e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Symmetric competition causes population oscillations in an
individual-based model of forest dynamics
Paul Caplat a,b,∗ , Madhur Anand a , Chris Bauch b
a
b
Department of Environmental Biology, University of Guelph, Guelph, ON, N1G2W1, Canada
Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G2W1, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
Individual-based modelling is a promising tool for scaling from the individual to the popula-
Received 24 May 2007
tion and community levels that allows a wide range of applied and theoretical approaches.
Received in revised form
Here, we explore how intra-specific competition affects population dynamics using FOR-
27 September 2007
SITE, an individual-based model describing tree–tree interactions in a spatial and stochastic
Accepted 3 October 2007
context. We first describe FORSITE design and submodels following the ODD (Overview,
Published on line 28 November 2007
Design concepts and Details) guideline for individual-based models. We then use simulation to study how competition symmetry (i.e., the way individual size affects resource
Keywords:
partitioning) changes temporal and spatial population dynamics. We compare our results
Plant model
to those of an earlier deterministic (analytical) model of annual plants which found that
Time-series analysis
(i) under asymmetric competition (i.e., advantaging tall individuals), population dynamics
Moran’s I
converge quickly to a stable equilibrium and (ii) under symmetric competition, some val-
Boreal forest
ues of competition strength and population growth rate make population dynamics exhibit
long-term oscillations. We find generally similar results, despite the existence of overlapping generations in trees. A thorough analysis of stage structures in the model allows us to
explain this behaviour. We also show that decreasing tree dispersal distances, in the case
of symmetric competition, results in a wave-like spatial pattern, caused by desynchronized
sub-populations. Finally, we link the results obtained with FORSITE to different types of
resource limitation observed in northern temperate and sub-boreal forests, emphasizing
the implications of such difference on long-term biome dynamics. We note that FORSITE is
a flexible platform that can be easily adapted for other ecological modelling studies.
© 2007 Elsevier B.V. All rights reserved.
1.
Introduction
Scaling up processes from individual to community and landscape levels has long been a critical issue in ecology (Levin,
1992; Wiens et al., 1993). Individual-based modelling (IBM)
(DeAngelis and Mooij, 2005; Grimm and Railsback, 2005) is a
promising tool for scaling that allows a wide range of applied
and theoretical approaches. Recent studies use individualbased models to generate results that could probably not have
∗
Corresponding author. Tel.: +1 519 824 4120.
E-mail address: [email protected] (P. Caplat).
0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2007.10.002
been obtained through analytical models alone (Uchmanski,
1999, 2000). The proliferation of IBMs, however, has not
yielded substantial advances in ecological theory (Berger and
Hildenbrandt, 2000) and Grimm et al. (2006) call for a generalized approach to IBM description.
Recent individual-based plant models have found that taking into account dispersal mechanisms (e.g., Higgins and Cain,
2002), and the effects of environmental heterogeneity (Yu and
Wilson, 2001; Snyder and Chesson, 2003) at the individual level
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e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
are critical for understanding community dynamics. Competition is obviously also critical in understanding community
dynamics and has been carefully detailed in spatially explicit
models of vegetation dynamics (e.g, SORTIE model, Pacala
et al., 1996; Berger and Hildenbrandt, 2003). How competition between individuals plays a role in population dynamics,
however, is less clear. Intra-specific competition has been
extensively studied in community ecology (see Calcagno et al.,
2006 for a review) but within this field the mechanism underlying the individual relationships has rarely been explicitly
taken into account (Adler and Mosquera, 2000).
In forest ecosystems, competition for light explains a large
part of the community structure (Pacala et al., 1996) and for
that reason many models have tried to incorporate, explicitly, competition for light at the individual level (Moravie and
Robert, 2003; Bauer et al., 2004). However, nutrients like nitrogen, or water supply, can also be limiting factors in forests
(Pacala et al., 1996; Wilson, personal communication, 2007).
This is particularly true in boreal forests, where tree density
and the angle of light incidence make competition for light
almost negligible (Pham et al., 2005). In that context, belowground competition may become relatively more important.
