American Institute of Mathematical Sciences

Advanced
July  2019, 18(4): 1613-1636. doi: 10.3934/cpaa.2019077

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

Liang Kong 1,, , Tung Nguyen 1, and  Wenxian Shen 2,

1. 

Department of Mathematical Sciences, University of Illinois at Springfield, Springfield, IL 62703, USA
2. 

Department of Mathematics and Statistics, Auburn University, AL 36849, USA

* Corresponding author

Received  March 2018 Revised  October 2018 Published  January 2019

This paper is concerned with the asymptotic dynamics of two species competition systems of the form
$ \begin{equation*} \begin{cases} u_t(t,x) = \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in {\mathbb{R}} \cr v_t(t,x) = \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in {\mathbb{R}} \end{cases} \end{equation*} $
where
$ (\mathcal{A}u)(t,x) = u_{xx}(t,x) $
, or
$ (\mathcal{A}u)(t,x) = \int_{ {\mathbb{R}} }\kappa(y-x)u(t,y)dy-u(t,x) $
 (
$ \kappa(\cdot) $
 is a smooth non-negative convolution kernel supported on an interval centered at the origin),
$ a_i(t+T,x) = a_i(t,x) $
,
$ b_i(t+T,x) = b_i(t,x) $
,
$ c_i(t+T,x) = c_i(t,x) $
, and
$ a_i $
,
$ b_i $
, and
$ c_i $
 (
$ i = 1,2 $
) are spatially homogeneous when
$ |x|\gg 1 $
, that is,
$ a_i(t,x) = a_i^0(t) $
,
$ b_i(t,x) = b_i^0(t) $
,
$ c_i(t,x) = c_i^0(t) $
 for some
$ a_i^0(t) $
,
$ b_i^0(t) $
,
$ c_i^0(t) $
, and
$ |x|\gg 1 $
. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. In particular, we study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds. We also study a relevant problem, that is, the continuity of the spreading speeds of time periodic two species competition systems with respect to time periodic perturbations, and prove that the spread speeds of such systems are lower semicontinuous with respect to time periodic perturbations.
Keywords: Two species competition system, localized spatial variation, persistence, spreading speeds.
Mathematics Subject Classification: 35K58, 45G15, 92D25.
Citation: Liang Kong, Tung Nguyen, Wenxian Shen. Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077
References:
[1]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.

[2]

X. Bao, W.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.

[3]

H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Annali di Matematica, 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0.

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.

[5]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

[6]

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.

[7]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Functional Analysis, 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[8]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for Lotka-volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.

[9]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.

[10]

G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.

[11]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain Journal of Mathematics, 43 (2013), 489-513.  doi: 10.1216/RMJ-2013-43-2-489.

[12]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448. 

[13]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.

[14]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.

[15]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.

[16]

L. Kong and W. Shen, Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media, Methods Appl. Anal., 18 (2011), 427-456.  doi: 10.4310/MAA.2011.v18.n4.a5.

[17]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.

[18]

L. Kong, N. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.

[19]

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

[20]

W.-T. Li, L Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.

[21]

T. Nguyen and Nar Rawal, Coexistence and extinction in time-periodic Volterra-Lotka type systems with nonlocal dispersal, Discrete Contin. Dyn. Syst., Series B, 23(9) (2018), 3799-3816. doi: 10.3934/dcdsb.2018080.

[22]

Nar Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.

[23]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, Journal of Differential Equations, 249 (2010), 749-795.  doi: 10.1016/j.jde.2010.04.012.

[24]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.

[25]

G. 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: 10.1016/j.matpur.2010.11.005.

show all references

References:
[1]

X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), 2402-2435.  doi: 10.1016/j.jde.2013.06.024.

[2]

X. Bao, W.-T. Li and W. Shen, Traveling wave solutions of Lotka-Volterra competition systems with nonlocal dispersal in periodic habitats, J. Differential Equations, 260 (2016), 8590-8637.  doi: 10.1016/j.jde.2016.02.032.

[3]

H. Berestycki, F. Hamel and L. Rossi, Liouville-type results for semilinear elliptic equations in unbounded domains, Annali di Matematica, 186 (2007), 469-507.  doi: 10.1007/s10231-006-0015-0.

