June 2018, 23(4): 1559-1579. doi: 10.3934/dcdsb.2018059

Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity

1. 

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

2. 

Department of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

* Corresponding author: S. Guo

Received  April 2017 Revised  August 2017 Published  February 2018

Fund Project: The second author is supported by NSF of China (Grants No. 11671123)

This paper is concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global attractivity of the semi-trivial solution. In addition, an attracting region was obtained by means of the method of upper and lower solutions.

Citation: Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
References:
[1]

W.G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101 (1990), 139-153. doi: 10.1016/0025-5564(90)90019-U.

[2]

S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, Journal of 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, John Wiley & Sons, 2003.

[4]

S. Chen and J. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, Journal of Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031.

[5]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications, 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.

[6]

H. I. Freedman and X. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, Journal of Differential Equations, 137 (1997), 340-362. doi: 10.1006/jdeq.1997.3264.

[7]

P. Georgescu and Y.-H. Hsieh, Global dynamics of a predator-prey model with stage structure for the predator, SIAM Journal on Applied Mathematics, 67 (2007), 1379-1395. doi: 10.1137/060670377.

[8]

S.A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal on Applied Mathematics, 65 (2004), 550-566. doi: 10.1137/S0036139903436613.

[9]

S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, Journal of Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006.

[10]

S. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864-908. doi: 10.1093/imamat/hxx018.

[11]

S. GuoY. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, Journal of Differential Equations, 244 (2008), 444-486. doi: 10.1016/j.jde.2007.09.008.

[12]

S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with dirichlet boundary condition, Journal of Nonlinear Science, 26 (2016), 545-580. doi: 10.1007/s00332-016-9285-x.

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, 2013.

[14]

S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817. doi: 10.1016/j.jde.2015.09.031.

[15]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition--diffusion system I: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.

[16]

X. He and W. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity I, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596.

[17]

W. Huang, Global dynamics for a reaction--diffusion equation with time delay, Journal of Differential Equations, 143 (1998), 293-326. doi: 10.1006/jdeq.1997.3374.

[18]

K.-Y. Lam and W. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM Journal on Applied Mathematics, 72 (2012), 1695-1712. doi: 10.1137/120869481.

[19]

D. Li and S. Guo, Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247.

[20]

J. Li and Z. Ma, Stability switches in a class of characteristic equations with delay-dependent parameters, Nonlinear Analysis: Real World Applications, 5 (2004), 389-408. doi: 10.1016/j.nonrwa.2003.06.001.

[21]

Z. LiM. Han and F. Chen, Global stability of a predator-prey system with stage structure and mutual interference, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 173-187.

[22]

Y. Lin, X. Xie, F. Chen and T. Li, Convergences of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, Advances in Difference Equations, 2016 (2016), 19pp.

[23]

S. LiuL. ChenG. Luo and Y. Jiang, Asymptotic behaviors of competitive Lotka-Volterra system with stage structure, Journal of Mathematical Analysis and Applications, 271 (2002), 124-138. doi: 10.1016/S0022-247X(02)00103-8.

[24]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321 (1990), 1-44.

[25]

C. V. Pao, Coupled nonlinear parabolic systems with time delays, Journal of Mathematical Analysis and Applications, 196 (1995), 237-265. doi: 10.1006/jmaa.1995.1408.

[26]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, Journal of Mathematical Analysis and Applications, 198 (1996), 751-779. doi: 10.1006/jmaa.1996.0111.

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.

[28]

Y. Qu and J. Wei, Bifurcation analysis in a predator--prey system with stage-structure and harvesting, Journal of the Franklin Institute, 347 (2010), 1097-1113. doi: 10.1016/j.jfranklin.2010.03.017.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[30]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Mathematicae Applicatae Sinica, 18 (2002), 423-430. doi: 10.1007/s102550200042.

[31]

H.R. Thieme and X. Zhao, A non-local delayed and diffusive predator--prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7.

[32]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of the American Mathematical Society, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[33]

J. Wu, Theory and Applications of Partial Functional Differential Equations, volume 119. Springer Science & Business Media, 1996.

[34]

M. Xiao and J. Cao, Stability and Hopf bifurcation in a delayed competitive web sites model, Physics Letters A, 353 (2006), 138-150.

[35]

S. Yan and S. Guo, Bifurcation phenomena in a {L}otka-{V}olterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos, 27 (2017), 1750105, 24pp.

[36]

S. Yan and S. Guo, Stability analysis of a stage structure model with spatiotemporal delay effect, Computers & Mathematics with Applications, 73 (2017), 310-326. doi: 10.1016/j.camwa.2016.11.029.

[37]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case, Journal of Differential Equations, 245 (2008), 3376-3388. doi: 10.1016/j.jde.2008.03.007.

show all references

References:
[1]

W.G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Mathematical Biosciences, 101 (1990), 139-153. doi: 10.1016/0025-5564(90)90019-U.

[2]

S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, Journal of 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, John Wiley & Sons, 2003.

[4]

S. Chen and J. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, Journal of Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031.

[5]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications, 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182.

[6]

H. I. Freedman and X. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, Journal of Differential Equations, 137 (1997), 340-362. doi: 10.1006/jdeq.1997.3264.

