• Previous Article
    Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities
  • CPAA Home
  • This Issue
  • Next Article
    Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential
May 2018, 17(3): 1179-1200. doi: 10.3934/cpaa.2018057

Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay

1. 

Department of Mathematics, China University of Petroleum (East China), Qingdao, 266580, China

2. 

Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Received  February 2017 Revised  August 2017 Published  January 2018

Fund Project: W. Zuo is partially supported by the NSFC of China (Nos.11401584,11671236) and the Fundamental Research Funds for the Central Universities(Nos.16CX02053A, 16CX02015A, 14CX02220A), J. Shi is partially supported by NSF grants DMS-1313243 and DMS-1715651

The existence of traveling wave solutions and wave train solutions of a diffusive ratio-dependent predator-prey system with distributed delay is proved. For the case without distributed delay, we first establish the existence of traveling wave solution by using the upper and lower solutions method. Second, we prove the existence of periodic traveling wave train by using the Hopf bifurcation theorem. For the case with distributed delay, we obtain the existence of traveling wave and traveling wave train solutions when the mean delay is sufficiently small via the geometric singular perturbation theory. Our results provide theoretical basis for biological invasion of predator species.

Citation: Wenjie Zuo, Junping Shi. Traveling wave solutions of a diffusive ratio-dependent Holling-Tanner system with distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1179-1200. doi: 10.3934/cpaa.2018057
References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.

[2]

R. ArditiL. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent predation models, Amer. Nat., 138 (1991), 1287-1296.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.

[4]

P. AshwinM. V. BartuccelliT. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.

[5]

S.-S. Chen and J.-P. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.

[6]

S.-S. Chen and J.-P. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonlinear Anal. Real World Appl., 14 (2013), 1871-1886.

[7]

A. Ducrot and M. Langlais, A singular reaction-diffusion system modelling prey-predator interactions: Invasion and co-extinction waves, J. Differential Equations, 253 (2012), 502-532.

[8]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.

[9]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $\textbf{R}^{4}$, Trans. Amer. Math. Soc., 286 (1984), 557-594.

[10]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.

[11]

W. F. FaganM. A. LewisM. G. Neubert and P. Van Den Driessche, Invasion theory and biological control, Ecol. Lett., 5 (2002), 148-157.

[12]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.

[13]

W. FengW.-H. Ruan and X. Lu, On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 815-836.

[14]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.

[15] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, New York, 1979.
[16]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.

[17]

L. R. Ginzburg and H. R. Akçakaya, Consequences of ratio-dependent predation for steady-state properties of ecosystems, Ecology, 73 (1992), 1536-1543.

[18]

S. A. Gourley, Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays, Math. Comput. Modelling, 32 (2000), 843-853.

[19]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, In Nonlinear Dynamics and Evolution Equations, Volume 48 of Fields Inst. Commun., pages 137-200, Amer. Math. Soc., Providence, RI, 2006.

[20] B. D. HassardN. D. Kazarinoff and Y.-H Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, Cambridge-New York, 1981.
[21]

A. HastingsK. Cuddington and K. F. Davies, The spatial spread of invasions: new developments in theory and evidence, Ecol. Lett., 8 (2005), 91-101.

[22]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.

[23]

C.-H. HsuC.-R. YangT.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.

[24]

J.-H. HuangG. Lu and S.-G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.

[25]

W.-Z. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.

[26]

W.-Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equations, 24 (2012), 633-644.

[27]

W.-Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224.

[28]

W.-Z. Huang and M.-A. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.

[29]

Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl., 418 (2014), 163-184.

[30]

C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical Systems (Montecatini Terme, 1994), volume 1609 of Lecture Notes in Math., pages 44-118, Springer, Berlin, 1609.

[31]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.

[32]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.

[33]

G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.

[34]

G. LinW.-T. Li and M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.

[35]

X.-B. LinP.-X. Weng and C.-F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dynam. Differential Equations, 23 (2011), 903-921.

[36]

S.-W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.

