Download Modeling species fitness in competitive environments

Ecological Modelling 275 (2014) 31–36
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Ecological Modelling
journal homepage: www.elsevier.com/locate/ecolmodel
Modeling species fitness in competitive environments
Nina Šajna a , Primož Kušar b,∗
a
b
Department of Biology, FNM, University of Maribor, Koroška c. 160, SI-2000 Maribor, Slovenia
Department of Complex Matter, Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
a r t i c l e
i n f o
Article history:
Received 26 June 2013
Received in revised form 2 December 2013
Accepted 4 December 2013
Available online 8 January 2014
Keywords:
Fitness
Competition modeling
Resource accumulation
Coexistence
Multiple resource competition
a b s t r a c t
Using a model of resource acquisition, we studied species competition in a case where resources limit
population growth. Our model is based on calculations of the distribution of individuals of single or
multiple species over consumed resources. Calculations show that, as equilibrium is reached in purely
resource competitive systems, the density of resources is lowered to the lowest sustainable level, directly
leading to low levels of fitness among species. In the case of competition between species with different
lowest sustainable levels, the density of the more successful must be limited by some cause other than
the resource in question for all species to coexist. We explore two cases of such coexistence.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Competition for resources between subjects of the same species
and between species is one of the most important factors in ecology
and evolution. With competition, many evolutionary and ecological questions can be addressed. The ability of species to compete
increases their chance of survival in the existing local ecosystems
(Begon et al., 2005) and on the global scale influencing evolution
directly (Darwin, 1859). Many numerical models are used for modeling the competition, such as the basic Lotka–Volterra equations
(Volterra, 1931; Lotka, 1932) describing predation and competition,
the Monod model (Monod, 1942, 1950; Herbert et al., 1956) used
to describe different species competing for the same resources and
Droop’s model describing the growth of populations where nutrient quantities are growth limiting factors (Droop, 1974, 1975). All
of these models have been widely used and improved for more realistic use in specific cases: for example, to study competition under
multiple nutrient limitation (Tilman, 1982; Cherif and Loreau,
2010) and stability of ecological systems (Tilman, 1996; Huisman
and Weissing, 1999; Lehman and Tilman, 2000; Mougi and Kondoh,
2012). The above models use the density of competing species and
resources as observed/modeled variables and describe the dynamics of the system by coupled differential equations between them.
Co-existence and biodiversity has been studied as a function of
number of limiting resources (Tilman, 1982), temporal and spacial
gradient of parameters (Tilman, 1999; Lehman and Tilman, 2000)
∗ Corresponding author. Tel.: +386 51 412 050; fax: +386 1 477 39 98.
E-mail address: [email protected] (P. Kušar).
0304-3800/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ecolmodel.2013.12.007
or as a consequence of chaotic behavior of the models for certain
sets of parameters (Huisman and Weissing, 1999, 2002).
Alternatively there is a growing interest in individual-based
models, where individuals are followed and their behavior is
modeled depending on external parameters (DeAngelis and Mooij,
2005; Grimm et al., 2006; Railsback and Grimm, 2011; Martin et al.,
2013). This is especially suitable for modelling small populations
where we can follow individuals and their properties and consumption. Monte-Carlo methods are used to study system evolution
under different circumstances and population survival probability
can be studied.
In this article we propose a model that combines part of the benefits from both approaches. In this model we follow the temporal
evolution not only of a population density but also the distribution
of the population over successfully consumed resources. In this
way, we can also take into account the fitness of the population.
Information on fitness gives us further understanding of species
competitiveness in the ecosystem and possible vulnerabilities to
or advantages from ecosystem changes. We start by studying the
temporal behavior of a simple one-species system and proceed to
more complicated systems involving more species and resources.
