Supplementary Materials for

advances.sciencemag.org/cgi/content/full/7/6/eabe1100/DC1

Supplementary Materials for

Fairy circles reveal the resilience of self-organized salt marshes

Li-Xia Zhao, Kang Zhang, Koen Siteur, Xiu-Zhen Li, Quan-Xing Liu*, Johan van de Koppel*

*Corresponding author. Email: [email protected] (Q.-X.L.); [email protected] (J.v.d.K.) Published 5 February 2021, Sci. Adv. 7, eabe1100 (2021)

DOI: 10.1126/sciadv.abe1100

The PDF file includes:

Section S1

Legends for movies S1 to S3 Figs. S1 to S7

Tables S1 to S7

Other Supplementary Material for this manuscript includes the following:

(available at advances.sciencemag.org/cgi/content/full/7/6/eabe1100/DC1)

Movies S1 to S3

S1 Dimensionless models and stability analyses

S1.1 Sulfide-toxicity model I S1.1.1 Scaling of the model I

In the main text, we had built the coupled partial differential equations (PDEs, Eqs. (S1)) to describe the observed FCs formation based on a hypothesis of the sulfide-toxicity feedback. The model can be written as follows

@P

@t = rP (1 P

K) cP S + D_p P, (S1a)

@S

@t = ⇠("P d k_s

P + k_sS) + D_s S. (S1b)

For generality, we study this “fast-slow dynamics” system (S1) (also see Fig. 1G) with its nondimen- sionalized form as

@P

@T = ↵P(1 P) PS + D P, (S2a)

@S

@T = P 1

P + 1S + S. (S2b)

Here, T = d⇠t, X = q

d⇠

Dsx, P = _K¹P, and S = _K"^d S. The scaled plant density and sulfide concentration P and S, respectively, are functions of space, X, and time, T . The omitted non- dimensionalization means that parameters ↵, , and correspond to ratios of the ecological quanti- ties, and they are ↵ = _d⇠^r, = ^cK"_d2⇠, = ^K_ks, and D = ^D_D^p_s.

For system (S2), there exists one coexistence steady state, i.e. vegetated state, (P, S) = (P^⇤,S^⇤), where P^⇤ = ^{↵ +}

p4↵ +↵²+2↵ + ²

2 and S^⇤= ^{↵(1 P⇤)}. The Jacobian at (P^⇤,S^⇤)is then given by J =

"

↵P^⇤ P^⇤

1 +_{( P}^S_⇤^⇤₊₁₎2 1 P^⇤+1

# .

Furthermore, one can easily see that the positive equilibrium is linearly stable if and only if the real parts of eigenvalues are less than zero.

S1.1.2 Stability conditions of vegetated state

In this section, we study the stability of the equilibrium of the system (S2).

Lemma 1.1.

Lemma 1.1.

Lemma 1.1. The steady state (P^⇤,S^⇤)is existence and asymptotically stable if ↵ > 0, > 0, and

> 0.

Proof.

Proof.

Proof. It’s clear thatp

4↵ + ↵²+ 2↵ + ² ↵ =p

4↵ + (↵ + )² p

(↵ + )² > 0and (↵ + + 2 ) =p

(↵ + )²+ 4 (↵ + ) + (2 )² >p

4↵ + ↵²+ 2↵ + ²for ↵ > 0, > 0, and

> 0. So that, 0 < ^{↵ +}p

4↵ +↵²+2↵ + ²

2 < 1, and ^{↵(1 P⇤)} > 0. Therefore, we have P^⇤ > 0 and S^⇤ > 0.

From the Jacobian matrix at (P^⇤,S^⇤), the trace and determainant can be given as follows:

T_r = ₁+ ₂= ↵P^⇤ 1

P^⇤+ 1 < 0,

= _{1 2}= ( ↵P^⇤)( 1

P^⇤+ 1) ( P^⇤)(1 + S^⇤

( P^⇤+ 1)²) > 0.

The trace (Tr) is negative and the determainant ( ) is positive for ↵ > 0, > 0, > 0, P^⇤ > 0 and S^⇤ > 0. Therefore, 1 and 2 have the same sign and are negative. Thus, the steady state (P^⇤,S^⇤) is asymptotically stable.