Competition for light between large and small trees can be
defined as asymmetric, because competition for light favours
tall, wide-crowned trees rather than short ones (Bauer et al.,
2004). However, root competition is often symmetric (i.e., the
resource is equally partitioned between individuals independently of their size; Berntson and Wayne, 2000) or inversely
asymmetric (i.e., smaller individuals consume a larger portion of resource than taller ones; Schenk, 2006). The issue of
asymmetric vs. symmetric competition is thus, relevant when
dealing with forest populations, communities and landscapes.
The importance of competition type (asymmetric vs. symmetric) for plant community structure has been recognized
through simulation studies (Weiner, 1990; Pacala and Weiner,
1991; Bauer et al., 2004). Most studies have focussed on how
variability in individual growth rates affects size hierarchy
formation (Freckleton and Watkinson, 2001). In particular, it
has been shown that in populations characterized by symmetric competition, size hierarchy (i.e., the distinct density
of different size classes) increases with increasing competition. By comparison, when asymmetric competition occurs
size hierarchy does not respond to differences in densities.
In his review on asymmetric competition in plant populations, Weiner (1990) recognized that the effect of competition
on size hierarchy should have deep implications for population dynamics. Indeed, Pacala and Weiner (1991) showed
how adding an asymmetric competition term in an analytical model of plant population dynamics produces convergence
to a stable equilibrium, whereas symmetric competition produces sustained oscillations in population sizes under certain
conditions.
We are not aware of any individual-based model explicitly focusing on the effect of competition type on long-term
dynamics of plant populations. To address this issue, we
built FORSITE, a generic individual-, rule-based model that
details how dispersal and competition modes affect tree
spatial and temporal population dynamics. The aim of this
paper is two-fold: (1) to present FORSITE following the
ODD (Overview, Design concepts and Details) guideline for
individual-based models suggested by Grimm et al. (2006)
and (2) to show how the type of competition can dramatically affect the long-term behaviour of a single-species tree
population.
We were particularly interested in Pacala and Weiner’s
(1991) analytical demonstration of sustained oscillations
occurring under symmetric competition and specific conditions. As shown by Uchmanski (1999, 2000), introducing
stochastic interactions between individuals can dramatically
change the results obtained by an analytical model. Inspired
by Pacala and Weiner’s study, we expect FORSITE to reveal
(i) that under asymmetric competition, population dynamics converge quickly to a stable equilibrium and (ii) that
under symmetric competition, when growth rate decays
exponentially and when competition increase, population
dynamics exhibit long-term oscillations. However, we experiment with various additional parameters to establish the
range of this behaviour. Particularly, FORSITE aims to simulate
forest dynamics, while Pacala and Weiner’s model simulated
annual plants. It seems, therefore, interesting to see how the
difference in life cycle assumptions, among other things, can
influence the model outputs.
Consequently, we adopt a sequential simulation plan that
allows testing of these hypotheses and deconstructs the
model behaviour by increasing its complexity step by step.
This approach allows us to explain thoroughly the relationships between spatial and temporal dynamics that underlie
the simulated dynamics.
2.
Model description
2.1.
Purpose
The aim of FORSITE is to simulate the effects of different
types and degrees of competition on tree interactions without
incorporating details of the underlying physiological mechanisms, in order to model large scale phenomena. The acronym
emphasizes that it is a Forest model with Random, Spatial,
Individual and Temporal effects, acknowledging RITES (Random Individual and Temporal effects) that Clark et al. (2004)
show to be key in ecological dynamics by focusing explicitly
on the scale and timing of individual interaction. Unlike most
theoretical plant simulation approaches, in which species are
simply ranked as “inferior” and “superior” competitors (as in
the widely used lottery model; see Snyder and Chesson, 2003),
we simulate competition as the outcome of several mechanisms operating at the individual level. However, competition
is designed to allow flexibility in theory testing, and thus, does
not include empirically linked physiological details that can be
found in classical “gap” models (see Kobe et al., 1995; Pacala
et al., 1996) or detailed neighbourhood models (Bauer et al.,
2004).
2.2.
Implementation
FORSITE is based on the modelling platform CORMAS
(Bousquet et al., 1998) using the VisualWorks® environment.
This platform allows flexible designs, and allows implementation of individual-based stochastic dynamics in a spatial,
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dynamical context. The code is available upon request by contacting the corresponding author.
2.3.
