Interspecific Competition Models Derived from Competition Among Individuals

Abstract

This paper demonstrates how discrete-time models describing population dynamics of two competing species can be derived in a bottom-up manner by considering competition for resources among individuals and the spatial distribution of individuals. The competition type of each species is assumed to be either scramble, contest, or an intermediate between them. Individuals of two species are distributed over resource sites or patches following one of three distribution functions. According to the combination of competition types of the two species and the distribution of individuals, various interspecific competition models are derived. Furthermore, a general interspecific competition model that includes various competition models as special cases is derived for each distribution of individuals. Finally, this paper examines dynamics of some of the derived competition models and shows that the likelihood of coexistence of the two species varies greatly, depending on the type of spatial distribution of individuals.

Appendices

Appendix A

Interaction functions (47) and (48) can be derived as described below. Consider a site containing k and ℓ individuals of species 1 and 2, respectively, and an amount of resource R. When we focus on a given individual of species 1 in this site, the probability that exactly r−1 and ℓ′ individuals of species 1 and 2, respectively, are competitively superior to this individual is given by

$$ {k-1 \choose r-1} {\ell \choose \ell'} Q_1(u)^{r-1} \bigl[1-Q_1(u) \bigr]^{k-r} Q_2(u)^{\ell'} \bigl[1-Q_2(u) \bigr]^{\ell-\ell'}, $$

(64)

where u is the value of the competitive ability U of the individual under consideration, and

$$ Q_i(u) = \int_{u}^{\infty} du'\, q_i \bigl(u' \bigr) $$

(65)

is the probability that a given individual of species i is competitively superior to the individual under consideration. In this situation, because each individual tries to obtain s _i resources in the order of competitive ability, the probability that the individual under consideration can obtain s ₁ is

$$ \mbox{Prob} \bigl[s_1 r + s_2 \ell' \le R \bigr] = c_1^r c_2^{\ell'}, $$

(66)

where the probability on the left-hand side has been calculated with the distribution density of R, Eq. (6). By combining Eqs. (64) and (66) and summing up all possible situations, the interaction function for species 1 can be written as

(67)

(68)

where the last expression indeed yields Eq. (47) with Eq. (49). The interaction function (48) for species 2 can also be derived in the same manner.

Appendix B

We confirm here that the model described by Eqs. (50), (51), and (59) indeed includes the model (40) and (41) as a special case. Because in Sect. 3.4 we assumed that individuals of species 1 were always competitively superior to individuals of species 2, we assume here a situation in which q ₁(u) and q ₂(u) do not overlap, but instead there is a value a that satisfies q ₁(u)=0 for u<a and q ₂(u)=0 for u>a, as shown in Fig. 5. In addition, the following is assumed:

(69)

Here, $1-\hat{c}_{2}(u)$ represents the intensity of scramble type competition that an individual with U=u experiences from an individual of species 2 with a smaller value of U. Equation (69) indicates that this intensity is $1-\hat{c}_{21}$ for an individual with u>a, i.e., species 1, and $1-\hat{c}_{2}$ for an individual with u<a, i.e. species 2, depending on the species of the individual with U=u. With the assumptions above, Eqs. (50) and (51) become

$$ x_{t+1}=b_1 x_t \int_0^1 d Q_1 \biggl\{ 1+D_{11}\frac{x_t}{\lambda} +D_{21} \frac{y_t}{\lambda} \biggr\}^{-\lambda-1}, $$

(70)

$$ y_{t+1}=b_2 y_t \int_0^1 d Q_2 \biggl\{ 1+D_{12}\frac{x_t}{\lambda} +D_{22} \frac{x_t}{\lambda} \biggr\}^{-\lambda-1}, $$

(71)

where $D_{11}=1-\hat{c}_{1}+(\hat{c}_{1}-c_{1})Q_{1}$, $D_{21}=1-\hat{c}_{21}$, D ₁₂=1−c ₁, $D_{22}=1-\hat{c}_{2} +(\hat{c}_{2}-c_{2})Q_{2}$, and $Q_{i}(u) =\int_{u}^{\infty} du'\, q_{i}(u')$, which is the proportion of the individuals of species i with U>u. Performing the integrals in Eqs. (70) and (71) indeed gives Eqs. (40) and (41).

