# American Institute of Mathematical Sciences

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  June 2018 Early access  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 and 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. Guo, Y. 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. Li, M. 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. Liu, L. Chen, G. 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.

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##### 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. Guo, Y. 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. Li, M. 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. Liu, L. Chen, G. 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.
Solutions of model (7) with $\tau = 0.3<\tau_*$ tend to the semi-trivial steady state solution $(\theta_{d_1,\alpha,0},0)$.
Solutions of model (7) with $\tau = 3>\tau^*$ tend to the semi-trivial steady state solution $(0, \theta_{d_2,r})$.
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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