Bulletin of Mathematical Biology (2009) 71: 1727–1744
DOI 10.1007/s11538-009-9422-x
O R I G I N A L A RT I C L E
A Model for Plant Invasions: the Role of Distributed
Generation Times
Vicenç Méndeza,∗ , Daniel Camposa , Andy W. Sheppardb
a
Grup de Fisica Estadistica, Dept. de Fisica, Universitat Autonoma de Barcelona, 08193
Bellaterra, Barcelona, Spain
b
CSIRO Entomology, GPO Box 1700, Canberra, ACT 2601, Australia
Received: 17 December 2008 / Accepted: 2 April 2009 / Published online: 2 May 2009
© Society for Mathematical Biology 2009
Abstract An analytical model consisting of adult plants and two types of seeds (unripe and mature) is considered and successfully tested using experimental data available
for some invasive weeds (Echium plantagineum, Cytisus scoparius, Carduus nutans and
Carduus acanthoides) from their native and exotic ranges. The model accounts for probability distribution functions (pdfs) for times of germination, growth, death and dispersal
on two dimensions, so the general life-cycle of individuals is considered with high level of
description. Our work provides for the first time, for a model containing all that life-cycle
information, explicit relationship conditions for the invasive success and expressions for
the speed of invasive fronts, which can be useful tools for invasions assessment. The expressions derived allow us to prove that the different phenotypes showed by the weeds
in their native (exotic) ranges can explain their corresponding non-invasive (invasive) behavior.
Keywords Biological invasions · Dispersal · Generation times
1. Introduction
Although there is a long history of models for biological invasions, only a few approaches
have provided general and explicit analytical expressions for some essential traits, such
as the invasion speed. Probably, the best known are those by Kot et al. (1996), Lewis et
al. (2006), Shigesada et al. (1995), Skellam (1951), van den Bosch et al. (1990). These
expressions are of great usefulness as a first predictive tool for many situations (Hastings
et al., 2005).
From a historical perspective, reaction-dispersal models (Murray, 1993) are one of
the most common analytical approaches to biological invasions. In the most simple case,
∗ Corresponding author.
E-mail address: [email protected] (Vicenç Méndez).
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Méndez et al.
one describes the variation of the density of individuals n(x, t) by the well-known Fisher
equation
∂n
∂ 2n
= f (n)n + D 2 .
∂t
∂x
(1)
The first term on the right-hand side of (1) is the population growth rate and the second
term accounts for the diffusion. D is the diffusion coefficient. This is usually known in
ecology as the Skellam model (Skellam, 1951). However, this equation has serious shortcomings when applied to plant invasions.
Firstly, adult plants do not diffuse or move themselves, as assumed in the model (1),
but use intermediate stages (seeds) that undertake this process. To overcome this, stagestructured models have been proposed, where the individuals pass through several stages
allowing life-cycle structure to be incorporated into the model (Caswell et al., 2003;
Hastings et al., 2005; Neubert and Caswell, 2000). By doing this, however, these models become more complex and hard to solve analytically.
Secondly, the diffusion term in Eq. (1) describes a physical transport process which
is easy to interpret at a micro-scale, but at a macro-scale the parameter D is hard to interpret to obtain a numerical value. The well-known expression D = x 2 /(2td) for a
d-dimensional space (where x 2 is the mean squared displacement) is usually applied
(Andow et al., 1990), but estimation of D in this way is difficult (or even impossible)
in practice. For example, the data collected must come purely from the diffusion process
(i.e., reproduction of individuals is not disturbing the observational data); so, estimates of
D based on the evolution of the invasion front position are not mathematically acceptable.
Moreover, the characteristic diffusion distance must be much shorter than the characteristic length of the invasion process. It is then evident that diffusion cannot account for
long-distance dispersal events (Clark, 1998).
The most common alternative to diffusion models are those based on integro-difference
equations (Allen et al., 1996; Andersen, 1991; Kot et al., 1996; Mistro et al., 2005), where
the migration process is introduced by means of a dispersal kernel Φ(x) which measures
the probability that the individuals perform a jump of distance |x| in 2D (x = (x, y)
is the position vector in 2D). In this approach, the dispersal process is introduced in a
clear manner, but it is well known that an accurate implementation of dispersal, especially for long distances, is required in order to fit experimental results (Kot et al., 1996;
Neubert and Caswell, 2000; Skarpaas and Shea, 2007). For this reason, the main research
in this field has been focused on the implementation of accurate dispersal kernels, either by looking for proper ways to interpret raw dispersal data (Bullock et al., 2002;
Turchin, 1998; Lewis et al., 2006) or by mechanistic approaches (Katul et al., 2005;
Nathan et al., 2002). Less attention has been given to the temporal variation of systems,
since integro-difference models, including the stage-structured models cited above, are
limited to discrete times. These models then provide quite a useful approach for species
exhibiting seasonal behavior, but their accuracy is not so clear for those species with
non-seasonal or complex temporal patterns, as, for example, in the case of multigeneration species (Campos et al., 2002). The introduction of delay effects in integro-difference
models has also been explored (Mistro et al., 2005) to account for seasonality, but in general the existence of complicated temporal patterns requires that continuous-time models
A Model for Plant Invasions: the Role of Distributed Generation
1729
are considered. For that reason, in the present work we claim the interest of integrodifferential approaches, where both space and time are considered as continuous variables, and so continuous distribution functions are introduced not just for dispersal but for
every life-cycle process.
