Spatial Dynamics with Heterogeneity
Denis D. Patterson∗1, Simon A. Levin2, A. Carla Staver3, and Jonathan D.
Touboul4,5
1High Meadows Environmental Institute, Princeton University, Princeton, NJ
(denispatterson@princeton.edu)
2Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ
(slevin@princeton.edu)
3Department of Ecology and Evolutionary Biology, Yale University, New Haven, CT
(carla.staver@yale.edu)
4 Department of Mathematics, Brandeis University, Waltham MA (jtouboul@brandeis.edu)
5Volen Centre for Complex Systems, Brandeis University, Waltham MA
May 10, 2023
Abstract
Spatial systems with heterogeneities are ubiquitous in nature, from precipitation, tempera-
ture and soil gradients controlling vegetation growth to morphogen gradients controlling gene
expression in embryos. Such systems, generally described by nonlinear dynamical systems, often
display complex parameter dependence and exhibit bifurcations. The dynamics of heterogeneous
spatially extended systems passing through bifurcations are still relatively poorly understood,
yet recent theoretical studies and experimental data highlight the resulting complex behaviors
and their relevance to real-world applications. We explore the consequences of spatial hetero-
geneities passing through bifurcations via two examples strongly motivated by applications.
These model systems illustrate that studying heterogeneity-induced behaviors in spatial sys-
tems is crucial for a better understanding of ecological transitions and functional organization
in brain development.
Keywords:
PDEs, integro-diﬀerential equations, ecology, savanna-forest, brain development
Mathematics Subject Classiﬁcation(s):
35B32, 45K05, 92D40, 92B05
∗Corresponding author.
Simon Levin and Denis Patterson thank the NSF for support via the grant DMS-1951358, and Simon Levin also
appreciates support from the Army Research Oﬃce Grant W911NF-18-1-0325. Carla Staver appreciates support from
NSF Grant DMS-1951394 and Jonathan Touboul appreciates support from NSF Grant DMS-1951369.
1
arXiv:2305.05007v1  [math.DS]  8 May 2023

1
Introduction
In a variety of spatial natural systems, nonlinear dynamics and spatial interaction are aﬀected by
heterogeneities.
Such systems have been widely studied, in a variety of settings.
Systems with
gradients that transition from one dominant state to another, and including a spatially transient
region of bistability, generally show a more-or-less sharp transition between the two regions; the
position of this transition point depends on the intrinsic nonlinear dynamics as well as the nature of
the spatial interactions. The existence of such solutions in bistable systems was recently rigorously
established for a wide class of competitive reaction-diﬀusion equations in the limit of vanishing
diﬀusion [49]; other aspects of sharp boundary formation in heterogeneous media have also been
considered by numerous authors [11, 40, 41, 64]. Systems with transient bistability are only the
simplest case of what we will refer to as spatially transient bifurcation scenarios, which we deﬁne as
spatially extended systems with heterogeneities whereby the parameter varying in space is associated
with the crossing of one or more bifurcations. The behavior of more general spatially transient
bifurcations remains largely open and, to the best of our knowledge, has not been addressed in a
systematic manner. We argue however that understanding these dynamics is essential to better
describe the behavior of complex biological systems in heterogeneous environments.
We study
here two particular systems paradigmatic for this problem: an ecological system describing the
competition between savannas and forests within a gradient of precipitation, as well as a recently
developed model of brain development featuring gradients of morphogens.
The models presented here exhibit a rich phenomenology that, in line with the topic of this
special issue, have strong potential both for applications and mathematical developments. In the
case of front pinning stemming from transient passage through a bistable regime, ﬁrst applications
of the so-called Maxwell point theory to ecology appeared recently [57], and were applied to the
so-called Staver-Levin forest-savanna model [53, 54] in the presence of a gradient in [63], or in a
similar forest-savanna model in [20]. There are also some classical results in mathematical ecology
regarding the ability of species to invade, survive and coexist in heterogeneous environments [3, 8, 7].
The recent work [2] on climate tipping considers a bistable nonspatial model subject to diﬀusion
and a non-monotonic heterogeneous forcing term, and is arguably closest to the mechanisms studied
here. Similarly, researchers in mathematical neuroscience have studied how the shape of the input
into neural ﬁeld models can induce interesting dynamics in the output function [16, 32, 31, 30].
However, in these cases, the complexity of the dynamics are induced by the shape of the gradients.
In contrast, we will focus on simpler monotonically varying heterogeneities and, as we will show,
complex dynamics can emerge simply from the interaction of the underlying non-spatial dynamics
and the slope of the gradient (or the length scale of the problem). Moreover, we aim to highlight the
importance of the speed (or size of the region of space) for which the heterogeneity crosses diﬀerent
dynamical regimes of the underlying non spatial model. As we illustrate, this crucially inﬂuences
the emergence or non emergence of new spatial patterns or spatio-temporal dynamics.
Turing pattern formation in heterogeneous systems (but not including transient passages through
Turing instabilities) has also been widely explored, and it was shown that dispersal can sharpen
boundaries [19]. Numerous authors have developed criteria for pattern formation in speciﬁc reaction-
diﬀusion models where some parameters are allowed to vary across the spatial domain in a stepwise
manner [5, 46]. However, we believe that smooth variation of the gradient to be a key ingredient
in many applications (cf. section 3); it changes the mathematical approaches to the problem of
characterizing patterns signiﬁcantly and also produces qualitatively diﬀerent solutions, e.g. blending
of spot, stripes and labyrinths, as well as multi-frequency patterns [17].
Researchers have also
pursued perturbative results about the homogeneous case. For example, Benson et al. [4] carried
out a remarkably detailed bifurcation analysis of a reaction-diﬀusion system with diﬀusion coeﬃcient
D(x) = D + ηx2 in the limit as η ↓0, illustrating that a rich array of dynamics can be spawned by
small heterogeneity. The dynamics of spiked solutions to pattern forming reaction diﬀusion systems
subject to spatially localized heterogeneity have also attracted considerable attention (see [61] and
2