State variables and scales
The model includes two hierarchical levels: individual trees
and habitat cells. Individuals are defined by the following
state variables: identity number, age, stage, and height. A tree
smaller than a recruitment height threshold is defined as a
“seedling”, and then becomes a “sapling” after recruitment;
when it reaches a reproduction height threshold it becomes an
“adult”. Trees that do not reach reproduction threshold after
a given time are defined as “krummholtz”.
Individual trees are distributed on a homogeneous closed
grid composed of hexagonal habitat cells. The choice of cell
diameter depends on the modelled system. In the present
study, it defines the scale of the competitive neighbourhood,
which we fixed at 20 m (i.e., the size of a large canopy tree
crown).
In its current version, FORSITE does not account for exogenous spatial heterogeneity. However, its structure allows for
this to be easily included.
Hereafter, we characterize tree population by its size and
stage structure.
2.4.
Process overview and scheduling
The model proceeds in annual time steps. Within each year
(Fig. 1) every tree consumes a theoretical resource from its
habitat, grows, reproduces (if it has reached sexual maturity),
and may die. Resource uptake is a function of the density
of competing individuals within the habitat cell; growth is a
function of resource uptake, and affects reproduction ability
through fecundity and dispersal distance. Death may occur
due to stresses from slow growth caused by competition or
may also occur at some constant rate due to other causes.
2.5.
Design concepts
With individual-based models, one can study dynamics at the
population level that emerge from, and are driven by, interactions defined at the individual level (Grimm and Railsback,
2005). Such emergence occurs in FORSITE as population
dynamics emerge from the behaviour of individuals, which
are entirely represented by empirical rules based on ecological mechanisms. These mechanisms interact mostly through
density-dependent, nonlinear, processes occurring in a spatial arena. The density of competing neighbours slows down
tree growth by reducing available resources, which increases
the individual probability of dying and decreases the fecundity. Unlike earlier IBMs of plant competition (e.g., Huston and
Smith, 1987), FORSITE takes dispersal explicitly into account.
Tree height also interacts with dispersal distance, resulting in
an interaction between local density and global distribution of
trees.
Another degree of complexity is introduced through
stochasticity: several individual parameters (Table 1) are normally distributed around a mean defined at the population
level, while mortality and dispersal distances are interpreted
as probabilities.
To analyse the model, different levels of observation
can be defined. For the present analysis, we recorded only
population-level variables (total size and stage-class sizes)
over time.
2.6.
Initialization
Different initializations can be used, combining
i. Homogeneous or heterogeneous distribution of individuals
over the grid, defined by the number initially present in
habitat cells.
ii. Stage structure of the initial population: seeds or adults at
different proportions.
2.7.
Inputs
Apart from the initialization process, no process is “forced”
during simulations and all population dynamics arise endogenously from individual interactions.
2.8.
Submodels
2.8.1.
Resource uptake/competition
A tree is affected by competition with its neighbours for
limited resources. For convenience, we define the maximum
resource available as equal to 1 (minimum = 0). The resource
available r for a given tree is affected by the density of taller
individuals (NH>Hi ) through the parameter h, and by the density of shorter individuals (NH≤Hi ) through the parameter s:
r = e−C
(1)
with
C=
NH≤Hi
NH>Hi
+
h
s
(2)
Parameters h and s allow us to vary the strength and
type of competition. The case s = h defines absolute symmetrical competition (Weiner, 1990) in which the resource
is equally partitioned among individuals irrespective of their
size, whereas cases where s is greater than h define asymmetrical competition where an individual is less affected by
smaller individuals than by larger individuals. Note that a
change in both h and s values may be used to simulate a change
in habitat quality (thus, changing tree resource uptake), but
for the present study we consider habitat quality as homogeneous, and changing one parameter while keeping the other
constant is best interpreted as a change in competitive effects.
2.8.2.
Growth
Growth follows a Gompertz function (Moravie and Robert,
2003; Zeide, 2004) defining the annual height increment g as:
g = RH ln
H
max
H
r
(3)
where H is the tree height, Hmax the tree maximum height, r
the available resource and R the tree growth rate at maximum
resource.
