Target species and basic assumptions
We developed and analyzed spatially explicit models that describe range expansion of an invader species’ population that consists of many sub-populations. Invasive species often expand their range by stratified dispersal using human-mediated long-distance dispersal in addition to local expansion of sub-populations20. Extending a model rigorously analyzed by Takahashi et al.31, which explicitly involves (1) short-distance dispersal that expands the area of current sub-populations, and (2) long-distance dispersal that establishes new sub-populations beyond existing sub-populations (Fig. 4), we consider spatially inhomogeneous factors that influence on the long-distance dispersal (e.g., vectors’ distribution). The parameters and functions used in these models are listed in Table 1.
A schematic of short- and long-distance dispersal modes. Human activities may influence the long-distance dispersal by: (1) changing the number of propagules starting the long-distance dispersal ((R cdot varphi (x,y))), and (2) introducing biases in their spatial locations ((psi (x,y))). By compositing these short- and long-distance dispersals, we predict population establishment at the next time step.
We estimated the population size of the invader species as the area covered by any of these sub-populations in the study area. This simple measure of the population size was based on our assumption that the inside of a sub-population is homogeneous and these sub-populations vary only in their positions, sizes, and shapes. This assumption may oversimplify spatial architectures of the sub-populations, because even clonal colonies of perennial plants often have concentric structures that can influence reproductive output40; but this simplification is applicable when the reproductive rate of the species is high enough to reach a constant carrying capacity quickly.
Short- and long-distance dispersals
We considered stratified dispersals of the invader species. Note that we explicitly considered invader species’ dispersal but their vectors’ movement (e.g., human traffics) was included only implicitly. The short-distance dispersal expands the species’ range only by a constant velocity7. Therefore, we modeled the short-distance dispersal as radial expansion of these sub-populations with a constant speed g. Meanwhile, long-distance dispersal introduces new sub-populations into the population out of its parent sub-population. Empirical observations showed that long-distance dispersal mediated by human activities introduces a new sub-population to an area at a long distance from its source population, e.g., vehicles can move plant seeds for more than hundreds of kilometers41. Long-distance dispersal diminishes the influence of the source location on the destination of a dispersal event, so as a simplifying approximation we assumed that the destination of the long-distance dispersal is independent of the source location of a sub-population.
The assumption of source location independence of the dispersal destination allows us to describe a process of long-distance dispersal by two functions on (S): (1) a function (varphi (x,y)) describing spatial variation in disperser production rates, and (2) a function (psi (x,y)) describing the probability that a coordinate ((x,y)) is selected as a destination of the disperser. Following the notation of Jongejans et al.18, we call (varphi (x,y)) and (psi (x,y)) the source and destination functions, respectively. Note that we define the spatial average of the source function to be one (i.e., ({{int_{S} {varphi (x,y),dxdy} } mathord{left/ {vphantom {{int_{S} {varphi (x,y),dxdy} } {|S|}}} right. kern-nulldelimiterspace} {|S|}} = 1)). We define R as the regional average of the disperser production rate per unit area (a spatially homogeneous component of the disperser production rate). Using this formulation, we calculate an expected disperser production rate of a given area by integrating (Rvarphi (x,y)) over the area. The spatial integration of the destination function over the study area is one because we assume the population of the invader species will reach full carrying capacity.
We have defined a population of the invader species as a spatial union of all sub-populations in the study area because we assume sub-populations are homogeneous. Within the study area, (rho_{t} (x,y)) is 1 if the coordinates ((x,y)) are within at least one of the sub-populations of the invader species at time (t) (and otherwise 0). The integration of (Rvarphi (x,y)) is the expected disperser production rate for a given (rho_{t} (x,y)), and the integration (Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy}) is the expected total production of dispersers from a population in the t-th time step. Thus, assuming that the total number of dispersers that start long-distance dispersal at time (t) (denoted by (n_{{{text{d,}}t}})) follow a Poisson distribution, we can write a probability that the population produces k dispersers in the t-th time, Eq. (1).
$$ Pr [n_{{{text{d}},t}} = k] = frac{{lambda^{k} e^{ – lambda } }}{k!}{, }lambda = Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy} . $$
(1)
The destination of the disperser is determined by the destination function (psi (x,y)), which determines probabilities of ending a dispersal event at ((x,y)), which results in a new sub-population being established at that location. In total, the spatial distribution of new sub-populations introduced by long-distance dispersal follows a Poisson point process of which intensities are given as (psi (x,y)Rint_{S} {rho_{t} (x,y)varphi (x,y),dxdy}).
Three model types
We compared three different types of models by varying interactions between the species’ long-distance dispersal and human activities. These included: (1) a source-mediated-dispersal model assuming that the source function (varphi (x,y)) varies spatially by human activity while the destination function (psi (x,y)) is uniform, (2) a destination-mediated-dispersal model assuming that the destination function varies spatially while the source function is uniform, and (3) a full model assuming that both source and destination functions vary spatially.
Let (h(x,y)) be a function representing intensity of human activities at coordinates ((x,y)). Without loss of generality, we can assume that the function (h(x,y)) satisfies (int_{S} {h(x,y),dxdy} = 1), the total intensity over the study area is scaled to one. In the source-mediated-dispersal model, the source function is proportional to the human-activity intensity, i.e., (varphi (x,y) = |S| cdot h(x,y)), and the destination function (psi (x,y)) is uniformly equal to ({1 mathord{left/ {vphantom {1 {|S|}}} right. kern-nulldelimiterspace} {|S|}}). On the contrary, the source function of the destination-mediated-dispersal model is uniform and the destination function is (psi (x,y) = h(x,y)). The full model is a combination of the source- and destination-mediated-dispersal models in which both source and destination functions vary with area. In this study, we assume that a single factor determines both the source and destination functions, i.e., (varphi (x,y) = left| S right| cdot h(x,y)) and (psi (x,y) = h(x,y)) (Table 2).
