Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model
arxiv(2023)
摘要
Reaction-diffusion models describing interactions between vegetation and
water reveal the emergence of several types of patterns and travelling wave
solutions corresponding to structures observed in real-life. Increasing their
accuracy by also considering the ecological factor known as autotoxicity has
lead to more involved models supporting the existence of complex dynamic
patterns. In this work, we include an additional carrying capacity for the
biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces
the presence of two asymptotically small parameters: $\varepsilon$,
representing the usual scale separation in vegetation-water models, and
$\delta$, directly linked to autotoxicity. We construct three separate types of
homoclinic travelling pulse solutions based on two different scaling regimes
involving $\varepsilon$ and $\delta$, with and without a so-called superslow
plateau. The relative ordering of the small parameters significantly influences
the phase space geometry underlying the construction of the pulse solutions. We
complement the analysis by numerical continuation of the constructed pulse
solutions, and demonstrate their existence (and stability) by direct numerical
simulation of the full PDE model.
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