# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020312
##### References:

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