Individual-Based Competition Between Species with Spatial Correlation and Aggregation

Abstract

In order to clarify the theoretical relationship between individual behavior and population-level competition between two species with spatial correlation, this paper describes how discrete-time competition equations for the two species can be derived from local resource competition among individuals. Competition type of each species is either scramble, contest, or modified contest, and for various combinations of two competition types, different competition models are derived. Simple competition models that can approximate the above models when competition is weak are also derived. Furthermore, the derived models are used to investigate how coexistence conditions and coexistence probability depend on spatial correlation and aggregation of individuals. For the weak competition models, spatial aggregation and non-correlation, in terms of measures adopted here, play exactly symmetric roles in promoting coexistence. In contrast, for the fully developed models, spatial aggregation generally exerts stronger effects than non-correlation on coexistence. Coexistence probability also depends greatly on competition types. For example, two species are generally more likely to coexist when they are of the same competition type than of different competition types. Coexistence probabilities from the mathematical analysis are in good agreement with those from individual-based simulations.

Appendices

Appendix 1

This appendix provides a derivation of the probability generating function (4) of individual distribution. Let $M(t_1, t_2)$ be the moment generating function (mgf) of the accessibilities $a_1$ and $a_2$, defined in terms of the joint probability density $p_A(a_1, a_2)$ as

$$\begin{aligned} M(t_1,t_2)\equiv \mathop \int \limits _{0}^{\infty }\mathop \int \limits _{0}^{\infty }da_1 da_2\, e^{t_1 a_1+t_2 a_2} \,p_A(a_1,a_2). \end{aligned}$$

(39)

Then, the mgf of Kibble–Moran’s bivariate gamma distribution with mean 1 is given by (Kotz et al. 2000)

$$\begin{aligned} M(t_1,t_2)=\left\{ 1-\frac{t_1}{\lambda }-\frac{t_2}{\lambda } +(1-\rho )\frac{t_1 t_2}{\lambda ^2} \right\} ^{-\lambda }, \end{aligned}$$

(40)

where $\lambda \,(>0)$ is a shape parameter proportional to the variance, and $\rho \,(0\le \rho \le 1)$ is the correlation coefficient between $a_1$ and $a_2$.

Letting $p_i(\ell _i|a_i)$ be the conditional probability of finding exactly $\ell _i$ individuals of species $i$ in a given site with accessibility $a_i$, we can write the joint distribution $p(\ell _1,\ell _2)$ as

$$\begin{aligned} p(\ell _1,\ell _2)=\mathop \int \limits _{0}^{\infty }\mathop \int \limits _{0}^{\infty }da_1 da_2\, p_1(\ell _1|a_1) p_2(\ell _2|a_2) p_A(a_1,a_2). \end{aligned}$$

(41)

When we assume that each adult of species $i$ lays eggs at a given site with a probability proportional to the site’s accessibility $a_i$, and that the number of sites is infinitely large, the above conditional probabilities are Poisson distributed with mean $a_1 x_t$ or $a_2 y_t$:

$$\begin{aligned} p_1(\ell _1 | a_1)&= \frac{1}{\ell _1!}(a_1 x_t)^{\ell _1} e^{-a_1 x_t}, \end{aligned}$$

(42a)

$$\begin{aligned} p_2(\ell _2 | a_2)&= \frac{1}{\ell _2!}(a_2 y_t)^{\ell _2} e^{-a_2 y_t}. \end{aligned}$$

(42b)

Substituting Eqs. (42) into (41), and then substituting the result into the definition of the pgf $G(z_1, z_2)$ of the distribution of individuals, Eq. (3), we have

Interchanging the orders of the summation and the integral, and then performing the summation gives

With Eq. (39), this equation can be written as $M((z_1-1)x_t, (z_2-1)y_t)$, giving

which is Eq. (4).

Appendix 2

With the interaction functions derived in Electronic supplementary material for scramble competition, Eq. (1) is written as

$$\begin{aligned} x_{t+1}&= b_1 \sum _{\ell _1, \ell _2=0}^{\infty } \ell _1 c_1^{\ell _1-1} \hat{c}_{21}^{\ell _2} \, p(\ell _1, \ell _2), \end{aligned}$$

(46a)

$$\begin{aligned} y_{t+1}&= b_2 \sum _{\ell _1, \ell _2=0}^{\infty } \ell _2 c_1^{\ell _1} c_2^{\ell _2-1} \, p(\ell _1, \ell _2), \end{aligned}$$

