# American Institute of Mathematical Sciences

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

 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.
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.

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##### 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.
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