# Coexistence theory | Wikipedia audio article

Coexistence theory is a framework to understand
how competitor traits can maintain species diversity and stave-off competitive exclusion
even among similar species living in ecologically similar environments. Coexistence theory explains
the stable coexistence of species as an interaction between two opposing forces: fitness differences
between species, which should drive the best-adapted species to exclude others within a particular
ecological niche, and stabilizing mechanisms, which maintains diversity via niche differentiation.
For many species to be stabilized in a community, population growth must be negative density-dependent,
i.e. all participating species have a tendency to increase in density as their populations
decline. In such communities, any species that becomes rare will experience positive
growth, pushing its population to recover and making local extinction unlikely. As the
population of one species declines, individuals of that species tend to compete predominantly
with individuals of other species. Thus, the tendency of a population to recover as it
declines in density reflects reduced interspecific competition (between-species) relative to
intraspecific competition (within-species), the signature of niche differentiation (see
Lotka-Volterra competition).==Types of coexistence mechanisms==
Two qualitatively different processes can help species to coexist: a reduction in average
fitness between species or an increase in niche differentiation between species. These
two factors have been termed equalizing and stabilizing mechanisms, respectively.
Equalizing mechanisms reduce fitness differences between species, or relative competitive ability
in the absence of niche differentiation. As its name implies, it works by making similar
species more equal in their competitive ability. For example, when multiple species compete
for the same resource, competitive ability is determined by the minimum level of resources
a species needs to maintain itself (known as an R*, or equilibrium resource density).
Thus, the species with the lowest R* is the best competitor and excludes all other species
in the absence of any niche differentiation. Any factor that reduces R*s between species
(like increased harvest of the dominant competitor) is classified as an equalizing mechanism.
For species to coexist, fitness differences must be overcome by stabilizing mechanisms.
Stabilizing mechanisms promote coexistence by concentrating intraspecific competition
relative to interspecific competition. There are large number of named stabilizing mechanisms
including classical hypotheses of species coexistence. Resource partitioning, whereby
interspecific competition is reduced because species compete primarily through different
resources, is a stabilizing mechanism. Similarly, if species are differently affected by environmental
variation (e.g., soil type, rainfall timing, etc.), this can create a stabilizing mechanism
(see the storage effect). Stabilizing mechanisms increase the low-density growth rate of all
species.A general way of measuring the effect of stabilizing mechanisms is by calculating
the growth rate of species i in a community as r i ^=b i ( k i − k
^ +
A ) {\displaystyle {\hat {r_{i}}}=b_{i}(k_{i}-{\hat
{k}}+A)} where: r i ^ {\displaystyle {\hat {r_{i}}}}
is the long-term average growth rate of the species i when at low density. Because species
are limited from growing indefinitely, viable populations have an average long-term growth
rate of zero. Therefore, species at low-density can increase in abundance when their long-term
average growth rate is positive. b i {\displaystyle b_{i}}
is a species-specific factor that reflects how quickly species i responds to a change
in competition. For example, species with faster generation times may respond more quickly
to a change in resource density than longer lived species. In an extreme scenario, if
ants and elephants were to compete for the same resources, elephant population sizes
would change much more slowly to changes in resource density than would ant populations. k i − k
^ {\displaystyle k_{i}-{\hat {k}}}
is the difference between the fitness of species i when compared to the average fitness of
the community excluding species i. In the absence of any stabilizing mechanisms, species
i will only have a positive growth rate if its fitness is above its average competitor,
i.e. where this value is greater than zero. A {\displaystyle A}
measures the effect of all stabilizing mechanisms acting within this community.===Example calculation: Species competing
for resource===In 2008 Chesson and Kuang showed how to calculate
fitness differences and stabilizing mechanisms when species compete for shared resources
and competitors. Each species j captures resource type l at a species-specific rate, cjl. Each
unit of resource captured contributes to species growth by value vl. Each consumer requires
resources for the metabolic maintenance at rate μi.In conjunction, consumer growth is
decreased by attack from predators. Each predator species m attacks species j at rate ajm.