Recently, some new powerful mathematical techniques have been developed that allow detailed description of wavefront solutions for some rather general situations (Fedotov and Méndez, 2002; Méndez et al., 2004). These techniques lie within the context
of Continuous-Time Random Walk (CTRW) models, where temporal and spatial probability distribution functions (pdfs) can be introduced in such a way that the details of
the life-cycle of the individuals can be taken into account (Metzler and Klafter, 2000;
Montroll and Weiss, 1965). To illustrate the relevance of these integro-differential techniques for ecological problems, we present a new model for biological invasions which
tries to capture the essential dynamics of the individuals. We focus here on a version of
the model adapted for plant invasions; however, further modifications of the model for
animal invasions or more complex situations are straightforward.
This class of models may be of interest to a broad audience because although these
models are rather complex at first sight, the results and expressions reached in most cases
are simple and have clear ecological meaning. This makes them attractive tools for management purposes. One of the results derived from the analysis of our model provides
a threshold condition to predict whether a species will show invasive behavior or not as
a function of their specific life-cycle. We show that this condition can be an alternative
to Caswell’s models (Caswell, 1989) based on projected population matrices. In these
models the population growth parameter (usually labeled as λ) above or below unity determines whether a population increases or decreases with time, respectively.
To demonstrate the applicability of the mathematical approach presented here we compare our model and population data taken from the literature for different invasive plants.
Data from the native and exotic ranges of Echium plantagineum, Cytisus scoparius and
Carduus nutans (Grigulis et al., 2001; Paynter et al., 1998; Sheppard et al., 1988, 2002;
Woodburn and Sheppard, 1996) were used to demonstrate that when the fecundity of these
species in their exotic ranges is considered, the model predicts their invasive behavior with
the opposite demonstrated from fecundity in their native ranges. Finally, the invasion rates
recently found experimentally (Jongejans et al., 2007) for Carduus acanthoides are shown
to closely fit the analytical predictions from our model.
2. Mathematical model
We present a stage-structured model where each process is governed by temporal and/or
spatial probability distributions in order to obtain as detailed and realistic description of
the process as possible. We assume a spatially homogeneous media. In order to capture
the essential dynamics of an invasion process, we consider that it is necessary to account
for only three life stages: unripe seeds on the plant (called seeds of type 1), mature seeds
(dispersible seeds, called seeds of type 2), and adult plants.
Figure 1 describes the plant invasion life-cycle, including mature seed germination,
plant death or growth, the ripening phase of unripe seeds and dispersal of the mature seeds.
When a mature seed has dispersed, germination begins. After a random time interval
distributed according to the pdf β1 (t), a plant germinates with probability α0 . This plant
1730
Méndez et al.
Fig. 1 The life-cycle graph for the model (2a)–(2f). The random times t1 , t2 and t3 of germination, growth
and waiting are distributed according to the pdfs β1 (t), β2 (t) and ϕ(t), respectively.
grows over a random time interval distributed according to the pdf β2 (t) to maturity (i.e.,
able to produce seeds), unless it first dies within a random time distributed according to
the pdf φ(t). When a plant matures it produces Y unripe seeds. Unripe seeds become
mature seeds after a ripening period distributed according to the pdf ϕ(t). Mature seeds
disperse a random distance distributed by the dispersal kernel Φ(x) to repeat the cycle.
We have not expressed seed death explicitly, as mortality effects are implicitly taken into
account through every seed generating a new plant with probability α0 .
The two-dimensional continuous-time mesoscopic equations for each stage are:
t
P (x, t) =
dt φ ∗ (t )p(x, t − t ),
0
t
p(x, t) = p0 (x, 0)δ(t) + α0
0
t
S1 (x, t) =
t
s1 (x, t) = s1 (x, 0)δ(t) + Y
P (x, t − t )
dt β1 (t )s2 (x, t − t ) 1 −
,
K
dt ϕ ∗ (t )s1 (x, t − t ),
0
S2 (x, t) =
(2a)
(2b)
(2c)
dt β2 (t )φ ∗ (t )p(x, t − t ),
(2d)
0
t
dt β1∗ (t )s2 (x, t − t ),
0
t
s2 (x, t) = s2 (x, 0)δ(t) +
dt ϕ(t )
(2e)
dx Φ(x )s1 (x − x , t − t ).
(2f)
0
P (x, t), S1 (x, t) and S2 (x, t) are the number of plants, unripe seeds and mature seeds
located at point x at time t , respectively, and δ() the Dirac function. p(x, t), s1 (x, t)
and s2 (x, t) are the number of plants, unripe and dispersible seeds arriving at point x
at time t , respectively. The distribution functions φ ∗ (t), β1∗ (t) and ϕ ∗ (t) are the survival
probabilities of plant death, germination and seed ripening, respectively, and are defined
by
φ ∗ (t) =
t
∞
φ(t ) dt ,
β1∗ (t) =
∞
t
β1 (t ) dt ,
ϕ ∗ (t) =
t
∞
ϕ(t ) dt . (3)
A Model for Plant Invasions: the Role of Distributed Generation
1731
Equation (2a), describing the number of plants present at point x at time t , is the sum
of plants arrived at x at time t before and those that have survived during the period t .
