# American Institute of Mathematical Sciences

November  2019, 18(6): 2983-2999. doi: 10.3934/cpaa.2019133

## Asymptotic spreading for a time-periodic predator-prey system

 Key Laboratory of Applied Mathematics and Complex Systems, School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  July 2018 Revised  January 2019 Published  May 2019

This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator.

Citation: Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133
##### 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 ed.), Lecture Notes in Math., 446, Springer, Berlin, 1975, pp. 5–49. [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [3] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: doi.org/10.1016/j.jfa.2008.06.030. [4] W. J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351. doi: 10.3934/dcds.2018189. [5] W. J. Bo and G. Lin, Asymptotic spreading of time periodic competition diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3901-3914. doi: doi:10.3934/dcdsb.2018116. [6] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [7] T. R. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, Discrete Contin. Dyn. Syst., 1 (1995), 103-117. doi: 10.3934/dcds.1995.1.103. [8] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357. doi: 10.1016/j.jde.2016.02.023. [9] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. [10] S. R. Dunbar, Traveling wave solutions in diffusive predator-prey systems: periodic orbits and point-to-periodic heteroclic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078. doi: 10.1137/0146063. [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. [12] J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [13] J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.4171/jems/556. [14] P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185. doi: 10.1016/0022-0396(81)90016-4. [15] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., vol. 247, Longman Scientific Technical, Harlow, UK, 1991. doi: 0003-889X/97/050388-10. [16] S. B. Hsu and X. Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), no. 2,776–789. doi: 10.1137/070703016. [17] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269. [18] X. Liang, Y. Yi and X. Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010. [19] 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. [20] G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Anal., 74 (2011), 2448-2461. doi: 10.1016/j.na.2010.11.046. [21] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58. doi: 10.1016/j.na.2013.10.024. [22] G. Lin and R. Wang, Spatial invasion dynamics for a time period predator-prey system, Math. Methods Appl. Sci., 41 (2018), 7621-7623. doi: 10.1002/mma.5224. [23] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019), ID: 291. doi: 10.3390/math7030291. [24] 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)90027-8. [25] J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. doi: 10.1007/b98869. [26] G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002. [27] 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. [28] 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. [29] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51. doi: 10.1016/j.aml.2017.05.014. [30] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford, New York, 1997. doi: 10.1002/(SICI)1520-6300(1998)10:5<683::AID-AJHB17>3.0.CO;2-4. [31] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2$^{nd}$ Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [32] Z. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1999), 339-352. doi: 10.1080/00036819908840745. [33] Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations, 179 (2002), 538-561. doi: 10.1006/jdeq.2001.4044. [34] Z. Teng and L. Chen, Global asymptotic stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095. doi: 10.1016/S0362-546X(99)00441-1. [35] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. [36] M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327. doi: 10.1016/j.cnsns.2014.11.016. [37] M. X. Wang, W. J. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329. doi: 10.1016/j.jmaa.2016.04.007. [38] X. J. Wang and G. Lin, Traveling waves for a periodic Lotka-Volterra predator-prey system, Appl. Anal., (2018), in press. doi: 10.1080/00036811.2018.1469007. [39] H. F. Weinberger, Long-time behavior of a class of biological model, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [40] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3. [41] 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. [42] S. L. Wu, C. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009. [43] Q. Ye, Z. Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2$^{nd}$ edn, Science Press, Beijing, 2011. [44] T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), no. 8, 3538–3572. doi: 10.1016/j.jde.2013.01.031. [45] G. Y. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671. doi: doi.org/10.1016/j.matpur.2010.11.005. [46] 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, 2009, pp. 332–363. [47] X. Q. Zhao, Dynamincal Systems in Population Biology, 2$^{nd}$ edn. Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

