# 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. Google Scholar [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. 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, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2$^{nd}$ Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [32] Z. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1999), 339-352. doi: 10.1080/00036819908840745. Google Scholar [33] Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations, 179 (2002), 538-561. doi: 10.1006/jdeq.2001.4044. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] Q. Ye, Z. Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2$^{nd}$ edn, Science Press, Beijing, 2011. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [47] X. Q. Zhao, Dynamincal Systems in Population Biology, 2$^{nd}$ edn. Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32. doi: 10.1007/BF00276112. Google Scholar [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. 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, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] X. Li and S. Pan, Traveling wave solutions of a delayed cooperative system, Mathematics, 7 (2019), ID: 269. doi: 10.3390/math7030269. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] J. Smoller, Shock Waves and Reaction Diffusion Equations, 2$^{nd}$ Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [32] Z. Teng, Uniform persistence of the periodic predator-prey Lotka-Volterra systems, Appl. Anal., 72 (1999), 339-352. doi: 10.1080/00036819908840745. Google Scholar [33] Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations, 179 (2002), 538-561. doi: 10.1006/jdeq.2001.4044. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] Q. Ye, Z. Li, M. X. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, 2$^{nd}$ edn, Science Press, Beijing, 2011. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [47] X. Q. Zhao, Dynamincal Systems in Population Biology, 2$^{nd}$ edn. Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3. Google Scholar
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