Equation (2b), describing the number of plants appearing at point x at time t is the sum
of the initial distribution plus the type 2 seeds which fall and germinate at point x by
time t . The factor 1 − P /K introduces a competition effect between the type 2 seeds
able to generate a new plant and plants already existing at point x. The origin of this
competition lies in the limited resources of the environment, so that K stands for the
carrying capacity. Equation (2c), describing the number of unripe seeds at point x at
time t , is the accumulated number of unripe seeds produced on the plants. Equation (2d),
describing the number of unripe seeds arriving at point x at time t , is the sum of unripe
seeds already present at t = 0 plus those produced by adult plants that survived since
they germinated at time t − t . The number of mature seeds at point x at time t are those
that arrived at point x at time t − t and have not germinated during this period. This is
described by Eq. (2e). Equation (2f) expresses the number of mature seeds arriving at the
point x at time t as those initially arrived plus those unripe seeds which appeared at x and waited a time t before dispersing.
2.1. Model analysis
In this subsection we want to find the final state reached after invasion, that is, the population densities behind the invasion front. The first step for the analysis of our model is to
find the equilibrium states of the system (2a)–(2f). Equilibrium steady states (P 0 , S10 , S20 )
fulfill the conditions (see Appendix A for details) P 0 = α0 Y a0 P 0 (1 − P 0 /K), S10 =
τ Y a0 P 0 /τm and S20 = τ1 S10 /τ, where
∞
a0 ≡
dt β2 (t)φ ∗ (t)
(4)
0
and τ , τ1 and τm are the mean seed ripening, seed germination and plant mortality times,
respectively. Note that a0 is nothing but the probability of a plant to reach maturity, that
is,
it dies. Also note that the condition a0 < 1 is always fulfilled, as
∞produce seeds before
∗
dt
β
(t)
=
1
and
φ
(t)
< 1 for t > 0.
2
0
Two different equilibrium states appear:
State 1: P 0 , S10 , S20 = (0, 0, 0),
(5)
1
τ
1
State 2: P 0 , S10 , S20 = K 1 −
, K Y a0 1 −
,
α0 Y a0
τm
α0 Y a0
τ1
1
K Y a0 1 −
.
(6)
τm
α0 Y a0
In order to have a final state with biological meaning, the population density must be
positive and this requires
Y > Ymin ≡
α0
∞
0
1
.
dt β2 (t)φ ∗ (t)
(7)
As we show in Appendix A, this condition is necessary to exist for an invasive front
joining both states 1 and 2. Equation (7) is a threshold condition for invasion success
1732
Méndez et al.
in terms of the number of fertile unripe seeds released per plant and it depends only
on the parameter α0 and how the random times for death (φ ∗ (t)) and maturity (β2 (t))
are distributed. According to the condition a0 < 1 mentioned above, it comes also that
Ymin > 1.
Expression (7) is equivalent to the Lotka’s equation (Lotka, 1956) for growth in agestructured populations. In our model, Lotka’s result arises as a natural consequence, without the need for phenomenological arguments as in the original derivation (Lotka, 1956;
van den Bosch et al., 1990). Equation (7) means that the number of seeds giving rise to
new plants at each life-cycle (α0 Y ) must be higher than the number of adult plants which
died within a life-cycle.
We have presented first a 3-stage model because the life-cycle is implemented more
intuitively in that way but, in fact, the model can be nicely written as a closed and simple
expression, which makes it more manageable. The model (2a)–(2f) can be reduced (see
Appendix A) to the following equation for P (x, t) valid in the vicinity of the trivial steady
state, that is, for small population densities
t
Y
P (x, t) = f (x, t) +
dt β(t ) dx Φ(x )P (x − x , t − t ),
(8)
Ymin 0
where f (x, t) integrates the terms where initial conditions appear, and β(t) can be regarded as the pdf of times between successive generations; their explicit expressions are
given in Eqs. (A.20) and (A.9) in Appendix A. The role of the pdfs β1 , β2 , φ and ϕ is
summarized within β(t) (see Eq. (A.9)).
The implications of Eq. (7) are the most interesting derivations here. Given a pdf for
the generation of seeds and plant mortality, we can find the minimum number of seeds
produced by a plant to successfully invade. This outcome can sometimes be an alternative
to the well-known model by Caswell (1989) based on population matrices for predicting the population growth characteristics. The product Y α0 is approximately equivalent
to the parameter λ, which usually represents the population growth in the matrix formalism. Compared with Caswell’s approach, where the condition λ ≷ 1 determines the
invasive character of a species, we find that this threshold depends on the distributions
of plant survival φ ∗ (t) and seed production β2 (t). Therefore, our model can be seen as a
generalization of Caswell’s result for the case where temporal pdfs for every process are
considered.
We now discuss some specific examples to show how simple expressions for Ymin can
be obtained for different situations:
(a) Seed production and plant mortality are exponentially distributed, in consequence
β2 (t) = τ2−1 e−t/τ2 and φ(t) = τm−1 e−t/τm , where τi accounts for the typical time scale
of the processes. In this case condition (7) leads to
1
τ2
Ymin =
1+
.