2. Method
In presented model we follow the distribution of a population over resource consumption. Since sufficient consumption is
needed for survival and even more for successful reproduction,
this can be used as a measure of fitness (Begon et al., 2005). We
take into account that only a limited amount of resources can be
consumed and accumulated (Tilman and Kilham, 1976), although
32
N. Šajna, P. Kušar / Ecological Modelling 275 (2014) 31–36
luxury intake is possible. Another important point is that consumed
and accumulated resources are also used over time; therefore,
only consumption during a certain relevant time window (RTW)
is important for the current fitness of individuals. Throughout this
article the most recent time window is used as RTW. RTW should
be chosen long enough to account for the possibility of the fully
fed individual to starve to death in case of lack of resources and
on the other side to allow for the full starvation-recovery all the
way to the excess accumulation. Although different parameters
are defining fitness sufficient resource intake is crucial for specimen survival and reproduction ability and we use the amount of
consumed resources over the RTW for a measure of fitness in this
article.
We mark the amount of resources consumed by an individual
over RTW with K. To describe the fitness of the entire population,
we can use the density distribution of individuals over K (S = S(K)).
In the model, used population parameters are minimum resource
consumption Ks that still allows for survival of the individual,
although the fitness in this case is too low for reproduction; Kr is the
minimum resource consumption needed for successful reproduction of the individual, and Kmax the estimated maximum possible
consumption of resources over the RTW in case the resource is
abundant. Reproduction rate and decay rate of the species are
K dependent, and we use reproduction rate sr to describe the
reproduction of individuals with K > Kr , while the others do not
reproduce. All the individuals with consumption below Ks die, and
we use sd to describe the death rate of the others caused by aging
and external factors.
Mathematically, the problem of density distribution with ‘forgetfulness/use of accumulated resources’ is not easy to solve;
therefore, we use a distribution over N discrete values of the
amount of resource consumed (K) over the last RTW. In this way we
use time steps of t = RTW/N, and in every time step an individual
either consumes K = Kmax /N amount of resource or it does not. The
probability (P) of an individual finding and consuming the resource
depends on resource density F(t) and on the space-covering speed
of individual v – a species – dependent parameter describing the
amount of space (surface or volume) an individual can cover over
the time. If we simply assume that finding a resource is sufficient
for its consumption, the density of individuals that do not find the
resource in t can be calculated by integration of dS = −vS(t)F(t)dt
over t. Here we use S as the density of the individuals that have
not yet consumed the resource. For those that have already consumed the resource we, assume that they digest for the rest of the
t. Now the probability of finding and consuming K in the time
step can be calculated as:
P=
[S(t) − S(t + t)]
.
S(t)
(1)
Resource density F(t) depends on resource growth/inflow,
resource decay/outflow and resource consumption by individuals.
The temporal evolution of F can be calculated by integration of
dF = [fg − fd F(t) − KvS(t)F(t)]dt. The growth term fg depends on
the particular resource and can depend on density (reproduction,
growth) or on other resources. Since we are interested in cases
where competition takes place and most of the resources are in
quasi-equilibrium, we simply take it as a constant. We use linear
approximation fd F(t) for the decay/outflow term. This is useful
because it also limits the growth of resources to the values below
Fmax = fg /fd . In our calculation most of the resource decrease is
through consumption by the species being studied (the last term
in the equation), and we can expect fd F(t) to be small. We can
generalize our equations for multiple species and resources to
coupled differential equations:
j
j
dS i = −vi,j S i (t)Fj (t)dt
(2)
Fig. 1. Step diagram of species density calculation.
dF j =
−
j j j
j
j
Ki vi S i (t)Fj (t) + fg (t) − fd Fj (t)
dt
(3)
i
Here index i indicates species and index j resource. These equations
can be integrated over a selected time unit t, and the probability for an individual to consume the resource can be calculated.
j
Here we assumed v to be K independent and use S i to be the full
j
density of the species (initial S i =
S (K)), but the equations can
K i
also be expanded to use S for different K. Resource density F is continuous function of time and is initialized to the final value from
the previous step. The equations as stated above are for limiting
resources. When two resources (j , j ) can be exchanged, single disj j
tribution S i should be used and will account for consumption of
all exchangeable resources. Here we should point out that other
models for the probability of resource consumption can be used
with the rest of the calculation unchanged.
With probability calculated, we can calculate the next iteration
of consumed resource distribution. This is performed in multiple
steps, as shown in diagram in Fig. 1.
