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The relative contribution of long-distance dispersal and local diffusion in the spread of invasive species has been a subject of much debate. Invasion of the intertidal mudflats by Spartina alterniflora is an ideal example of stratified diffusion, involving both long-distance dispersal of seeds and local diffusion due to clonal growth. In conjunction with experimental data on range radius-versus-time curve, a traveling wave equation-based model is used to investigate the sensitivity of the spread rate of exotic S. alterniflora to parameters of long distance dispersal (c, maximum colonial establishment rate) and local colony diffusion (r, intrinsic growth rate) at two tidal marshes, the Eastern Chongming and the Jiuduansha Islands, at the Yangtze River estuary. Both Eastern Chong ming and Jiuduansha Islands are now national natural reserves in China, which were established in 2005. However, the mudflats and salt marshes in the two reserves are now heavily infested with introduced S. alterniflora, which may threaten the estuarine ecosystems and their biodiversity. S. alterniflora was first found in 1995 on Chongming. For rapid sediment accretion in mudflats in the estuary, S. alterniflora was also intentionally introduced to Jiuduansha in 1997 and Chongming in 2001, which has led to a rapid range expansion in the estuary. Our results show that range expansion of species with stratified diffusion is affected by both long-distance dispersal and local colony diffusion, and that there is a critical c*, below which the spread rate is more influenced by long-distance dispersal than by local diffusion. After applying this model to the invasion of S. alterniflora in the Yangtze River estuary, we derive that c = 1.7 × 10^{-3}, c* = 0.126 and c = 4.8 × 10^{-3} km^{-2}·yr^{-1}, c* = 0.140 km^{-2}·yr^{-1} at Chongming and Jiuduansha (Shanghai), respectively. Our results suggest that the range spread of S. alterniflora in the Yangtze River estuary is more influenced by long-distance dispersal than local colony diffusion, and that S. alterniflora generates about 1.7 × 10^{-3} to 4.8 × 10^{-3} colonies per square kilometers per year. This study provides important information about dispersal dynamics of S. alterniflora that may be useful for finding optimal control strategies.
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Like other complex biological processes, the expanding process of invasive species is often numerically intractable [

Early models of the spread of animal and plant populations were based on the process of diffusion and predicted a simple linear rate of spread [

For many invading species, the key to understanding the expanding process is ascertaining rare long-distance dispersal events, such as the movement of Argentine ants and Phytophthora lateralis disease spores by cars and trucks, or of zebra mussels by boats [

Spread models can be used to explore these two issues of ecology [

Invasion of the intertidal mudflats of Pacific coast estuaries by Spartina alterniflora Loisel. (smooth cordgrass) is an ideal example of stratified diffusion, involving both a long-distance dispersal of seeds that leads to establishment of new colonies and a local reproduction due to clonal growth through rhizome [

In this study, we apply the concept of stratified diffusion to model the spread of exotic S. alterniflora and then fit the dispersal coefficient of S. alterniflora using aerial photographs data on range radius-versus-time curve of S. alterniflora at two tidal marshes on Chongming Island and Jiuduansha Islands respectively at the Yangtze River estuary for a period of nine to ten years after introduction in this area.

Like other estuaries, the Yangtze River estuary is an important ecoregion, because it provides great ecosystem services for humans and wildlife in the region. Maintaining the integrity of the native ecosystems and conserving biodiversity in this estuary is of national importance. Therefore, two national reserves were established in 2005 in the estuary, one on Chongming Island and the other on Jiuduansha Islands (see

The introduction of S. alterniflora to the tidal marshes on Chongming and Jiuduansha at the Yangtze River estuary is a prime example of a spatially structured invasion, for which model performance can be assessed against the actual survey of population spread. In particular, the stratified diffusion model is constructed so that the relative effects of dispersal and local growth on the range expansion can be explicitly modeled.

Spartina alterniflora was deliberately introduced from the Atlantic coast of North America into China in 1979 to check erosion and promote sediment accretion. Although S. alterniflora still exhibits some anticipated ecological benefits [

Invasion of the intertidal mudflats of estuaries by S. alterniflora is a primary example of a stratified diffusion in a relatively simple habitat [

The model used here is almost identical to the one used by [

Our objective is to calculate the rate of population spread from the following two functions: 1) patch establishment rate (per unit area per year), b(x), which continuously decreases with increasing distance from the wave front, x, a pattern consistent with aerial photo analysis by previous studies [

It is worth noting for these model assumptions: First, the model describes the average density (ramets per unit area) of S. alterniflora, m (a, z, t), of age a, at spatial location z, and at time t. Although natural extinction of colonies did occur, we ignored it in the model by considering only those patches that successfully established. This may make the patch establishment rate, b(x), smaller. Second, by definition, the wave front is located at the point where the average ramet density is approximate to carrying capacity, K. We do not use parameter K to limit the growth of individual patches because we intend to simulate population dynamics only in the transition zone where the average ramet density is below the carrying capacity. Third, the simplified assumption that the patch establishment rate depends only on the distance from the population front does not imply that all long-distance dispersal originates only from the wave front. Usually, isolated patches in the transition zone have low densities. Thus, their contribution to long-distance dispersal should be much smaller than the contribution of high-density populations that have reached the carrying capacity (e.g. plants in the meadow edge zone,

Thus, the following traveling wave equation can be used for estimating the spread rate of the front, v, if functions b(x) and n(a) are specified explicitly [

We assume that the ramets density of a patch, n(a), increases exponentially with colony age a,

where n_{0} is the number of initial ramets of S. alterniflora in a patch, and r is the intrinsic growth rate.