[37]

S. M. Merchant and W. Nagata, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.

[38]

K. Mischaikow and J. F. Reineck, Travelling waves in predator-prey systems, SIAM J. Math. Anal., 24 (1993), 1179-1214.

[39]

M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bull. Math. Biol., 63 (2001), 655-684.

[40]

R. Peng and M.-X. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.

[41]

S. PetrovskiiA. Morozov and B.-L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, Bull. Math. Biol., 67 (2005), 637-661.

[42]

J. A. Sherratt, Periodic traveling waves in a family of deterministic cellular automata, Phys. D, 95 (1996), 319-335.

[43] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997.
[44]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.

[45]

A. I. VolpertV. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. Amer. Math. Soc., (1994).

[46]

Z.-C. WangW.-T. Li and S.-G. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.

[47]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.

[48]

W.-J. Zuo and Y.-L. Song, Stability and bifurcation analysis of a reaction-diffusion equation with spatio-temporal delay, J. Math. Anal. Appl., 430 (2015), 243-261.

show all references

References:
[1]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.

[2]

R. ArditiL. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent predation models, Amer. Nat., 138 (1991), 1287-1296.

[3]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.

[4]

P. AshwinM. V. BartuccelliT. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.

[5]

S.-S. Chen and J.-P. Shi, Global stability in a diffusive Holling-Tanner predator-prey model, Appl. Math. Lett., 25 (2012), 614-618.

[6]

S.-S. Chen and J.-P. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonlinear Anal. Real World Appl., 14 (2013), 1871-1886.

[7]

A. Ducrot and M. Langlais, A singular reaction-diffusion system modelling prey-predator interactions: Invasion and co-extinction waves, J. Differential Equations, 253 (2012), 502-532.

[8]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.

[9]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in $\textbf{R}^{4}$, Trans. Amer. Math. Soc., 286 (1984), 557-594.

[10]

S. R. Dunbar, Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), 1057-1078.

[11]

W. F. FaganM. A. LewisM. G. Neubert and P. Van Den Driessche, Invasion theory and biological control, Ecol. Lett., 5 (2002), 148-157.

[12]

J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704.

[13]

W. FengW.-H. Ruan and X. Lu, On existence of wavefront solutions in mixed monotone reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 815-836.

[14]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.

[15] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, New York, 1979.
[16]

R. A. Gardner, Existence of travelling wave solutions of predator-prey systems via the connection index, SIAM J. Appl. Math., 44 (1984), 56-79.

[17]

L. R. Ginzburg and H. R. Akçakaya, Consequences of ratio-dependent predation for steady-state properties of ecosystems, Ecology, 73 (1992), 1536-1543.

[18]

S. A. Gourley, Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays, Math. Comput. Modelling, 32 (2000), 843-853.

[19]

S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, In Nonlinear Dynamics and Evolution Equations, Volume 48 of Fields Inst. Commun., pages 137-200, Amer. Math. Soc., Providence, RI, 2006.

[20] B. D. HassardN. D. Kazarinoff and Y.-H Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, Cambridge-New York, 1981.
[21]

A. HastingsK. Cuddington and K. F. Davies, The spatial spread of invasions: new developments in theory and evidence, Ecol. Lett., 8 (2005), 91-101.

[22]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.

[23]

C.-H. HsuC.-R. YangT.-H. Yang and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), 3040-3075.

[24]

J.-H. HuangG. Lu and S.-G. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), 132-152.

[25]

W.-Z. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.

[26]

W.-Z. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dynam. Differential Equations, 24 (2012), 633-644.

[27]

W.-Z. Huang, A geometric approach in the study of traveling waves for some classes of non-monotone reaction-diffusion systems, J. Differential Equations, 260 (2016), 2190-2224.

[28]

W.-Z. Huang and M.-A. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model, J. Differential Equations, 251 (2011), 1549-1561.

[29]

Y.-L. Huang and G. Lin, Traveling wave solutions in a diffusive system with two preys and one predator, J. Math. Anal. Appl., 418 (2014), 163-184.