[7]

P. Georgescu and Y.-H. Hsieh, Global dynamics of a predator-prey model with stage structure for the predator, SIAM Journal on Applied Mathematics, 67 (2007), 1379-1395. doi: 10.1137/060670377.

[8]

S.A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal on Applied Mathematics, 65 (2004), 550-566. doi: 10.1137/S0036139903436613.

[9]

S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, Journal of Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006.

[10]

S. Guo, Spatio-temporal patterns in a diffusive model with non-local delay effect, IMA Journal of Applied Mathematics, 82 (2017), 864-908. doi: 10.1093/imamat/hxx018.

[11]

S. GuoY. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, Journal of Differential Equations, 244 (2008), 444-486. doi: 10.1016/j.jde.2007.09.008.

[12]

S. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with dirichlet boundary condition, Journal of Nonlinear Science, 26 (2016), 545-580. doi: 10.1007/s00332-016-9285-x.

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, 2013.

[14]

S. Guo and S. Yan, Hopf bifurcation in a diffusive Lotka-Volterra type system with nonlocal delay effect, Journal of Differential Equations, 260 (2016), 781-817. doi: 10.1016/j.jde.2015.09.031.

[15]

X. He and W. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition--diffusion system I: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.

[16]

X. He and W. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity I, Communications on Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596.

[17]

W. Huang, Global dynamics for a reaction--diffusion equation with time delay, Journal of Differential Equations, 143 (1998), 293-326. doi: 10.1006/jdeq.1997.3374.

[18]

K.-Y. Lam and W. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM Journal on Applied Mathematics, 72 (2012), 1695-1712. doi: 10.1137/120869481.

[19]

D. Li and S. Guo, Bifurcation and stability of a Mimura-Tsujikawa model with nonlocal delay effect, Mathematical Methods in the Applied Sciences, 40 (2017), 2219-2247.

[20]

J. Li and Z. Ma, Stability switches in a class of characteristic equations with delay-dependent parameters, Nonlinear Analysis: Real World Applications, 5 (2004), 389-408. doi: 10.1016/j.nonrwa.2003.06.001.

[21]

Z. LiM. Han and F. Chen, Global stability of a predator-prey system with stage structure and mutual interference, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 173-187.

[22]

Y. Lin, X. Xie, F. Chen and T. Li, Convergences of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, Advances in Difference Equations, 2016 (2016), 19pp.

[23]

S. LiuL. ChenG. Luo and Y. Jiang, Asymptotic behaviors of competitive Lotka-Volterra system with stage structure, Journal of Mathematical Analysis and Applications, 271 (2002), 124-138. doi: 10.1016/S0022-247X(02)00103-8.

[24]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321 (1990), 1-44.

[25]

C. V. Pao, Coupled nonlinear parabolic systems with time delays, Journal of Mathematical Analysis and Applications, 196 (1995), 237-265. doi: 10.1006/jmaa.1995.1408.

[26]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, Journal of Mathematical Analysis and Applications, 198 (1996), 751-779. doi: 10.1006/jmaa.1996.0111.

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.

[28]

Y. Qu and J. Wei, Bifurcation analysis in a predator--prey system with stage-structure and harvesting, Journal of the Franklin Institute, 347 (2010), 1097-1113. doi: 10.1016/j.jfranklin.2010.03.017.

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[30]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Mathematicae Applicatae Sinica, 18 (2002), 423-430. doi: 10.1007/s102550200042.

[31]

H.R. Thieme and X. Zhao, A non-local delayed and diffusive predator--prey model, Nonlinear Analysis: Real World Applications, 2 (2001), 145-160. doi: 10.1016/S0362-546X(00)00112-7.

[32]

C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Transactions of the American Mathematical Society, 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.

[33]

J. Wu, Theory and Applications of Partial Functional Differential Equations, volume 119. Springer Science & Business Media, 1996.

[34]

M. Xiao and J. Cao, Stability and Hopf bifurcation in a delayed competitive web sites model, Physics Letters A, 353 (2006), 138-150.

[35]

S. Yan and S. Guo, Bifurcation phenomena in a {L}otka-{V}olterra model with cross-diffusion and delay effect, International Journal of Bifurcation and Chaos, 27 (2017), 1750105, 24pp.

[36]

S. Yan and S. Guo, Stability analysis of a stage structure model with spatiotemporal delay effect, Computers & Mathematics with Applications, 73 (2017), 310-326. doi: 10.1016/j.camwa.2016.11.029.

[37]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case, Journal of Differential Equations, 245 (2008), 3376-3388. doi: 10.1016/j.jde.2008.03.007.

Figure 1.  Solutions of model (7) with $\tau = 0.3<\tau_*$ tend to the semi-trivial steady state solution $(\theta_{d_1,\alpha,0},0)$.
Figure 2.  Solutions of model (7) with $\tau = 3>\tau^*$ tend to the semi-trivial steady state solution $(0, \theta_{d_2,r})$.
Figure 3.  Solutions of model (7) with $\tau = 1.485\in(\ln\dfrac{c\bar{\alpha}}{\bar{r}}, \ln\dfrac{\bar{\alpha}}{a\bar{r}})$ tend to a positive steady state.
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