[4]

C. Conley and R. Gardner, An application of the generalized Morse index to traveling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.

[5]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.  doi: 10.1007/BF00276112.

[6]

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.

[7]

J. Fang, X. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Functional Analysis, 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.

[8]

J.-S. Guo and X. Liang, The minimal speed of traveling fronts for Lotka-volterra competition system, J. Dynam. Differential Equations, 23 (2011), 353-363.  doi: 10.1007/s10884-011-9214-5.

[9]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Commun. Pure Appl. Anal., 11 (2012), 1699-1722.  doi: 10.3934/cpaa.2012.11.1699.

[10]

G. Hetzer and W. Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal., 34 (2002), 204-227.  doi: 10.1137/S0036141001390695.

[11]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations, Rocky Mountain Journal of Mathematics, 43 (2013), 489-513.  doi: 10.1216/RMJ-2013-43-2-489.

[12]

Y. Hosono, The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model, Bull. Math. Biol., 66 (1998), 435-448. 

[13]

W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0.

[14]

Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556.

[15]

Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I.

[16]

L. Kong and W. Shen, Positive stationary solutions and spreading speeds of KPP equations in locally spatially inhomogeneous media, Methods Appl. Anal., 18 (2011), 427-456.  doi: 10.4310/MAA.2011.v18.n4.a5.

[17]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity, J. Dynam. Differential Equations, 26 (2014), 181-215.  doi: 10.1007/s10884-014-9351-8.

[18]

L. Kong, N. Rawal and W. Shen, Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats, Math. Model. Nat. Phenom., 10 (2015), 113-141.  doi: 10.1051/mmnp/201510609.

[19]

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

[20]

W.-T. Li, L Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.

[21]

T. Nguyen and Nar Rawal, Coexistence and extinction in time-periodic Volterra-Lotka type systems with nonlocal dispersal, Discrete Contin. Dyn. Syst., Series B, 23(9) (2018), 3799-3816. doi: 10.3934/dcdsb.2018080.

[22]

Nar Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dynam. Differential Equations, 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.

[23]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, Journal of Differential Equations, 249 (2010), 749-795.  doi: 10.1016/j.jde.2010.04.012.

[24]

X. Yu and X.-Q. Zhao, Propagation phenomena for a reaction-advection-diffusion competition model in a periodic habitat, J. Dynam. Differential Equations, 29 (2017), 41-66.  doi: 10.1007/s10884-015-9426-1.

[25]

G. 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: 10.1016/j.matpur.2010.11.005.

[1]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[2]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[3]

Yuanshi Wang, Hong Wu. Transition of interaction outcomes in a facilitation-competition system of two species. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1463-1475. doi: 10.3934/mbe.2017076
[4]

Georg Hetzer, Wenxian Shen. Two species competition with an inhibitor involved. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 39-57. doi: 10.3934/dcds.2005.12.39
[5]

Xinyu Tu, Chunlai Mu, Pan Zheng, Ke Lin. Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156
[6]

Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187
[7]

Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087
[8]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208
[9]

Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 837-861. doi: 10.3934/dcdsb.2021067
[10]

Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621
[11]

Chiu-Ju Lin. Competition of two phytoplankton species for light with wavelength. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 523-536. doi: 10.3934/dcdsb.2016.21.523
[12]

Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659
[13]

Alan E. Lindsay, Michael J. Ward. An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139
[14]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
[15]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268
[16]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[17]

Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691
[18]

Kuang-Hui Lin, Yuan Lou, Chih-Wen Shih, Tze-Hung Tsai. Global dynamics for two-species competition in patchy environment. Mathematical Biosciences & Engineering, 2014, 11 (4) : 947-970. doi: 10.3934/mbe.2014.11.947
[19]

S.A. Gourley, Yang Kuang. Two-Species Competition with High Dispersal: The Winning Strategy. Mathematical Biosciences & Engineering, 2005, 2 (2) : 345-362. doi: 10.3934/mbe.2005.2.345
[20]

Xiaoqing He, Sze-Bi Hsu, Feng-Bin Wang. A periodic-parabolic Droop model for two species competition in an unstirred chemostat. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (210)
  • HTML views (198)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy