# 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 & Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077
##### References:

show all references

##### References:
 [1] Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441 [2] 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 & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 [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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2018, 38 (7) : 3617-3636. doi: 10.3934/dcds.2018156 [6] Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087 [7] 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 [8] Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208 [9] Hua Nie, Yuan Lou, Jianhua Wu. Competition between two similar species in the unstirred chemostat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 621-639. doi: 10.3934/dcdsb.2016.21.621 [10] Chiu-Ju Lin. Competition of two phytoplankton species for light with wavelength. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 523-536. doi: 10.3934/dcdsb.2016.21.523 [11] Jifa Jiang, Fensidi Tang. The complete classification on a model of two species competition with an inhibitor. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 659-672. doi: 10.3934/dcds.2008.20.659 [12] 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 & Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139 [13] Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213 [14] Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 [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 & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268 [16] Hua Nie, Sze-Bi Hsu, Jianhua Wu. Coexistence solutions of a competition model with two species in a water column. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2691-2714. doi: 10.3934/dcdsb.2015.20.2691 [17] 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 [18] 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 [19] Xiaoqing He, Sze-Bi Hsu, Feng-Bin Wang. A periodic-parabolic Droop model for two species competition in an unstirred chemostat. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4427-4451. doi: 10.3934/dcds.2020185 [20] Zhiguo Wang, Hua Nie, Jianhua Wu. Spatial propagation for a parabolic system with multiple species competing for single resource. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1785-1814. doi: 10.3934/dcdsb.2018237

2019 Impact Factor: 1.105

## Metrics

• HTML views (191)
• Cited by (0)

• on AIMS