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 and 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. 

[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.

[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.

[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. 

[5]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[15]

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

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

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

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. 

[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.

[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.

[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. 

[5]

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

[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. 

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[15]

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

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

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

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