  Previous work indicates that tropical forest can exist as an alternative
stable state to savanna. Therefore, perturbation by climate change or human
impact may lead to crossing of a tipping point beyond which there is rapid
forest dieback that is not easily reversed. A hypothesised mechanism for such
bistability is a feedback between fire and vegetation, where fire spreads as a
contagion process on grass patches. Theoretical models have largely implemented
this mechanism implicitly, by assuming a threshold dependence of fire spread on
flammable vegetation. Here, we show how the nonlinear dynamics and bistability
emerge spontaneously, without assuming equations or thresholds for fire spread.
We find that the forest geometry causes the nonlinearity that induces
bistability. We demonstrate this in three steps. First, we model forest and
fire as interacting contagion processes on grass patches, %using a cellular
automaton showing that spatial structure emerges due to two counteracting
effects on the forest perimeter: forest expansion by dispersal, and forest
erosion by fires originating in adjacent grassland. Then, we derive a
landscape-scale balance equation in which these two effects link forest
geometry and dynamics: forest expands proportionally to its perimeter, while it
shrinks proportionally to its perimeter weighted by adjacent grassland area.
Finally, we show that these perimeter quantities introduce nonlinearity in our
balance equation and lead to bistability. Relying on the link between structure
and dynamics, we propose a forest resilience indicator that could be used for
targeted conservation or restoration.