the references therein), while other work on this topic has allowed more general heterogeneous
structure by working in asymptotic regimes of the diﬀusivity strengths [59]. However, biological
problems (such as the one we present in section 3) often exhibit spatial heterogeneity on the same
scale of the spatial domain itself, rendering many asymptotic or perturbative results of relatively
limited utility in applications.
There has been recent and growing interest in revisiting and generalizing the classical Turing
pattern paradigm to account for general spatial heterogeneity [35, 58]. Although the literature has
focused mainly on heterogeneity in the diﬀusion coeﬃcient(s), several investigators have recently
employed series-approximation approaches to derive conditions for the onset of patterns with het-
erogeneity in the reaction dynamics [34, 37]. While these results are undoubtedly promising, the
vast bulk of the existing literature deals with reaction-diﬀusion systems. Many applications require
models incorporating more complex spatial operators, such as chemotaxis, nonlocal spatial operators
or advection (which can alter pattern forming dynamics even when present at very low levels [33]),
thus demanding the development of new mathematical machinery. The inclusion of heterogeneity in
pattern forming systems has also recently been reported by several authors to induce spatio-temporal
oscillations [12, 36], posing yet more interesting mathematical questions in this area.
In section 2, we begin our exploration of heterogeneity-induced dynamics by ﬁrst investigating the
forest-grass Staver-Levin model with nonlocal interactions and a rainfall gradient that increasingly
favors forest tree expansion. Turning next to a more detailed version of the model with four functional
types, we incorporate the impact of the rainfall on the various types of vegetation by allowing the
forest and savanna tree birth rates to vary spatially, since they are likely to exhibit the most spatial
variation and impact outcomes most strongly. Our approach is to adopt qualitatively appropriate
gradients, as opposed to trying to quantitatively estimate the impact of increasing rainfall on the
productivity of each of the functional types in the model. We are guided by a detailed bifurcation
analysis of the corresponding nonspatial model and this allows us to predict potential emergent
behaviors caused by the heterogeneity and illustrate interesting spatio-temporal dynamics. Through
a numerical bifurcation analysis of the system of integro-diﬀerential equations, we ﬁnd multistability
between several nonhomogeneous solutions for a signiﬁcant range of the dispersal parameter; we
ﬁnd front-pinned solutions (which connect stable solution branches from the nonspatial model), and
nonhomogeneous grass and forest dominated solutions. In the four-functional-type spatial Staver-
Levin model there is a much greater variety of solutions in the presence of heterogeneity.
We
highlight several plausible transition mechanisms for the empirically observed savanna to forest
transition with increasing rainfall. Once more, we uncover front-pinned solutions, including a front-
pinned solution predicting a grassland band mediating the savanna-forest transition, as well as
multistability and changes in stability as the dispersal parameters vary. Additionally, we observe
periodic waves of invasion for gradients that intersect a large region of parameter space that produces
stable oscillations in the nonspatial Staver-Levin model. Intriguingly, as we vary the width or speed
at which the gradient cuts through oscillatory region, we see period doubling of the waves and
eventually more complex solutions that appear to display the hallmarks of spatio-temporal chaos.
This illustrates how the speed at which the gradient crosses through diﬀerent dynamical regimes or
bifurcations crucially determines the resulting dynamics.
In section 3, we discuss a PDE model of arealization in the mammalian forebrain featuring
transcription factor gradients that mediate the competition between diﬀerent neural fates in early
embryonic development [17]. This model explained unexpected patterning and dislocation of the
typically sharp boundary between abutting cortical regions upon artiﬁcial manipulation of the tran-
scription factor gradients in mice. Mathematically, the dramatic breakdown of the boundary-forming
mechanism in the mice corresponds to a transient passage of the gradient through a region of pa-
rameter space with pattern-forming instabilities. For a 1D spatial domain, we demonstrate that the
structure of the resulting solutions of the mathematical model depend crucially on how the gradient
transits the pattern forming region and emphasize the qualitative diﬀerences when comparing these
3

solutions to those for a typical pattern forming system on a homogeneous domain.
2
The savanna-forest transition with a rainfall gradient
The ﬁrst example we shall study is a transient passage through a family of periodic orbits, in the
case of the transition between savannas and forest within a gradient of precipitation.
2.1
Spatially extended savanna-forest dynamics
The vegetation model that we shall use for this, hereafter referred to as the Staver-Levin (SL)
model [54], describes the interaction between savanna trees with an age structure (S for savanna
saplings, T for adult savanna trees), forest trees (F), and grass patches (G). Grass patches are
locations that carry ﬁres that limit the expansion of forest trees and delay the maturation of savanna
saplings, but are also the locations where new trees of both types can grow. In [47], we introduced
a spatially explicit stochastic model accounting for birth, death and interaction rules between these
ecological species proved that the fractions of the diﬀerent components at location x and time t1
satisfy the following system of nonlinear integro-diﬀerential equations in an appropriate mean-ﬁeld
limit:
∂tG(x, t) = µS + νT + φ
Z
Ω
w(x −y)G(y, t) dy

F −αG
Z
Ω
JF (x −y)F(y, t) dy
−β G
Z
Ω
JT (x −y)T(y, t) dy,
(1a)
∂tS(x, t) = −µS −ω
Z
Ω
w(x −y)G(y, t) dy

S −αS
Z
Ω
JF (x −y)F(y, t) dy
+ β G
Z
Ω
JT (x −y)T(y, t) dy,
(1b)
∂tT(x, t) = −νT + ω
Z
Ω
w(x −y)G(y, t) dy

S −αT
Z
Ω
JF (x −y)F(y, t) dy,
(1c)
∂tF(x, t) = α(G + S + T)
Z
Ω
JF (x −y)F(y, t) dy −φ
Z
Ω
w(x −y)G(y, t) dy

F,
(1d)
for each (x, t) ∈Ω× R+ for some Ω⊂R2.
Since the system of equations given by (1) describes the evolution of probability densities, we
have
G(x, t) + S(x, t) + T(x, t) + F(x, t) = 1 for each (x, t) ∈Ω× R+.
(2)
In this model, the constants µ and ν are the mortality rates of saplings and savanna trees respec-
tively. The functions φ and ω represent the burning rates of forest trees and saplings due to ﬁre;
they are theoretically predicted, and empirically observed, to have sharp threshold or sigmoidal
proﬁles as functions of the available ﬂammable cover (grass in this framework) [51]. For numerical
investigations, we employ the following smooth approximations to a sigmoid for φ and ω:
ω(G) = ω0 +
ω1 −ω0
1 + e−(G−θ1)/s1 ,
φ(G) = φ0 +
φ1 −φ0
1 + e−(G−θ2)/s2
for G ∈[0, 1].
(3)
with parameter values as given in Table 1 below.
The kernel function w measures the ability of ﬁre to spread spatially from a point that is already
burning. The constants α and β account for the strength of forest-tree and savanna-tree invasion via
1Also representing the probability for a site at location x to be of a given type at time t.
4

seed dispersal, with the spatial distribution of these seeds captured by the kernels JF and JT . The
inclusion of nonlocal or long-range interactions is considered by many most appropriate for spatial
vegetation models as dispersal of seeds is often long range or even heavy-tailed [43, 55]. The spatial
interaction (ﬁre spread and seed dispersal) are assumed isotropic so all kernels are of convolution
type and the model (1) is thus (for now) posed on a homogeneous spatial domain. For simplicity,
and in all numerical results, we use zero mean Gaussian kernels with diﬀerent standard deviations
(to reﬂect the relative length scales of the diﬀerent spatial process). In particular, we have
G(x, σ) :=
1
√
2πσ2 e−x2/2σ2,
σ > 0,
x ∈Ω,
with w(x) = G(x, σW ), JT (x) = G(x, σT ), and JF (x) = G(x, σF ). In the main text, we will always
consider the system with reﬂecting boundary conditions, which are appropriate for both the nonlocal
operators and the heterogeneous nature of the models (see Appendix A.2). The results are quali-
tatively similar for open boundary conditions and the Supplementary Materials contain simulations
with open boundaries for comparison purposes.
In this work, we allow α and β to be functions of spatial position, x, to mimic heterogeneous eﬀects
due to environmental gradients observed in reality, particularly the pronounced rainfall gradients
in sub-Saharan Africa and the Amazon [6, 62].
We will neglect the inﬂuence of environmental
variation on the ﬁre and natural mortality processes in the model since we expect these eﬀects to be
less impactful on the dynamics. However, we note that ﬁre frequency and intensity tend to increase
and then decrease with increasing rainfall [22] and that tree mortality tends to decrease and then
increase with increasing rainfall and varies substantially with soil properties [50].
Table 1 below summarizes the parameters of the SL model, along with their ecological inter-
pretations and default numerical values. Further details on the parameter gradients and numerical
schemes can be found in Appendix A.1.
Table 1: Summary of parameters for the SL model
Ecological interpretation
Expression
Default Value
Forest tree birth rate
α(x)
[piecewise linear]
Savanna saplings birth rate
β(x)
[piecewise linear]
Savanna sapling-to-adult recruitment rate
ω
ω(G) = ω0 +
ω1−ω0
1+e−(G−θ1)/s1
ω0 = 0.9, ω1 = 0.4,
θ1 = 0.4, s1 = 0.01
Forest tree mortality rate
φ
φ(G) = φ0 +
φ1−φ0
1+e−(G−θ2)/s2
φ0 = 0.1, φ1 = 0.9,
θ2 = 0.4, s2 = 0.05
Savanna sapling mortality rate
µ
0.1
Adult savanna tree mortality rate
ν
0.05
Forest tree seed dispersal parameter
σF
-
Savanna tree seed dispersal parameter
σT
-
Fire spread (dispersal) parameter
σW
-
In addition to the four-functional-type SL model stated above, numerous other related models
with the same (or very similar) underlying interaction rules have been studied in the literature [14,
13, 25, 39, 51, 56, 63]. One important special case is that in which only the grass and forest types are
present, essentially reducing the model to a direct competition between ﬁre and seed dispersal eﬀects.
In this case, allowing α to vary in space, the system (1) reduces to the single integro-diﬀerential
5

equation
∂tG(x, t) = φ
Z
Ω
w(x −y)G(y, t) dy

(1 −G(x, t))−α(x)G(x, t)