First, we calculate the density distribution for RTW + t (time
interval of (N + 1)t):
S (K, t) = PS(K − K, t) + (1 − P)S(K, t).
(4)
Now we must transform it back to distribution over RTW. For
this we need to know for every S(K) the part of it that consumed
K of resource RTW ago – c(K). By subtraction of this part, we again
have the distribution over the most recent RTW (last Nt):
S (K, t) = (1 − c(K))S (K, t) + c(K + K)S (K + K, t).
(5)
Finally, we account for those born and those that died:
S(K, t + t) = (1 − d(K))S (K, t) + b(K, t).
(6)
The death rate here depends on K (those with K < Ks starve to
death). In this paper we do not count the death rate due to external factors as K dependent, but implementation of this would be
straightforward. In the case of multiple resources, we assume no
correlation between consumption of different resources and we
calculate the deaths through lack of resources accordingly. The
birth rate, on the other hand, depends
on the subjects with K > Kr
giving us the full growth of b = sr K>K S (K, t). Again, no correr
lation is assumed between different resource distributions, and in
the case of multiple resources, we take into account that fitness
N. Šajna, P. Kušar / Ecological Modelling 275 (2014) 31–36
33
Table 1
Overview of symbols and parameters.
Symbol
Symbol meaning
S
RTW
K
S(K)
Kr
Ks
Kmax
Kbto
F
N
t
K
P
fg
fd
sr
sd
S
density of the population
relevant (memorized) time window
quantity of consumed resource over the last RTW
density of the population with K
minimum K needed for successful reproduction
minimum K needed for survival
maximal K over memorized RTW
initial K attributed to newly born
resource density
number of memorized time steps
time step used for calculation of consequent densities
model resource quantity consumed in used time step
probability of consuming K in used time step
growth of the resource
decay rate of the resource
birth rate of the fit part of the population
death rate of the population
density of the population that did not consume resource over t
individuals’s covering speed
part of the population that consumed resource RTW ago
growth of the population over t
v
c
b
through all resources must be sufficient to reproduce. In our model
we can choose the initial distribution of newly born individuals. It
can be chosen so that the relative vulnerability of offspring is taken
into account, with K only suitably larger than Ks .
Most of the above calculation steps are straightforward, except
for the step where we account for resource consumption RTW ago.
For this, we need information on what portion of individuals consumed the resource RTW ago (c) for every S(K). Since resource
density and species density change with time, consequently so does
consumption probability, death rate and birth rate, making information on resource consumption over the period of the relevant
past time hard to obtain. We calculate it by memorizing distributions over all possible paths of resource consumption (2N densities).
This way we know the distribution for every possible path leading to to current state and from that we can calculate the part of
the species density that consumed resource in the first memorized
step – c(K). Programmatically, it can be done by writing indices in
binary form with digits at the nth position representing resource
consumption (bit set) or non-consumption (bit is 0) nt ago. For
example S0011. . .011 did not consume any resource during the last
two time steps but it did in the two steps before that. It also consumed at steps N − 1 and N time steps ago. The sum of set bits
gives us K and S(K) can be calculated as a sum of all perturbations with K set bits. For next step we simply perform bit rotation
removing the last bit and setting the first one to 1 for those that consumed resources in the last step and to 0 for those that did not. For
example S0011. . .011 goes to either S10011. . .01 or S00011. . .01 . For every
S. . . we perform Eqs. (3)–(5). Because of 2N scaling, this method is
only useful up to N ≈ 25, owing to computational barriers. This is
nevertheless enough for qualitative study of competition (Table 1).
3. Results
The simplest case for our model involves a single-species
population competing for a single resource with constant growth.
In order for species to survive, the resource density must be
sufficiently high. This density has been marked as R* (Tilman,
1982, 1999). A rough estimate of R* in our model can be derived
from the assumption that fitness on average should be approximately Kr for the population to sustain itself at the same density,
giving us an estimate for the probability of individual claiming
the resource in every time step to be on average (P ≈ Kr /Kmax ).