S1.1.3 Dispersion relation

For we seek plane wave perturbations of the form

P = P^⇤+ Ae^{jk·x+ t} (S3a)

S = S^⇤+ ¯Ae^{jk·x+ t}, (S3b)

where j =p

1, A and ¯Ais a vector of of the conjugate constant coefficients, k is the corresponding wavenumber. Substituting the solution form (S3a) in linearized Eq. (S2), we obtain the linear system

J (|k|²) I_2⇥2A = 0, (S4)

where we define

J (|k|²) =

"

↵P^⇤ P^⇤

1 +_{( P}^S_⇤^⇤₊₁₎2 1 P^⇤+1

#

|k|²

 D 0

0 1.0 .

For convenience, we denote q = |k|². We require det(J(q) I) = 0, which yields a quadratic equation in , i.e., defining

H(q) = det(J (q) I) = ² tr(J (q)) det(J (q)). (S5) To check for stability of the spatial perturbations, recall that a quadratic polynomial H(q) = 0 has all roots in the left half complex plane.

S1.2 Nutrient-depletion model II S1.2.1 Scaling of the model II

In the main text, we had built the nutrient-depletion model as follows

@P

@t = rcP N

k1+ N dP + D_p P, (S6a)

@N

@t = Iin+ sdP cP N

k₁+ N + Dn N. (S6b)

For without loss of generality, we study above model (S6) with its nondimensionalized form

@P

@T = ↵P N

1 +N P + D P, (S7a)

@N

@T = +P P N

1 +N + N . (S7b)

Here, T = dt, X = q

d

Dnx, P = _k^s₁P, and N = _k¹₁N. The scaled plant density and nutrient concen- tration P and N , respectively, are functions of space, X, and time, T . The omitted nondimension- alization means that parameters ↵, , correspond to ratios of the ecological quantities. They are

↵ = ^rc_d, = _dk^Iin₁, = _ds^c , and D = _D^Dp_n.

System (S7) exists one coexistence steady state, vegetated state, (P, N ) = (P^⇤,N^⇤), where P^⇤= ^↵_↵ and N^⇤= _{↵ 1}¹ . The Jacobian at (P^⇤,N^⇤)is then given by

J =

"

0 ₍₁₊^↵P_N^⇤_⇤)2

1 _{↵ (1+}^P_N^⇤_⇤)2

# .

S1.2.2 Stability conditions of vegetated state

In this section, we study the stability of the equilibrium of the system (S7).

Lemma 1.2.

Lemma 1.2.

Lemma 1.2. The steady state (P^⇤,N^⇤) is existence and asymptotically stable if > ↵ > 1 and

> 0.

Proof.

Proof.

Proof. It’s clear that ^↵_↵ > 0 and _{↵ 1}¹ > 0 for > ↵ > 1 and > 0. Therefore, P^⇤ > 0 and N^⇤ > 0.

From the Jacobian at (P^⇤,N^⇤), the trace and determainant can be given as follows:

T_r = ₁+ ₂ = P^⇤

(1 +N^⇤)² < 0,

= _{1 2}= (1

↵) ↵P^⇤

(1 +N^⇤)² > 0.

The trace (Tr) is negative for > 0, P^⇤ > 0 and N^⇤ > 0. The determainant ( ) is positive for

> ↵ > 1, P^⇤ > 0and N^⇤ > 0. Therefore, 1 and 2 have the same sign and are negative. Further, the steady state (P^⇤,N^⇤)is asymptotically stable.

S1.3 Scale-dependent sulfide-plant model III

As mentioned in the main text, Model I is not a scale-dependent feedback model that described the FCs patterns on salt-marsh ecosystems. There is no quasi-stable pattern for long-term evolution.

Here we make a modified version of the model I with the same mechanism but introduced an inhibition process to plant growth. Then the dynamics system became an activator-inhibitor scheme (namely Model III) and written as

@P

@t = r S

P²

k²_p+ P² cP S + D_p P, (S8a)

@S

@t = s_in+ "P² dS + D_s S. (S8b)

In terms of equations (S8) the crucial conditions for pattern formation are that the scale-dependent feedbacks. It reproduces the classical Turing-like fairy circles patterns were shown in Fig. S5B. Here the growth of plants in model III is a Hill function with an exponential coefficient of 2.0. The sulfide concentrations have inhibited the growth of seedlings and enhance the mortality rate of plants. r is the plant maximal growth rate and kp is the half-saturation constant to control the positive facilitation of plant growth. Parameter " is the sulfide production for every unit of plant growth, and d describes the detoxification coefficient. sin describes the production rate of sulfide concentration arising from bare mudflat. We ran simulations of this model (S8) using the same parameters of model I as are shown in Table S1 but Dp = 0.05, Ds= 10.0, kp = 2.0, c = 10.5, sin= 0.05, and " = 0.01.

S1.3.1 Scaling of the model III

With a similar way above, we can obtain the nondimensionalized form of model (S8) as

@P

@T = ↵P²

S(1 + P²) PS + D P, (S9a)

@S

@T = +P² S + S. (S9b)

Here, T = dt, X = q

d

Dsx, P = _k¹_pP, and S = _"k^d2

pS; and the parameters are ↵ = _"k^r3

p, = ^c"k_d2^2p,

= _"k^sin₂

p, and D = ^D_D^p_s.

There exists one coexistence steady state, vegetated state, (P, S) = (P^⇤,S^⇤), where P^⇤ is the root of P⁶+ (2 + )P⁴+ ² + 2 P² ↵P + ² = 0and S^⇤ = +P^⇤2. The Jacobian at (P^⇤,S^⇤)is then given by

J = 2

4 ^S

⇤2(P^⇤2+1)^2+2↵P^⇤ S^⇤(P^⇤2+1)²

↵P^⇤2

S^⇤2(P^⇤2+1) P^⇤

2P^⇤ 1

3 5.

We can obtain the stability of positive equilibrium (P^⇤,S^⇤) with numerical phase-plane portraits be- cause of its complexity. Hence, the bifurcation and numerical solutions were shown in Fig. 4C with pde2path package in the main text.

Note that here we don’t give the completely theoretical analyses and the stability of the trivial state with no vegetation because of ecological significance. Here the positive solutions are only considered.

S2 Supplementary Movies

Movie S1: Fairy circle patterns develop from the plant-sulfide model I. Three typical seedlings clumps were assumed on intertidal mudflats.

Movie S2: Fairy circle and concentric ring patterns develop from plant-nutrient model II. Three typical seedlings clumps were assumed on intertidal mudflats.

Movie S3: Persistent fairy circle patterns result from the scale-dependent feedback model III (also called Turing-patterns) with interactions between sulfide and plants.

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50°S 0°

50°N

120°W 60°W 0° 60°E 120°E

Longitude

Latitude

Labels

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● A B C D E F

G H I J K

Shanghai ●●

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29°N 29.5°N 30°N 30.5°N 31°N 31.5°N 32°N 32.5°N 33°N

119°E 119.5°E 120°E 120.5°E 121°E 121.5°E 122°E 122.5°E 123°E

A B C

D E F

G H J

Figure S1. (Continued on the following page.)

I

Figure S1. (Previous page.) Overview of FC patterns in tidal salt marshes all over the world.

(A) Nanhui shoal, Shanghai (30.996886ºN, 121.942381ºE), species Spartina alterniflora. (B) Nanhui shoal, Shanghai (31.000412ºN, 121.944415ºE), species Scirpus mariqueter. (C) FC patterns in tidal saltmarshes in China, Jiangyanansha Island (31.238313ºN, 121.816176ºE; source: Google Earth), species Zizania latifolia. (D) Chongming Island, Shanghai (31.625022ºN, 121.777107ºE), species Spartina alterniflora. (E) South of Hangzhou bay, Zhejiang (30.380342ºN, 121.195967ºE), species Phragmites australis. (F) South of Hangzhou bay, Zhejiang (121.101975ºE, 30.344667 ºN), species Scirpus triqueter and Spartina alterniflora. (G) Ring-type and spotted patterns in Spartina on Polvora Island (Rio Grande, Brazil; -32.017535ºS, -52.107005ºW; source: Google Earth). (H) Ring-type patterns in salt marshes mostly dominated by species of Spartina densiflora in the Bah´ıa Blanca Estuary, Argentina (-38.759201ºS, -62.319251ºW; source: Google Earth). (I) Ring-type patterns in Spartina in Ellewoutsdijk, The Netherlands (51.386283ºN, 3.820968ºE). Photos, left panel, courtesy of Johan v.d. Koppel; right panel, source: Google Earth. (J) Ring-type patterns in northeastern Rockdedundy Island, USA (31.367161ºN, -81.343698ºW; source: Google Earth). (K) FC patterns of seagrass meadows in Pollenca bay, Mallorca Island, Western Mediterranean, Spain (39.8675ºN, 3.1095ºE), species Posidonia oceanica and Cymodocea nodosa. Photo Credit: (A) by Quan-Xing Liu and (I) by Johan van de Koppel.