1 −
Z
Ω
JF (x −y)G(y, t) dy

,
(4)
where the normalization condition G(x, t) + F(x, t) = 1 and the assumption that JF and w are
probability density functions on Ω(and hence have unit integrals) enable this reduction.
2.2
Heterogeneities
In the tropics, empirical data show that while savanna landscapes tend to dominate at low rainfall
levels, there is evidence of bistability between savanna-dominated and forest-dominated biomes at
intermediate rainfall, and forest naturally tends to dominate at suﬃciently high rainfall levels [52,
53]. Consequently, any model purporting to explain the savanna-forest transition should explicitly
account for the spatial heterogeneity arising from the signiﬁcant diﬀerences in rainfall levels across
the domain. To this end, and for the sake of tractability, we consider a one-dimensional bounded
spatial domain Ω= [0, 1] throughout and allow both the forest-tree birth rate α and the savanna-tree
birth rate β to vary with position in the domain; in particular, we choose both to be increasing linear
functions of x ∈Ωso that the domain becomes progressively wetter (and more forest favored) as
x increases (see [54] and the references therein). Intuitively, we expect the slope of the forest birth
rate to be larger than that of the savanna trees since eventually the forest trees must out-compete
the savanna trees at high rainfall levels.
We primarily study the dynamics of the heterogeneous model in the small-dispersal limit since
the seed and ﬁre dispersal scale is several orders of magnitude smaller than the scale on which
the rainfall gradient is signiﬁcant. We focus on the case of centered Gaussian dispersal kernels as
a representative class of kernels which illustrate our main qualitative conclusions. However, our
conclusions should hold for a broad class of nonnegative even kernels, although spreading speeds
will diﬀer for heavy-tailed kernels and it would undoubtedly be interesting to investigate the impact
of heavy-tailed dispersal on the nonequilibrium solutions shown in section 2.5.
2.3
Front pinning in the grass-forest model
Consider the grass-forest model given by equation (4) on the 1D spatial domain Ω= [0, 1].
If
we assume completely localized interactions, our model reduces to a family of ordinary diﬀerential
equations indexed by the spatial variable x:
˙G = φ(G)(1 −G) −α(x)G(1 −G),
x ∈[0, 1],
(5)
Equation (5) thus describes the evolution of the grass cover proportion in an isolated patch of
landscape (i.e. one without spatial interactions). We study the corresponding spatial model, given
by (4), subject to a rainfall gradient which spans from a grass-dominated regime at the left end-
point of the domain x = 0 to a forest-dominated regime at the right-most point of the domain
x = 1; we reﬂected this in the model by choosing α(x) linear with positive slope. Such a gradient in
the forest tree birth rate reﬂects the increasing productivity of forest trees at progressively higher
rainfall. Figure 1 B shows the time-evolution of grass cover in the spatial grass-forest model (4) with
dispersal parameter σ = 0.01 with rainfall gradient chosen as outlined above. Figure 1 A shows the
ﬁnal steady-state solution proﬁle from Figure 1 B (dashed green line) overlaid with the bifurcation
diagram from the corresponding nonspatial model (5) (red for stable equilibria, black for unstable).
Solutions to equation (5) and their stability are computed for each value of x in XPP using
AUTO [15]. Solutions to the corresponding spatial model (4) are computed using a ﬁnite-diﬀerence
spatial discretization of the integrodiﬀerence equation so that it can be solved via time stepping
6

schemes and with a ﬁxed point approach in MATLAB for bifurcation analysis. We recall that all
simulations shown in the main text are with Gaussian kernels and for reﬂecting boundary conditions
(see Appendix A.2). However, the results are very similar for other centered univariate kernels and
for other types of boundary conditions since in the small to moderate dispersal cases we study,
boundary eﬀects are minimal. We also systematically explored the space of initial conditions to
uncover multistability, which was observed in a number of cases. Grid convergence and time-step
convergence studies were carried out to verify that the stability of computed solutions was robust.
By overlaying the nonspatial bifurcation diagram onto the spatial solution in Figure 1 A, we see
that the spatial solution essentially interpolates between the two stable solution branches from the
nonspatial model with a sharp transition between the grass and forest dominated parts of the domain.
As per Figure 1 A, the gradient begins before the ﬁrst saddle node bifurcation of the nonspatial
model and hence the lower solution branch is not a candidate solution across the whole domain in
the spatial model, as we might have expected. Likewise, the gradient extends beyond the upper
nonspatial saddle-node bifurcation and so the upper solution branch is similarly not a candidate
spatial solution either (at least for small dispersal). This is a completely new type of solution not
present without spatial interactions and heterogeneity. Numerous authors have proposed this type
of mechanism as an explanation for the empirically observed sharp savanna-forest boundaries in
sub-Saraharan Africa, the Amazon, and other tropical regions with similarly abrupt savanna-forest
transitions [9, 20, 21, 62].
A
B
Figure 1: A: ODE bifurcation diagram with space as the bifurcation parameter (stable equilibrium
solution curves in red and unstable equilibria in black) overlaid with a stable steady-state solution
of the IDE for dispersal σ = 0.01 on the heterogeneous domain (dashed green line). B: Space time
plot of convergence to the steady-state solution plotted in A; the speed of the wave of invasion forest
into grass slowly tends to zero before equilibrium is reached.
Convergence to the steady-state in Figure 1 B proceeds via a wave of invasion that fails to
propagate beyond a point in the spatial domain and where a sharp front forms in the solution.
This phenomenon has been termed “front pinning” or “range pinning” in the literature and has
been demonstrated in many bistable PDE-based ecological models [9, 57, 63], and other applied
contexts [38]. The point in the spatial domain at which the front forms in the solution is referred to
as the Maxwell point, and is a generically stable conﬁguration in heterogeneous systems. Front pinned
solutions can also be stable in certain spatially homogeneous systems [41, 40]. Heuristically, pinning
is harder to observe in a homogeneous system because waves of invasion will select some constant
wave speed c with which to propagate on a homogeneous domain. If the system parameters are set to
exactly generate c = 0, then front-pinning can occur, but this parameter set will have measure zero
7