Using this probability and a time independent resource density
Fig. 2. Intraspecific competition: (a) population density (dashed) and resource density (solid) as a function of time. (b) Resource consumption distribution (S(k, t) –
fitness) as a function of time. The line represents the average value of fitness.
(F(t) = R* ) in Eqs. (1) and (2) we calculate the minimal sustainable density of resource to be R∗ ≈ ln(1 − Kr /Kmax )/vt. We are
only interested in the case where density of resources without
consumption is higher than this (F > R* when S = 0). Accordingly,
fg is taken large enough and constant in time. Results of the
calculation are shown in Fig. 2. We start our calculation for
single species single resource with an initial population density lower than the maximal sustainable density of the system
with all the individuals fully fed (luxury). Unless noted, we use
following parameters: N = 21, sr = 0.03, sd = 0.005, fg =
10, fd = 0.03, v = 0.1, K = 1, Kmax = NK = 21, Ks =
0.3Kmax Kr = 0.7Kmax and Kbto = 8. The choice for offspring
distribution at birth (Kbto ) is K = 8 consumed (N = 21), with all
resources consumed in the oldest positions; therefore, after birth,
every time step t without resource consumption shifts the
individual to lower values of K (lower fitness), accounting for
vulnerability of offspring. In Fig. 2a we see natural growth in the
area where there is resource in abundance (at the beginning).
There all the individuals are fully fed and able to reproduce, and
all of the offspring have enough resource for growth (Fig. 2b).
This ends once the average resource reachable for individual
becomes too low for sustainable reproduction. As this happens
most of the adult individuals still have enough accumulated
resources to breed; therefore, species density increases even after
that point. However, simultaneously resource density drops below
R* consequently leading to decrease of the average fitness of the
individuals to levels below sustainable reproduction level, and
more individuals die than are born, causing a decrease in population density. Finally, after some damped oscillations in resource and
population density, equilibrium is reached. At this point the average fitness of the individuals is low, and reproduction is in balance
with mortality (Fig. 2b). The first dip in average fitness is a consequence of newborns, which are positioned at low fitness (Preuss
et al., 2009; Martin et al., 2013). Single-species single resource
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N. Šajna, P. Kušar / Ecological Modelling 275 (2014) 31–36
Fig. 4. Intraspecific competition for one to four limiting resources: (a) Distribution
of population over N = 1–4 limiting resources. All parameters are equal for different
resources. (b) and (c) Equilibrium distribution of species over resource consumption
for the case of two limiting resources (I and II) as a function of resource growth ratio
(fgII /fgI ) for resource I in (b) and resource II in (c). Lines are the average values (green
line – I, black line – II). (For interpretation of references to colour in this legend, the
reader is referred to the web version of this article.)
Fig. 3. Fitness of species in equilibrium as a function of the quotient of death rate
to birth rate for a single species, single resource case: (a) total density of population
and density of resources and (b) Fitness distribution of the species. Average fitness
of the population is shown with a green line. (For interpretation of references to
colour in this figure legend, the reader is referred to the web version of this article.)
competition eventually leads to a population with low fitness levels independently of the resource growth rate or initial density and
the population is consequently potentially vulnerable to external
factors like invasive species or climate change.
The dependance of fitness on the death-to-birth rate ratio sd /sr
is shown in Fig. 3. Here we can see that fitness increases in case of a
larger death-to-birth rate ratio. This rate in the model accounts for
natural deaths as well as predation rate. The lowering of population
through external factors might therefore increase the fitness of the
population, and some species even develop self limiting mechanisms like intra-specific aggression, cannibalism, and competition
for enemy-free space to lower their density (Berryman, 1999).