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Table 1 – Overview of the model parameters and default values used in the present analysis
Level
Parameter
Description
Mean value
Model
x
Nc
Cell diameter
Number of habitat cells
20
400
Species
R
Hmax
Growth rate (m yr−1 )
Maximum height (m)
0.1 (±0.01)
25 (±2.5)
h
s
M0
Md
Mc
Hrep
fmax
p
˛
c
ˇ
Strength of taller tree competition
Strength of shorter tree competition
Probability of mortality at zero growth
Decay of growth-dependent mortality
Intrinsic mortality
Height threshold for reproduction (m)
Species maximum fecundity (seedlings yr−1 )
Probability of short-distance dispersal
Mode of the short dispersal kernel (m)
Exponent in the height/dispersal function
Mode of the long dispersal kernel
Defined by D and c
Distance increment (m) by height increment for
the height/dispersal function
100
100
0.7
0.8
0.01
2
5
0.95
10
1.5
(500)
Individual
tree
D
Reference
(1)
(1)
(2)
(3)
(4)
(4)
(4)
(4)
2.5
References: (1) Zeide, 2004, (2) Pacala et al., 1996, (3) Keane et al., 2001, (4) Higgins and Cain, 2002.
2.8.3.
Reproduction
speed):
A tree becomes adult if it reaches a threshold height (Hrep ).
An adult tree produces f seeds. The individual fecundity f is
defined for a given step t as:
f = fmax
H
Hmax
(4)
where fmax is the species maximum fecundity.
Each seed is dispersed to a given habitat cell according to
a dispersal kernel mitigated by the height of the seed source.
The dispersal kernel is defined as a double exponential
(mixed) kernel, with a typical fat-tailed shape that allows us
to take account of the rare but important long-distance events
(Clark et al., 1998, Higgins et al., 2000). The density of probability for a single seed is given for d (distance from the seed
source) by the following:
P(d) =
p (−d/˛) (1 − p) (−d/ˇ)
e
+
e
2˛
2ˇ
ˇ = [DH]c
(6)
where ˇ is the mean long-distance dispersal, and parameters
D and c are to be defined. The parameter ˛ (mode of the shortdistance dispersal kernel) is a constant.
We assume no directionality in dispersal (Higgins and
Richardson, 1996).
As individual height is affected by direct competition, from
Eqs. (4) and (6) we see that a dense neighbourhood decreases
individual fecundity as well as dispersal distances.
Seeds germinate the following year and start to grow from
an initial height of 2 cm following Eq. (1). Growth parameters
for seedlings follow the same rules as adults.
2.8.4.
Mortality
Trees die according to two processes:
(5)
The first term gives the probability of seeds being dispersed
at short-distance (proportion p of the seeds), while the second
represents long-distance dispersal (proportion (1 − p)). This
function was shown by Higgins and Cain (2002) to allow great
flexibility in dispersal modelling: with p = 1 one can simulate a
local dispersal case, whereas with p = 0 and b bigger than the
size of the grid one can simulate a global dispersal case (which
follows the historical occupancy model’s assumptions, Adler
and Mosquera, 2000).
In the case of long-distance dispersal, the actual dispersal distance follows a power–law relationship to the
seed-source height (which is the relationship found by Greene
and Johnson, 1996, between wind speed and tree height; we
assume that dispersal distance is directly linked to wind
i. Intrinsic mortality represents all causes of mortality which
are not related to growth (senescence, disturbances occurring at the individual level; Keane et al., 2001). Each
individual dies with a constant probability Mc per year.
ii. Growth dependent mortality affects trees that grow too
slowly. This effect has been found in forest ecosystems due
to limits in the individual ability to cope with suppressed
growth (Keane et al., 2001):
m(g) = M0 e−Md g
(7)
where M0 is the probability of mortality at zero growth, Md
the decay of growth dependent mortality, g the annual height
increment, and m(g) the probability of death per year.
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
2.9.
495
Model analysis
We began with a sensitivity analysis in which we vary all
the parameters over a large range to assess the robustness
of the model’s qualitative behaviour. We also tested the sensitivity of the dynamics to the initial conditions. Then, for
each simulation presented we computed the coefficient of
variation in population size between 10 replicates (simulation
runs).
To assess the effects of competition type on long-term
population dynamics, we determined in what regimes the
parameters Nc , Md , h, and s produce oscillations in the dynamics. We investigated the effects of varying h and s since these
two parameters control whether competition is symmetric or
asymmetric. We investigate the effects of varying Md because
it appears to be a key parameter differentiating species in
northern forests (Pacala et al., 1996). We started with a nonspatial context (Nc = 1), with Md taking its default value (0.8)
and h = s = 800, characterizing pure symmetric competition.