in parameter space and hence a pinned front solution is not generic (and thus physically unrealistic).
In our model, we have nonconstant wave speeds c(x) due to the heterogeneity in the system and
bistability persists for a large parameter region in the absence of spatial interactions (see Figure 1),
so stable front-pinned solutions are expected for appropriate gradient choices. Various authors have
considered calculation of the Maxwell point for reaction diﬀusion versions of the Staver-Levin model
and other ecological models [63, 20]. However, this approach appears technically limited to scalar
systems with diﬀusion as the dominant spatial interaction, meaning the method is approximate for
models with nonlocal interactions. Moreover, the Maxwell point calculations are only approximate
when the values of the nonspatial steady states vary as a function of the gradient (as they do in our
model, see Figure 1 A). The existence of front-pinned solutions has been established rigorously in
reaction-diﬀusion models via asymptotic analysis in the small diﬀusion limit [49], but, to the best
of our knowledge, this question remains open for systems involving nonlocal operators.
A
B
C
D
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0.5
1
0.75
B
C
D
Figure 2: A: Bifurcation diagram as the dispersal parameter σ = σF = σW is varied with the
L1 norm of the grass component of the solution on the y-axis (reﬂecting boundary conditions and
Gaussian kernels for both seeds and ﬁre). Red lines: Stable solutions, Black lines: Unstable solutions,
B/C/D: Solutions from the grass dominated branch, the front-pinned branch and the forest branch,
respectively.
The existence and stability of these solutions are functions of the spatial interactions. In Figure
2 we investigate the impact of the dispersal parameter σ on front pinning and multistability in the
system2. The standard deviations of the Gaussian ﬁre and seed kernels, σW and σF , are chosen
equal to a common value σ, which serves as the bifurcation parameter. We observe that there is a
large dispersal range (or range of spatial scales) for which the model supports three stable solutions,
a grass-dominated solution (Figure 2 B1), a forest-dominated solution (Figure 2 B3) and a front-
pinned solution with a sharp transition between forest and grass (Figure 2 B2). This latter solution
disappears for high dispersal values through a saddle-node bifurcation. Heuristically, when dispersal
2Note that varying the dispersal parameter in this way is equivalent to varying the size of the spatial domain (with
a smaller dispersal parameter corresponding to a larger domain).
8

is large compared to the spatial heterogeneity, it eﬀectively rapidly homogenizes the system and the
system can no longer support such a heterogeneous solution and instead converges to more well-
mixed solutions. We further observe that the forest-dominated solution branch exists and remains
stable for all but very low dispersal values, eventually disappearing in a saddle-node bifurcation
around σ ≈0.026. The grass-dominated solutions disappear at low dispersal rates, in favor of mixed
equilibria associated with unavoidable forests in most humid regions and grass at driest regions. The
all-grass solution (i.e. G(x) = 1 for all x ∈[0, 1]) is a solution for every value of σ but is always
unstable.
2.4
Front-pinning in the four-functional-type forest-savanna model
stable oscillations 
forest (1 stable FP)
forest/savanna bistable
GH
GH
GH
4
5C
5B
5A
0
0.4
0.8
1.2
1.2
1.6
0.15
0
1
0.5
Grass
5A
5B
5C
S=T=0
savanna
4
A
B
S=T=0
forest (2 stable FPs)
0.4
0.8
GH
1.6
Figure 3: A: One-parameter bifurcation diagrams corresponding to each of the rainfall gradients
studied. Stable/unstable equilibria denoted by red/black, stable/unstable limit cycles in green/blue
and heteroclinics denoted by dashed black lines. B: Two-parameter bifurcation diagram in α and
β for the nonspatial Staver-Levin model (i.e. completely localized interactions). Transcritical bi-
furcation curves in blue, saddle node curves in magenta, supercritical Hopf curves in purple and
subcritical Hopf curves in dark green, with switching points from one type to the other at Bautin
(Generalized Hopf, label GH) points.
9

We introduce the eﬀects of a rainfall gradient phenomenologically by allowing both α and β to be
linearly increasing in x, meaning that both kinds of trees become more productive as we move along
the gradient, or moving rightwards in the 1D spatial domain Ω= [0, 1]. To reﬂect the empirical
observation that forest trees dominate at high rainfall, the slope of α will be larger than that of
β in all scenarios. Estimating the real quantitative impact of sub-Saharan African rainfall on each
of these functional types is a formidable and outstanding challenge, but we can glean considerable
insight into the range of dynamics and savanna-forest transitions with our more qualitative approach.
Previous studies have noted the array of behaviors that emerge upon introducing a realistic rainfall
gradient into this model [63], but it can be diﬃcult to motivate and understand the resulting spatial
dynamics without reference to the already complex dynamics of the nonspatial version of (1). Figure
3 B shows the two-parameter bifurcation diagram for the (nonspatial) four-functional-type Staver-
Levin model as a function of the forest tree birth rate α and the savanna tree birth rate β. We
immediately see that a rainfall gradient tracing a linear path in α-β space may connect regions with
very diﬀerent dynamics in the absence of spatial interactions, including stable limit cycles involving
all four functional types (see [56] for more details). The top row of subﬁgures in Figure 3 A shows
various one parameter bifurcation diagrams (only the grass component of the solution is shown)
along gradients in α-β space for which we discuss the corresponding spatial dynamics below.
A
B
C
S=T=0
D
Figure 4: A: For x ∈[0, 0.4], the system is in a savanna state, for x ∈[0.4, 0.7] there is a grass
dominated state with a small number of forest trees and then after the Maxwell point around x = 0.7,
the system transitions to a forest dominated state. B: Alternative stable state with savanna on the
left side of the domain with grass dominant on the right of the domain. C: Alternative stable state
with forest dominant across the entire domain. Dispersal parameters are set to σ = 0.025 for all
panels and boundary conditions are reﬂecting. D: Nonspatial bifurcation diagram for (1) with the
spatial variable x as the bifurcation parameter.
Figure 4 shows three multi-stable solutions for a ﬁxed parameter set and a linear rainfall gradient
(shown in α-β space in Figure 3) in the savanna-forest model given by (1); the dispersal parameters
are identically σ = 0.025 in all panels. Panels A and B of Figure 4 show a pair of stable solutions in
which all four functional types are present. The solution in 4 A follows the upper stable branch in
the nonspatial bifurcation diagram (Figure 4 D) until a Maxwell point around x ≈0.7 where there
is a rapid transition to forest dominance (and the lower stable branch in Figure 4). This solution
has both a stable savanna and a stable forest domain separated by a grass band suﬃciently high to
suppress forest trees; this is a novel prediction in terms of the savanna-forest transition in the tropics
and does not seem to appear in the existing literature, but anecdotally is consistent with the low
tree biomass and density observed in, e.g., the Bateke Plateau bordering the Congo rainforest [44].
The grass band is caused by a transcritical bifurcation in the nonspatial model (marked by the
vertical pink dashed line), up to which, grass steadily increases at the expense of savanna trees; this
high level of grass exceeds the ignition threshold for ﬁre (at least locally) and hence keeps forest at
a low level. At the transcritcal, the savanna loses stability and the system enters the forest-grass
subsystem, in the grass-dominated state. Figure 4 panel C shows a forest dominated solution that
10

