W.Y. Lin, Eds.; World Scientific: Singapore, 2009,332–363. doi: 10.1142/9789812834744_0015.  Google Scholar

show all references

##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, In Partial Differential Equations and Related Topics, J.A. Goldstein Eds., Lecture Notes in Mathematics, Vol. 446. Springer, Berlin, German, (1975), 5–49.  Google Scholar [2] X. Bao and W.-T. Li, Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats, Nonlinear Anal. Real World Appl., 51 (2020), 102975, 26 pp.  doi: 10.1016/j.nonrwa.2019.102975.  Google Scholar [3] X. Bao, W.-T. Li, W. Shen and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), 3048-3091.  doi: 10.1016/j.jde.2018.05.003.  Google Scholar [4] P. W. Bates, On some nonlocal evolution equations arising in materials science, In: Nonlinear Dynamics and Evolution Equations (Ed. by H. Brunner, X.Q. Zhao, X. Zou), Fields Inst. Commun., 48 (2006), 13–52, AMS, Providence.  Google Scholar [5] J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 725-755.  doi: 10.1017/S0308210504000721.  Google Scholar [6] W. Ding and X. Liang, Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media, SIAM J. Math. Anal., 47 (2015), 855-896.  doi: 10.1137/140958141.  Google Scholar [7] A. Ducrot, Convergence to generalized transition waves for some Holling-Tanner prey-predator reaction-diffusion system,, J. Math. Pures Appl., 100 (2013), 1-15.  doi: 10.1016/j.matpur.2012.10.009.  Google Scholar [8] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differ. Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar [9] A. Ducrot, J. S. Guo, G. Lin and S. Pan, The spreading speed and the minimal wave speed of a predator-prey system with nonlocal dispersal, Z. Angew. Math. Phys., 70 (2019), Art. 146, 25 pp. doi: 10.1007/s00033-019-1188-x.  Google Scholar [10] S. R. Dunbar, Travelling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.  Google Scholar [11] W. F. Fagan and J. G. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens, Amer. Nat., 155 (2000), 238-251.  doi: 10.1086/303320.  Google Scholar [12] J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar [13] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, In: Trends in Nonlinear Analysis (Ed. by M. Kirkilionis, S. Kr$\ddot{o}$mker, R. Rannacher, F. Tomi), 153–191, Springer: Berlin, 2003.  Google Scholar [14] L. Hopf, Introduction to Differential Equations of Physics, Dover: New York, 1948.  Google Scholar [15] Y. Jin and X. Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity, 22 (2009), 1167-1189.  doi: 10.1088/0951-7715/22/5/011.  Google Scholar [16] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269.  Google Scholar [17] X. Li, S. Pan and H. B. Shi, Minimal wave speed in a dispersal predator-prey system with delays, Boundary Value Problems, 2018 (2018), Paper No. 49, 26 pp. doi: 10.1186/s13661-018-0966-2.  Google Scholar [18] X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar [19] G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Analysis, 74 (2011), 2448-2461.  doi: 10.1016/j.na.2010.11.046.  Google Scholar [20] G. Lin, Asymptotic spreading fastened by inter-specific coupled nonlinearities: A cooperative system, Physica D, 241 (2012), 705-710.  doi: 10.1016/j.physd.2011.12.007.  Google Scholar [21] G. Lin and W. T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, Eur. J. Appl. Math., 23 (2012), 669-689.  doi: 10.1017/S0956792512000198.  Google Scholar [22] G. Lin, W. T. Li and S. Ruan, Spreading speeds and traveling waves of a competitive recursion, J. Math. Biol., 62 (2011), 165-201.  doi: 10.1007/s00285-010-0334-z.  Google Scholar [23] G. Lin, S. Pan and X. P. Yan, Spreading speeds of epidemic models with nonlocal delays, Mathe. Biosci. Eng., 16 (2019), 7562-7588.  doi: 10.3934/mbe.2019380.  Google Scholar [24] G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dyn. Differ. Equ., 26 (2014), 583-605.  doi: 10.1007/s10884-014-9355-4.  Google Scholar [25] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019), 291.  doi: 10.3390/math7030291.  Google Scholar [26] R. Lui, Biological growth and spread modeled by systems of recursions. Ⅰ. Mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar [27] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar [28] J. D. Murray, Mathematical Biology, II. Spatial Models and Biomedical Applications., Third edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag: New York, 2003.  Google Scholar [29] M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bull. Math. Biol., 63 (2001), 655-684.  doi: 10.1006/bulm.2001.0239.  Google Scholar [30] S. Pan, Asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407 (2013), 230-236.  doi: 10.1016/j.jmaa.2013.05.031.  Google Scholar [31] S. Pan, Convergence and traveling wave solutions for a predator-prey system with distributed delays, Mediterr. J. Math., 14 (2017), Art. 103, 15 pp. doi: 10.1007/s00009-017-0905-y.  Google Scholar [32] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar [33] S. Pan, G. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.  Google Scholar [34] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press: Oxford, UK, 1997. Google Scholar [35] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS: Providence, RI, USA, 1995.  Google Scholar [36] 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 [37] H. F. Weinberger, Long-time behavior of a class of biological model, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [38] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar [39] P. Weng and X. Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar [40] Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction Diffusion Equations, Science Press, Beijing, 2011.   Google Scholar [41] Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems, IMA J. Appl. Math., 76 (2011), 493-513.  doi: 10.1093/imamat/hxq048.  Google Scholar [42] G. Zhang, W. T. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Model., 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.  Google Scholar [43] X. Q. Zhao, Spatial dynamics of some evolution systems in biology, In Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, Y. Du, H. Ishii, W.Y. Lin, Eds.; World Scientific: Singapore, 2009,332–363. doi: 10.1142/9789812834744_0015.  Google Scholar
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