Appendix C

This appendix describes the derivation of Lotka–Volterra competition models which approximate some of the competition models derived in this paper. We start with the following discrete-time competition model:

(72)

(73)

When population densities x _t and y _t are small, if we approximate the per capita growth rates f(x _t ,y _t ) and g(x _t ,y _t ) by their Taylor series around x _t =y _t =0 up to the order of x _t and y _t , Eqs. (72) and (73) become

(74)

(75)

where f ₀, f ₁, f ₂, g ₀, g ₁, and g ₂ are constants. If the differences between population densities in two successive generations are always small, long term population dynamics will be able to be described approximately by some differential equations. Here, in order to do so, a new time τ≡tΔτ is introduced, where Δτ≪1. Rewriting Eqs. (74) and (75) as

(76)

(77)

indicates that the long term dynamics will be described approximately by the following differential equations:

(78)

(79)

which are Lotka–Volterra competition equations. Comparing these equations with

(80)

(81)

we can read intrinsic per capita growth rate r _i , carrying capacity K _i , and interspecific competition coefficient α _ji , which represents the relative effect of competition of species j on species i.

Parameters of Lotka–Volterra equations corresponding to some of the models derived in this paper are presented in the following. The intrinsic percapita growth rate r _i is given by (b _i −1)/Δτ for any competition models, and other parameters depend on the type of each model. For Eqs. (40) and (41), which describe a model for intermediate vs. intermediate competition when p _k,ℓ is the negative trinomial distribution, we obtain

$$ K_i^{-1}=\frac{2b_i}{b_i-1} \biggl(1+\frac{1}{\lambda} \biggr) (1+\beta_i) (1-c_i), $$

(82)

$$ \alpha_{21} = \frac{2\beta_{21}}{1+\beta_1} \frac{1-c_2}{1-c_1}, $$

(83)

$$ \alpha_{12} = \frac{2}{1+\beta_2} \frac{1-c_1}{1-c_2}. $$

(84)

When p _k,ℓ is the independent negative binomial distribution, we have

$$ K_i^{-1}=\frac{2b_i}{b_i-1} \biggl(1+\frac{1}{\lambda_i} \biggr) (1+\beta_i) (1-c_i), $$

(85)

$$ \alpha_{21} = \biggl(1+\frac{1}{\lambda_1} \biggr)^{-1} \frac{2\beta_{21}}{1+\beta_1} \frac{1-c_2}{1-c_1}, $$

(86)

$$ \alpha_{12} = \biggl(1+\frac{1}{\lambda_2} \biggr)^{-1} \frac{2}{1+\beta_2} \frac{1-c_1}{1-c_2}. $$

(87)

For a general model for intermediate vs. intermediate competition described by Eqs. (50), (51), and (59) when p _k,ℓ is the negative trinomial distribution, we obtain

$$ K_i^{-1} = \frac{b_i}{b_i-1} \biggl(1+\frac{1}{\lambda} \biggr) \int du\, q_i(u) D_i(u), $$

(88)

$$ \alpha_{ji}= \frac{\int du\, q_i(u) D_j(u)}{\int du\, q_i(u) D_i(u)}. $$

(89)

When p _k,ℓ is the independent negative binomial distribution, we have

$$ K_i^{-1} = \frac{b_i}{b_i-1} \biggl(1+\frac{1}{\lambda_i} \biggr) \int du\, q_i(u) D_i(u), $$

(90)

$$ \alpha_{ji}= \biggl(1+\frac{1}{\lambda_i} \biggr)^{-1} \frac{\int du\, q_i(u) D_j(u)}{\int du\, q_i(u) D_i(u)}. $$

(91)

Note that Eq. (91) indicates that α _ji , which represents the effect of competition of species j on species i, does not depend on 1/λ _j , the degree of spatial aggregation of species j, while depending on 1/λ _i .