is stable for all levels of the dispersal parameter. As in the forest-grass front-pinning example, the
front-pinned solutions eventually lose stability at suﬃciently large levels of dispersal (equivalent to
a smaller spatial domain); this occurs around σ ≈0.05 for the gradient chosen in this particular
example. In the Supplementary Materials, we show that this example is robust to stochasticity and
nonlinearity in the rainfall gradient (see SM1.).
2.5
Transient passage through a family of periodic orbits
C
B
A
Figure 5: A: Simple (period 1) waves of invasion of savanna into forest led by a grass/ﬁre front. B:
More complex waves of invasion (period 2). C: Solution from the chaotic regime.
Figure 5 shows solutions for three diﬀerent rainfall gradients which intersect the stable oscillations
region of the nonspatial bifurcation diagram shown in Figure 3 B. The solution in Figure 5 A
varies from forest ≈0.5 on the extreme left of the domain up to forest ≈1 on the right but this
forest dominance is punctuated by waves of invasion by the other functional types. These waves
are triggered by a wave of grass (carrying ﬁre), which burns the forest trees and allows savanna
to outcompete grass in the wake of the initial wave.
The savanna then becomes vulnerable to
replacement by forest as it eliminates the grass, which carried ﬁre to suppress the forest. This cycle
repeats in a similar manner to the simple periodic oscillations observed in the nonspatial Staver-
Levin model [56]. The left and right endpoints of the gradient anchor the system in the savanna and
forest states respectively, although the “savanna state” has unusually high forest tree presence. The
solution shown in Figure 5 B is a period-two wave that appears to be caused when we enlarge the
region of space which the gradient spends in the stable oscillation region of the nonspatial system.
When this region is large enough, waves take suﬃciently long to reach the right-hand termination
point that another wave has already begun on the left side of the domain, eventually leading to
period-doubling due to the interaction of the two waves. Finally, in Figure 5 C, we enlarge the
oscillating region that the gradient passes through yet further and observe what appears to be
a complicated quasi-periodic behavior or spatio-temporal chaos in the solution. The solutions in
Figure 5 were computed using reﬂecting boundary conditions but the solutions are qualitatively
similar with open boundary conditions so the phenomena shown are not sensitive to this choice
11

(see the Supplementary Materials for the corresponding simulations). To further investigate the
0
0.5
1
Precipitation
B
C
E
F
0
0.5
1
Figure 6: A-E: Forest component of each solution with the same gradient as Figure 5B but for
varying slopes (σ = 0.02 in all cases). The slopes in panels A-E are: A - 1.5, B - 2, C - 2.5, D - 3.4,
E - 3.42. F: Illustration of how the gradient slopes change across the other panels (blue gradient
with slope 1, red gradient with slope 2, and yellow gradient with slope 3.5).
phenomenon of wave generation shown above, we considered the gradient from Figure 5 B once
more, but this time we allowed the slope of the gradient to vary in order to change the region of the
spatial domain in the oscillatory regime (slope changes are illustrated in Figure 6 F). Panels A, B
and C of Figure 6 show the forest component of the solution for slopes of 1, 2 and 2.5 respectively.
As the slope of the gradient increases, we soon revert from period-two waves to simpler period-one
waves. The left vertical red dashed line in panels A-F of Figure 6 indicates the onset of oscillations
without spatial interactions, corresponding to the start of the fold of limit cycles associated with
the subcritical Hopf curve marked by a solid dark green line in Figure 3 B; the right vertical red
dashed line indicates the oﬀest of oscillations, corresponding to the heteroclinic-to-saddle connection
marked by a dashed light blue line in Figure 3 B. Even as the region of oscillations predicted by the
nonspatial model shrinks, the waves continue to travel remarkably far past the right hand red line
marking the non-spatial heteroclinic. Figures 6 D and E show solutions for slope values of 3.4 and
3.5, with oscillations ﬁnally abating in panel E as the oscillatory region ﬁnally becomes too small to
support stable oscillations in the spatial model. This example illustrates the phenomenon of transient
passage through a family of periodic orbits in a spatial model; it shows that real systems may pass
through regimes that support oscillations without spatial interactions but that these oscillations will
only be seen in spatially-extended models if the gradient passes through this region of parameter
space suﬃciently slowly.
It is evidently of both theoretical and applied interest to understand
in more detail when transient passage through oscillations will result in periodic solutions in the
spatial system and how these periodic solutions may further bifurcate, as they do above in Figure 5.
Immediate questions and challenges in this domain include identifying criteria to determine the onset
or oﬀset of spatio-temporal oscillations and characterizing transitions between diﬀerent oscillatory
regimes (e.g. period doubling, quasi-periodic and chaotic behaviors) for representative classes of
spatial operators.
12

3
Transient patterns at the frontiers: Applications to brain
development
Physiological formation of cortical regions relies on the precise positioning of sharp and regular
boundaries during embryonic development, which are thought to be guided by the presence of
positional cues (typically, gradients of morphogens) [18, 29]. Failures to forming cortical regions
with sharp and regular boundaries at speciﬁed location in brain were implicated in many serious
pathologies [26, 48, 60]. Each territory is characterized by the expression of a speciﬁc combination
of molecular marks that compete with each other and according to the gradients of morphogens
promoting or repressing speciﬁc genes [45]. As an example, in the mammalian forebrain, cells arising
from divisions of a common populations of progenitors are fated to become medial entorhinal cortex
(MEC) cells or neocortical (NC) cells based on their exposure to patterning transcription factors
(TFs) expressed in a graded fashion. In [17], it was shown that changes in morphogen gradients
not only led to simple boundary shifts, but also to the possibility of a shattered boundary with the
formation of regular patterns of ectopic cortical domains. A mathematical model introduced in this
paper proposed that this phenomenon could be related to a slow passage through a pattern-forming
instability that we explore here in more detail.
3.1
A model of brain arealization
The model proposed in Feng et al. [17] was based principally on the following three experimentally
and biologically motivated mechanisms:
(I.) Competition: The diﬀerentiation into MEC or NC neuronal identity is a competitive process
(e.g.,competition on the genetic resources).
(II.) External cues (heterogeneity): Extracellular TF gradients favor diﬀerentiation into MEC cells
on the posterior side and into NC cells on the anterior side of the brain.
(III.) Aggregation/diﬀerential adhesion mechanisms: Neurons, having a tendency to diﬀuse slowly,
also have the ability to aggregate preferentially with cells of their own type (Feng et al. [17]
showed that diﬀerential adhesion was the primary driver of cells preferentially aggregating
with other cells of their own type).
We thus arrive at the following continuum mathematical model of MEC and NC identity marker
levels across the cortex:
∂tE(x, t) = E(1 −E −k1N) + DE∆E −χ1 ∇· (Φ(E) ∇CE) + ρE(x),
(6a)
∂tCE(x, t) = E −CE + DCE∆CE,
(6b)
∂tN(x, t) = N(1 −N −k2E) + DN∆N −χ2∇· (Φ(N)∇CN) + ρN(x),
(6c)
∂tCN(x, t) = N −CN + DCN ∆CN,
(x, t) ∈Ω× R+,
(6d)
where the species E accounts for MEC cell fate markers, N accounts for NC cell fate markers, ∇
denotes the gradient operator, ∇· denotes the divergence operator and ∆is the diﬀusion operator.
The competition between E and N is reﬂected by classical Lotka-Volterra competitive reaction
dynamics; we choose k1 = k2 = 2 to inhabit a bistable regime mirroring the common lineage of
MEC and NC progenitor cells when spatial interactions are neglected. External signals promote
the expression of each gene: E is promoted at a rate ρE(x) at position x (COUP-TFI eﬀects and
other signals promoting MEC fate) and N is promoted at a rate ρN(x) (TF eﬀects promoting NC
fate) at position x. We account for cell aggregation via two monitor species, CE and CN, generated
respectively by cells expressing E or N markers and subject to degradation and diﬀusion. CE and CN
attract E and N cells respectively with χ1 and χ2 denoting the strengths of the aggregation forces.
13