(9)
α0
τm
(b) Seed production occurs at a fixed time τ2 after germination, so β2 (t) = δ(t − τ2 ), and
mortality is as in case (a). This matches plants producing seeds at approximately a
fixed age, e.g., an annual species. This case leads to
Ymin =
1 τ2 /τm
e
.
α0
(10)
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1733
(c) Seed production occurs periodically with a period given by τ2 , so β2 (t) = δ(t − τ2 ) +
δ(t − 2τ2 ) + δ(t − 3τ2 ) + · · · , and mortality over time is again exponentially distributed. This choice for β2 (t) allows us to take into account that a given plant can
generate seeds many times (that is, many generations) before dying and so a seasonal
behavior is introduced. Condition (7) leads to
Ymin =
1 τ2 /τm
−1 ,
e
α0
(11)
where the periodical behavior of β2 (t) makes the condition (7) less restrictive compared to case (b)).
(d) Seed production occurs periodically, as in (c), and lifespan is fixed at τm , so φ(t) =
δ(t − τm ). This case has some theoretical interest, as we now need to introduce a
parameter n defined as n ≡ Int [τ2 /τm ] (“Int” means the integer part) to define the
number of times that the plants produce seeds before they die. From this definition,
the condition (7) has the form
Ymin =
1
.
α0 n
(12)
3. Model applications
3.1. Invasion success of weeds
To illustrate the usefulness of the expression (7), we compared published data of three
different invasive species in their native and exotic ranges. In the exotic range, species can
exhibit invasive behavior. This comparison of population ecology has been made to assist
management strategies. Our model should show that predicted Ymin values are higher than
the actual plant fecundities in the native range and lower than actual plant fecundities in
the exotic range. Although the three species studied show important differences in their
life-cycle, by introducing suitable pdfs, our model accounts for these differences quite
easily.
3.1.1. E. plantagineum
This is an annual plant native to the western Mediterranean basin that was introduced
and became invasive in pastures in Australia. Grigulis et al. (2001) compared its population dynamics in Evora, Portugal (native) with Canberra, Australia (invasive). From
the published data we accurately estimated the parameters needed to evaluate Ymin . As
E. plantagineum is an annual, all surviving plants produce seed once at age 1 year and
die. This presents the case in (12) with n = 1.
To estimate the parameters Y and α0 , we averaged the data obtained by Grigulis et al.
(2001) over all the populations considered. If we assume that seeds produced can survive
in the seed bank up to 6 years, then
5
j
(Sb ) ,
α0 = GSi h 1 +
j =1
(13)
1734
Méndez et al.
Table 1 Parameter estimates from our model of the invasiveness of E. plantagineum, C. scoparius and C.
nutans in their native and exotic range
Plant
Range
α0
a0
E. plantagineum
Nativea
Exoticb
7.1 × 10−3
1.2 × 10−2
0.18
0.66
783
124
322
261
C. scoparius
Nativec
Exoticb
0.0314
0.023
0.06
0.16
531
272
81.6
300
C. nutans
Nativec
Exoticb
0.01
0.12
0.109
7.2 × 10−3
Ymin
917
1161
Actual Y
125
1950
a Portugal
b Australia
c France
where G is the seedling establishment fraction, Si is the seed-bank incorporation rate, h
is the fraction of the seed bank germinating every year and Sb is the seed-bank survival
rate. The averaged results found from our analysis are given in Table 1. Our hypothesis
for Ymin holds, as only those populations where Y > Ymin exhibit invasive behavior.
3.1.2. C. scoparius
This leguminous shrub is native to western Europe and has become an invasive weed in
some Australian ecosystems, suppressing native species and increasing fire risk (Hoskins
et al., 1998). We compare the population dynamics of this species in a native habitat in
southern France (Paynter et al., 1998) with those from an invasive population in Australia
(Sheppard et al., 1994).
In the native range, the parameters were estimated as follows. The average percentage
seed germination was used to estimate α0 (see Fig. 2 in Paynter et al., 1998). Fecundity
was estimated from average seed production per unit area (38.3 seeds m−2 ), taken from
the total study area (400 m2 ) and that 7% of plants produced any seeds by the end of the
experiment (so, Y = 81.6 seeds per plant). The survival distribution is taken from Fig. 4
within the same reference (Paynter et al., 1998), where from the second year sampling,
survival decay is approximately a linear function of time.
Finally, we estimated the germination distribution β2 (t) using that 18% of the flowering plants produced seeds in their first-flowering year (at 3 years of age), as noted by
Sheppard et al. (1994). We assume that after the second-flowering year all surviving plants
(they can survive up to 20 years) could produce seeds, and so the pdf for seed production
over time reads
β2 (t) =
tf
γl δ(t − l)
(14)
l=ti
where ti = 3 years, tf = 20 years, γ3 = 0.18 and γl>3 = 1.
Estimation of the parameters for C. scoparius in the exotic range was carried out in a
similar way as for the native range, as the published data are very similar. The averaged
value α0 = 0.023 is already given in the text by Sheppard et al. (1994) and fecundity data
are also presented in graphical form. The age at the first seed production is also studied
A Model for Plant Invasions: the Role of Distributed Generation
1735
in the original work, and from that we estimated β2 (t) as in (14) but with ti = 2 years,
tf = 20 years, γ2 = 0.003, γ3 = 0.068, γ4 = 0.706, γ5 = 0.990 and γl>5 = 1.
