Influence of density-dependent diffusion on pattern formation in a refuge G. G. Piva1 C. Anteneodo12 1Department of Physics Pontifical Catholic University of Rio de Janeiro

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Inﬂuence of density-dependent diﬀusion on pattern formation in a refuge

G. G. Piva1, C. Anteneodo1,2

1Department of Physics, Pontiﬁcal Catholic University of Rio de Janeiro,

PUC-Rio, and 2National Institute of Science and Technology for Complex Systems,

INCT-CS, Rua Marquˆes de S˜ao Vicente 225, 22451-900, Rio de Janeiro, RJ, Brazil

We investigate a nonlocal generalization of the Fisher-KPP equation, which incorporates logistic

growth and diﬀusion, for a single species population in a viable patch (refuge). In this framework,

diﬀusion plays an homogenizing role, while nonlocal interactions can destabilize the spatially uni-

form state, leading to the emergence of spontaneous patterns. Notably, even when the uniform

state is stable, spatial perturbations, such as the presence of a refuge, can still induce patterns.

These phenomena are well known for environments with constant diﬀusivity. Our goal is to investi-

gate how the formation of winkles in the population distribution is aﬀected when the diﬀusivity is

density-dependent. Then, we explore scenarios in which diﬀusivity is sensitive to either rarefaction

or overcrowding. We ﬁnd that state-dependent diﬀusivity aﬀects the shape and stability of the pat-

terns, potentially leading to either explosive growth or fragmentation of the population distribution,

depending on how diﬀusion reacts to changes in density.

I. INTRODUCTION

A remarkable property of biological systems is the for-

mation of spatial structures. Patterns can emerge by self-

organization as a result of speciﬁc interactions between

individuals, without the need for external drivers [1–3].

In particular, in the framework of Fisher-type dynamics,

which includes logistic growth and diﬀusion [4], when

competitive interactions are spatially extended (nonlo-

cal), they can give rise to self-organized spatial oscil-

lations, with the characteristic wavelength determined

by the range of these interactions [5–7]. For instance,

plant competition for water, known for generating spa-

tial patterns, can be considered a nonlocal process due to

root spatial structure or water diﬀusive dynamics [8, 9].

The nonlocality of other elementary processes, such as re-

production and random dispersal, has a less central role

but can interfere constructively or destructively with the

possibility of pattern formation [10]. Furthermore, while

within Fisher dynamics diﬀusion can be detrimental to

pattern formation, since it promotes homogenization, it

can still inﬂuence the shape and stability of the resulting

patterns, depending on the kind of diﬀusion (e.g., normal

or anomalous) [11].

Although patterns can emerge solely from interactions

among individuals, they are naturally aﬀected or even

induced by environmental conditions, as these control

the rates of biological processes [12, 13]. Such eﬀects

can be observed at ecological scales in nature, as in the

case of vegetation patterns induced by spatial hetero-

geneities [14–16], as well as artiﬁcially produced in the

laboratory, such as when bacterial colonies are subjected

to adverse conditions such as ultraviolet light, except for

a protected area (refuge) [17]. In fact, growth-rate het-

erogeneity can induce pattern formation, even under con-

ditions that will not give rise to patterns in homogeneous

media [18].

Furthermore, in real environments, not only growth

rates, as in the case of a refuge, but also mobility can be

heterogeneous, e.g., in active suspensions where move-

ment is inﬂuenced by nutrient gradients [19], or due to

structural features of the environment, as the movement

of bacteria in porous media [20, 21]. Furthermore, het-

erogeneous diﬀusion can also emerge as a reaction [22]

that the individuals manifest, for instance, in response to

overcrowding or sparsity of a population, favoring, or not,

the random motion among other individuals [4, 11, 23–

35]. The speciﬁc way organisms respond to concentra-

tion depends on several conditions and vary from species

to species [4, 32–34]. For instance, in populations of

grasshoppers, the diﬀusion coeﬃcient is enhanced at high

densities, where encounters between individuals are more

frequent, but in other species, this occurs at low den-

sities [4]. Another source of heterogeneous diﬀusion,

accompanied by bias, is chemotaxis [36–39]. Density-

dependent factors are also present in migratory disper-

sal [24, 40, 41]. State-dependent diﬀusivity in biological

population dynamics has been previously considered in

the context of critical conditions for survival [11, 40, 42],

and also with regard to pattern formation [43, 44].

Let us mention an important study on the distinct roles

on pattern formation of Fokker-Planck and Fick’s laws

of diﬀusion, for spatially varying coeﬃcient of diﬀusion,

D(x) [45]. Although we will address systems with non-

locality as pattern-formation mechanism, let us mention

that there are works showing that the eﬀects of nonho-

mogeneous environments cannot be neglected in systems

with Turing instabilities [46, 47].

