# American Institute of Mathematical Sciences

May  2021, 26(5): 2599-2623. doi: 10.3934/dcdsb.2020197

## Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system

 1 College of Mathematics, Hunan University, Changsha, Hunan 410082, China 2 School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  November 2019 Revised  April 2020 Published  May 2021 Early access  June 2020

Fund Project: The first and second authors are supported by NSF of China (Grant Nos. 11671123 and 11801089) and by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan)

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue.

Citation: Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
##### References:
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(9), 121 (2019), 47–82. doi: 10.1016/j.matpur.2018.06.010. [21] L. Ma and S. Guo, Bifurcation and stability of a two-species diffusive Lotka-Volterra model, Comm. Pure Appl. Anal., 19 (2020), 1205-1232.  doi: 10.3934/cpaa.2020056. [22] L. Ma, S. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 18pp. doi: 10.1142/S0218127418500335. [23] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction diffusion equations, J. Math. Biol., 18 (1983), 169-184.  doi: 10.1007/BF00280665. [24] W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.fm. [25] H. Qiu, S. Guo and S. Li, Stability and Bifurcation in a Predator-Prey System with Prey-Taxis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 25pp. doi: 10.1142/S0218127420500224. [26] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview, Philos. Trans. R. Soc. B., 357 (2002), 1153-1170.  doi: 10.1098/rstb.2002.1117. [27] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [28] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dynam. Differential Equations, 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [29] D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003. [30] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. [31] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [32] L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225.  doi: 10.1016/S0960-0779(02)00068-1. [33] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006. [34] P. Zhou and X.-Q. Zhao, Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 264 (2018), 4176-4198.  doi: 10.1016/j.jde.2017.12.005. [35] R. Zou and S. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B. (2020) in press doi: 10.3934/dcdsb.2020093.

show all references

##### References:
 [1] L. C. Birch, Experimental background to the study of the distribution and aboundance of insects: Ⅰ. The influence of temperature, moisture and food on the innate capacity for increase of three grain beetles, Ecology, 34 (1953), 698-711.  doi: 10.2307/1931333. [2] S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  doi: 10.1006/jdeq.1996.0003. [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [4] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.  doi: 10.1080/00036817408839081. [5] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031. [6] X. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80.  doi: 10.1512/iumj.2012.61.4518. [7] Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006. [8] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182. [9] J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0. [10] S. Guo, Patterns in a nonlocal time-delayed reaction-diffusion equation, Z. Angew. Math. Phys., 69 (2018), 31pp. doi: 10.1007/s00033-017-0904-7. [11] S. Guo and S. Li, On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition, Appl. Math. Lett., 103 (2020), 7pp. doi: 10.1016/j.aml.2019.106197. [12] S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 26 (2016), 545-580.  doi: 10.1007/s00332-016-9285-x. [13] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6. [14] S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, J. Differential Equations, 260 (2016), 781-817.  doi: 10.1016/j.jde.2015.09.031. [15] X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546.  doi: 10.1016/j.jde.2012.08.032. [16] R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay, J. Differential Equations, 250 (2011), 2779-2806.  doi: 10.1016/j.jde.2011.01.011. [17] S. Li and S. Guo, Stability and Hopf bifurcation in a Hutchinson model, Appl. Math. Lett., 101 (2020), 7pp. doi: 10.1016/j.aml.2019.106066. [18] W.-T. Li, X.-P. Yan and C.-H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos Solitons Fractals, 38 (2008), 227-237.  doi: 10.1016/j.chaos.2006.11.015. [19] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.  doi: 10.1016/j.jde.2005.05.010. [20] Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusionadvection system in heterogeneous environments, J. Math. Pures Appl. (9), 121 (2019), 47–82. doi: 10.1016/j.matpur.2018.06.010. [21] L. Ma and S. Guo, Bifurcation and stability of a two-species diffusive Lotka-Volterra model, Comm. Pure Appl. Anal., 19 (2020), 1205-1232.  doi: 10.3934/cpaa.2020056. [22] L. Ma, S. Guo and T. Chen, Dynamics of a nonlocal dispersal model with a nonlocal reaction term, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 18pp. doi: 10.1142/S0218127418500335. [23] J. D. Murray and R. P. Sperb, Minimum domains for spatial patterns in a class of reaction diffusion equations, J. Math. Biol., 18 (1983), 169-184.  doi: 10.1007/BF00280665. [24] W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.fm. [25] H. Qiu, S. Guo and S. Li, Stability and Bifurcation in a Predator-Prey System with Prey-Taxis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 25pp. doi: 10.1142/S0218127420500224. [26] R. M. Sibly and J. Hone, Population growth rate and its determinants: An overview, Philos. Trans. R. Soc. B., 357 (2002), 1153-1170.  doi: 10.1098/rstb.2002.1117. [27] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196. [28] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dynam. Differential Equations, 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [29] D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003. [30] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. [31] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [32] L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225.  doi: 10.1016/S0960-0779(02)00068-1. [33] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006. [34] P. Zhou and X.-Q. Zhao, Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 264 (2018), 4176-4198.  doi: 10.1016/j.jde.2017.12.005. [35] R. Zou and S. Guo, Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B. (2020) in press doi: 10.3934/dcdsb.2020093.
Solutions of (58) with parameters (59) and $r = 1.1$ tend to a spatially nonhomogeneous boundary steady-state
Solutions of (58) with parameters (60) and $r = 2.1$ tend to a spatially nonhomogeneous positive steady-state
The Corresponding Table of Bifurcation Parameter and Open Set
 Theorem Bifurcation Value Open Set $\Lambda$ Theorem 2.3 $\mu_1$ $(\mu_1-\delta_1, \mu_1)\bigcup(\mu_1, \mu_1+\delta_1)$ Theorem 2.4 $\mu_2$ $(\mu_2-\delta_2, \mu_2)\bigcup(\mu_2, \mu_2+\delta_2)$ Theorem 2.5 $\mu_\ast$ $(\mu_\ast-\delta_3, \mu_\ast)\bigcup(\mu_\ast, \mu_\ast+\delta_3)$ Theorem 2.6 $(\mathbf{i})$ $\mu_1$ $(\mu_1-\delta_5, \mu_1)\bigcup(\mu_1, \mu_1+\delta_5)$ Theorem 2.6 $(\mathbf{ii})$ $\mu_2$ $(\mu_2-\delta_6, \mu_2)\bigcup(\mu_2, \mu_2+\delta_6)$
 Theorem Bifurcation Value Open Set $\Lambda$ Theorem 2.3 $\mu_1$ $(\mu_1-\delta_1, \mu_1)\bigcup(\mu_1, \mu_1+\delta_1)$ Theorem 2.4 $\mu_2$ $(\mu_2-\delta_2, \mu_2)\bigcup(\mu_2, \mu_2+\delta_2)$ Theorem 2.5 $\mu_\ast$ $(\mu_\ast-\delta_3, \mu_\ast)\bigcup(\mu_\ast, \mu_\ast+\delta_3)$ Theorem 2.6 $(\mathbf{i})$ $\mu_1$ $(\mu_1-\delta_5, \mu_1)\bigcup(\mu_1, \mu_1+\delta_5)$ Theorem 2.6 $(\mathbf{ii})$ $\mu_2$ $(\mu_2-\delta_6, \mu_2)\bigcup(\mu_2, \mu_2+\delta_6)$
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