Finally, the survival data showed the best fit to a power-law of the form
φ ∗ (t) ≈
1
.
1.607 + 0.844t 2.37
(15)
The estimated values of Y and Ymin for the native and exotic range in Table 1 were as
expected for non-invasive and invasive populations, respectively.
3.1.3. C. nutans
This thistle of European origin has become a weed in many parts of the world. Sheppard
et al. (1988) have published demographic data of C. nutans for some native populations
in France studied over more than 2 years reported in 3-monthly intervals. The parameters
corresponding to the exotic range have been extracted from analogous published data
from one site in Australia (Woodburn and Sheppard, 1996).
Fecundity Y was estimated from seed density and flowering plant density. The probability α0 was estimated from the product of the proportion of surviving mature seeds
entering the seed bank and the proportion of the seed bank that recruited. The values are
presented in Table 1, where lower α0 values in the native range reflect the important effect
of predispersal predation (Sheppard et al., 1994). C. nutans is monocarpic, so flowering is
followed by death. However, as the age at flowering can vary among individuals, we could
use the same assumption as used for E. plantagineum. In this case we needed to produce
a survival distribution, which could be done accurately from the published demographic
data and leads to
Native:
Exotic:
1
,
1 + 9t 1.47
1
φ ∗ (t) ≈
.
1 + 106t 2.69
φ ∗ (t) ≈
The data indicate individual survival is greater in the native range than in the exotic habitat, as is also evident from a0 values in Table 1. Nevertheless, the importance of the predispersal predation on the seed bank and the great differences in fecundity between the
two ranges are the factors that determine the invasive behavior of C. nutans in Australia
(Woodburn and Sheppard, 1996), as confirmed by comparing Y and Ymin parameters in
Table 1.
3.2. Invasion speed of weeds
The overall radial rate v at which an invasion front spreads can be computed by using the
recently explored Hamilton–Jacobi method (Fedotov and Méndez, 2002; Méndez et al.,
2004). The invasion speed v is given by the formula
v=
H (p)
p
with
∂H
H
= ,
∂p
p
where H (p) is given by
1=
Y
(H )Φ(p)
β
Ymin
(16)
1736
Méndez et al.
with
Φ(p)
= 2π
∞
r dr Φ(r)I0 (pr),
0
where I0 (·) is the modified Bessel function of order zero. The details of how the speed is
obtained are given in Appendix B. Here, we focus our attention to apply our theoretical
predictions to real situations. Recent experimental work by Jongejans et al. (2007) for another invasive thistle (C. acanthoides) was used to test our theoretical predictions for the
invasion rate. Rosettes of C. acanthoides were introduced into uninvaded plots in Maryland (USA) where each rosette was considered as a founder individual for new invasive
thistle populations.
The cumulative probability distribution for jump lengths ψ(r) was measured for different years and different treatments (named 0x, 1x and 2x clippings). The relation ber
tween Φ(r) and ψ(r) is given by ψ(r) = 2π 0 r dr Φ(r ) or Φ(r) = ( 2π1 r ) d ψ(r)/dr.
To compare our predictions for the invasion rate with the observed results by Jongejans et
al. (2007), it is necessary to know the value for the quotient Y /Ymin . The field data are not
in a fine enough spatial scale, however, to make this comparison possible with a desirable
accuracy, except for the case of the invasion in 1995, where the number of seed released
per plot was approximately 1111 seeds per plot for any treatment.
As Ymin is not known, we have estimated it by fitting our theoretical prediction with the
observed value. Fitting the cumulative distribution ψ(r) to the experimental data for 0x
clipping in 1995, we estimated (with R 2 = 0.961) the dispersal kernel to be of the form
Φ(r) =
k
e−kr
−kr
2πr e min − e−krmax
(17)
where k = 4.63 m−1 is the inverse of the characteristic jump length and rmin = 0.04 m and
rmax = 0.44 m are the minimum and maximum jump lengths. So, infinite dispersal lengths
are not allowed in our kernel (17) in order to avoid unrealistic effects on the invasion rate.
Introducing (17) into (B.5a) but integrating from rmin to rmax and considering one year as
the fixed time between two successive generations (i.e., β(t) = δ(t − τg ) with τg = 1 yr),
we find Ymin = 44.4 which will be used for comparing with 1x and 2x clippings.
Fitting the dispersal kernel to the data from the 1x clipping for 1995 we obtain the same
kernel as in (17) but with R 2 = 0.977, k = 3.57 m−1 , rmin = 0.036 m and rmax = 0.55 m.
Introducing these values into (17), (B.5a) we compute from (B.6) that the invasion rate is
0.45 m/yr which is very close to the observed result of 0.49 m/yr. By fitting the dispersal
kernel to the data for 2x clipping in 1995, we obtain (with R 2 = 0.982)
Φ(r) =
kr + k1
2πr
with k = 2.87 m−1 , k1 = 0.14, rmin = 0 and rmax = 0.30 m. In this case we obtained
an invasion rate of 0.26 m/yr which is again in agreement with the observed result of
0.27 m/yr.
