American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2020197

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

Dan Wei 1, and  Shangjiang Guo 2,,

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  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.

Keywords: Stability, reaction-diffusion-advection, Lyapunov-Schmidt reduction, Lotka-Volterra system.
Mathematics Subject Classification: Primary: 92D25; Secondary: 35K57.
Citation: Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020197
References:
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[16]

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[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.  Google Scholar

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[24]

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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.  Google Scholar

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J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.1093/biomet/38.1-2.196.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30] Q. X. Ye and Z. Y. Li, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Solutions of (58) with parameters (59) and $ r = 1.1 $ tend to a spatially nonhomogeneous boundary steady-state
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Figure 2.  Solutions of (58) with parameters (60) and $ r = 2.1 $ tend to a spatially nonhomogeneous positive steady-state
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Table 1.  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) $
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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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