(46b)

where $c_i=e^{-s_i/\bar{R}_i}$, and $\hat{c}_{21}=e^{-\hat{s}_{21}/\bar{R}}$. Comparing these equations with the definition of $G(z_1, z_2)$, Eq. (3), we observe that they can be written as

$$\begin{aligned} x_{t+1}&= b_1 \left. \frac{\partial }{\partial z_1}G(z_1,z_2) \right| _{z_1=c_1, \, z_2=\hat{c}_{21}}, \end{aligned}$$

(47a)

$$\begin{aligned} y_{t+1}&=b_2 \left. \frac{\partial }{\partial z_2}G(z_1,z_2) \right| _{z_1=c_1, \, z_2=c_2}. \end{aligned}$$

(47b)

Then, substituting Eq. (4) for $G(z_1, z_2)$ in the above equations finally gives Eq. (9) in Table 1.

Appendix 3

This appendix describes the derivation of Eq. (24). First, the pgf of the distribution of individuals $G(z_1,z_2)$ defined by Eq. (3) is approximated by its second-order Taylor series around $z_1=z_2=1$:

$$\begin{aligned}&G(z_1,z_2)\simeq G_* +(z_1-1)\frac{\partial }{\partial z_1}G_* +(z_2-1)\frac{\partial }{\partial z_2}G_* \nonumber \\&\quad +\frac{1}{2} (z_1-1)^2\frac{\partial ^2}{\partial z_1^2}G_* +\frac{1}{2} (z_2-1)^2\frac{\partial ^2}{\partial z_2^2}G_* \nonumber \\&\quad +(z_1-1)(z_2-1) \frac{\partial ^2}{\partial z_1 \partial z_2}G_*, \end{aligned}$$

(48)

where $*$ denotes that the functions are evaluated at $z_1=z_2=1$. Next, from the definition of $G(z_1, z_2)$ in Eq. (3), the derivatives of $G(z_1, z_2)$ above can be written as

$$\begin{aligned} \frac{\partial }{\partial z_i}G_* = \langle \ell _i \rangle , \end{aligned}$$

(49a)

$$\begin{aligned} \frac{\partial ^2}{\partial z_i^2}G_* = \langle \ell _i (\ell _i-1) \rangle , \end{aligned}$$

(49b)

$$\begin{aligned} \frac{\partial ^2}{\partial z_1 \partial z_2}G_* = \langle \ell _1 \ell _2 \rangle , \end{aligned}$$

(49c)

from which Eq. (48) can be written as

$$\begin{aligned}&G(z_1,z_2)\simeq 1 + (z_1-1)\langle \ell _1\rangle + (z_2-1)\langle \ell _2\rangle \nonumber \\&\quad +\frac{1}{2} (z_1-1)^2 \langle \ell _1(\ell _1-1)\rangle +\frac{1}{2} (z_2-1)^2 \langle \ell _2(\ell _2-1)\rangle \nonumber \\&\quad +(z_1-1)(z_2-1) \langle \ell _1\ell _2\rangle . \end{aligned}$$

(50)

Finally, using this approximation instead of Eq. (4) in deriving competition models gives the following approximate model:

$$\begin{aligned} x_{t+1}&=b_1 \left\{ \langle \ell _1\rangle -\Gamma _{11}\langle \ell _1(\ell _1-1)\rangle -\Gamma _{21}\langle \ell _1\ell _2\rangle \right\} ,\end{aligned}$$

(51a)

$$\begin{aligned} y_{t+1}&=b_2 \left\{ \langle \ell _2\rangle -\Gamma _{22}\langle \ell _2(\ell _2-1)\rangle -\Gamma _{12}\langle \ell _1\ell _2\rangle \right\} , \end{aligned}$$

(51b)

where $\Gamma _{ij}$ for each competition case is defined in Sect. 3.2.

Appendix 4

This appendix presents some functions needed to evaluate the coexistence probability $p_c$ for the contest vs. contest competition with the uniform distributions of competitive ability $q_i(u)$ described in Fig. 2. First, the competition model described by Eq. (11) with

$$\begin{aligned} D_i(u)=Q_i(u)\equiv \mathop \int \limits _u^\infty du' \,q_i(u') \end{aligned}$$