The Md value was chosen as an “intermediate” decay compared to field values found by Pacala et al. (1996); h and s values
were chosen in order to produce a high number of trees in
the population, thus, avoiding stochastic extinction. Then we
manipulated each parameter: Md taking values 0.25 and 0.8,
Nc taking values 50 and 250, h taking values 200 and 800, and s
taking values 800, 2000 and 40,000. Note that except for Nc , Md ,
h and s, all parameters took their default value as expressed
in Table 1.
To help visualize individual dynamics in the model, we
plotted individual resource intake r over time for a cohort of
60 individuals chosen in one grid cell (cf. Fig. 6.16 in Grimm
and Railsback, 2005).
Finally, for illustrative purpose we tested the effect of the
short-distance dispersal probability p on the population spatial structure, for both competition types. For that we plotted
Moran’s I correlograms (Ellingson and Andersen, 2002) of tree
density obtained with a specific simulation, in which the trees
colonize an empty 5 × 50 cells grid from a single initial seed
source. With p = 1 (cf. Eq. (5)) trees disperse only at short-
Fig. 2 – One thousand years time-series of the adult
population with different initial conditions. Thick grey line:
initialization of 200 adults, thin black line: initialization of
20 seeds. Nc = 10, Md = 0.25, (a) h = 200 and s = 40,000, and (b)
h = 800 and s = 800.
distance, and we expected to find a different spatial structure
than the one produced with long-distance dispersal events.
3.
Results
3.1.
Model behaviour
FORSITE exhibits a strong robustness in terms of qualitative
response to changes in parameters or initialization though
non-robust behaviour was observed when the simulated population was too small, producing stochastic extinction events.
In the case of alternative initialization, after a transient phase
of about 100 years the dynamics were very similar, as can
be seen in Fig. 2. In the case of asymmetric competition
(Fig. 2a), populations obtained with different initial density are
almost indistinguishable, whereas with symmetric competition (Fig. 2b), the behaviour is similar but the two curves are
out of phase. The simulations presented here all show a strong
quantitative robustness as the average coefficient of variation
in the number of adult trees, computed for 10 repetitions of
the same parameter set, equals 2%.
3.2.
Fig. 1 – Annual life history of the model trees (activity
diagram). Actions (large letters) depend on variables (in
brackets).
Effects of parameters on population dynamics
At constant grid size Nc , mortality decay Md and taller tree
effect h, a change in the effect of smaller individuals s dras-
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e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
Fig. 3 – Effect of model parameters on 1000-year
time-series of the adult population. Each curve is an
average of 10 simulation runs on a 5 × 10 grid (Nc = 50).
Thick grey line: symmetric competition (h = 100, s = 100,
Md = 0.8); dashed line (top): asymmetric competition
(h = 100, s = 40,000, Md = 0.8); thin black line: asymmetric
competition with low h (h = 20, s = 40,000, Md = 0.8) and
dotted line: symmetric competition with low mortality
decay (h = 100, s = 100 and Md = 0.2).
tically changes the dynamics exhibited by the time-series
(Fig. 3). An increase in s produces an increase in population size (by reducing the competition), but also changes the
qualitative dynamics of the model. When s = h (pure competition symmetry) the population exhibits strong oscillations,
whereas for very large values of s relative to h (high competition asymmetry) the population appears to fluctuate randomly
about a steady-state. The same behaviour can be found when
decreasing h so that the population average level remains
similar between the symmetric and asymmetric case (Fig. 3).
In every case time-series begin with a transient dynamics
exhibiting strong oscillations, but this phase does not last
more than 100 years in the case of asymmetric competition.
Changes in Md conserve the effect of competition symmetry,
but very low values of Md decrease the population level and
the oscillations exhibited by the time-series (Fig. 3).
The stage structure (Fig. 4), in the case of symmetric competition, is characterized by a high number of seeds, while
saplings exhibit very low densities, reaching zero for long periods of time. Peaks of krummholtz and saplings coincide with
the beginning of each seed source cycle, while peaks in seed
density follows closely peaks in adult density. The same ratios
between the densities of different stages can be observed
for asymmetric competition (not shown), although no cycles
occur in that case.
