Symmetric competition kernels

Recently the only available symmetric competition kernel is the Gaussian one:

where B_i and B_j are the resource use related trait values of the two species, while σ_b determines how steeply decrease the strength of competition with increasing difference in resource use (Figure @ref(fig:gauss)).

Shape of Gaussian competirion kernel with different σ_b values

For B_i = B_j, α_ij = 1 irrespectively to value of σ_b. If σ_b = 0 the strength of competition is zero any case of B_i ≠ B_j. If σ_b = ∞, α_ij = 1 for all species pairs.

Gaussian competition kernel can be used by setting competition.kernel="Gaussian.competition,kernel" (which is the default value of this parameter). Value of σ_b has to be set by parameter sigma.b.

According to MacArthur & Levins (1967), this competition kernel can be deduced as overlap of Gaussian resource use curves. Their general formula for overlap is

where U is the resource use function, and x is the quality of the resource (e.g. seed size or rooting depth). Let both U_i and U_j be density function of normal (Gaussian) distribution with same standard deviation (σ), and let rescale x to be expected values equals to zero and d = B_i − B_j, respectively:

Since $x^2+(x-d)^2=2x^2+d^2-2xd=2\left(x-\frac{d}{2}\right)^2+\frac{d^2}{2}$

Note that σ_b in equation @ref(eq:eq1) equals to 2σ² in equation @ref(eq:eq4)

Asymmetric competition kernels

Recently two types of asymmetric competition kernels are available via asymmetric.competition.kernel function:

• Kisdi’s convex-concave function

• smooth function suggested by Nattrass et al. (2012)

Kisdi’s convex-concave function

It is a function defined by equation (2) in Kisdi (1999), however the parametrization are slightly modyfied:

Contrary to the Gaussian competition kernel, It has three parameters (C, v, σ_b) instead of the only one parameter of Gaussian competition kernel. Note that in the R function these parameters are called ac.C, ac.v and sigma.b, respectively. C and v have to be positive, while σ_b ≠ 0. Possible values of the function ranges from zero to C. If σ_b > 0 it is a decreasing sigmoid (convex-concave) function (Figure @ref(fig:kisdi1)) of trait difference (B_i − B_j) with inflection point at B_i − B_j = σ_bln v , where the strength of competition is C/2 .

Shape of Kisdi’s convex-convave function with different values of v (C=1,σ_b=0.1)

Strength of competition between functionally equivalent individulas (i.e. if B_i = B_j) is $\alpha_{ij}=C\left(\frac{v}{1+v}\right)$. If the other two parameter fixed, absolute value of parameter σ_b determines the steepness of the curve around its inflection point (Figure @ref(fig:fig3) ).

Shape of Kisdi’s convex-convave function with different values of σ_b (C=1,v=1)

Smooth function poropsed by Nattrass et al. (2012)

This is also a sigmoid function defined by a formula similar to Kisdi’s function:

It ranges from 1 − C to 1 + C. Position of its inflection point is B_i − B_j = 0, where its value is 1, irrespective to the parameter values. When its range (i.e. value of parameter C) is fixed, σ_b determines the steepness of the curve at the inflection point: lower σ_b results in steeper curve.

Shape of smooth function by Nattrass et al. with different values of σ_b (C=1)

However equation @ref(eq:eq6) is not defined when sigma_b = 0 and B_j = B_i, following suggestion of Nattrass et al. (2012) asymmetric.competition.kernel function set the strength of competition to 1 in this case.

SeedProduction function

This function calculates number of produced seeds for each individual. The number of seeds is random number from Bernoulli (zero or one seed) or Poisson distribution (unlimited number of seeds). The expected value of produced seeds (irrespectively to the distribution) for individual i in local community k depends on the competition for resources:

Where: b₀ is the maximum probability of reproduction in competition free conditions; K is level of competition above which probability of reproduction becomes zero; α_ij = competitive effect of individual j on individual i, calculated from resource acquisition traits by the competition kernel functions.