July  2017, 16(4): 1405-1426. doi: 10.3934/cpaa.2017067

Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property

South China Normal University, Guangzhou, China

* Corresponding author: P. X. Weng

Received  July 2015 Revised  August 2016 Published  April 2017

Fund Project: Research is supported by the Natural Science Foundation of China (11171120), and the The Natural Science Foundation of Guangdong Province (2016A030313426)

A diffusive competing pioneer and climax system without cooperative property is investigated. We consider a special case in which the system has no co-existence equilibrium. Under the appropriate assumptions, we show the linear determinacy and the existence of single spreading speed. Furthermore, we obtain the existence of traveling wave solution which connects two boundary equilibria, and also confirm that the spreading speed coincides with the minimal wave speed. The results in this article reveals a phenomenon of strongly biological invasion which implies that the invasion of a new species will leads to the extinction of the resident species.

Citation: Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067
References:
[1]

J. R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species, in Differential Equations with Applications to Biology (Ruan S. G., Gail Wolkowicz S. K., J. H. Wu eds.), Fields Inst. Commun., 21 (1999), 51-63. Google Scholar

[2]

J. R. Buchanan, Turing instability in pioneer/climax species interactions, Math. Biosci., 194 (2005), 199-216. doi: 10.1016/j.mbs.2004.10.010. Google Scholar

[3]

S. BrownJ. Dockery and M. Pernarowski, Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species, Math. Biosci., 194 (2005), 21-36. doi: 10.1016/j.mbs.2004.10.001. Google Scholar

[4]

J. M. Cao and P. X. Weng, Linear stability of equilibria for a diffusive pioneer-climax species model, Journal of South China Normal University (Natural Science Edition), 46 (2014), 16-22. Google Scholar

[5]

J. M. Cushing, Nonlinear matrix models and population dynamics, Nat. Resource Modeling, 2 (1988), 539-580. Google Scholar

[6]

Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in Nonlinear Dynamics and Evolution Equations (Brunner H., Zhao X. Q., X. F. Zou eds.), Fields Inst. Commun., 48 (2006), 95-135. Google Scholar

[7]

J. E. Franke and A. -A. Yakubu, Pioneer exclusion in a one-hump discrete pioneer-climax competing system, J. Math. Biol., 32 (1994), 771-787. doi: 10.1007/BF00168797. Google Scholar

[8]

S. B. Hsu and X. -Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Appl., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[9]

M. A. LewisB. Li and H. F. Weinberger, Spread speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144. Google Scholar

[10]

X. Liang and X. -Q. Zhao, Asymptotic speeds of spread and teaveling waves for montone semiflows with applications, Commum. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 294-314. doi: 10.2307/2001590. Google Scholar

[12]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Spriner-Verlag, New York, 2002. Google Scholar

[13]

J. D. Murray, Mathematical Biology: Ⅱ. Spatial Models and Biomedical Applications, SprinerVerlag, New York, 2003. Google Scholar

[14]

J. F. Selgrade and G. Namkong, Planting and harvesting for pioneer-climax models, Rocky Mountain J. Math., 24 (1994), 293-310. doi: 10.1216/rmjm/1181072467. Google Scholar

[15]

J. F. Selgrade and G. Namkong, Stable periodic behavior in a pioneer-climax models, Natur. Resource Modeling, 4 (1990), 215-227. Google Scholar

[16] N. Shigesada and K. Kawasaki, Biology Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. Google Scholar
[17]

S. Sumner, Hopf bifurcation in pioneer-climax species models, Math. Biosci., 137 (1996), 1-24. doi: 10.1016/S0025-5564(96)00065-X. Google Scholar

[18]

S. Sumner, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing, Natur. Resource Modeling, 11 (1998), 155-171. Google Scholar

[19]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. Google Scholar

[20]

H. F. WeinbergerM. A. Lewis and B. T. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6. Google Scholar

[21]

H. F. WeinbergerK. Kawasaki and N. Shigesada, Spreading speed for a partially cooperative 2-species reaction-diffusion model, Discrete Contin. Dyn. Syst., 23 (2009), 1087-1098. doi: 10.3934/dcds.2009.23.1087. Google Scholar

[22]

P. X. Weng and X. F. Zou, Minimal wave speed and spread speed of competing pionner and climax species, Applicable Analysis, 93 (2014), 2093-2110. doi: 10.1080/00036811.2013.868442. Google Scholar

[23]

Z. H. Yuan and X. F. Zou, Co-invasion waves in a reaction diffusion model for competing pioneer and climax species, Nonlinear Analysis: RWA, 11 (2010), 232-245. doi: 10.1016/j.nonrwa.2008.11.003. Google Scholar

[24]

X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser B., 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117. Google Scholar

show all references

References:
[1]

