Introduction Columba livia 2003 Acacia 1973 Cuculus canorus 2000 2002 2007 1987 2002 2003a 1977 1978 2000 2003b 2001 2002 2005 2003 2004 2007 Temnothorax albipennis 2000 2001 1974 2006 2002 2005 2002 2005 2006 2006 2005 1978 2001 2002 To maximise fitness, the colony should emigrate as quickly as possible to avoid predation and other hazards. Therefore, during house hunting, a fast build up of recruiters is essential. Why then do ants mix fast carrying with slow reverse tandem running, when they already have forward tandem running at their disposal? In this paper, the role of reverse tandem running is theoretically investigated. In particular, through the use of mathematical models, we explore under what circumstances reverse tandem running can have a positive influence on emigration speed. Materials and methods 2002 2003a 1 2002 2006 N FN A F N P Discussion S R μ k λ Q φ 2001 Fig. 1 FTR RTR 1 Both tandem running and social carrying involve a pair of ants from two different classes. Hence, recruitment can only occur if ants of both participating classes are available; Once the quorum has been met, recruiters cannot carry and perform reverse tandem runs simultaneously (we also assume recruiters are not involved in other activities than these two). X Y XY X Y A R λRA A R f f 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\lambda RS/(S+R).$$\end{document} 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\lambda RC/(C+R).$$\end{document} P f φRP P R f \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar f$\end{document} 1 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar f = \min\{S f, f\}$\end{document} C f f \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar f$\end{document} S C \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar f$\end{document} l c r \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}c} {l{\left( {\lambda ,R,Q,A} \right)} = }{\left\{ {\begin{array}{*{20}l} {{\lambda \frac{{RA}}{{R + A}}} \hfill} & {{{\text{if}}\,R < Q,} \hfill} \\ {0 \hfill} & {{{\text{otherwise}},} \hfill} \\ \end{array} } \right.} \\ {c{\left( {\phi ,R,Q,P} \right)} = }{\left\{ {\begin{array}{*{20}l} {{\phi \frac{{RP}}{{R + P}}} \hfill} & {{{\text{if}}\,R \geqslant Q,} \hfill} \\ {0 \hfill} & {{{\text{otherwise}},} \hfill} \\ \end{array} } \right.} \\ \end{array}$$\end{document} B A P \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r(\lambda,R,Q,B) = \left\{ \begin{array}{ll} \lambda \frac{RB}{R+B}&\quad\text{if } R \ge Q,\\ 0 & \quad\text{otherwise}. \end{array} \right.$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll} \dot A &= -\mu A - l(\lambda,R,Q,A),\\ \dot S &= \mu A - kS - f r(\lambda,R,Q,S)\\ \dot R &= kS + l(\lambda,R,Q,A) + f r(\lambda,R,Q,S)\\ \dot P &= -(1-f) c(\phi,R,Q,P),\\ \dot C &= (1-f) c(\phi,R,Q,P), \end{array} \right.$$\end{document} A S R P C FN ε ε ε F N \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll} \dot A &= -\mu A -l(\lambda,R,Q,A),\\ \dot S &= \mu A - kS,\\ \dot R &= kS + l(\lambda,R,Q,A) + f r(\lambda,R,Q,C),\\ \dot P &= -(1-f) c(\phi,R,Q,P),\\ \dot C &= (1-f) c(\phi,R,Q,P) - f r(\lambda,R,Q,C), \end{array} \right.$$\end{document} f r λ R Q C ε l r c ε We also explored a number of other models in which some assumptions were relaxed. These are briefly discussed in the final section of this paper. μ F k Q f P A 1965 μ F k 2002 2005 1 Table 1 1 Parameters Description Value/range N Colony size 250 F Fraction of active ants [0.05,0.5] Q Quorum threshold n.a. f Fraction of post-quorum reverse tandem running time n.a. μ  − 1  − 1 [0.01,0.2] λ  − 1  − 1 0.1 φ  − 1  − 1 0.2 k  − 1  − 1 {0.0001,0.001} λ φ N 2005 Results 2 k μ F 2 f Q F μ f Q k 2 Fig. 2 top figures bottom figures left two columns right two columns F μ k 1 k k Note that, although models 1 and 2 broadly give similar predictions, they differ in the amount of post-quorum time spent on reverse tandem runs. In model 1, this reaches a full 100% in model 1, but never so in model 2. k F μ F old 3 2001 F 2 F N Fig. 3 top row bottom row 2 k Discussion 2003a 2003b k F μ not 2004 2003a 2003a Critique on model 2 2006 4 2005 3 Fig. 4 μ F Reverse tandem activity 2 F μ F FN F F F 2 F F 3 Nonlinearities in the models and divisions between active and passive ants T. albipennis 2002 2006 F μ k 2002 2006 Hypothesised explanations 2002 2002 2001 2002 2001 2006 2003a The first models in which reverse tandem runs have been explicitly incorporated to analyse their role have yielded clear predictions: under a range of conditions, we expect a negative correlation between levels of early and late recruitment. This finding lends itself well to simple experiments, and we aim to present those in the near future. The build up of recruiter numbers serves two purposes: to decide on a nest and to increase the number of ants actively involved in transport. The decision-making process and the implementation of this decision are thus conflated. This in itself is a side-effect of the distributed nature of this system. Reverse tandem running may thus be a logical extension to overcome this inherent problem. This suggests that such additional backup behaviours could be a common feature of decentralised collective decision-making systems.