Figure 1.
Solutions of model (7) with $\tau = 0.3<\tau_*$ tend to the semi-trivial steady state solution $(\theta_{d_1,\alpha,0},0)$ .
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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 |
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.
Keywords:
Spatial heterogeneity,
stage structure,
Lotka-Volterra competition,
steady state solution.
Mathematics Subject Classification:
Primary: 92D25, 92D40; Secondary: 35K57.
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:
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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. |
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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. |
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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. |
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. 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. |
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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