These results can be discussed in the light of those obtained in a recent work (Skarpaas
and Shea, 2007), where the invasion speeds for C. acanthoides and C. nutans were studied too. In that article, the authors obtained some predictions for the front speeds by using
A Model for Plant Invasions: the Role of Distributed Generation
1737
nonparametric (Clark et al., 2001) and mechanistic estimates of the dispersal kernel. Their
work was specially focused on the role of the dispersal patterns and the influence of longdistance dispersal on the invasion speed, while in our approach we have rather highlighted
the importance of the pdfs governing the life-cycle of individuals. The present results are
restricted to local dispersal (the work by Jongejans used as a reference was restricted to
dispersal distances of up to 4 m) and, according to that, we have introduced the thresholds
rmin and rmax in order to remove the problem of data extrapolation (Clark et al., 2001,
2003). These important differences prevent us from performing a direct comparison between both works. However, we stress that they both agree with the idea that models
based on pdfs can fit quite well the behavior of weed invasions if the dispersal kernel can
be implemented accurately.
4. Conclusions
We have presented a new model for plant invasions that integrates the most essential features of the life-cycle of individuals, and we have shown that general analytical
expressions for the invasion threshold and for the invasion rate can be obtained. Although stage-structured and similar models existing in the literature (Caswell et al., 2003;
Neubert and Caswell, 2000) enable to implement also in detail the life-cycle of individuals, the analysis presented here provides also a high accuracy (especially for nonseasonal or complex temporal patterns) and useful analytical tools, owing to the continuous approach based on temporal pdfs. Our model is based on the Continuous-Time
Random Walk framework (Montroll and Weiss, 1965), which has been subject to extensive research in the last years (Fedotov and Méndez, 2002; Méndez et al., 2004;
Metzler and Klafter, 2000) by physicists and mathematicians; the main aim of the present
work consisted of turning all that research into a comprehensive and manageable model
for biological invasions in order to attract the interest of ecologists on the field of integrodifferential models.
The model presented here was specifically formulated for invasive plants, because their
stage-structured dynamics is clear, as is the meaning of the developed pdfs. The model
may be easily modified, however, for animal invasions. For example, animals can have
three different life-cycle stages; juveniles, free adults (able to disperse) and parents. The
temporal pdfs that one could introduce for animals might read: (i) the probability that
juveniles become adult in a time t , (ii) the probability that a free adult becomes a parent
after a time t , and (iii) the probability that parents become free adults again after a time t ,
along with the pdfs for mortality. Therefore, the ideas and techniques studied in this paper
can have a potential application for the analysis of a wide range of models and biological
contexts.
The plant invasion model presented here provides some simple analytical expressions
which may be of general interest within the context of invasion assessment. Specifically,
Eq. (7) determines a minimum seed production threshold necessary for invasion processes
to occur. Proper threshold estimation requires known probability distributions for seed
generation and survival over time. Such information is often available, however, especially for short-lived species where detailed experimental data are easy to obtain. We
used three such cases to analytically assess the invasive character of three different weeds
(E. plantagineum, C. scoparius and C. nutans) in their exotic ranges, by comparing with
1738
Méndez et al.
similar data from their native ranges for assumed non-invasive behavior. Species invasion rates could also be examined in detail, and successful comparison were made with
experimentally observed rates for C. acanthoides.
Analytical models based on pdfs have been criticized because the accurate estimation
of these probability functions is not experimentally easy, and may lead to some undesirable effects, especially if data extrapolation is performed (Clark et al., 2003). We show,
however, that these kinds of models can fit the experimental results when a proper estimation of the pdfs is made. Such models may provide attractive mathematical management
tools for predicting the invasive status of populations. Integro-differential stage-structured
models provide an essential improvement on simpler models where intermediate life
stages are ignored, since pdfs can be introduced for each of the life stages to realistically
capture species dynamics within the life-cycle. We have shown in this study that building
some biological complexity and reality into analytical models can be achieved, countering arguments that such approaches are an oversimplification of real invasion processes.
Within a proper context, these approaches provide useful tools for analyzing invasions.
Acknowledgements
This research has been partially supported by the Departament d’Educació i Universitats
de la Generalitat de Catalunya by Grant 2006-BP-A-10060 (DC), and by Grants Nos. FIS
2006-12296-C02-01 and SGR 2005-00087 (VM).
Appendix A: Steady states and their stability analysis
From our general model (2a)–(2f), the system of equations governing the asymptotic temporal behavior of spatially homogeneous states can be written as
t
dt φ ∗ (t )p(t − t ),
P (t) =
0
t
p(t) = α0
0
t
P (t − t )
dt β1 (t )s2 (t − t ) 1 −
,
K
dt ϕ ∗ (t )s1 (t − t ),
S1 (t) =
0
t
s1 (t) = Y
∗
dt β2 (t )φ (t )p(t − t ),
0
t
S2 (t) =
(A.1)
0
t
s2 (t) =
dt β1∗ (t )s2 (t − t ),
dt ϕ(t )s1 (t − t ).
0
To compute the densities of the different steady states we take first the asymptotic
limit or large time limit t → ∞ in (A.1). Defining the quantities P (t → ∞) = P 0 ,
A Model for Plant Invasions: the Role of Distributed Generation
1739
p(t → ∞) = p 0 , S1 (t → ∞) = S10 , s1 (t → ∞) = s10 , S2 (t → ∞) = S20 and S2 (t →
∞) = S20 , one finds from (A.1) the following set of algebraic equations: P 0 = p 0 τm ,
p0 = α0 s20 (1 − P 0 /K), S10 = s10 τ, s10 = Yp 0 a0 , S20 = s20 τ1 and s20 = s10 . This set can be
solved to get the solutions (5) and (6).