Considering our intended application, it makes sense to always consider (6) with no-ﬂux boundary
condition for Ω⊂Rn with n ≤3. It should be stressed that our model is phenomenological in
nature and we have not aimed at a detailed biophysical description of the physical and cellular
process involved, but rather we have tried to capture their eﬀects qualitatively in a parsimonious
and interpretable model.
The model (6) follows the classical framework of Keller-Segel chemotaxis systems with cell aggre-
gation limited by saturation eﬀects at high cell density by Φ : R 7→R [24, 27, 28]. Mathematically,
this makes the model more stable and avoids the potential for ﬁnite-time blow-up of solutions, as
often arises in other Keller-Segel-like models (cf. [23]). We employ a Ricker-type saturation func-
tion, i.e. Φ(z) = αze−αz with α > 0, but other choices appropriate for capturing saturation produce
qualitatively similar results to that shown below and in Feng et al. [17].
Table 2 summarizes the parameters, their biological interpretations, and their default values in
the brain patterning model.
Table 2: Summary of parameters for the brain arealization model
Biological interpretation
Expression
Default Value
Entorhinal marker level
E
[dynamic]
Entorhinal monitor concentration
CE
[dynamic]
Neocortex marker level
N
[dynamic]
Neocortex monitor concentration
CN
[dynamic]
COUP-TFI eﬀects
ρE(x)
[piecewise linear]
Patterning TF eﬀects promoting neocortex
ρN(x)
[piecewise linear]
Entorhinal competition
k1
2
Neocortex competition
k2
2
Aggregation saturation parameter
α
1.2
Entorhinal adhesion strength
χ1
1.5
Neocortex adhesion strength
χ2
1.5
Entorhinal cell diﬀusion
DE
0.2
Entorhinal monitor diﬀusion
DCE
0.2
Neocortex cell diﬀusion
DN
0.2
Neocortex monitor diﬀusion
DCN
0.2
3.2
Transient passage through a pattern forming instability
Our primary goal in this paper is to illustrate the mathematical mechanism by which the model (6)
explains the experimental observations of ectopic MEC from Feng et al. [17] and hence we refer the
interested reader to that paper for further biological details. Indeed, with appropriately chosen TF
gradients (i.e. ρE(x) and ρN(x)), the system (6) qualitatively matches all observed brain patterning
phenotypes from the Feng experiments [17]. The only modiﬁcation we make to the model (6) is to
choose χ1 and χ2 as ﬁxed constants. It turns out that these coeﬃcients are also inﬂuenced somewhat
by COUP-TFI and this dual role of COUP-TFI is at the heart of the complex boundary shattering
phenomenon observed in vivo, but this is not necessary to explain the dynamics of interest here from
a mathematical perspective.
First, consider the system (6) posed on a homogenous domain, i.e.
ρE(x) ≡ρE ∈R+ and
ρN(x) ≡ρN ∈R+.
Homogeneous solutions ( ¯E, ¯CE ¯N, ¯CN) to (6) obey the following system of
14

nonlinear equations:
0 = ¯E(1 −¯E −2 ¯N) + ρE,
(7a)
0 = ¯N(1 −¯N −2 ¯E) + ρN,
(7b)
with ¯CE = ¯E and ¯N = ¯CN. The system (7) has between one and three solutions depending on the
values of ρE and ρN. In the absence of spatial interactions, the stability of these equilbria is shown as
a function of ρE and ρN in Figure 7 panels A and B. Figure 7 A tracks the stability of homogeneous
solutions as a function of ρE with ρN = 0.1 and shows that there is a region of bistability (in light
blue) with two stable homogeneous solutions separated by an unstable equilibrium. In Figure 7 B
we allow both ρE and ρN to vary and we observe that the two saddle-node bifurcations from panel A
collide in a codimension 2 cusp bifurcation at (0.25, 0.25) in ρE-ρN space. Between these saddle-node
curves there is a signiﬁcant region of bistability (in light blue) while outside of this region there is
a single solution that is N dominated above the line ρE = ρN and E dominated below that line.
B
0.2
0.2
0
0
0.2
0.2
0
0
0.1
0.1
0.1
0.1
0.3
0.3
A
0.3
0.3
Figure 7: A: One-parameter bifurcation diagram for solutions of (6) without spatial interactions
with ρN = 0.1. B: Two-parameter bifurcation diagram for (6) without spatial interactions. Magenta
lines are curves of saddle-nodes which collide in a cusp bifurcation.
Turning now to the full spatial model, we can linearize about the homogeneous equilibria dis-
cussed above and classical linear stability analysis reveals pattern forming instabilities will emerge
for a range of values of the diﬀusion coeﬃcients and chemotactic strength parameters (see Feng et
al. [17] supplementary information for details). Figure 8 D shows the potential instability region in
ρE-ρN-space via a heatmap of the maximum of the principal eigenvalues of the linearized operators
about each of the homogeneous equilibria that were stable in the absence of spatial interactions;
this red region is typically referred to as the Turing space [42]. Numerical simulations of solutions
to (6) conﬁrm the presence of patterns in the red regions of Figure 8 D, and reveal the nature and
diversity of these patterns (spots, mixed spot-stripe and labyrinths patterns were observed for this
system in 2D domains).
The analysis above is valid for ρE and ρN equal to some constant ﬁxed values but, biologically,
they both vary across the cortex. In our idealized model, we assume ρE simply promotes E and ρN
promotes N. Moreover, E should dominate at one side of the spatial domain, while N dominates
at the other. Thus a reasonable choice of the gradient must start above the line ρE = ρN (which
represents perfectly balanced competition) and end below it. One such choice is shown in Figure 8 D
by the solid black line giving a linear gradient varying from points P1 to P2 and cutting through the
red instability region. However, as alluded to earlier in the discussion of spatio-temporal oscillations
in section 2, the key question in terms of the resulting spatial dynamics is: how quickly does this
gradient cross the pattern forming region? In Figure 8 E, we illustrate how we can vary the slope
of the gradient or in other words, vary the length of the region R2 to adjust how long the gradient
15

spends in the pattern forming regime from the homogeneous domain problem. Figure 8 panels A,
B and C show three solutions of the model (6) using the piecewise linear gradient from Figure 8 D
on the 1D spatial domain Ω= [0, 40] with the edges of the region R2 marked by red vertical lines
in each case. In Figure 8 A, we don’t observe standard pattern formation but rather a type of front
pinned solution similar to those discussed earlier for the SL model (see section 2). In panel B, we
see the solution form a couple of small amplitude spikes, coming closer to standard pattern forming
behavior, and in panel C, the solution now supports a multiple spikes and resembles the expected
solution for a pattern forming system in the subdomain R2.
0
40
0
0.5
1
E
A
C
0
0.4
0.8
0
0.4
0.8
-0.8
-0.4
0
0.2
P1
P2
D
R1
R3
R2
0
40
B
20
20
0
40
0
0.5
1
20
0
40
0
0.5
1
20
R1
R3
R2
Figure 8: A-C: Model simulations with a linear morphogen gradient in the region marked by red
vertical lines; constant morphogen levels ﬁxed at the gradient endpoints outside the red marked
region. All solutions shown at time t = 200. D: Heatmap of the maximum principal eigenvalue
obtained when linearizing around all homogeneous equilibria that are stable without spatial inter-
actions (diﬀusion coeﬃcients all 0.2 and χ1 = χ2 = 1.5). E: Illustration of how we vary the gradient
in ρE-ρN-space across the 1D spatial domain Ω= [0, 40].
Notably, the spikes in Figure 8 F do not have a single wavelength or frequency. Multiple wave-
lengths appear to be present as we cross the instability region; this is distinct from a standard Turing
pattern with a single dominant wavelength, and analogous to the model and experiments reported
in [17] where there appears to be a blend of spot, stripe and labyrinthine patterns, mirroring exactly
the phenomenology observed in vivo. Clearly, the width of R2 crucially inﬂuences the ability of the
system to form patterns and the nature of those patterns. Moreover, the wavelength of the pat-
terns supported by the system posed on a homogeneous domain relative to the width of R2 appears
important in determining whether patterns can “ﬁt” into potential patterning region.
To the best of our knowledge, the biologically motivated dynamics above, which we term “tran-
sient passage through a pattern forming instability”, do not ﬁt into any of the established frameworks
that provide criteria for the onset of Turing-like patterns, not even those designed to incorporate
16

