Pattern Formation in Mesic Savannas

Abstract

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. 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.

Keywords: Integro-differential equations; Pattern formation; Savanna; Spatial modeling; Vegetation dynamics.

Fig. 1

Two-parameter bifurcation diagram in $\alpha$ (forest tree birth rate) and $\beta$ (savanna tree birth rate) for the nonspatial Staver-Levin model (1). Transcritical bifurcation curves in blue, saddle node curves in magenta, supercritical Hopf curves in purple and subcritical Hopf curves in dark green. Points labeled GH denote codimension 2 Bautin (or Generalized Hopf) bifurcation points at which the Hopf bifurcations change criticality. The all-grass state is the only stable state in the dark green shaded region; the all-grass state and a forest dominated state are bistable in the orange shaded region (Color figure online)

Fig. 2

A, B One-parameter bifurcation diagrams for the resource limited model for various values of the resource constraint r; stable fixed points in red and unstable fixed points in black. C Two-parameter bifurcation diagram in r and $\alpha$ (forest tree birth rate); curve of transcritical bifurcations in blue, curves of saddle node bifurcations in magenta (Color figure online)

Fig. 3

A Red shaded regions satisfy the necessary condition for an instability of a spatially homogeneous solution, i.e. $max\left(N,(,\alpha ,,,r,)\right)>0$, where the max is taken over all stable equilibria of the nonspatial system. B Plots of the dispersion relations for the spatial model for two parameter sets varying $\alpha$ and r (the system is bistable in one case and monostable in the other). C Maximum of the principal eigenvalue of the linearized system with red regions denoting pattern-forming parameter regimes. Parameters: Gaussian kernels with ${\sigma }_W=0.05$, ${\sigma }_F=0.1$ and ${\sigma }_R=0.4$ (Color figure online)

Fig. 4

A Heatmap of the maximum dispersion relation obtained by linearizing about each (nonspatially) stable homogeneous equilibrium. The saddle-node curves of the nonspatial system are denoted by the green dashed lines and the magenta stars indicate the positions of the parameters for the solutions shown in panels B and C. B, C Solutions of the spatially extended system with resource limitation on the 1D spatial domain $\mathrm{\Omega }=[-\pi ,\pi ]$ with periodic boundary conditions. D Selection of solutions on a 2D domain, $\mathrm{\Omega }=[-\pi /2,\pi /2]\times [-\pi /2,\pi /2]$, with periodic boundary conditions ($\alpha =2.15$ and $r=1$). Other parameters: Gaussian kernels with ${\sigma }_W=0.01$, ${\sigma }_F=0.1$ and ${\sigma }_R=0.4$ (Color figure online)

Fig. 5

Top row parameter regime: ${\sigma }_F=0.1$, ${\sigma }_W=0.025$, ${\sigma }_R=0.15$. Bottom row parameter regime: ${\sigma }_F=0.1$, ${\sigma }_W=0.025$, ${\sigma }_R=0.05$. A, D Heatmap of the maximum dispersion relation obtained by linearizing about each (nonspatially) stable homogeneous equilibrium. The dashed horizontal magenta line corresponds to the one-parameter bifurcation diagrams in (B, E) and the magenta star indicates the position of the solutions shown in (C, F) respectively. B, E One-parameter bifurcation diagrams for the spatial model varying the forest tree birth rate ($\alpha$); the positions of the solutions shown in panel C are indicated with a star (in panel B the color of the star matches of the color of the solution curve in panel C). C, F Steady-state profiles of stable heterogeneous solutions with two bistable solutions shown in panel C (Color figure online)

Fig. 6

A–C Heatmaps showing the regions where the dispersion relation is positive for at least one homogeneous equilibrium in the absence of spatial interactions, i.e. the pattern forming region, as a function of ${\sigma }_R$ (the standard deviation of the resource competition kernel) for different values of $\alpha$ (the forest tree birth rate). The green dashed lines in panel A indicate the position of saddle node bifurcation points (and hence the bistable region) in the non-spatial model. Other parameter values: ${\sigma }_F=0.1$ and ${\sigma }_W=0.01$ (Color figure online)

Fig. 7

Two-parameter bifurcation diagram in $\alpha$ (forest tree birth rate) and $\beta$ (savanna tree birth rate) for the nonspatial resource-limited Staver-Levin model (14) with $r=0.84$. Transcritical bifurcation curves in blue, saddle node curves in magenta, supercritical Hopf curves in purple and subcritical Hopf curves in dark green. Points labeled CP denote codimension 2 Cusp bifurcations and points labeled BT denote codimension 2 Bogdanov-Takens bifurcations. Points labeled GH denote codimension 2 Bautin (or Generalized Hopf) bifurcation points which are points at which the Hopf bifurcations change criticality. The all-grass state is the only stable state in the dark green shaded region (bottom left corner) (Color figure online)