The dynamics of one cohort (equal age trees within one cell,
Fig. 5) exhibits a difference between the cases of symmetric
and asymmetric competition. In the case of asymmetric competition (Fig. 5, top), the resource available for one individual
increases with time and reaches a plateau, until the individual
dies. After frequent deaths in the first stages, tree deaths are
randomly distributed in time. In the case of symmetric competition (Fig. 5, bottom), on the other hand, individual resource
decreases when the adult population is at a peak, and then
Fig. 4 – Stage structure on 1000-year time-series of the
adult population with symmetric competition. Each curve
is an average of 10 simulation runs on a 5 × 10 grid (Nc = 50).
Thick black line: adults; thick grey line: seedlings; dashed
line (bottom): krummholtz; thin black line: saplings and
dotted line (top): seeds.
increases when the adult population decreases. Because of
symmetric competition, all individuals within one cell receive
the same amount of resource. Death events coincide with
rapid declines in adult dynamics.
3.3.
Spatial patterns
The spatial spread of the trees from one initial source can
be seen in snapshots of the grid (Fig. 6) obtained with two
Fig. 5 – Evolution of resource uptake within one cohort of
60 individuals, with asymmetric competition (top) and
symmetric competition (bottom). The thick black line is a
rescaled adult time-series. Thin lines are individual
resource uptakes, reaching 0 when the individual dies.
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
497
Fig. 6 – Snapshots of the model spatial grid (Nc = 250) characterized by their Moran’s I correlograms. Spatial grids (top): adult
trees (grey dots) colonize the grid from an initial source on the left side. Cell darkness increases with tree density. (a and b)
With long-distance dispersal (p = 0.95); (c and d) short-distance dispersal only (p = 1); (a and c) t = 100 years; (b and d) t = 2000
years. Correlograms (bottom): on x-axis, number of 20 m distance lags; on y-axis, Moran’s I value; solid circles denote
statistically significant points (>0.95%).
modes of dispersal (long-distance with p = 0.95, Fig. 6a and
b; short-distance with p = 1, Fig. 6c and d), after 100 years
and 2000 years of simulation (respectively, a and c, and b
and d), with symmetric competition. Below each snapshot
is shown the corresponding Moran’s I correlograms of adult
density.
In the case of long-distance dispersal, at t = 100 years
(Fig. 6a), the population has spread further from the point
source than for the case of short-distance dispersal at t = 100
years (Fig. 6c). The Moran’s I values of adult density reflect
these observations and are typical of a clustered process,
with values decreasing along distance (going from significantly positive to significantly negative numbers). The flat
curve in Fig. 6c is produced by a large part of the grid being
empty.
In the case of long-distance dispersal, after 2000 years
(Fig. 6b) the population appears to be almost homogeneously
distributed across the grid. No significant autocorrelation can
be found in the adult population and the Moran I values are
flat. However, in the case of short-distance dispersal after 2000
years (Fig. 6d), there is significant spatial heterogeneity. The
Moran’s I values exhibits cycles, oscillating between positive
and negative values over distance, suggesting that neighbouring regions on the grid tend to be negatively correlated in
terms of adult density. This is confirmed by visual inspection
of the grid snapshot (Fig. 6d).
With asymmetric competition (not shown), we found the
same patterns except for the case with short-distance dispersal after 2000 years, as the Moran’s I values do not exhibit
cycles.
4.
Discussion
FORSITE is a flexible but robust model of forest dynamics that
captures how variation in individual life-traits gives rise to differing long-term dynamics. Its level of detail is intermediate
between models designed for testing broad ecological theories
(e.g., occupancy models, see Chave et al., 2002) and models
which represent in great detail the plant–environment interactions to simulate real ecosystems (e.g., SORTIE, Pacala et al.,
1996). It is, strictly speaking, an individual-based model (sensu
Grimm and Railsback, 2005), as no variable is forced at the
global scale.
In the present study, we focused on the effects of competition type on long-term population dynamics, within a
single-species system. In our simulations, a tree population
characterized by asymmetric competition (s > > h) exhibits relatively small fluctuations about a steady-state after a short
transient dynamics, at all values of the mortality parameters.