To analyze the stability of the equilibrium states in our model, we introduce in (A.1)
the following definitions: P (t) = P 0 + ε0 (t), p(t) = p 0 + δ0 (t), S1 (t) = S10 + ε1 (t),
s1 (t) = s10 + δ1 (t), S2 (t) = S20 + ε2 (t) and s2 (t) = s20 + δ2 (t), where εi and δi are small
perturbations. The linearized equations are
∞
ε0 (t) =
dt φ ∗ (t )δ0 (t − t ),
(A.2)
0
s20 t dt β1 (t )ε0 (t − t )
K 0
t
P0
+ α0 1 −
dt β1 (t )δ2 (t − t ),
K
0
t
dt ϕ ∗ (t )δ1 (t − t ),
ε1 (t) =
δ0 (t) = −α0
(A.3)
(A.4)
0
t
δ1 (t) = Y
t
ε2 (t) =
dt β2 (t )φ ∗ (t )δ0 (t − t ),
(A.5)
0
dt β1∗ (t )δ2 (t − t ),
(A.6)
dt ϕ(t )δ1 (t − t ).
(A.7)
0
δ2 (t) =
t
0
Looking for exponential solutions (eλt ) leads to the characteristic equation
1 + α0
Y
s20
P0
1 (λ)φ∗ (λ) =
(λ),
β
1−
β
K
Ymin
K
(A.8)
where we have defined β(t) as
1
β(t) =
a0
t
t−t1
dt1 β1 (t1 )
0
dt2 β2 (t2 )φ ∗ (t2 )ϕ(t − t1 − t2 ),
(A.9)
0
the pdf of generation time β(t), instead of using β1 (t) and β2 (t) separately, as this is ecologically adequate. Focusing now on the trivial steady state defined in (5), the character(λ) = Ymin /Y. The left-hand side of this equation is a monotonically
istic equation reads β
(λ = 0) = 1, there is always a positive λ value
decreasing function with λ toward 0. As β
where β (λ) reaches the value Ymin /Y . This guarantees the instability of the trivial steady
state provided that Y > Ymin . A similar analysis can be performed for the non-trivial steady
state defined in (6), but here the right-hand side of (A.8) never equals the left-hand side
for positive λ values, and this shows that the non-trivial steady state cannot be unstable.
The system (2a)–(2f) then has two equilibrium states of different stability which can be
connected by a traveling wave, namely the invasion front.
1740
Méndez et al.
Linearizing the system (2a)–(2f) around the unstable trivial steady state, we get (A.1)
but replacing the factor 1 − P /K by 1. Performing the Fourier–Laplace transform they
turn into the system
P (k, s) = φ∗ (s)p(k, s),
(A.10)
1 (s)s2 (k, s),
p(k, s) = p0 (k, 0) + α0 β
(A.11)
S1 (k, s) = s1 (k, s)ϕ∗ (s),
(A.12)
∗
s1 (k, s) = s1 (k, 0) + Y β
2 φ (s)p(k, s),
(A.13)
S2 (k, s) = s2 (k, s)β1∗ (s),
(A.14)
ϕ (s)s1 (k, s).
s2 (k, s) = s2 (k, 0) + Φ(k)
(A.15)
In order to reduce this system to a new one for P , S1 and S2 , we combine the equations
in the following way. First isolate s2 (k, s) from (A.14) and introduce it into (A.11). The
resulting equation for p(k, s) must be substituted into (A.10) to get
P (k, s) = p0 (k, 0)φ∗ (s) + α0
1 (s)
φ∗ (s)β
S2 (k, s).
β1∗ (s)
(A.16)
Now isolate p(k, s) from (A.10) and substitute into (A.13) and the resulting equation for
s1 (k, s) has to be introduced in (A.12) to find
S1 (k, s) = s1 (k, 0)ϕ∗ (s) + Y
∗
∗ (s)
β
2 φ (s)ϕ
P (k, s).
φ∗ (s)
(A.17)
Finally, solve s1 (k, s) from (A.12) and introduce the result into (A.15). The final expression for s2 (k, s) is introduced into (A.14) to get
ϕ (s)β1∗ (s)
Φ(k)S1 (k, s).
S2 (k, s) = s2 (k, 0)β1∗ (s) +
ϕ∗ (s)
(A.18)
To get a closed equation for P (x, t) one introduces S1 (k, s) from (A.17) into (A.18) and
the resulting equation for S2 (k, s) can be substituted into (A.16) to obtain
t
Y
dt β(t ) dx Φ(x )P (x − x , t − t ),
(A.19)
P (x, t) = f (x, t) +
Ymin 0
where the term f (x, t) has the Fourier–Laplace transform
1 (s)s2 (k, 0)
f (k, s) = p0 (k, 0)φ∗ (s) + α0 φ∗ (s)β
1 (s)
ϕ (s)Φ(k)s1 (k, 0).