J. R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species, in Differential Equations with Applications to Biology (Ruan S. G., Gail Wolkowicz S. K., J. H. Wu eds.), Fields Inst. Commun., 21 (1999), 51-63. Google Scholar

[2]

J. R. Buchanan, Turing instability in pioneer/climax species interactions, Math. Biosci., 194 (2005), 199-216. doi: 10.1016/j.mbs.2004.10.010. Google Scholar

[3]

S. BrownJ. Dockery and M. Pernarowski, Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species, Math. Biosci., 194 (2005), 21-36. doi: 10.1016/j.mbs.2004.10.001. Google Scholar

[4]

J. M. Cao and P. X. Weng, Linear stability of equilibria for a diffusive pioneer-climax species model, Journal of South China Normal University (Natural Science Edition), 46 (2014), 16-22. Google Scholar

[5]

J. M. Cushing, Nonlinear matrix models and population dynamics, Nat. Resource Modeling, 2 (1988), 539-580. Google Scholar

[6]

Y. H. Du and J. P. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, in Nonlinear Dynamics and Evolution Equations (Brunner H., Zhao X. Q., X. F. Zou eds.), Fields Inst. Commun., 48 (2006), 95-135. Google Scholar

[7]

J. E. Franke and A. -A. Yakubu, Pioneer exclusion in a one-hump discrete pioneer-climax competing system, J. Math. Biol., 32 (1994), 771-787. doi: 10.1007/BF00168797. Google Scholar

[8]

S. B. Hsu and X. -Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Appl., 40 (2008), 776-789. doi: 10.1137/070703016. Google Scholar

[9]

M. A. LewisB. Li and H. F. Weinberger, Spread speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144. Google Scholar

[10]

X. Liang and X. -Q. Zhao, Asymptotic speeds of spread and teaveling waves for montone semiflows with applications, Commum. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 294-314. doi: 10.2307/2001590. Google Scholar

[12]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Spriner-Verlag, New York, 2002. Google Scholar

[13]

J. D. Murray, Mathematical Biology: Ⅱ. Spatial Models and Biomedical Applications, SprinerVerlag, New York, 2003. Google Scholar

[14]

J. F. Selgrade and G. Namkong, Planting and harvesting for pioneer-climax models, Rocky Mountain J. Math., 24 (1994), 293-310. doi: 10.1216/rmjm/1181072467. Google Scholar

[15]

J. F. Selgrade and G. Namkong, Stable periodic behavior in a pioneer-climax models, Natur. Resource Modeling, 4 (1990), 215-227. Google Scholar

[16] N. Shigesada and K. Kawasaki, Biology Invasions: Theory and Practice, Oxford University Press, Oxford, 1997. Google Scholar
[17]

S. Sumner, Hopf bifurcation in pioneer-climax species models, Math. Biosci., 137 (1996), 1-24. doi: 10.1016/S0025-5564(96)00065-X. Google Scholar

[18]

S. Sumner, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing, Natur. Resource Modeling, 11 (1998), 155-171. Google Scholar

[19]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. Google Scholar

[20]

H. F. WeinbergerM. A. Lewis and B. T. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222. doi: 10.1007/s00285-007-0078-6. Google Scholar

[21]

H. F. WeinbergerK. Kawasaki and N. Shigesada, Spreading speed for a partially cooperative 2-species reaction-diffusion model, Discrete Contin. Dyn. Syst., 23 (2009), 1087-1098. doi: 10.3934/dcds.2009.23.1087. Google Scholar

[22]

P. X. Weng and X. F. Zou, Minimal wave speed and spread speed of competing pionner and climax species, Applicable Analysis, 93 (2014), 2093-2110. doi: 10.1080/00036811.2013.868442. Google Scholar

[23]

Z. H. Yuan and X. F. Zou, Co-invasion waves in a reaction diffusion model for competing pioneer and climax species, Nonlinear Analysis: RWA, 11 (2010), 232-245. doi: 10.1016/j.nonrwa.2008.11.003. Google Scholar

[24]

X. -Q. Zhao and W. D. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser B., 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117. Google Scholar

Figure 1.  Equilibria of system (4) under (A1) or (A2)
[1]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[2]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[3]

Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053

[4]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020047

[5]

Carlos Castillo-Chavez, Bingtuan Li, Haiyan Wang. Some recent developments on linear determinacy. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1419-1436. doi: 10.3934/mbe.2013.10.1419

[6]

Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087

[7]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883

[8]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126

[9]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663

[10]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355

[11]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405

[12]

Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267

[13]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[14]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

[15]

Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451

[16]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455

[17]

Guo Lin, Wan-Tong Li, Mingju Ma. Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 393-414. doi: 10.3934/dcdsb.2010.13.393

[18]

Junfan Lu, Hong Gu, Bendong Lou. Expanding speed of the habitat for a species in an advective environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 483-490. doi: 10.3934/dcdsb.2017023

[19]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591

[20]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]