By comparison, a population subject to symmetric competition (s ∼ h) exhibits strong, regular oscillations in population
size when mortality is low (high mortality decay Md ), although
oscillations disappear at large mortality values (low Md ). These
results confirm the predictions of Pacala and Weiner’s (1991)
analytical model while bringing additional insight into the
conditions that determine the various behaviours for this type
of system. Simulating the growth of a monoculture of annual
plants, Pacala and Weiner (1991) found that asymmetric competition results in stable population dynamics, regardless of
the magnitude of competition strength or functional form
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e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
(that is, the form of the growth rate decay when competition increase). On the other hand, they showed that with
exponential growth rate decay and “sufficiently large” growth
rate, symmetric competition results in oscillations in population size. Though designed for trees (thus, with overlapping
generations), our simulations confirm these results, since we
show how high individual mortality (which would produce
a low population growth rate in Pacala and Weiner’s model)
prevents the population from oscillating even in the case of
symmetrical competition. However, our use of an individualbased approached allows incorporation of mechanistic
details underlying this result, through the analysis of stage
structure.
In the case of symmetric competition with low growthrelated mortality, the number of seedlings can accumulate
rapidly along with the number of adult trees (Fig. 4). This
creates very high competitive pressure (decrease in resource
uptake in Fig. 5), thus, slowing down the growth of all individuals but seedlings in particular. Because growth-dependent
mortality is low, while the rate of adult replacement is also
very low, the density of seedlings still increases since adults
still produce seeds. This large cohort of seedlings prevents the
development of saplings due to growth limitations from the
competition. Therefore, when the adults start to die off due
to intrinsic mortality (which can be seen in Fig. 5), there are
no saplings to replace them and it takes some time before
the seedlings becomes saplings and hence adults. This causes
the steep decline in adult population size. Once the saplings
become adults, the population of adults starts to increase
again and the cycle repeats. This can be seen as an example of
delayed density dependence which is known to induce oscillations in various model systems (Vandermeer, 2006). When
growth mortality is high, seedlings die in sufficient numbers to
make room for saplings to provide a steady source of replacements for dying adults. Hence, no oscillations are observed.
By comparison, in the case of asymmetric competition,
seedlings do not compete as strongly with saplings, hence,
the saplings can persist. When the adults start to die due to
intrinsic mortality, they can again provide a ready source of
replacement.
It is interesting to compare these results with those of
Bauer et al. (2002) who found the same relationships between
cycle dampening and high mortality within a plant model. In
their model, a “field-of-neighbourhood” (based on basal area)
allows for taking into account the strength of competition
exerted on one plant when it overlaps with other individual’s
area. Although this mechanism produces asymmetric competition at the population level (Berger and Hildenbrandt, 2000),
it produces larger influence on small individuals than would
pure competition asymmetry. We believe this leads to a mechanism similar to the one we describe, since high mortality
increases the regulation of small individuals, limiting density
dependence exerted on taller ones.
The implication of temporal cycles on spatial pattern is
revealed in the analysis of the spatial structure exhibited by
a spreading population characterized by symmetric competition (Fig. 6). When different sub-populations (distinguished
by the time of settlement) are desynchronized due to the
absence of long-distance dispersal, their desynchronized temporal cycles result in waves in the spatial structure: while
one sub-population reaches its peak, neighbouring populations may be at their minima. In comparison, for the case
of long-distance dispersal, spatial structure is only visible
during the first few decades of simulation (Fig. 6a). The grid
is quickly saturated and the whole population is synchronized and constant over time, resulting in a flat spatial signal
(Fig. 6b), which was found also in the case of asymmetric
competition. Besides revealing FORSITE’s abilities to simulate
spatial dynamics, this analysis shows how a spatial pattern
can be caused by a temporal process. It confirms the importance of long-term studies and careful approaches in pattern
analysis.
4.1.
Model limitations and perspectives
FORSITE is characterized by simple relationships. We have
chosen to design the model with flexible functions instead
of detailed – and perhaps more taxon-specific – mechanisms.
For this reason its application to nonforest ecosystems would
require only a different parameterization of the submodels.