+ α0 φ∗ (s)β
(A.20)
Appendix B: The Hamilton–Jacobi formalism
This method was originally introduced by Freidlin (1996) and consists in finding the
Hamilton–Jacobi equation for the system (2a)–(2f). To this end, one starts by introducing
A Model for Plant Invasions: the Role of Distributed Generation
1741
the hyperbolic scaling x → x/, t → t/ to get
t/
P (x, t) =
dt φ ∗ (t )p (x, t − t )
(B.1a)
0
p (x, t) = p0 (x/, 0)δ(t/)
t/
P (x, t − t )
dt β1 (t )s2 (x, t − t ) 1 −
+ α0
,
K
0
t/
dt ϕ ∗ (t )s1 (x, t − t ),
S1 (x, t) =
0
s1 (x, t)
t/
= s1 (x/, 0)δ(t/) + Y
dt β2 (t )φ ∗ (t )p (x, t − t ),
(B.1b)
(B.1c)
(B.1d)
0
t/
S2 (x, t) =
dt β1∗ (t )s2 (x, t − t ),
(B.1e)
0
s2 (x, t) = s2 (x/, 0)δ(t/)
t/
dt ϕ(t ) dx Φ(x )s1 (x − x , t − t ),
+
(B.1f)
0
where we have defined the new fields P (x, t) = P (x/, t/), p (x, t) = p(x/, t/),
S1,2
(x, t) = S1,2 (x/, t/), s1,2
(x, t) = s1,2 (x/, t/). We are interested in studying the
asymptotic limit for (2a)–(2f), which is equivalent to taking → 0 in the above equations. Then, it is expected that the new fields take only two values, 0 and 1, as → 0.
This means that the solutions to (2a)–(2f) converge to the indicator function and whose
boundary may be considered as a moving front position separating the stable state 2 and
the unstable state 1. Since the new fields are positive, we can make use of the transfor
(x, t) =
mations P (x, t) = A1 exp[−G (x, t)/], p (x, t) = A2 exp[−G (x, t)/], S1,2
A3,4 exp[−G (x, t)/], s1,2 (x, t) = A5,6 exp[−G (x, t)/] where G (x, t) ≥ 0. The new
function G (x, t) will determine the location of the invasive front in the limit → 0. The
system (B.1a)–(B.1f) turns then into
t/
A1 = A2
dt φ ∗ (t ) exp
0
G (x, t) − G (x, t − t )
,
(B.2a)
A2 = p0 (x/, 0)δ(t/)eG (x,t )/
t/
G (x, t) − G (x, t − t )
+ A6 α0
dt β1 (t ) exp
0
e−G (x,t−t )/
× 1 − A1
,
K
t/
G (x, t) − G (x, t − t )
dt ϕ ∗ (t ) exp
A 3 = A5
,
0
(B.2b)
(B.2c)
1742
Méndez et al.
A5 = s1 (x/, 0)δ(t/)eG (x,t )/
t/
G (x, t) − G (x, t − t )
∗ + A2 Y
dt β2 (t )φ (t ) exp
,
0
t/
G (x, t) − G (x, t − t )
dt β1∗ (t ) exp
A4 = A6
,
0
t/
dt ϕ(t ) dx Φ(x )
A6 = s2 (x/, 0)δ(t/)eG (x,t )/ + A5
× exp
(B.2d)
(B.2e)
0
G (x, t) − G (x − x , t − t )
.
(B.2f)
Taking the limit → 0 and introducing the concept of derivative, the system (B.2a)–(B.2f)
can be cast in the form M · A = 0 where A = (A1 , . . . , A6 ), and G(x, y, t) =
lim→0 G (x, y, t). If we introduce the Hamiltonian function H = −∂t G and the generalized momentums px = ∂x G, py = ∂y G, then
⎛
−φ̂ ∗ (H )
1
0
∗
−Y β
2 φ (H )
0
0
1
⎜0
⎜
⎜0
M =⎜
⎜0
⎜
⎝0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
−ϕ̂ ∗ (H )
1
0
x , py )
−ϕ̂(H )Φ(p
⎞
0
−α0 β̂1 (H ) ⎟
⎟
⎟
0
⎟.
⎟
0
⎟
β̂ ∗ (H ) ⎠
1
1
In order to have a non-trivial solution, det(M) = 0 must be required. So that
1=
Y
(H )Φ(p
x , py )
β
Ymin
(B.3)
where (H ) stands for the Laplace transform with parameter H and (p) is the bilateral
transform with parameter p, defined as
∞
∞
dx
dy epx x+py y Φ(x, y).
(B.4)
Φ(px , py ) =
−∞
−∞
Equation (B.3) can be regarded as the Hamilton–Jacobi equation. The integral in (B.4)
can be computed by assuming an isotropic dispersal kernel Φ(x, y) = Φ(r) where
r 2 = x 2 + y 2 and changing to polar coordinates x = r cos θ , y = r sin θ . Then Eq. (B.4)
turns into
∞
∞
2π
dθ
r dr Φ(r)epx r cos θ+py r sin θ = 2π
r dr Φ(r)I0 (pr) (B.5a)
Φ(p)
=
0
0
0
with p 2 = px2 + py2 and I0 (·) the modified Bessel function of order 0. Equation (B.3) is
important since it allows us to find the overall invasion speed from the relation
H
H
∂H
= .
(B.6)
v = min
with
p
∂p
p
A Model for Plant Invasions: the Role of Distributed Generation
1743
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