[30]

C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical Systems (Montecatini Terme, 1994), volume 1609 of Lecture Notes in Math., pages 44-118, Springer, Berlin, 1609.

[31]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.

[32]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.

[33]

G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 96 (2014), 47-58.

[34]

G. LinW.-T. Li and M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393-414.

[35]

X.-B. LinP.-X. Weng and C.-F. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dynam. Differential Equations, 23 (2011), 903-921.

[36]

S.-W. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.

[37]

S. M. Merchant and W. Nagata, Wave train selection behind invasion fronts in reaction-diffusion predator-prey models, Phys. D, 239 (2010), 1670-1680.

[38]

K. Mischaikow and J. F. Reineck, Travelling waves in predator-prey systems, SIAM J. Math. Anal., 24 (1993), 1179-1214.

[39]

M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bull. Math. Biol., 63 (2001), 655-684.

[40]

R. Peng and M.-X. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 149-164.

[41]

S. PetrovskiiA. Morozov and B.-L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, Bull. Math. Biol., 67 (2005), 637-661.

[42]

J. A. Sherratt, Periodic traveling waves in a family of deterministic cellular automata, Phys. D, 95 (1996), 319-335.

[43] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, 1997.
[44]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.

[45]

A. I. VolpertV. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. Amer. Math. Soc., (1994).

[46]

Z.-C. WangW.-T. Li and S.-G. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.

[47]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.

[48]

W.-J. Zuo and Y.-L. Song, Stability and bifurcation analysis of a reaction-diffusion equation with spatio-temporal delay, J. Math. Anal. Appl., 430 (2015), 243-261.

Figure 1.  Hopf bifurcation curves of (3.4). Left: $A = 1.9,~B = 0.3,~a = 1$ (satisfying (H3)); Right: $A = 1.75,~B = 0.3,~a = 1$ (satisfying (H3'))
Figure 2.  Wave profiles for prey (A) and predator (B) of the traveling wave solutions of (1.2) with $d = 0.1,~a = 1,~A = 0.4,~B = 0.01$
Figure 3.  Wave profiles for prey (A) and predator (B) of the traveling wave solutions of (1.1) with $d = 0.1,~a = 1,~A = 0.4,~B = 0.01$ and $\tau = 1$
Figure 4.  Wave profiles for prey (A) and predator (B) of the traveling wave solutions of (1.1) with $d = 0.1,~a = 1,~A = 0.4,~B = 0.01$ and $\tau = 1.5$
[1]

Canan Çelik. Dynamical behavior of a ratio dependent predator-prey system with distributed delay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 719-738. doi: 10.3934/dcdsb.2011.16.719

[2]

Zhicheng Wang, Jun Wu. Existence of positive periodic solutions for delayed ratio-dependent predator-prey system with stocking. Communications on Pure & Applied Analysis, 2006, 5 (3) : 423-433. doi: 10.3934/cpaa.2006.5.423

[3]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

[4]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747

[5]

Inkyung Ahn, Wonlyul Ko, Kimun Ryu. Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey. Conference Publications, 2013, 2013 (special) : 11-19. doi: 10.3934/proc.2013.2013.11

[6]

Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321

[7]

Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303

[8]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

[9]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267

[10]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227

[11]

Zhijun Liu, Weidong Wang. Persistence and periodic solutions of a nonautonomous predator-prey diffusion with Holling III functional response and continuous delay. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 653-662. doi: 10.3934/dcdsb.2004.4.653

[12]

Meng Fan, Qian Wang. Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 563-574. doi: 10.3934/dcdsb.2004.4.563

[13]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

[14]

Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure & Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481

[15]

Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173

[16]

Roger Lui, Hirokazu Ninomiya. Traveling wave solutions for a bacteria system with density-suppressed motility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-10. doi: 10.3934/dcdsb.2018213

[17]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[18]

Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161

[19]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[20]

Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (51)
  • HTML views (283)
  • Cited by (0)

Other articles
by authors

[Back to Top]