A Spots B Ring-type patterns C Concentric rings

D Spatiotemporal evolution of spotted and fairy-circle patterns

E

Figure S2. Three typically examples of spatial self-organized patterns observed in salt- marshes ecosystems at our studies areas. The FC patterns exist in both Scirpus mariqueter and Spartina alterniflora: (A) spots, (B) fairy circles (FCs, also named ring-type patterns), and (C) concentric rings. (D) Spatiotemporal evolution of spot and fairy circle patterns with one growth sea- son in 2019. (E) Large-scale fairy circle patterns established in 2018 at Nanhui shoal, Shanghai (source: Google Earth). Photo Credit: (A-D) by Li-Xia Zhao.

Sulfide

DOC

SRB SOB

Salinity Wet NH4

Clay

Elevation

Center

On ring Outer

−0.5 0.0 0.5

−0.8 −0.4 0.4

CA 1 (79.4% variation)

CA 2 (20.6% variation)

0.0

●

●

●

●

●

●

0.097

0.00058 0.16

30 35 40 45

Center Inter On ring

In−situ location of patches

Total carbon of plant (%)

●

●

ring spots Pattern: p=0.54 Location: p<0.01 P × L: p=0.65 NS

NS NS

A B

Figure S3. Correspondence analysis of several potential physical, chemical, and sediment factors on FCs formation. (A) Correspondence analysis (CA) of the effect of environmental vari- ables on the plant biomass of the center, on ring, and outer areas, respectively. SRB, sulfate-reducing bacteria; SOB, sulfur-oxidizing bacteria. (B) In-situ measurement of total carbon percentage of the plant of S. mariqueter leaves. Both rings and spots show the same increasing trend of the carbon from the center of patches to the outer edge of patches with a significant difference at p < 0.001, whereas total nitrogen percentage, was shown in Fig. 2E, displays a different trend between the ring-type and spot patterns (n=6). Error bars indicate standard deviation.

Transition from spot to ring type

Transition from ring to concentric circle 6 m

20190511 20190913

20190511 20190913

A

B

C

D

Figure S4. Spatiotemporal dynamics and FC development in field observations and two hy- potheses models. (A) Pattern transition from spots to ring-type patterns. (B) Pattern transition from ring-type to concentric circles. (C) Sulfide feedback model and (D) nutrient limitation model. Both simulations were started from an isolated point. The sulfide mechanism shows only FC formation re- sulting from spotted patterns, whereas the nutrient limitation mechanism reproduces spot, ring-type or FC, and concentric ring patterns. The color bars depict the plant biomass per square meter (g/m²).

Photo Credit: (A-B) by Quan-Xing Liu.

Figure S5.Exotic Spartina alterniflora (green color) invades native Scirpus mariqueter (brown color) on intertidal mudflats at Nanhui shoal, Shanghai. FC patterns can prevent from invasion by exotic Spartina, reflecting a feature of Allee effects. Photo Credit: Xiu-Zhen Li.

plant biomass [g/m²] plant biomass [g/m²]

A B

Figure S6. Comparing the transient patterns of model I with self-organized patterns of scale- dependent feedback model III. (A) Transient spatial pattern from plant-sulfide model I shown in Fig. 1G. (B) Constancy Turing-like FCs from the scale-dependent feedback model III with parameters D_p= 0.05, Ds= 10.0, kp = 2.0, c = 10.5, Iin= 0.05, and " = 0.01. Both numerical simulations started from random initial conditions. The color bars depict the plant biomass per square meter.

a a

0 10 20 30

center on_ring outer

Clay (%)

a

a a

0 250 500 750 1000

center on_ring outer

DOC (μM)

a

a a

0 1000 2000 3000

center on_ring outer

Ecp (mS/m)

Kruskal−Wallis,p = 0.86

a a

a

−4

−2 0 2

center on_ring outer

Elevation change (cm)

a

b

0 50 100 150 200

center on_ring outer

Ammonium (μM) ^{a a}

a

0 1 2 3

center on_ring outer SOB (104 Copies/μL)

a

a Kruskal−Wallis, p = 0.3

a

0.0 2.5 5.0 7.5 10.0

center on_ring outer SRB (106 Copies/μL)

a

b Kruskal−Wallis,p = 0.014

b

0 5 10 15

center on_ring outer

Sulfide (μM)

a

0 20 40 60 80

center on_ring outer

Wet (%Vol.)

