| Abstract: |
| We introduce a three-component extension of the classical Klausmeire-Gray-Scott model, termed the WPAE model, to explore how water, vegetation, and allelopathic inhibition interact to generate complex spatial patterns in ecosystems. Departing from the classical two-component Klausmeier model, where limited resources tend to induce vegetation aggregation, our modified model demonstrates that allelopathic inhibition serves as a critical constraint, localizing vegetation populations specifically in water-abundant regimes. Moreover, we find that elevating the reaction rate of the allelopathic inhibitor leads to the emergence of robust traveling multi-spike solutions. Through systematic analysis, we observe a spectrum of complex spike interactions, including merging, repulsive bouncing, and more sophisticated dynamics involving splitting and competition. This work aims to unravel the underlying mechanisms driving these phenomena. Our approach involves first constructing a single stationary spike solution using matched asymptotic expansions and examining its spectral stability. We further derive explicit traveling spike solutions and obtain governing equations for their propagation speed. The validity of our theoretical results is confirmed via numerical simulations, which show strong consistency with our analytical findings. |
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