heterogeneity (e.g. [34, 37, 58]). Moreover, it would evidently be of interest to characterize the
nature of the patterns further to understand if diﬀerent qualitative patterns are possible (for two
or three dimensional domains), as these may have diﬀerent implications for applications in biology.
Thus, this applied example provides strong motivation for the continued development of mathemat-
ical tools for studying pattern formation in the presence of spatial heterogeneities, particularly for
non-reaction-diﬀusion systems and large continuously-varying heterogeneities.
4
Conclusions and Discussion
Despite the prevalence of pattern formation in heterogeneous domains in nature, the abundant theo-
retical literature in pattern formation has still not developed the tools to address these questions for
general transient passages through instabilities. The two biologically motivated examples introduced
here do not ﬁt into any of the established frameworks that provide criteria for the onset of Turing-
like patterns, not even those designed to incorporate heterogeneity. We hope that highlighting these
examples will motivate new mathematical theory to better understand the underlying dynamics in
each case.
Existing work to allow heterogeneity in pattern forming systems has achieved great progress
in characterizing small heterogeneous perturbations of homogeneous systems [4] or bifurcations of
heterogeneous systems with steady states that are explicitly derived. There has also been signiﬁcant
work in the neuroscience literature studying more complex heterogeneities as inputs to neural ﬁeld
equations [16, 32, 31, 30], and some recent work in theoretical ecology highlighting the complexity
of dynamics that non-monotonic gradients can generate in bistable systems [2]. In contrast, we
are concerned with characterizing the properties of solutions to spatially heterogeneous systems as
they relate to a bifurcating behavior in the underlying homogeneous (typically non-spatial) system.
In other words, the mechanism of interest here is rather spatial models with multistable behavior
which have spatial gradients (heterogeneities) connecting regions with diﬀerent underlying dynam-
ical behaviour. Thus, the complexity in the dynamics emerges from the rate at which (monotonic)
gradients connect regions with diﬀerent dynamic behaviors. For example, a much studied paradigm
for the two species SL model (forest-grass) presented in Section 2.3 is one in which there are es-
sentially three regions of the spatial domain: a region where the grassland state is stable, a region
of bistability between forest and grass, and a region where forest is stable. Similarly, in the brain
arealization model, there is a spatial gradient crossing from a region of enthorinal fate dominance to
a region of neocortical fate dominance. However, in this case, the complex dynamics emerge when
we increase the adhesion strength suﬃciently to introduce a third distinct region in which pattern
formation occurs, leading to what we term transient passage through a pattern forming instability.
Mathematically, we expect some asymptotic regimes to be amenable to analysis. In particular,
we expect that situations with very localized interactions (e.g., vanishing diﬀusion for instance) will
closely match the non-spatial dynamics away from bifurcations and in regions of space with unique
homogeneous stable attractors, with rapid transitions between distinct patterns at bifurcations or
within multi-stable regions. Existing mathematical works in this domain have focused on compe-
tition models that may include bistability, and only recently qualitative properties in asymptotic
regimes were derived [49]. These situations are in fact only among the simplest cases of the general
question of pattern formation in heterogeneous domains and through bifurcations. Mathematically,
we expect that tools from perturbative analysis, geometric (singular) perturbation theory [1], or
changes of parameterization of solutions as used in [49] to derive viscosity solutions could allow
rigorous characterizations of those regimes. Regimes with ﬁxed diﬀusion but very slow variations
of the environment in space are likely similar, and will sometimes be precisely matching regimes of
very localized interactions in space through appropriate changes of variables. Another asymptotic
regime that will likely be amenable to analysis is systems with sigmoidal gradients of heterogeneity
in the limit of very sharp gradients. In these regimes, we expect to observe a convergence towards
17

the solution of a system with patchy heterogeneity (Heaviside step function) with a single transition
between the leftmost and rightmost regimes and no impact of the transient patterns, as in Figures
6E and 8A, and as studied in [34, 46]. For all these questions, systems with multiple spatially ho-
mogeneous solutions (and possibly transient equilibria in space) will constitute the ﬁrst models to
analyze.
Systems with spatially transient oscillations or patterns will be associated with a richer phe-
nomenology. In systems with spatially transient Turing-like patterns, we expect not only the diﬀu-
sion properties to play a role in the emergence of a pattern at the transition, but also the intrinsic
properties of the pattern itself (as its length scale or the modes of instability associated). Indeed, we
expect that contrasting with transient spatially homogeneous regimes, the emergence of a pattern
at the transition will also depend on whether or not multiple patterns “ﬁt” in the region of insta-
bility, and possibly not fully expressed patterns may emerge as well as in Figure 8 A. In multiple
dimensions, the nature of the pattern and progressive morphing of it along the gradient will also
arise, and this already plays a role in the patterns observed both from numerical simulations of the
model and the corresponding experiments in [17]. In systems with spatially transient oscillations,
not only the question of the mere existence of an oscillation at the transition arises, but problems
related to the waves generated, their regularities and their bifurcations have a strong impact on
the qualitative behavior of the solution. The prospects for theoretical results in this case seems
somewhat more pessimistic for nonlocal spatial models, such as the SL model studied here, but it
is likely that this phenomenon can fruitfully be studied for reduced form phase models with simple
coupling structures. Moreover, while the theoretical predictions related to the wave-like solutions of
the SL model presented here are not easily testable, similar mechanisms could be tested in smaller
scale experimental systems, such as chemical oscillators [10].
A
The Staver-Levin Model
A.1
Parameters and numerics
All codes to generate the ﬁgures from this paper are maintained on Github at
github.com/patterd2/SL model rainfall gradient.
The parameter gradients used in the paper for α and β are linear, i.e.
α(x) = αc + αsx,
β(x) = βc + βsx,
x ∈Ω= [0, 1].
(8)
Parameter
αc
αs
βc
βs
Figures 1 & 2
0.5
1.25
N/A
N/A
Figure 4
0.8
0.5
0.15
0.1
Figure 5A
0.2
0.8
1.9
0.1
Figures 5B & 6
0.2
0.8
1.5
0.1
Figure 5C
0.2
0.8
0.4
0.1
Table 3: Parameter values for the gradients for each ﬁgure from the main text.
In Figure 6 of the main text we adjust the speed at which we move along the rainfall gradient in
the following way. We choose:
α(x) = αc + αsP(x),
β(x) = βc + βsP(x),
x ∈Ω= [0, 1].
(9)
18

where for slope parameter Ps ≥1, we deﬁne
P(x) =





0,
x ∈[0, (1 −1/Ps)/2) ,
Ps (x −0.5),
x ∈((1 −1/Ps)/2, (1 + 1/Ps)/2) ,
1,
x ∈((1 + 1/Ps)/2, 1] .
We refer to Ps as the “slope parameter” in the main text. The solution shown in Figure 6 A has
Ps = 1.5, panel B is for Ps = 2, panel C is for Ps = 2.5, panel D is for Ps = 3.4 and panel E is with
Ps = 3.42.
We computed approximate numerical solutions to the spatial two-species SL model, given by
equation (4), by ﬁrst discretizing time with an explicit Euler scheme to obtain
G(x, n + 1) = G(x, n) + h