Given predation and resource capture, the density of species i, Ni, grows at rate 1 N j d N j d
t=∑ l c j
l v l R l − ∑ m a j
m P m − μ j {\displaystyle {\frac {1}{N_{j}}}{\frac {dN_{j}}{dt}}=\sum
_{l}c_{jl}v_{l}R_{l}-\sum _{m}a_{jm}P_{m}-\mu _{j}} where l sums over resource types and m sums
over all predator species. Each resource type exhibits logistic growth with intrinsic rate
of increase, rRl, and carrying capacity, KRl=1/&alpha;Rl, such that growth rate of resource
l is 1 R l d R l d
t=r l R ( 1
− α l R R l ) − ∑ j c j
l N j . {\displaystyle {\frac {1}{R_{l}}}{\frac {dR_{l}}{dt}}=r_{l}^{R}\left(1-\alpha
_{l}^{R}R_{l}\right)-\sum _{j}c_{jl}N_{j}.} Similarly, each predator species m exhibits
logistic growth in the absence of the prey of interest with intrinsic growth rate rPm
and carrying capacity KPm=1/αPm. The growth rate of a predator species is also increased
by consuming prey species where again the attack rate of predator species m on prey
j is ajm. Each unit of prey has a value to predator growth rate of w. Given these two
sources of predator growth, the density of predator m, Pm, has a per-capita growth rate 1 P m d P m d
t=r m P (
1 − α m P P m )
+ ∑ j w N j a j
m {\displaystyle {\frac {1}{P_{m}}}{\frac {dP_{m}}{dt}}=r_{m}^{P}(1-\alpha
_{m}^{P}P_{m})+\sum _{j}wN_{j}a_{jm}} where the summation terms is contributions
to growth from consumption over all j focal species. The system of equations describes
a model of trophic interactions between three sets of species: focal species, their resources,
and their predators. Given this model, the average fitness of a
species j is k j=1 b j ( ∑ l
=1 c j
l v l K l R − ∑ m a j
m K m P − μ i ) {\displaystyle k_{j}={\frac {1}{b_{j}}}\left(\sum
_{l=1}c_{jl}v_{l}K_{l}^{R}-\sum _{m}a_{jm}K_{m}^{P}-\mu _{i}\right)} where the sensitivity to competition and predation
is b j=( ∑ l c j
l 2 v l K l R r l R + ∑ m a j
m 2 w K m P r m P ) . {\displaystyle b_{j}={\sqrt {\left(\sum _{l}{\frac
{c_{jl}^{2}v_{l}K_{l}^{R}}{r_{l}^{R}}}+\sum _{m}{\frac {a_{jm}^{2}wK_{m}^{P}}{r_{m}^{P}}}\right)}}.} The average fitness of a species takes into
account growth based on resource capture and predation as well as how much resource and
predator densities change from interactions with the focal species.
The amount of niche overlap between two competitors i and j is ρ
=( ∑ l c i
l v l c j
l α l R r l R + ∑ m a i
m w a j
m α m P r m P ) / b i b j , {\displaystyle \rho=\left(\sum _{l}{\frac
{c_{il}v_{l}c_{jl}}{\alpha _{l}^{R}r_{l}^{R}}}+\sum _{m}{\frac {a_{im}wa_{jm}}{\alpha _{m}^{P}r_{m}^{P}}}\right)/b_{i}b_{j},} which represents the amount to which resource
consumption and predator attack are linearly related between two competing species, i and
j. This model conditions for coexistence can
be directly related to the general coexistence criterion: intraspecific competition, αjj,
must be greater than interspecific competition, αij. The direct expressions for intraspecific
and interspecific competition coefficients from the interaction between shared predators
and resources are α j
j=s j / k j {\displaystyle \alpha _{jj}=s_{j}/k_{j}} and α i
j=
ρ s j / k i . {\displaystyle \alpha _{ij}=\rho s_{j}/k_{i}.} Thus, when intraspecific competition is greater
than interspecific competition, α j
j>α i
j=s j k j>
ρ s j k i {\displaystyle \alpha _{jj}>\alpha _{ij}={\frac
{s_{j}}{k_{j}}}>\rho {\frac {s_{j}}{k_{i}}}} which, for
two species leads to the coexistence criteria ρ