In this work, we analyze two classes of heterogeneous

diﬀusivity: state-dependent (where the diﬀusivity re-

sponds to the population density) and space-dependent

(where diﬀusivity is associated to the quality of the envi-

ronment). In both cases there might be a feedback that

mitigates or reinforces pattern formation. Moreover, we

focus on the eﬀects that diﬀusive heterogeneities have

on the spatial distribution of a population inhabiting a

refuge immersed in an adverse environment. We consider

as starting point the description of a single-population

dynamics given by the spatially-exteded (nonlocal) form

of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP)

arXiv:2210.11638v4 [q-bio.PE] 18 Jan 2025

dynamics [4], which includes random movements (nor-

mal diﬀusion) and logistic growth with nonlocal compe-

tition. Then we generalize this equation by substituting

normal diﬀusion by each form of heterogeneous diﬀusion

considered. For state-dependent diﬀusivity, we focus on

two functional forms, namely, decay or increase with the

population density, reﬂecting enhanced mobility in re-

sponse to sparseness and overcrowding, respectively. For

this purpose, exponential dependencies are studied as

paradigm. For the spatial-dependent case, we consider

that diﬀusivity is associated to the quality of the envi-

ronment, which is diﬀerent inside and outside the refuge.

By numerical integration of the eﬀective dynamics

equation, we ﬁnd that heterogeneous diﬀusivity does not

aﬀect signiﬁcantly the critical conditions for pattern for-

mation within the refuge, in agreement with the theoret-

ical linear-stability analysis, but heterogeneity does af-

fect the shape and stability of the patterns. This study

may bring insights, for instance, on observations made

in experiments with bacteria [17], where puzzling results

cannot be explained by considering only the simple form

of the FKPP equation.

II. MODEL

We consider a single-species population living in a focal

patch of size Limmersed in a large hostile environment

in one dimension, scenario mimicked by a positive growth

rate rin >0 inside the refuge, and a negative one rout <0

outside, namely

r(x) = rin + (rout −rin)Θ(|x| − L/2) ,(1)

being Θ the Heaviside step function.

Then, the generalized FKPP dynamics, with hetero-

geneity and spatially-extended competition, for the pop-

ulation density u(x, t), becomes

∂tu=∂x(D(u, x)∂xu) + r(x)u−u(γ ⋆ u),(2)

where the symbol “⋆” stands for the convolution oper-

ation that provides nonlocality through an interaction

kernel γ, which for simplicity we consider to be a nor-

malized rectangular shape of width 2w. This kind of

system was considered before, for constant diﬀusion co-

eﬃcient D, that is, for homogeneous diﬀusivity [18]. The

extension we propose below, inspired in previous litera-

ture [1–3, 26, 48], assumes that the diﬀusivity can depend

on the density and/or on the spatial coordinate directly,

D(u, x), reﬂecting a reaction of the mobility in response

to the distribution of other individuals or to a hostile

medium.

A. State-dependent diﬀusivity

We are mainly interested in variations of the diﬀusiv-

ity that are self-generated, as response to the population

level. First in Sec. IV, we will investigate a decreasing

function of the population density, namely,

D1(u) = dexp(−u/σ),(3)

where dand σare positive parameters, such that σcon-

trols the decay with density, recovering a homogeneous

diﬀusivity proﬁle in the limit σ→ ∞. This choice was

motivated by previous work assuming that density has

a negative impact on diﬀusion [26, 48]. This functional

form of D1(u) reﬂects a reaction to sparsity, with greater

mobility the more rareﬁed the population is. In our case

of a refuge within a hostile environment, the functional

form of D1implies a lower diﬀusivity inside, where the

population is more dense since rout < rin.

For the opposite possibility of enhanced response to

overcrowding, we will use as counterpart of Eq. (3),

D2(u) = d[(1 −exp(−u/σ)],(4)

for which homogeneity is obtained in the opposite limit

σ→0.

B. Space-dependent diﬀusivity

As another relevant case, we will consider a diﬀusivity

proﬁle D(x) taking the values Din and Dout inside and

outside the refuge, respectively, namely

D(x) = Dout + (Din −Dout)Θs[L/2− |x|],(5)

where Θsis a smoothed Heaviside step function, and the

jump has width s, such that the usual Heaviside is recov-

ered in the limit s→0. Similar settings have been used

to study the role of space-dependent diﬀusion on the crit-

ical patch size [42, 49, 50]. In Sec. V, we will investigate

its impact on pattern formation.

III. METHODS

Results for the diﬀerent scenarios described above, fo-

cusing on pattern formation and mode stability, will be

shown in the next sections. In all cases, Eq. (2) was nu-

merically integrated using a forward-time centered-space

algorithm (typically, ∆x= 0.02 and ∆t= 10−5), with

periodic boundary conditions in a grid much larger than

the refuge width, starting from the initial condition cor-

responding to the homogeneous solution plus small ran-

dom ﬂuctuations, namely, u(x, t = 0) ≃u0+ξ(x), being

ξan uncorrelated uniformly distributed variable of am-

plitude much smaller than u0=rin. In numerical simu-

lations, the refuge spans the interval (−L/2, L/2), with

ﬁxed size L= 10. The width of the rectangular kernel

γis also ﬁxed (w= 1). The numerical results are com-

plemented by analytical considerations based on linear

stability analysis.

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Influenceofdensity-dependentdiffusiononpatternformationinarefugeG.G.Piva1,C.Anteneodo1,21DepartmentofPhysics,PontificalCatholicUniversityofRiodeJaneiro,PUC-Rio,and2NationalInstituteofScienceandTechnologyforComplexSystems,INCT-CS,RuaMarquˆesdeS˜aoVicente225,22451-900,RiodeJaneiro,RJ,BrazilWeinvestiga...

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Influence of density-dependent diffusion on pattern formation in a refuge G. G. Piva1 C. Anteneodo12 1Department of Physics Pontifical Catholic University of Rio de Janeiro

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