(1 −G(x, n)) φ
Z
R
w(x −y) G(y, n) dy

−G(x, n)

1 −
Z
R
JF (x −y) G(y, n) dy
 
,
h > 0,
(x, n) ∈[0, 1] × Z+.
(10)
We then discretize the integrals in (10) using the 1D trapezoidal rule and approximate the solution
on the evenly spaced grid {0, ∆, 2∆, . . . , 5} for some ∆> 0. In practice, we found that a step size
h < 0.1 and 200 spatial grid points was suﬃcient to ensure numerical stability of the scheme and
consistency of the qualitative dynamics, and the scheme remained stable as we decreased the time
step and increased the number of grid points. Solutions shown in the main text are for 400 grid
points. A similar discretization was used for the four-species SL model.
A.2
Boundary Conditions for nonlocal operators
To ﬁx ideas, consider the (ill-posed) nonlocal operator given by
K[u](x, t) =
Z
R
J(x −y)u(y, t) dy,
t ∈R+,
(11)
where the function u is deﬁned on [0, L] × R+. From an applied perspective, the value of K[u](x, t)
represents the cumulative impact of seed dispersal (or ﬁre signal) on u(x, t), the vegetation present
at position x at time t. The value of the integrand in this formula, J(x−y)u(y, t), is dispersal (either
seed or ﬁre transmission) from vegetation present at position y ∈R but for numerical simulations
the solution is only deﬁned on a ﬁnite domain, i.e. u(y, t) is only deﬁned for y ∈[0, L] for each
t ∈R+. Hence we must ask: what choice of boundary conditions extends u from a function on [0, L]
to a function on R and respects the character of the ecological application at hand?
Periodic boundary conditions: Nonlocal operators similar in structure to (1) feature extensively
in the neuroscience literature in so-called neural mass or neural ﬁeld models, virtually always with
periodic boundary conditions. With periodic boundary conditions one takes the standard periodic
extension of u given by
˜u(x, t) = u(x mod L, t),
x ∈R.
This type of boundary is typically favored in a spatially homogeneous problem in which boundary
eﬀects on the dynamics are not expected be signiﬁcant.
Although we do not expect signiﬁcant
boundary eﬀects in our problem, periodic boundaries will not be a appropriate for the heterogeneous
medium problem we wish to consider; it will also eﬀectively make the heterogeneity itself spatially
periodic and this can lead to signiﬁcant undesirable boundary eﬀect (e.g. forest at high rainfall
invading into savanna at low rainfall via the boundary, cf. Figure 9).
Open boundary conditions: The most intuitively appealing option is to take inspiration from the
real-world and assume that seeds (resp. ﬁre transmission) which exit the domain boundaries at 0
19

and L are simply removed from the system, i.e. the boundaries are “open”. Additionally, no seeds or
ﬁres are transmitted into the domain from outside. We refer to this as an open boundary condition
and it implies that we extend the deﬁnition of u as follows:
˜u(x, t) =
(
u(x, t),
x ∈[0, L],
0,
x /∈[0, L].
This type of boundary condition has the advantage of being physically realistic for both seed dispersal
and ﬁre transmission but will introduce noticeable boundary eﬀects in the solutions. If the dispersal
kernels are suﬃciently localized relative to the domain, these boundary eﬀects will not qualitatively
impact which stable solutions are selected. For suﬃciently long range dispersal, stability of solutions
can be eﬀected by boundary eﬀects with this boundary condition but this will not be relevant at the
length scales we consider for our applications.
Reﬂecting boundary conditions: This is a particular type of periodic extension of the solution
which we employ in some of our numerical investigations because it minimizes boundary eﬀects for
even (symmetric) kernels. First extend u from a function on [0, L] to [−L, L] by reﬂection:
uR(x, t) =
(
u(x, t),
x ∈[0, L],
u(−x, t),
x ∈[−L, 0].
Now let ˜u be the standard 2L periodic extension of uR (as deﬁned above) so that ˜u is deﬁned on all
of R. The reﬂection symmetry introduced in this extension, plus the choice of even kernels, ensures
that the solutions to the IDE system (1) closely approximate those of the underlying nonspatial
model in the limit as dispersal tends to zero. This choice also has the virtue of putting “wetter”
regions next to “wetter” regions in the context of the rainfall gradient model (see Figure 9) and
hence we choose to show solutions using this boundary condition in the main text.
A.3
Non-monotonic gradients and robustness in the SL model
In Figure 10 we show simulations corresponding to the setup of Figure 4 of the main text, but with
non-monotonic rainfall gradients to emphasize the robustness of the multistability observed in this
example. In addition to our default linear gradient, we also show qualitatively similar solutions for a
(stochastic) noisy version of the linear gradient (row B) and a nonlinear gradient with the same start
and end points (row C). These simulations were carried out with reﬂecting boundary conditions.
20

0
L
Rainfall
  level
Periodic boundary conditions
0
L
-L
.....
.....
2L
Open boundary conditions
0
L
-L
.....
.....
2L
Reflecting boundary conditions
0
L
-L
.....
.....
2L
Figure 9: Comparison of the practical eﬀects of the diﬀerent boundary conditions on the heteroge-
neous medium structure in the rainfall gradient model.
Rainfall gradient
A
B
C
Figure 10: Row A shows the solutions from Figure 4 along with the corresponding linear rainfall
gradient.
Row B shows solutions for the same parameters but with noise added to the rainfall
gradient. Row C shows solutions for a similar parameter regime but with a nonlinear, but still
monotonic, rainfall gradient that starts and ends at the same points in α-β space.
21

B
Wave-like solutions to the SL model under diﬀerent bound-
ary conditions
Figure 11 below shows more detail on the potentially chaotic solutions observed in Figure 6 of the
main text. In particular, the spatial averages of the solution components show no periodic character,
even over a very long time interval.
Figure 11: Row 1 shows the solutions from Figure 6C over a longer time horizon. Row 2 shows
the dynamics of the spatial averages of the solution components versus time (saplings component
omitted).
The simulations in Figure 12 are for the same parameters as Figure 6 but in this case with an
“open” boundary condition, as opposed to the reﬂective boundary condition used in the main text.
These results show that the complex dynamics shown in Figure 6 are not dependent on the choice
of boundary condition, but some small boundary eﬀects are noticeable at the right-hand boundary
in the third solution.
22

Figure 12: Solutions corresponding to those shown in Figure 6 for an “open” boundary condition.
Figure 13 shows a wider range of gradients intersecting the region of stable oscillations in α-β
space. We vary the parameter βc from 0.2 (top solution) to 2 (bottom solution) and observe a
transition from chaotic dynamics to increasing regular waves.
23

Figure 13: Solutions for a range of gradients between those shown in Figures 6A and 6C from the
main text.
24

C
The Brain Patterning Model
C.1
Numerical parameters and schemes
The PDEs were solved numerically using the open source ﬁnite element solver FreeFEM (version
4.2.1 - available at https://freefem.org). We used a semi-implicit discretization scheme to evolve
the weak formulation of the PDEs forward in time. We used a ﬁxed step-size scheme, a uniformly
spaced mesh and piecewise linear continuous ﬁnite elements (P1 elements in FreeFEM notation).
For the 1D spatial simulations shown in the main text, we chose Ω= [0, 40] and a uniformly spaced
mesh with 400 points, along with a stepsize of h = 0.1 (convergence was observed for h < 0.2).
The FreeFEM routines for solving the PDEs and the accompanying MATLAB code for processing
output is available at github.com/Touboul-Lab.
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