Mathematics of Planet Earth by Hans G. Kaper & Fred S. Roberts

Author:Hans G. Kaper & Fred S. Roberts
Language: eng
Format: epub
ISBN: 9783030220440
Publisher: Springer International Publishing

Fig. 7.10Bifurcation diagram for the Australian ecosystem model (7.2.1) in 1d with weak shading feedback (small R W and R H in (7.2.3)). The diagram shows a small bistability range, P f < P < P T, of low-biomass uniform vegetation (black line) and a periodic pattern (green line) and a subrange of homoclinic snaking. The inset (a) shows the periodic solution while the insets (b–e) show localized pattern solutions or hybrid states

Homoclinic snaking can also be found in a bistability range of low-biomass uniform vegetation and periodic spot pattern, as Fig. 7.10 shows [11], implying the feasibility of hybrid-state transitions and gradual shifts in fluctuating environments. However, when the periodic-pattern solution branch extends to the stability range of the bare-soil state, homoclinic snaking breaks down in what appears to be a Belyakov–Devaney transition [38]. In that case shifts from periodic patterns to bare soil, or desertification, are found to be abrupt [95]. While most model studies predict wide bistability ranges of spot patterns and bare soil, and thus the likelihood of abrupt desertification, these studies have been confined to a single species. Quite often dryland landscapes consist of woody and herbaceous species, forming a bistability range of woody spot patterns and uniform herbaceous vegetation [25]. Studies of two-species models do predict homoclinic snaking [47], suggesting that the degradation of woody spot patterns may be gradual rather than abrupt.

In the tristability range of bare soil, periodic patterns, and uniform vegetation [see Fig. 7.7] many front types are expected to coexist, pinned, or moving; fronts separating domains of uniform vegetation and bare soil, domains of uniform vegetation and periodic patterns, and domains of periodic patterns and bare soil. The dynamics, interactions, and stability properties of these front solutions, and the implications for regime shifts have hardly been studied [96].

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