When there are more limiting resources for a population, its
individuals need to consume all the resources in sufficient quantities to survive and reproduce (Tilman and Kilham, 1976). In Fig. 4,
the results of our model for a single species and multiple limiting
resources are shown. In our model there is no correlation in fitness
distribution for different resources. The necessity of achieving sufficient fitness for each limiting resource decreases the density of
the population and leads to better fitness over every resource, as
shown for 1-4 limiting non-exchangeable, but otherwise equivalent resources in Fig. 4a. A similar result is shown in Fig. 4b and
c for two resources as a function of the ratio between the growth
of two resources (all other parameters are equal). If either of the
resources is more scarce, it is automatically growth-limiting, the
average fitness over this resource is lower and, in the case of relative
abundance of another resource, is brought to the lowest sustainable
fitness level. In our model this is equivalent to the lowest sustainable resource density R* ; consequently, the density of the population is limited by the more scarce resource. Consumption of the
more abundant resource is now limited by the density of species;
therefore, density of this resource is at a higher level than it would
be if it were the only limiting resource. Since the density of more
abundant resource is now higher than R* of the better competitor,
this might enable species with higher R* (weaker competitors) to
be maintained when the more scarce resource is not limiting for it
(Tilman, 1977). In Fig. 5a–c we show results of our model for this
Fig. 5. Near equilibrium distribution of S for two competing populations: (a)–(c)
two populations (1 and 2) share one limiting resource I, while population 2 also
depends on the additional limiting resource II but simultaneously has the advantage in acquiring the common resource (v2 /v1 = 1.5). (a) Distribution of population
1 over the common resource as a function of fgII /fgI . (b) Population 2 over common
resource and (c) over the resource II. Lines are the average consumption for population 1 (green) and 2 (black for common resource and grey for resource II). (d) and
(e) Two species (1 and 2) share one limiting resource I. The first has an advantage in
acquiring the resource (v1 /v2 = 2), but the second can substitute the resource I with
an alternate resource II. (d) Distribution of population 1 over resource I as a function
of fgII /fgI . (e) Distribution of population 2 over combined resources (I+II). Lines in (d)
and (e) are the relevant average consumptions, with a green line for population 1,
and black for 2.
N. Šajna, P. Kušar / Ecological Modelling 275 (2014) 31–36
case. While growth of a resource II – essential only to species 2 – is
low (lower values of fgII /fgI in Fig. 5a-c) it limits the density of this
species. And although this species is better competitor for resource
I (R2∗ < R1∗ ) species limited density allows for levels of a common
resource I at values higher than R1∗ and R2∗ consequently enabling
survival of a weaker competitor 1. But once the growth rate of the
resource II is sufficiently high species 2 will lover the density of the
common resource bellow R1∗ causing extinction of species 1.
Another possibility for stable coexistence – even for similar species having the same limiting resources – is developing
organs or mechanisms that enable competing species to consume
resources which are not easy to consume for the original species:
for example, developing a larger mouth in the case of bacterivorous
protist species (Violle et al., 2011). Although growing optimized
organs might lower the competitiveness of species for a particular
type of resource, it can in turn enable consumption of otherwise
non-reachable resources, again allowing stable co-existence of the
two species. The fitness of competing species (1 and 2) in this case
is shown in Fig. 5d and e as a function of the ratio between the
growth of resources reachable only by the species 2 (resource II)
to the growth of resources reachable by both (resource I), but with
species 2 being a weaker competitor for the common resource I.
While growth rate of the resource II is low it enables the survival of
species 2, while species 1 limits the density of the common resource
I. But once growth rate of the resource II is increased the density
of species 2 increases to the levels where its consumption of the
common resource simply by chance is sufficient to lower resource
density below the level needed for species 1 to survive.
4. Discussion
Growth of populations increases resource consumption, and if
there is no other growth limitation, resource density is eventually reduced to the level where not all the specimens can obtain
amounts sufficient to live normally and reproduce. Fitness of population is lowered to the level where the most successful population
can still sustain itself at non lowering density, although some
individuals do not reach the fitness level needed for successful
reproduction. The resource density that still enables equilibrium
of a species population has been defined as R* and is species- and
resource-dependent (Tilman, 1982, 1999). The growth dynamics
before this level is reached are shown in Fig. 2. For lower population densities, there is a natural/exponential growth, but before
equilibrium is reached, population density attains values higher
than sustainable density, which in turn brings the resource density below R* . This is possible because the resource is accumulated
while its consumption and growth are not yet in equilibrium and
because the fitness does not immediately represent the density of
the resource and still enables individuals to reproduce, although
already at the point where there are insufficient resources for all to
survive.
