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doi: 10.3934/dcdsb.2022108
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Spreading speed in a non-monotonic Ricker competitive integrodifference system

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

*Corresponding author: Guo Lin

Received  December 2021 Revised  April 2022 Early access June 2022

Fund Project: The first author is supported by NSF of China (No. 11971213) and Natural Science Foundation of Gansu Province of China (No. 21JR7RA535)

This article studies the propagation threshold in a competitive system without the classical comparison principle, which models the invasion-coexistence process between two competitors of which one is the invader and another is the aborigine. We obtain a linearly determined threshold that is the spreading speed of the invader in initial value problems. To estimate the spreading speed, we construct several auxiliary scalar equations and use the method of generalized upper solutions. In population dynamics, our results imply that even in a nonmonotone competitive system, the invader may invade the habitat of the aborigine at an almost constant speed and two species could coexist eventually.

Citation: Guo Lin, Yahui Wang. Spreading speed in a non-monotonic Ricker competitive integrodifference system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022108
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, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5–49.

[2]

A. BourgeoisV. LeBlanc and F. Lutscher, Dynamical stabilization and traveling waves in integrodifference equations, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3029-3045.  doi: 10.3934/dcdss.2020117.

[3]

A. BourgeoisV. LeBlanc and F. Lutscher, Spreading phenomena in integrodifference equations with nonmonotone growth functions, SIAM J. Appl. Math., 78 (2018), 2950-2972.  doi: 10.1137/17M1126102.

[4]

C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol., 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4.

[5]

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.

[6]

J. HofbauerV. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570.  doi: 10.1007/BF00276199.

[7]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[8]

M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.

[9]

M. A. LewisB. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.

[10]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 389-399.  doi: 10.3934/dcdsb.2009.12.389.

[11]

K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497.  doi: 10.1016/j.jmaa.2011.11.055.

[12]

K. Li and X. Li, Travelling wave solutions in integro-difference competition system, IMA J. Appl. Math., 78 (2013), 633-650.  doi: 10.1093/imamat/hxs002.

[13]

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.

[14]

G. Lin, Y. Niu, S. Pan and S. Ruan, Spreading speed in an integrodifference predator-prey system without comparison principle, Bull. Math. Biol., 82 (2020), 28 pp. doi: 10.1007/s11538-020-00725-y.

[15]

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.

[16]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics, 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.

[17]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.

[18]

S. Pan, Monotone traveling waves of nonmonotone integrodifference equations, J. Appl. Anal. Comput., 11 (2021), 477-485.  doi: 10.11948/20200069.

[19]

S. PanG. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.

[20]

S. Pan and J. Liu, Bistable traveling wave solutions in a competitive recursion system with Ricker nonlinearity, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1-9.  doi: 10.14232/ejqtde.2014.1.7.

[21]

B. Ryals and R. J. Sacker, Global stability in the 2D Ricker equation revisited, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 585-604.  doi: 10.3934/dcdsb.2017028.

[22] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. 
[23]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266.  doi: 10.3934/dcdsb.2012.17.2243.

[24]

H. F. WeinbergerM. 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.

[25]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.

[26]

Z. Yuan and X. Zou, Co-invasion waves in a reaction diffusion model for competing pioneer and climax species, Nonlinear Anal. Real World Appl., 11 (2010), 232-245.  doi: 10.1016/j.nonrwa.2008.11.003.

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, Lecture Notes in Math., 446, Springer, Berlin, 1975, 5–49.

[2]

A. BourgeoisV. LeBlanc and F. Lutscher, Dynamical stabilization and traveling waves in integrodifference equations, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3029-3045.  doi: 10.3934/dcdss.2020117.

[3]

A. BourgeoisV. LeBlanc and F. Lutscher, Spreading phenomena in integrodifference equations with nonmonotone growth functions, SIAM J. Appl. Math., 78 (2018), 2950-2972.  doi: 10.1137/17M1126102.

[4]

C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol., 50 (2005), 161-188.  doi: 10.1007/s00285-004-0284-4.

[5]

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.

[6]

J. HofbauerV. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25 (1987), 553-570.  doi: 10.1007/BF00276199.

[7]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.

[8]

M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.

[9]

M. A. LewisB. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.

[10]

B. Li, Some remarks on traveling wave solutions in competition models, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 389-399.  doi: 10.3934/dcdsb.2009.12.389.

[11]

K. Li and X. Li, Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system, J. Math. Anal. Appl., 389 (2012), 486-497.  doi: 10.1016/j.jmaa.2011.11.055.

[12]

K. Li and X. Li, Travelling wave solutions in integro-difference competition system, IMA J. Appl. Math., 78 (2013), 633-650.  doi: 10.1093/imamat/hxs002.

[13]

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.

[14]

G. Lin, Y. Niu, S. Pan and S. Ruan, Spreading speed in an integrodifference predator-prey system without comparison principle, Bull. Math. Biol., 82 (2020), 28 pp. doi: 10.1007/s11538-020-00725-y.

[15]

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.

[16]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Interdisciplinary Applied Mathematics, 49, Springer, Cham, 2019. doi: 10.1007/978-3-030-29294-2.

[17]

J. D. Murray, Mathematical Biology. II. Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.

[18]

S. Pan, Monotone traveling waves of nonmonotone integrodifference equations, J. Appl. Anal. Comput., 11 (2021), 477-485.  doi: 10.11948/20200069.

[19]

S. PanG. Lin and J. Wang, Propagation thresholds of competitive integrodifference systems, J. Difference Equ. Appl., 25 (2019), 1680-1705.  doi: 10.1080/10236198.2019.1678597.

[20]

S. Pan and J. Liu, Bistable traveling wave solutions in a competitive recursion system with Ricker nonlinearity, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1-9.  doi: 10.14232/ejqtde.2014.1.7.

[21]

B. Ryals and R. J. Sacker, Global stability in the 2D Ricker equation revisited, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 585-604.  doi: 10.3934/dcdsb.2017028.

[22] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. 
[23]

H. Wang and C. Castillo-Chavez, Spreading speeds and traveling waves for non-cooperative integro-difference systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2243-2266.  doi: 10.3934/dcdsb.2012.17.2243.

[24]

H. F. WeinbergerM. 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.

[25]

T. YiY. Chen and J. Wu, Unimodal dynamical systems: Comparison principles, spreading speeds and travelling waves, J. Differential Equations, 254 (2013), 3538-3572.  doi: 10.1016/j.jde.2013.01.031.

[26]

Z. Yuan and X. Zou, Co-invasion waves in a reaction diffusion model for competing pioneer and climax species, Nonlinear Anal. Real World Appl., 11 (2010), 232-245.  doi: 10.1016/j.nonrwa.2008.11.003.

Figure 1.  Snapshots of the solution of model (11) with $ r_1 = 1.01 $
Figure 2.  Snapshots of the solution of model (11) with $ r_1 = 1.01 $
Figure 3.  Snapshots of $ v_n(x) $ for (11) with $ r_1 = 2.4 $
Figure 4.  Snapshots of (11) with $ r_1 = 2.4 $
Figure 5.  Dynamics of (11) with $ r_1 = 4.4 $
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