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September  2021, 26(9): 4767-4788. doi: 10.3934/dcdsb.2020312

Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases

1. 

School of Mathematics, Tianjin University, Tianjin 300350, China

2. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author: Yuxiang Zhang

Received  April 2020 Revised  July 2020 Published  September 2021 Early access  October 2020

Fund Project: The first author is supported by NSF of China (11701415). The second author is supported by NSF of China (11571187)

In this paper, we study the invasion dynamics of a diffusive pioneer-climax model in monotone and non-monotone cases. For parameter ranges in which the system admits monotone properties, we establish the existence of spreading speeds and their coincidence with the minimum wave speeds by monotone dynamical system theories. The linear determinacy of the minimum wave speeds is also studied by constructing suitable upper solutions. For parameter ranges in which the system is non-monotone, we further determine the existence of spreading speeds and traveling waves by the sandwich technique and upper-lower solution method. Our results generalize the existing results established under monotone assumptions to more general cases.

Citation: Yuxiang Zhang, Shiwang Ma. Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4767-4788. doi: 10.3934/dcdsb.2020312
References:
[1]

A. Alhasanat and C. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Diff. Eqns., 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003.

[2]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.  doi: 10.1017/S0305004100053494.

[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. R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species, Fields Inst. Commun., 21 (1999), 51-63. 

[5]

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

[6]

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

[7]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci, 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[8]

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

[9]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Diff. Eqns., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.

[10]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Diff. Eqns., 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[11]

M. Ma and C. Ou, Linear and nonlinear speed selection for mono-stable wave propagations, SIAM J. Math. Anal., 51 (2019), 321-345.  doi: 10.1137/18M1173691.

[12]

M. Olinick, An Introduction to Mathematical Models in the Social and Life Sciences, Addison-Welsey, Reading, MA, 1978.

[13]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.

[14]

J. F. Selgrade and G. Namkoong, Stable periodic behavior in a pioneer-climax model, Nat. Resour. Model., 4 (1990), 215-227.  doi: 10.1111/j.1939-7445.1990.tb00098.x.

[15]

J. F. Selgrade and G. Namkoong, Population interactions with growth rates dependent on weighted densities, Differential equation models in biology, epidemiology and ecology, Lecture Notes Biomath., 92 (1991), 247-256.  doi: 10.1007/978-3-642-45692-3_18.

[16]

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

[17]

S. Sumner, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing, Nat. Resour. Model., 11 (1998), 155-171.  doi: 10.1111/j.1939-7445.1998.tb00306.x.

[18]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783.  doi: 10.1007/s00332-011-9099-9.

[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]

P. Weng and J. Cao, Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property, Comm. Pur. Appl. Anal., 16 (2017), 1405-1426.  doi: 10.3934/cpaa.2017067.

[21]

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

[22]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Eqns., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[23]

Z. Yuan and X. 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. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.

show all references

References:
[1]

A. Alhasanat and C. Ou, Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, J. Diff. Eqns., 266 (2019), 7357-7378.  doi: 10.1016/j.jde.2018.12.003.

[2]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431-433.  doi: 10.1017/S0305004100053494.

[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. R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species, Fields Inst. Commun., 21 (1999), 51-63. 

[5]

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

[6]

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

[7]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci, 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.

[8]

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

[9]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Diff. Eqns., 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.

[10]

S. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Diff. Eqns., 237 (2007), 259-277.  doi: 10.1016/j.jde.2007.03.014.

[11]

M. Ma and C. Ou, Linear and nonlinear speed selection for mono-stable wave propagations, SIAM J. Math. Anal., 51 (2019), 321-345.  doi: 10.1137/18M1173691.

[12]

M. Olinick, An Introduction to Mathematical Models in the Social and Life Sciences, Addison-Welsey, Reading, MA, 1978.

[13]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Bd. Can., 11 (1954), 559-623.  doi: 10.1139/f54-039.

[14]

J. F. Selgrade and G. Namkoong, Stable periodic behavior in a pioneer-climax model, Nat. Resour. Model., 4 (1990), 215-227.  doi: 10.1111/j.1939-7445.1990.tb00098.x.

[15]

J. F. Selgrade and G. Namkoong, Population interactions with growth rates dependent on weighted densities, Differential equation models in biology, epidemiology and ecology, Lecture Notes Biomath., 92 (1991), 247-256.  doi: 10.1007/978-3-642-45692-3_18.

[16]

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

[17]

S. Sumner, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing, Nat. Resour. Model., 11 (1998), 155-171.  doi: 10.1111/j.1939-7445.1998.tb00306.x.

[18]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), 747-783.  doi: 10.1007/s00332-011-9099-9.

[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]

P. Weng and J. Cao, Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property, Comm. Pur. Appl. Anal., 16 (2017), 1405-1426.  doi: 10.3934/cpaa.2017067.

[21]

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

[22]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Diff. Eqns., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

[23]

Z. Yuan and X. 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. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.

Figure 1.  Typical fitness functions $ f $ and $ g $ in model (1)
Figure 2.  Nullclines and the structure of equilibria of (2) under (3)
Figure 3.  The graph of functions $ \overline{g}(w) $ and $ \underline{g}(w) $ under (H1$ ' $)
Figure 4.  The observed pioneer invasion waves for $ u $ and $ v $
Figure 5.  The observed climax invasion waves for $ u $ and $ v $
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