PRUNUS: a spatially explicit demographic model to study plant invasions in stochastic, heterogeneous environments

Abstract

To model the invasion of Prunus serotina invasion within a real forest landscape we built a spatially explicit, non-linear Markov chain which incorporated a stage-structured population matrix and dispersal functions. Sensitivity analyses were subsequently conducted to identify key processes controlling the spatial spread of the invader, testing the hypothesis that the landscape invasion patterns are driven in the most part by disturbance patterns, local demographical processes controlling propagule pressure, habitat suitability, and long-distance dispersal. When offspring emigration was considered as a density-dependent phenomenon, local demographic factors generated invasion patterns at larger spatial scales through three factors: adult longevity; adult fecundity; and the intensity of self-thinning during stand development. Three other factors acted at the landscape scale: habitat quality, which determined the proportion of the landscape mosaic which was potentially invasible; disturbances, which determined when suitable habitats became temporarily invasible; and the existence of long distance dispersal events, which determined how far from the existing source populations new founder populations could be created. As a flexible “all-in-one” model, PRUNUS offers perspectives for generalization to other plant invasions, and the study of interactions between key processes at multiple spatial scales.

Appendix 1: further information about the PRUNUS model

Life cycle graph and transition matrices

After Sebert-Cuvillier et al. (2007), we retained the following 11 stages to account for the life cycle of Prunus serotina (Fig. 6):

Fig. 6

Life cycle graph of Prunus serotina as considered in the model (adapted from Sebert-Cuvillier et al. 2007)

• seed stages: to incorporate seed dormancy, which can attain 2 years (Marquis 1975), we distributed the annual seed production per mature tree among three life stages: seed-2, seed-1 and seedling, corresponding to seeds germinating on the 3rd, 2nd and 1st spring following release, respectively. The proportion of seeds that never germinate to reach the seedling stage, which includes seeds without embryo and seeds that die at seed-1 or seed-2 stage, was removed from the total number of produced seeds.

• sapling stages: sterile plants without cotyledons and taller than 10 cm were considered as saplings. Under dense shade (e.g., in the understorey of a closed-canopy forest), saplings become rapidly suppressed and form a long-living sapling bank (Starfinger 1997; Closset-Kopp et al. 2007). Such suppressed saplings (i.e., saplings remaining at sapling1 stage) are called ‘Oskars’ since they develop a “sit and wait” strategy (Starfinger 1997). Conversely, in full light conditions (e.g., in a gap or a clearcut), sapling1 can grow to reach superior sapling stages and ultimately the canopy. As time to reach the canopy averages 7 years, saplings were distributed among 7 stages: sapling1, sapling2, sapling3, sapling4, sapling5, sapling6, and sapling7, corresponding to saplings high of <25, 25–50, 50–150, 150–250, 250–350, 350–450, and 450–550 cm, respectively. Individuals can neither stay more than 1 year in a given sapling stage nor skip any intermediate stage, except for ‘Oskars’ which are able to remain at the sapling1 stage for several decades.

• adultstage: tall shrubs and trees that are sexually mature, survive several decades during which they produce seeds annually, were grouped into a single adult stage.

To account for the resprouting capacity of Prunus serotina individuals through stump and root suckers, we included further transition probabilities for stages sapling2, sapling3, sapling4, sapling5, sapling6, sapling7 and adult to go back to the sapling1 stage after their aerial parts have suffered dieback, especially under closed canopy conditions.

This 11-stage life cycle has been mathematically translated into four 11 × 11 transition matrices (Sebert-Cuvillier et al. 2007):

• Matrix A (Table 2): shaded environmental conditions—This matrix includes two positive elements on the diagonal, corresponding to individuals remaining in the same stage (i.e., sapling1-stage and adult-stage), and six null rows, since saplings1 cannot reach the next stage under shade conditions. The total number of seeds that are yearly produced by an adult tree averages 6,011, but only 42% are viable and thus able to become seedlings (Closset-Kopp et al. 2007). After Marquis (1975), we distributed 2.5, 26 and 13.5% of seeds among seed-2, seed-1 and seedling stages, respectively. Moreover, to incorporate post-release seed predation, from which only 55.4% of the released seeds escape (unpublished data), we set the mean annual number of seeds entering seed-2, seed-1 and seedling stages to 6011 × 0.025 × 0.554 = 83.1, 6,011 × 0.26 × 0.554 = 865.9, and 6011 × 0.135 × 0.554 = 449.3, respectively. Both probabilities for seed-2 individuals to reach the seed-1 stage in a year, and for seed-1 individuals to reach the seedling stage in a year, were set to 1 since the mortality has already been taken into account. The probability for seedlings to reach the sapling1 stage was set to 0.1105.

  Table 2 The transition matrix A accounting for shaded environmental conditions

• Matrix B (Table 3): treefall-induced canopy gap—The new probability for sapling1 individuals to remain in this stage in full light conditions equals 0.1. The 7 transition coefficients on the sub-diagonal describe the self-thinning process due to competition for space and light during the aggradation phase. All 7 coefficients of the sapling1 row equal 0.1 to account for the resprouting of overtopped saplings during the aggradation phase, which lasts 7 years in our model.

  Table 3 The transition matrix B accounting for canopy gap conditions

• Matrix C (Table 4): canopy closure—This matrix is applied when environment conditions shift from full light to shade, on the 8th year after gap creation. All individuals of the seven sapling stages suffer from aerial part death but actively resprout from stumps and/or roots. Matrix C is nearly the same as matrix A, but with six additional coefficients in the sapling1 row, all equalling 0.8. As adults already reached the canopy, they do not resprout.