Kruskal−Wallis,p = 0.87 Kruskal−Wallis,p = 0.89

Kruskal−Wallis,p = 0.34 Kruskal−Wallis,p = 0.044

c

Kruskal−Wallis,p = 0.68 Kruskal−Wallis,p = 0.79

a

a a

A B C

D

G H

E F

I

Figure S7. The changes of physical, chemical, and sediment properties were sampled along a spatial gradient in the radial direction from the center of patches to outer bare areas, namely in the order center, inter, on-ring, and outer with respect to FC patterns. All data were in-situ field measurements. Different letters in each panel indicate significant differences (p < 0.05) among locations. Error bars indicate standard error.

Table S1. Definitions and interpretation of variables and parameters in sulfide and nutrient models respectively.

Symbol Value^⇤ Unit Definition and interpretation Sulfide model

Variables

S µM Concentration of sulfide within the mud sediment

P g/m² Biomass (dry weight) of plants

t day Time

[—] = @_xx+ @_yy, spatial diffusion terms following Fick’s laws Parameters

r 0.80 g/day Growth rate of plants

K 10.0 g Carrying capacity of plants in the studies area c 10.0 g/day/g Maximum death rate caused by sulfide per day

" 0.004 µM/g The effective generation rate of hydrogen sulfide production arising from plant biomass

d 0.50 1/day Detoxification rate of sulfide

k_s 0.20 µM Half-saturation constant of the plants biomass to sulfide detoxification

⇠ 0.10 [—] Non-negative dimensionless parameter controls the orders of magnitude between sulfide and plants

D_p 0.002 m²/day Diffusion constant of plant seeds or roots

Ds 0.1Dp m²/day Diffusion constant of sulfide within mud sediment Nutrient model

N µM Nutrient concentration

k₁ 5.0 µM Half-saturation constant of nutrient concentration to plants growth rate

d 0.25 1/day Mortality of plants included the losses by tidal waves I_in 0.15 µM/day Nutrient input by tides

s 0.02 [—] The conversion factor from dead plants biomass to nutrient c 1.0 1/day Maximum uptake rate of nutrient by plants

D_n 0.02 m²/day Diffusion coefficient of nutrient

Table S2. Significance of pattern and location effects on stoichiometric changes of total nitrogen (TN) and C:N ratio of Scirpus mariqueter.

Variable Factors Parameters

F P

TC Pattern 0.39 0.54

Location 6.11 0.01**

Pattern × Location 0.45 0.65

TN Pattern 33.25 3.73e-05***

Location 2.04 0.17

Pattern × Location 4.27 0.03*

C:N ratio Pattern 21.57 0.000318***

Location 0.29 0.75

Pattern × Location 1.12 0.35

Table S3. Significance of spatial location effects on stoichiometric changes of total nitrogen (TN) and C:N ratio of Scirpus mariqueter at varied location respectively.

Variable Location Parameters

F P

TN Center 8.35 0.03*

Inter 284.2 1.34e-05***

On ring 0.73 0.43

C:N ratio Center 7.84 0.04*

Inter 45.38 0.00109**

On ring 1.02 0.36

Table S4. The Fisher’s Least Significant Difference for multiple comparisons of total nitrogen (TN) and C:N ratio of Scirpus mariqueter ring-type patterns from patch center to outer.

Variable Contrast Estimate SE df L.CL U.CL t.ratio P

TN Center-Inter 0.0029 0.32 13 -0.69 0.70 0.009 0.99

Center-On ring -0.73 0.31 13 -1.39 -0.06 -2.37 0.03 Inter-On ring -0.73 0.31 13 -1.39 0.07 -2.38 0.03 C:N ratio Center-Inter -2.36 3.4 13 -9.7 4.98 -0.70 0.50 Center-On ring 3.39 3.26 13 -3.64 10.43 1.04 0.32 Inter-On ring 5.76 3.26 13 -1.28 12.79 1.77 0.10

Table S5. The Fisher Least Significant Difference for comparisons of sulfide and ammonium concentration distribution from patch center to outer.