For instance, for simulating competition light incidence and
tree crown structure are not explicitly taken into account as
they are in other models (Pacala et al., 1996; Bauer et al.,
2004). Hence, the two-parameter competition approach allows
simulation of a wide range of behaviours at different spatialscales. Other mechanisms (e.g., disturbances) that have been
shown to be essential in plant dynamics (especially in forest ecosystems) have not been included in this version of
FORSITE, but their inclusion is under study. We chose to simulate growth-dependent mortality with an exponential decay
as used in SORTIE (Pacala et al., 1996). However, we tested in
a preliminary study other approaches like growth thresholddependent mortality used in several models (Grimm and
Railsback, 2005), but did not find a significant difference in
the model outcomes. Also, though less detailed in terms of
individual mechanisms than other models (e.g., Jabowa/Foret
used in Huston and Smith, 1987), FORSITE details dispersal
processes more thoroughly than most of them. Particularly,
the integration of long-distance dispersal events should allow
simulating large scale colonization dynamics, which is of particular interest when considering that most spatially explicit
individual-based models are limited to the stand scale (Gratzer
et al., 2004).
Another limiting assumption in the present study is the
absence of spatial heterogeneity. Though it would require a
specific study in the future, we can predict from our analysis of the spatial structure that environmental heterogeneity
resulting in desynchronized populations (i.e., with non-viable
gaps larger than dispersal events) would decrease oscillations at large scale. However, as it has often been shown
that extremely weak coupling (here, dispersal) is sufficient to
result in population synchronization (Vandermeer, 2006), it is
likely that oscillations could occur as soon as the conditions
needed in our model are reached (low mortality, symmetric
competition). The parameters used in the present study were
inspired by those used by Pacala et al. (1996) but the different
scales used by FORSITE and SORTIE did not allow for a direct
comparison. As we focused on qualitative results, within a theoretical approach, this low level of realism does not appear
to be an issue. However, due to FORSITE’s flexible structure
e c o l o g i c a l m o d e l l i n g 2 1 1 ( 2 0 0 8 ) 491–500
it is possible to contemplate the integration of field-based
parameters.
One could ask of the realism of the competitive relationships portrayed here, as many forest models, although much
more detailed than FORSITE, do not take into account the
issue of competition type. Symmetric competition in plants
seems rare (Weiner, 1990), but it can still be observed. It
has been demonstrated that whereas competition for light
in plants is asymmetric, belowground competition (between
roots for nutrient uptake) is often symmetric (Berntson and
Wayne, 2000) or inversely asymmetric (i.e., smaller individuals consume a larger portion of resource than taller ones
Schenk, 2006). Describing competition type through “per
gram” effects (how resource depletion is dependent on size),
Kochy and Wilson, 2000) suggest that within the same growth
form, competition type is likely to be similar, while different growth forms exhibit strong differences. They show how
this phenomenon is of special importance for the study of
forest/grassland ecotones, where woody plants are advantaged by their size for light competition, while herbaceous
species are advantaged by their root system. Consequently,
tree seedlings compete with herbaceous plants in a symmetric mode, whereas adult trees compete with and herbaceous
plants asymmetrically. This idea brings an exciting prospect
of what can be done with FORSITE to study more realistic
dynamics. Furthermore, even if pure symmetric competition
is unlikely in plants (especially taking seeds into account),
it is more likely in animal populations between juveniles
and adults, since juveniles require significant resources and
parental care. FORSITE is designed for modelling tree interactions, but the consequences of competition type detailed here
can probably be generalized over different biological types.
For instance it is well known that predator–prey relationships can produce strong oscillations in animal populations
(Vandermeer, 2006) and it would be interesting to know how
the type of competition/interaction could affect these dynamics.
The appearance of cycles in ecological models and data is
indeed widespread. Most studies, like the one here, focus on
the causes for the appearance of cycles. However, the ecological consequences of cycles in population dynamics are
numerous when considering disturbances and interspecific
interactions, but have not been given much attention. Oscillations can be especially critical for biological invasions, as
they can provide a periodic opportunity for an invasive species
to enter a system (Vandermeer, 2006; Caplat, Anand, Bauch,
unpublished data). We believe that emphasizing key structuring mechanisms, associated with the proper interactions
(Clark et al., 2004) offers exciting perspectives on such phenomena.
Acknowledgements
We gratefully acknowledge funding from the Natural Sciences
and Engineering Research Council of Canada (NSERC) (M.A.
and C.B.), Canadian Foundation for Innovation, Ontario Ministry for Research and Innovation, Inter-American Institute
for Global Change Research and the Canada Research Chairs
program (M.A.).
499
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