A mathematical model for the territorial competition of the subterranean termites Coptotermes formosanus and Reticulitermes flavipes (Isoptera: Rhinotermitidae)

Abstract

The foraging territories of two subterranean termites, Coptotermes formosanus Shiraki and Reticulitermes flavipes (Kollar), were simulated using a two-dimensional model to explore how territorial competition changes according to two variables characterizing territory formation: the total number of territories, and the blocking probability. Meanwhile, the blocking probability quantitatively describes the likelihood that a tunnel will be terminated when another tunnel is encountered. In our previous study, we introduced an interference coefficient γ to characterize territorial competition, and obtained γ as a function of the total number of territories and the blocking probability for a single termite species by model simulation. In the field, the territorial competition of more than two termite species is frequently observed. Here, we extended the γ function to be able to explain the competition between the two species by applying statistical regression to the simulation data. Further, we statistically checked the extended γ function by comparing the γ function for a single species. We also discuss another approach to mathematically derive the extended γ function, which can be easily generalized for use in cases of territorial competition involving more than two termite species.

Appendices

Appendix 1

To mathematically derive the γ function, we selected a rational function as the best fitting function for the simulation data, because the model is known for describing complicated structures having nonlinearities (Blondel and Gevers 1993; Bultheel et al. 1999). The function has a simple form for the numerator and denominator with respect to N ₁ , N ₂ , and p _block, as written below.

$$ \gamma (\,N_{1} ,N_{2} ,p_{\text{block}} ) = \;\frac{{(a_{1} N_{1} + a_{2} N_{2} )p_{\text{block}} + (a_{3} N_{1} + a_{4} N_{2} ) + a_{5} p_{\text{block}} + a_{6} }}{{(b_{1} N_{1} + b_{2} N_{2} )p_{\text{block}} + (b_{3} N_{1} + b_{4} N_{2} ) + b_{5} p_{\text{block}} + b_{6} }} $$

(19)

Equation (19) is a generalized form of the equation used for single-species termite’s territory competition. See Appendix 2 of the previous study (Jeon and Lee 2011).

When p _block is zero, γ should become zero regardless of N ₁ and N ₂ because there is no interference among the territories. Hence, a ₃ , a ₄ , and a ₆ should be zero. We then obtain the following relation:

$$ \gamma (\,N_{1} ,N_{2} ,p_{\text{block}} ) = \;\frac{{(a_{1} N_{1} + a_{2} N_{2} )p_{\text{block}} + a_{5} p_{\text{block}} }}{{(b_{1} N_{1} + b_{2} N_{2} )p_{\text{block}} + (b_{3} N_{1} + b_{4} N_{2} ) + b_{5} p_{\text{block}} + b_{6} }} $$

(20)

To determine the values of the coefficients, we considered the following extreme conditions: N ₁ or N ₂ tends to infinity or zero. When N ₁ tends to infinity for fixed N ₂ while P is not zero, no territory can grow because of the extremely high density of the territories in the finite space; thus, γ should become 1. This condition can be written as follows:

$$ \mathop {\lim }\limits_{{N_{1} \to \infty }} \gamma \, = \;\frac{{a_{1} p_{\text{block}} }}{{b_{1} p_{\text{block}} + b_{3} }} = 1\quad{\text{for}}\,{\text{all}}\quad p \ne 0 $$

(21)

To satisfy the conditions irrespective of the non-zero p _block, a ₁ should be equal to b ₁ , and b ₃ should be equal to zero. Similarly, when N ₂ tends to infinity for fixed N ₁ while p _block is not zero, γ should become 1. This condition can be written as follows:

$$ \mathop {\lim }\limits_{{N_{2} \to \infty }} \gamma \, = \;\frac{{a_{2} p_{\text{block}} }}{{b_{2} p_{\text{block}} + b_{4} }} = 1\quad{\text{for all}}\quad p \ne 0 $$

(22)

To satisfy the conditions, a ₂ should be equal to b ₂ , and b ₄ should be equal to zero.

In addition, when both N ₁ and N ₂ approach zero, γ should become 0, because there is no interference between territories, as written in Eq. (23).

$$ \mathop {\lim }\limits_{\begin{subarray}{l} N_{1} \to 0 \\ N_{2} \to 0 \end{subarray} } \gamma \, = \;\frac{{a_{5} p_{\text{block}} }}{{b_{5} p_{\text{block}} + b_{6} }} = 0\quad{\text{for all}}\quad p \ne 0 $$

(23)

From Eq. (23), a5 should be equal to 0. Then, we obtain Eq. (24) from Eq. (20).

$$ \gamma (N_{1} ,N_{2} ,p_{\text{block}} ) = \;\frac{{(a_{1} N_{1} + a_{2} N_{2} )p_{\text{block}} }}{{(a_{1} N_{1} + a_{2} N_{2} + b_{5} )p_{\text{block}} + b_{6} }} $$

(24)

Finally, (i) by using the definitions of N ₁  = αN and N ₂  = (1−α) N and (ii) by dividing both the numerator and denominator by the [·]-bracket term, the interference formula for dual-species termite’s territory competition is derived in the generalized form of Eq. (25):

$$ \gamma (N,p_{\text{block}} ;\alpha ) = \frac{{\{ a_{1} \alpha + a_{2} (1 - \alpha )\} Np_{\text{block}} }}{{[\{ a_{1} \alpha + a_{2} (1 - \alpha )\} N + b_{5} ]p_{\text{block}} + b_{6} }} = \frac{{\frac{N}{N + f(\alpha )}p_{\text{block}} }}{{p_{\text{block}} + \frac{g(\alpha )}{N + f(\alpha )}}} $$

(25)

where f(α) and g(α) will be numerically obtained from the simulations for various α.

Appendix 2

Pseudo-code for termite territory simulation model