If more than one species is competing for the same resource,
all lower the resource density, bringing it down to the level of the
lowest R* and leading all species with higher R* to extinction. If
the species have similar R* , this might take a long time, but since
the resource density is approximately constant and below R* of
the weaker competitor, the fitness of this competitor is constant
and below sustainable level. This causes an exponential decay of
this species. In cases where competition over the same limiting
resources is the only growth-limiting factor, the competition is governed by the most scarce resource that is brought to the lowest
R* if other resources are abundant (see Fig. 4). However when all
resources are brought close to their R* , all resources need to be
taken into account, and the overall fitness of the species defines
growth dynamics. In this case our model shows its strength, since
35
having distributions of resource consumption enables us to define
necessary fitness with combined resource consumption. We use
the simplest method for uncorrelated resource consumption and
uncorrelated resource requirements, but the model can be modified for interdependence in different resource requirements and
even consumption if a suitable correlation is assumed. Coexistence
of different species competing for exactly the same resources is
possible if their R* s are the same, or in some quasi-stable condition
where densities are in exactly the right configurations to achieve
sustainable oscillations, however with each deviation of parameters taking the system out of the equilibrium. In order to have
stable coexistence, we need a system where densities of resources
are self-sustainably kept at levels higher than or equal to R* of all
species: for example, if two species differ from each other through
either non-common limiting resource or the ability of one species
to acquire resources not reachable by the other. In the first case
development of ability or organs adds requirements for additional
resources that, in turn, keep the density of this otherwise better
competitor low, consequently keeping the resources to levels sufficient for the other species to survive (Fig. 5a–c). In more complex
systems, this can also be caused by other external factors like predators, temperature etc. that keep the competitors at lower levels
(Estes et al., 2011). In the second case, development of specialized
organs might adversely affect the ability of species to compete but,
on the other hand, allow for substitutions involving resources not
reachable by the other species (for example, the Giraffe’s neck or
microorganisms in Violle et al. (2011)).
It has been shown that temporal variation in available resources
and a higher number of competing species increase system stability
(Lehman and Tilman, 2000; Mougi and Kondoh, 2012). Alternatively Huisman and Weissing (1999) showed that using existing
models quasi stable coexistence of many populations can be established through oscillations of population and resource densities.
With presented model we were unable to reproduce such oscillations with more populations than number of limiting resources.
The reason for this might be that the ‘memory’ effect of accumulated resources is damping the oscillations. The model, however,
shows that when the purely resource-competitive system is close
to equilibrium, the fitness of species is lowered to the lowest
sustainable level, potentially making the species more vulnerable to external events like disease and predators. Anyhow in
nature equilibrium is hardly ever achieved and inhomogeneity
of the environment and dispersal ability of the species can act
as the generator of stability in many species coexistence. Different conditions favoring different species enable survival for
all, while their dispersion enables their presence even in regions
where otherwise they could not survive locally. This can also be
modeled by presented model, since different conditions can be
modeled separately, simultaneously taking diffusion of species into
account.
The distribution in our model depends on all the parameters,
and while we can arbitrary control most of them, we are limited in
the choice of distribution size N because computational intensity
scales with N to the power of two. This limits the choice of the smallest resource intake K and time step t, and also the maximum
size of luxury intake Kmax . A careful choice of these parameters is
therefore needed. A small distribution size results in broader distribution (Kreyszig, 1993). However, in real systems distribution is
also broadened by species and environment variations for which
we do not account in our model, making the results of the model
somehow closer to real world distributions, although not in a fully
controllable way.
In conclusion, we propose a model for resource competition
that takes into account distribution of populations over consumed
resources. By calculation of consumption distribution, the fitness of
species can be modeled. We show that, although stable coexistence
36
N. Šajna, P. Kušar / Ecological Modelling 275 (2014) 31–36
of species is possible in competing ecosystems, systems where
competition is the governing growth mechanism will unavoidably
lower the average fitness of species close to the lowest sustainable
level.
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