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##### 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 ed.), Lecture Notes in Math., 446, Springer, Berlin, 1975, pp. 5–49. [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population dynamics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [3] H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189. doi: doi.org/10.1016/j.jfa.2008.06.030. [4] W. J. Bo, G. Lin and S. Ruan, Traveling wave solutions for time periodic reaction-diffusion systems, Discrete Contin. Dyn. Syst., 38 (2018), 4329-4351. doi: 10.3934/dcds.2018189. [5] W. J. Bo and G. Lin, Asymptotic spreading of time periodic competition diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3901-3914. doi: doi:10.3934/dcdsb.2018116. [6] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [7] T. R. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, Discrete Contin. Dyn. Syst., 1 (1995), 103-117. doi: 10.3934/dcds.1995.1.103. [8] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357. doi: 10.1016/j.jde.2016.02.023. [9] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. [10] S. R. Dunbar, Traveling wave solutions in diffusive predator-prey systems: periodic orbits and point-to-periodic heteroclic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078. doi: 10.1137/0146063. [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. [12] J. Fang and X. Q. Zhao, Existence and uniqueness of traveling waves for non-monotone integral equations with applications, J. Differential Equations, 248 (2010), 2199-2226. doi: 10.1016/j.jde.2010.01.009. [13] J. Fang and X. Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.4171/jems/556. [14] P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), 168-185. doi: 10.1016/0022-0396(81)90016-4. [15] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., vol. 247, Longman Scientific Technical, Harlow, UK, 1991. doi: 0003-889X/97/050388-10. [16] S. B. Hsu and X. Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), no. 2,776–789. doi: 10.1137/070703016. [17] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269. [18] X. Liang, Y. Yi and X. Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010. [19] 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. [20] G. Lin, Spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Anal., 74 (2011), 2448-2461. doi: 10.1016/j.na.2010.11.046. [21] G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58. doi: 10.1016/j.na.2013.10.024. [22] G. Lin and R. Wang, Spatial invasion dynamics for a time period predator-prey system, Math. Methods Appl. Sci., 41 (2018), 7621-7623. doi: 10.1002/mma.5224. [23] X. L. Liu and S. Pan, Spreading speed in a nonmonotone equation with dispersal and delay, Mathematics, 7 (2019), ID: 291. doi: 10.3390/math7030291. [24] 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)90027-8. [25] J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. doi: 10.1007/b98869. [26] G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262. doi: 10.1016/j.matpur.2009.04.002. [27] 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. [28] 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. [29] S. Pan, Invasion speed of a predator-prey system, Appl. Math. Lett., 74 (2017), 46-51. doi: 10.1016/j.aml.2017.05.014. [30] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford, New York, 1997. doi: 10.1002/(SICI)1520-6300(1998)10:5<683::AID-AJHB17>3.0.CO;2-4. [31] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2$^{nd}$ Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [32] Z. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1999), 339-352. doi: 10.1080/00036819908840745. [33] Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations, 179 (2002), 538-561. doi: 10.1006/jdeq.2001.4044. [34] Z. Teng and L. Chen, Global asymptotic stability of periodic Lotka-Volterra systems with delays, Nonlinear Anal., 45 (2001), 1081-1095. doi: 10.1016/S0362-546X(99)00441-1. [35] M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013. [36] M. X. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327. doi: 10.1016/j.cnsns.2014.11.016. [37] M. X. Wang, W. J. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329. doi: 10.1016/j.jmaa.2016.04.007. [38] X. J. Wang and G. Lin, Traveling waves for a periodic Lotka-Volterra predator-prey system, Appl. Anal., (2018), in press. doi: 10.1080/00036811.2018.1469007. [39] H. F. Weinberger, Long-time behavior of a class of biological model, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [40] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548. doi: 10.1007/s00285-002-0169-3. [41] 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. [42] S. L. Wu, C. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009. [43] Q. Ye, Z. Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2$^{nd}$ edn, Science Press, Beijing, 2011. [44] T. Yi, Y. Chen and J. Wu, Unimodal dynamical systems: comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), no. 8, 3538–3572. doi: 10.1016/j.jde.2013.01.031. [45] G. Y. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671. doi: doi.org/10.1016/j.matpur.2010.11.005. [46] 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, 2009, pp. 332–363. [47] X. Q. Zhao, Dynamincal Systems in Population Biology, 2$^{nd}$ edn. Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.
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