  We analyze a spatially extended version of a well-known model of
forest-savanna dynamics, which presents as a system of nonlinear partial
integro-differential equations, and study necessary conditions for
pattern-forming bifurcations. Analytically, we show that homogeneous solutions
dominate the dynamics of the standard forest-savanna model, regardless of the
length scales of the various spatial processes considered. However, several
different pattern-forming scenarios are possible upon including spatial
resource limitation, such as competition for water, soil nutrients, or
herbivory effects. Using numerical simulations and continuation, we study the
nature of the resulting patterns as a function of system parameters and length
scales, uncovering subcritical pattern-forming bifurcations and observing
significant regions of multistability for realistic parameter regimes. Finally,
we discuss our results in the context of extant savanna-forest modeling efforts
and highlight ongoing challenges in building a unifying mathematical model for
savannas across different rainfall levels.