Dispersal data are frequently fit with a negative exponential curve or a negative power function [

1) Constant decreasing function

We suppose that b(x) decrease linearly with the distance x, namely,

where x_{max} is the maximum distance at which the patch of S. alterniflora could establish (namely, the width of the transition zone) and c = b(0) > 0, is the dispersal coefficient, the maximum patch establishment rate in the transition zone. Equation (1) yields the following equation,

namely,

where V = v/x_{max}, the relative rate of population spread (proportional to the width of the transition zone, x_{max}). Thus we can use numerical method to estimate V.

2) Decelerated decreasing function

We suppose that the establishment rate of new colony decelerates decreases with the distance away from the front, namely,

And we obtain the traveling wave equation as follows,

3) Accelerated decreasing function

We suppose that the establishment rate of new colony accelerates decreases with the distance away from the front, namely,

Similarly, we obtain the traveling wave equation as following,

The model did not differ qualitatively in behavior when each of the three functions of b(x) was chosen (

The spread rate of the front depends mainly on parameter c and r (Equation (5)). We cannot solve the Equation (5) analytically, and thus use numerical methods to analyze. The values of n_{0}/K quantify the initial density of individuals in the transition zone. We expect this value to be very small (<0.01) in most local populations, because establishment of new colonies is a rare event on tidal mudflats and the initial number of individuals is usually small [

We used available data on the range spread of S. alterniflora from two sites in the Yangtze River estuary: Jiuduansha Shoals Wetlands (30˚10'N, 122˚01'E) and Chongming Dongtan Wetlands (31˚38'N, 121˚58'E) [

Parameter x_{max}: the mean maximum distance from the wave front where isolated colonies become established.

Notation | Definition | Value |
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x | Distance away from the wave front | |

x_{max} | Width of the transition zone | 2000 (m) |

b(x) | Colonial establishment rate | |

n(a) | Number of individuals in a colony of age a | |

m(a, t, z) | Average density per unit area of colonies, of age a, at spatial location z, at time t | |

Z(t) | Location of the wave front at time t | |

N(z, t) | Average density of individuals at spatial location z, at time t | |

K | Environmental capacity | 262 (ind. m^{−2}) |

r | Intrinsic growth rate | 1.5 |

c | Maximum colonial establishment rate | |

c^{*} | Critical value of c | |

n_{0} | Number of initial individuals in a colony | 3 (ind. colony^{−1}) |

v | Speed of population spread | |

V | Relative rate of population spread, V = v/x_{max} |

This distance has been reported to be a maximum about 2 km based on aerial photographs take in the Yangtze River estuary [_{max} on the rate of spread, we assumed that x_{max} = 2000 m.

Parameter n_{0}: the initial number of established seedlings in isolated populations. We assumed that a colony may start from a small number of seedlings (<10) based on the observation of widely scattered distribution of seedlings (not tightly clumped). Three levels of n_{0 }were initially tested within a reasonable range (1, 3and 10). Due to the similarity in the outcome, only results for n_{0} = 3 are presented.

Parameter K: carrying capacity. In this paper, we use mean ramet density in meadow as an estimation of K for S. alterniflora in the Yangtze River estuary. The value has been reported to be 262 ramet m^{−2} [^{−2}.

Parameter r: the rate of ramet number increasing with colonies. Some studies estimated the average intrinsic rate of increase in S. alterniflora colonies from samples collected in several sites in the Yangtze River estuary [

In this study the velocity of spread has been estimated from time series of aerial photos [

The relative spread rate V almost increased linearly with parameter r and c respectively (^{*} in the sensitivity analysis (^{*}, the parameter c played a more important role in the relative spread rate than r (dV/dc > dV/dr); when c > c^{*}, r weighted more than the parameter c on spread rate (dV/dc < dV/dr). In this study, the fitted values of c = 1.7 × 10^{−3} and 4.8 × 10^{−3} km^{−2}∙yr^{−1}, and c^{*} = 0.126 and 0.140 km^{−2}∙yr^{−1} at Chongming and Jiuduansha, respectively. More generally, within a biologically reasonable scope of parameter r (0.5 - 1.5) for S. alterniflora, the values of the critical dispersal coefficient (c^{*}) were far below 1 (^{*} by approximately three orders of magnitude. Thus, we suggest that the range spread of S. alterniflora in the Yangtze River estuary

is more influenced by long-distance dispersal than local growth. Our results support the proposal that, for species that spread through stratified diffusion, the distance and rate at which new foci are created may be more important than the rate of spread through local diffusion from established foci [

For both Chongming and Jiuduansha, range expansion of S. alterniflora was linear (^{−3} and 6.3 × 10^{−3} km^{−2}∙yr^{−1} at Chongming and Jiuduansha, respectively. The accuracy was 28% and 76% respectively. The range of predictions made by our model and coalescing colony model of stratified diffusion has the same order of magnitude [

The extent to which species spread by stratified diffusion may influence the implementation of control strategies. For example, the effectiveness of control measures can be greatly increased by preventing the establishment of new foci or by eliminating new foci rather than focusing efforts on established invasion fronts [

We are grateful to Prof. Cangming Fang, Hanji Shang, and Bing Zhao for constructive comments on the earlier versions of this manuscript. This study was financially supported by the National Natural Science Foundation of China (31370433) and the National Basic Research Program (Grant No. 2006CB403305).

Based on the traveling wave Equation (5), the derivatives of V with respect to r and c are given as following.