Variable Patterns Location Mean Std Min Max Groups

Sulfide Ring-type Center 7.76 1.19 6.07 9.37 a

Inter 10.71 1.31 9.24 12.54 b

On ring 11.66 2.27 8.32 14.52 b

Outer 9.97 0.88 8.71 10.69 ab

Concentric ring Center 7.13 1.49 5.02 8.45 a

Inter 7.59 0.77 7.0 8.71 a

On ring 6.90 0.50 6.2 7.39 a

Outer 7.52 1.23 6.86 9.37 a

Ammonium (NH

)

Ring-type Center 183.52 44.23 153.67 234.33 a

Inter 290.02 179.06 155.83 493.33 ab

On ring 47.5 6.84 42.67 52.33 ab

Outer 108.2 11.78 96.22 119.78 b

Concentric ring Center 4.59 0.50 3.96 5.04 a

Inter 5.05 0.56 4.21 5.44 ab

On ring 5.56 1.16 4.48 7.16 ab

Outer 5.98 0.12 5.84 6.06 b

Table S6. The Fisher Least Significant Difference for multiple comparisons of sulfide and ammonium concentration distribution from patch center to outer.

Variable Patterns Contrast Estimate SE df L.CL U.CL

Sulfide Ring-type Center-Inter -2.95 0.96 17 -4.97 -0.94 -3.09 0.007

Center-On ring -3.9 0.96 17 -5.92 -1.88 -4.08 0.008

Center-Outer -2.2 1.06 17 -4.44 0.03 -2.08 0.05

Inter-On ring -0.95 0.91 17 -2.87 0.98 -1.04 0.31

Inter-Outer 0.75 1.02 17 -1.4 2.9 0.73 0.47

On ring-Outer 1.69 1.02 17 -0.46 3.85 1.66 0.11

Concentric ring Center-Inter -0.46 0.76 11 -1.58 0.67 -0.9 0.39

Center-On ring 0.23 0.76 12 -2.11 1.19 -0.61 0.55

Center-Outer -0.4 0.76 12 -1.42 1.88 0.31 0.77

Inter-On ring 0.69 0.76 12 -2.04 1.25 -0.52 0.61

Inter-Outer 0.07 0.76 12 -0.96 2.34 0.92 0.38

On ring-Outer -0.63 0.76 12 -1.58 1.71 0.09 0.93

Ammonium (NH

)

Ring-type Center-Inter -106.5 80.7 7 -297.29 84.3 -1.32 0.23

Center-On ring 136 90.2 7 -77.3 349.3 1.51 0.18

Center-Outer 75.3 80.7 7 -115.48 266.1 0.93 0.38

Inter-On ring 242.5 90.2 7 29.2 455.8 2.69 0.03

Inter-Outer 181.8 80.7 7 -8.98 372.6 2.25 0.06

On ring-Outer -60.7 90.2 7 -274.02 152.6 -0.67 0.52

Concentric ring Center-Inter -0.46 0.51 11 -1.58 0.67 -0.9 0.39

Center-On ring -0.97 0.51 11 -2.09 0.16 -1.89 0.09

Center-Outer -1.39 0.55 11 -2.6 -0.17 -2.5 0.03

Inter-On ring -0.51 0.51 11 -1.63 0.62 -1.0 0.34

Inter-Outer -0.93 0.55 11 -2.14 0.29 -1.68 0.12

On ring-Outer -0.42 0.55 11 -1.63 0.8 -0.76 0.46

Table S7. AIC weights, BIC, adjusted R², and significant level of the fitted functions for relations between the growth rate and patch-size radii.

Model AIC BIC Adjusted R² P values

Linear 1571.7 1586.9 0.47 <0.001 (t = 14.94)

Exponential model 1361.8 1376.9 0.43 <0.001 (t = 23.27) Allee-effect model^⇤ 1631.9 1652.1 0.49 <0.001 (t = 8.70)

⇤The observed data are preferred fitted by an Allee-like effect model because there exists an obvious critical path size to survival at early colonization in the intertidal salt-marsh ecosystems.

AIC, Akaike information criterion; BIC, Bayesian Information Criterion.