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, c_jl. Each unit of resource captured contributes to species growth by value v_l. Each consumer requires resources for the metabolic maintenance at rate μ_i.^[8]

In conjunction, consumer growth is decreased by attack from predators. Each predator species m attacks species j at rate a_jm.

Given predation and resource capture, the density of species i, N_i, grows at rate

${\displaystyle {\frac {1}{N_{j}}}{\frac {dN_{j}}{dt}}=\sum _{l}c_{jl}v_{l}R_{l}-\sum _{m}a_{jm}P_{m}-\mu _{j}}$

where l sums over resource types and m sums over all predator species. Each resource type exhibits logistic growth with intrinsic rate of increase, r^R_l, and carrying capacity, K^R_l = 1/α^R_l, such that growth rate of resource l is

${\displaystyle {\frac {1}{R_{l}}}{\frac {dR_{l}}{dt}}=r_{l}^{R}\left(1-\alpha _{l}^{R}R_{l}\right)-\sum _{j}c_{jl}N_{j}.}$

Similarly, each predator species m exhibits logistic growth in the absence of the prey of interest with intrinsic growth rate r^P_m and carrying capacity K^P_m = 1/α^P_m. 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 a_jm. 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, P_m, has a per-capita growth rate

${\displaystyle {\frac {1}{P_{m}}}{\frac {dP_{m}}{dt}}=r_{m}^{P}(1-\alpha _{m}^{P}P_{m})+\sum _{j}wN_{j}a_{jm}}$

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

${\displaystyle k_{j}={\frac {1}{s_{j}}}\left(\sum _{l=1}c_{jl}v_{l}K_{l}^{R}-\sum _{m}a_{jm}K_{m}^{P}-\mu _{i}\right)}$

where the sensitivity to competition and predation is

${\displaystyle s_{j}={\sqrt {\left(\sum _{l}{\frac {c_{jl}^{2}v_{l}K_{l}^{R}}{r_{l}^{R}}}+\sum _{m}{\frac {a_{jm}^{2}wK_{m}^{P}}{r_{m}^{P}}}\right)}}.}$

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

${\displaystyle \rho =\left(\sum _{l}{\frac {c_{il}v_{l}c_{jl}}{\alpha _{l}^{R}r_{l}^{R}}}+\sum _{m}{\frac {a_{im}wa_{jm}}{\alpha _{m}^{P}r_{m}^{P}}}\right)/s_{i}s_{j},}$

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, α_jj, must be greater than interspecific competition, α_ij. The direct expressions for intraspecific and interspecific competition coefficients from the interaction between shared predators and resources are

${\displaystyle \alpha _{jj}=s_{j}/k_{j}}$

and

${\displaystyle \alpha _{ij}=\rho s_{j}/k_{i}.}$

Thus, when intraspecific competition is greater than interspecific competition,

${\displaystyle \alpha _{jj}>\alpha _{ij}={\frac {s_{j}}{k_{j}}}>\rho {\frac {s_{j}}{k_{i}}}}$

which, for two species leads to the coexistence criteria

${\displaystyle \rho <{\frac {k_{1}}{k_{2}}}<{\frac {1}{\rho }}.}$

Notice that, in the absence of any niche differences (i.e. ρ = 1), species cannot coexist.