  Table 4 The transition matrix C accounting for the canopy closure phase

• Matrix D (Table 5): clearcut-induced full light conditions—The clearcut affects all individuals at both sapling and adult stages, so that their aerial parts die back. Hence, matrix D has seven null rows. The resprouting probabilities are the same as in matrix C, but here adults also resprout.

  Table 5 The transition matrix D accounting for clearcut conditions

Habitat suitability index

A vector ‘habitat suitabiliy’ V has been elaborated by assigning a habitat suitability index V _i to each cell of the matrix. For this purpose, we followed the approach proposed by Chabrerie et al. (2007b) and already applied in Sebert-Cuvillier et al. (2008), with the notable exception that all cells were ecologically homogeneous. We retained two environmental variables that were available as maps in a geographic information system (GIS): soil types and soil drainage classes. Partial indices were derived for each of them:

• The partial soil type index (I _soiltype) ranged from 0.2 to 1.8 (Table 6), a value inside the interval [0,1] indicating a cell limiting Prunus serotina establishment, while a value inside the interval [1,2] characterizes a cell promoting it;

  Table 6 Values of the partial soil type index

• The partial soil drainage index (I _drainage) ranged from 0 to 1 (Table 7), drainage classes greater than 2 preventing from establishment, while classes 1 and 2 have no influence.

  Table 7 Values of the partial drainage coefficient

We then combined those two partial indices into a single index V _i quantifying the overall cell suitability for Prunus serotina establishment as mature trees. For each cell, this index is computed as follow:

• If I _drainage = 1, then V _i  = I _soiltype

• Else, $V_{i} = \sqrt {I_{\text{soiltype}} \times I_{\text{drainage}} }$

Hence, when a cell has a partial soil drainage index equal to 0, the tree cannot develop whatever the soil type. The habitat suitability index V _i ranged from 0 to 2. The time-independent vector V has 170,425 components that can be mapped (Fig. 7). At each time step, in each cell i the number of individuals at the Sapling-2 stage is multiplied by V _i .

Fig. 7

Discretization of the Compiègne-Laigue forest complex into a habitat suitability map. The scale on the right indicates the habitat suitability index V _i

Characteristics of forest-management related periodic disturbances

Four deterministic parameters were specified for each cell i, depending on its dominant canopy tree species (Table 8): the clearcut-return interval (Cl _i ), the percentage of light due to a small-scale thinning (Pth _i ), the time between two small-scale thinnings (Th _i ) and the duration of the disturbance induced by the clearcut (t _i ).

Table 8 Values (in years) of clearcut-return interval (Cl _i ), duration of the perturbation induced by the clearcut in the cell (t _i ), percentage of light due to a small-scale thinning (Pth _i ) and time between two small-scale thinnings (Th _i ) for each dominant canopy tree species in the Compiègne-Laigue forest complex

Dispersal functions

Dispersal functions incorporated both the mass action of local dispersal and the stochastic nature of long-distance dispersal. At each time step n and for each cell i, P _f G _i (n) and P _b G _i (n) seeds are dispersed outside cell i by foxes and birds, respectively, according to the dispersal functions. Hence, (1 − P _f  − P _g )G _i (n) seeds are dispersed by gravity inside cell i.

The probability for a seed coming from a cell i to be dispersed by a vector v (birds or foxes) at the distance x was given by the dispersal lognormal function f _v that depended on two parameters: the shape parameter S _v and the scale parameter L _v :

$$ {f_{v} (x) = {\frac{2\pi x}{{(2\pi )^{1.5} S_{v} x^{2} }}}\exp \left( { - {\frac{{\left( {\ln \left( {{\frac{x}{{L_{v} }}}} \right)} \right)^{2} }}{{2S_{v}^{2} }}}} \right)} $$

(1)

where x is the distance (in grid units).

Following the recommendations of Greene et al. (2004), we chose S _v  = 1, and two values were selected for L _v that approximate the mean dispersal distance by birds and foxes. After Deckers et al. (2005), Pairon et al. (2006) and personal field observations, we retained a mean dispersal distance of 100 m for birds (i.e. L _b  = 2 grid units) and 918 m for foxes (i.e. L _f  = 18.36 grid units).

The integral ${\int_{0}^{x} {\tilde{f}_{v} (r){\text{dr}}} },$ where the integration variable is the distance r and ${\tilde{f}}_{v}$ is the dispersal function (${\tilde{f}_{v} = \tilde{f}_{b} }$ for birds and ${\tilde{f}_{v} = \tilde{f}_{f} }$ for foxes), which has been scaled such as ${\int_{0}^{\infty } {\tilde{f}_{v} (r){\text{dr}}} } = 1,$ gave the probability for a seed to be dispersed by the vector v at a distance inferior to x at each time step. This probability, also called “distribution function”, is an increasing continuous function. Hence, we have two distribution functions, one for birds and another for foxes.

For each seed picked by a vector, we randomly selected a number P between 0 and 1. We computed the unique antecedent R of this number by the following vector distribution function: $x \mapsto {\int_{0}^{x} {\tilde{f}_{v} (r){{dr}}} .}$ R gave the distance at which the seed would be dispersed. Next, a direction ϑ was randomly chosen between 0 and 360° to determine which cell will receive this seed. The seed was thus added to a “sink” cell j (j ≠ i) at the distance R from “source” cell i following direction ϑ.