(52)

can be written as

$$\begin{aligned}&x_{t+1}=b_1 x_t \mathop \int \limits _{0}^{1} dQ_1\, \Bigl (1+(1-\rho )Q_2(Q_1)\frac{y_t}{\lambda } \Bigr )\nonumber \\&\quad \cdot \Bigl \{ 1+Q_1\frac{x_t}{\lambda } +Q_2(Q_1)\frac{y_t}{\lambda } +(1-\rho )Q_1 Q_2(Q_1)\frac{x_t y_t}{\lambda ^2} \Bigr \}^{-\lambda -1}, \end{aligned}$$

(53a)

$$\begin{aligned}&y_{t+1}=b_2 y_t \mathop \int \limits _{0}^{1} dQ_2\, \Bigl (1+(1-\rho )Q_1(Q_2)\frac{x_t}{\lambda } \Bigr )\nonumber \\&\quad \cdot \Bigl \{ 1+Q_1(Q_2)\frac{x_t}{\lambda } +Q_2\frac{y_t}{\lambda } +(1-\rho )Q_1(Q_2) Q_2\frac{x_t y_t}{\lambda ^2} \Bigr \}^{-\lambda -1}, \end{aligned}$$

(53b)

where $Q_i(Q_j)$ denotes $Q_i(u)$ as a function of $Q_j(u)$, given by

$$\begin{aligned} Q_1(Q_2)&= {\left\{ \begin{array}{ll} 1-\delta _1 + \frac{\delta _1}{\delta _2}Q_2 &{} (0<Q_2 \le \delta _2), \\ 1 &{} (\delta _2 < Q_2 < 1), \end{array}\right. } \end{aligned}$$

(54a)

$$\begin{aligned} Q_2(Q_1)&= {\left\{ \begin{array}{ll} 0 &{} (0<Q_1 \le 1-\delta _1), \\ \frac{\delta _2}{\delta _1}(Q_1-1+\delta _1) &{} (1-\delta _1 < Q_1 < 1). \end{array}\right. } \end{aligned}$$

(54b)

When $x_t=0$ or $y_t=0$, the integrals in Eq. (53) can be evaluated easily. When we express the growth rate of species $i$ as $g_i(x, y)=b_i G_i(x,y)$, the specific forms of $G_2(0, y^*)$ and $G_1(0, y^*)$, which are relevant to the invasibility of species 1, are given by

$$\begin{aligned} G_2(0,y^*)&=\frac{1}{y^*} \Bigl \{ 1-\Bigl (1+\frac{y^*}{\lambda }\Bigr )^{-\lambda } \Bigr \}, \end{aligned}$$

(55)

$$\begin{aligned} G_1(0,y^*)&= \frac{\delta _1}{\delta _2 y^*} \left[ \rho \Bigl \{ 1-\Bigl (1+\delta _2\frac{y^*}{\lambda }\Bigr )^{-\lambda } \Bigr \} \right. \nonumber \\&\quad + \left. (1-\rho )\frac{\lambda }{\lambda -1} \Bigl \{ 1-\Bigl (1+\delta _2\frac{y^*}{\lambda }\Bigr )^{-\lambda +1} \Bigr \} \right] +(1-\delta _1). \end{aligned}$$

(56)

Similarly, the specific forms of $G_1(x^*,0)$ and $G_2(x*,0)$, which are relevant to the invasibility of species 2, are given by

$$\begin{aligned} G_1(x^*,0)&=\frac{1}{x^*} \Bigl \{ 1-\Bigl (1+\frac{x^*}{\lambda }\Bigr )^{-\lambda } \Bigr \}, \end{aligned}$$

(57)

$$\begin{aligned} G_2(x^*,0)&= \frac{\delta _2}{\delta _1 x^*} \left[ \rho \Bigl \{ \Bigl (1+(1-\delta _1)\frac{x^*}{\lambda }\Bigr )^{-\lambda } -\Bigl (1+\frac{x^*}{\lambda }\Bigr )^{-\lambda } \Bigr \} \right. \nonumber \\&\quad + \left. (1-\rho )\frac{\lambda }{\lambda -1} \Bigl \{ \Bigl (1+(1-\delta _1)\frac{x^*}{\lambda }\Bigr )^{-\lambda +1} -\Bigl (1+\frac{x^*}{\lambda }\Bigr )^{-\lambda +1} \Bigr \} \right] \nonumber \\&\quad + (1-\delta _2) \Bigl (1+(1-\rho )\frac{x^*}{\lambda }\Bigr ) \Bigl (1+\frac{x^*}{\lambda }\Bigr )^{-\lambda -1}. \end{aligned}$$

(58)

Functions (55)–(58) are used to numerically evaluate the likelihood of two-species coexistence in Sect. 4.2.