Asymptotic growth rate of a population
Spatial dimension introduces complexity, though rigorous mathematical analysis is still viable for a small population with few small sub-populations. This situation may arise with an accidentally transferred population. Here, we consider infinitesimally small populations to be rare for invasive species and derive an asymptotic value of the growth rate to the size of the area inhabited by the population.
Each sub-population includes age, so we incorporated age-structured population dynamics29,31 into the model, described by the differential equation,
$$ frac{partial n(a,t)}{{partial t}} + frac{partial n(a,t)}{{partial a}} = 0, $$
(2)
where (n(a,t)) represents the frequency of sub-populations with age a at time t. Note that Eq. (2) assumes no extinction of sub-populations. The equation has two boundary conditions: (1) (n(a,0)) represents an age distribution of the initial population, and (2) (n(0,t)) represents the number of new sub-populations (i.e., age 0 sub-populations) introduced by the long-distance dispersal at time t.
To determine the number of new sub-populations, we need to determine how many long-distance dispersers will emerge from a given population by including spatial heterogeneities. Recall that a sub-population expands outward by a constant speed g. Therefore, if we ignore overlaps among sub-populations each sub-population keeps a circular shape of radius proportional to age. In addition, if a sub-population is young, i.e., its size is small, we can regard the value of the source function (varphi (x,y)) inside the sub-population as uniform. Let ((x_{i} ,y_{i} )) and (a_{i}) be the position of the center and age of the (i)-th sub-population, respectively. With the above approximations, we can simply derive the expected number of long-distance dispersers that start dispersal from the (i)-th sub-population as (pi R cdot (ga_{i} )^{2} varphi (x_{i} ,y_{i} )).
On the other hand, existing sub-populations also originate from long-distance dispersal. Therefore, a position of the sub-population also follows the destination function (psi (x,y)). Building on the expected number of new sub-populations we described at the last paragraph, we calculate an average over the study area to calculate a mean-field approximation of the number of long-distance dispersers from an age a sub-population as (pi Rint_{S} {psi (x,y)(ga)^{2} varphi (x,y),dxdy}).
We assume that a population consists of a few small sub-populations in this formulation and a disperser will always establish outside existing sub-populations. Therefore, the total number of a new sub-population (i.e., (n(0,t))) is a summation of new sub-populations produced by each of the existing sub-populations,
$$ begin{gathered} n(0,t) = pi Rint_{0}^{t} {n(a,t)left[ {int_{S} {psi (x,y)(ga)^{2} varphi (x,y),dxdy} } right],da} = pi Rleft( {int_{S} {psi (x,y)varphi (x,y),dxdy} } right)int_{0}^{t} {(ga)^{2} n(a,t),da} . end{gathered} $$
(3)
With Eq. (3), the asymptotic growth rate of the Eq. (2) can be calculated as follows29,31,
$$ left( {2pi Rg^{2} int_{S} {psi (x,y)varphi (x,y),dxdy} } right)^{1/3} . $$
(4)
The asymptotic growth rate indicates that the integration (int_{S} {psi (x,y)varphi (x,y),dxdy}) describes the influence of the source and the destination functions in the early phases of population growth. Therefore, hereafter we call (int_{S} {psi (x,y)varphi (x,y),dxdy}) a spatial factor (F_{{text{h}}}) of the long-distance dispersal. Note that the spatial factor reduces to 1 for both source- and destination-mediated dispersal models. For the full models of which source and destination functions are (varphi (x,y) = left| S right|h(x,y)) and (psi (x,y) = h(x,y)), respectively, we can reduce the spatial factor (F_{{text{h}}}) to (int_{S} {left( {sqrt {left| S right|} h(x,y)} right)^{2} dxdy}), equivalent to Simpsons’ diversity index.
Numerical analysis
We evaluated the effects of the dispersal vector on distribution with an individual-based approach that describes colonies in a population as groups of individuals within a circular shape of various sizes (Fig. 1a,c,e for typical model outputs, see supplemental information SI 1 for detailed settings). To evaluate the effect of spatial heterogeneity on population dynamics, for each model type we generated 100 of (h(x,y)) randomly (see shaded area of Fig. 1a,c,e, and SI 2 for the algorithm used) and ran 100 independent realizations for each (h(x,y)). For each realization, we split the time course into three phases: establishment, expansion, and naturalization. The phases were based on the proportion of area inhabited by the population (Fig. 2a; less than 5%, 5% to 95%, and 95% to complete occupation of the total area, respectively), and measured the length of each phase. We linearly interpolated the time course based on area covered for each phase.
Using the same set of realized time courses, we estimated the asymptotic growth rate as the peak of a distribution of the logarithmic value of the instantaneous growth rate, defined as a difference of logarithmic values of covered area that are adjacent in a time course. We excluded periods that a population covers less than 1% or more than 50% of the total area to avoid strong demographic stochasticity of initial dynamics and a deceleration phase of S-shaped growth, respectively. We gathered these logarithmic values of instantaneous growth rates from 100 realizations with the same (h(x,y)) and dispersal type, then estimated the density distribution using Gaussian kernel estimation. Finally, we determined the maximum point of the estimated density distribution as the estimated asymptotic growth rate.
Source: Ecology - nature.com
