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## Charles Cox: Final Report |
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Introduction
Army ants exhibit complex group behavior which appears to be a function of some centralized intelligence. The ants form raids and trail networks, the complexity of which belies recognized ant intelligence. It has been the subject of research, therefore, to explain how such simple insects may exhibit complex group behavior. In his 1940 publication, Schneirla describes how a particular species, Eciton burchelli, operates with detailed accounts of the insects~R activity during 24 hour periods [2]. Keshet, Watmough and Ermentrout present a complex computer program to simulate simple ant functionality and show that basic individual properties may, in fact, lead to the phenomenon of trail formation [3]. With similar inspiration, Keshet and Watmough provide a framework, in one dimension, for modeling army ant movement [7]. The work done by Keshet and Watmough was integral for this project~Rs attempts to model army ant propagation with partial differential equations and expand into two dimensions. The Subject
Eciton burchelli is a swarm raiding army ant which forms a tree-like raid structure [2]. The front portion of the ant raid is characterized by a high density of ants with multiple contributory trails. The trails lead back to the bivouac site which is literally a mass of ants interlocking limbs to form a nest structure. Of particular interest to this project, as an area of focus, is the period when a raid first begins. Schneirla explains that as dawn approaches the ants begin to emerge from the nest site. As the sun warms the earth to a sufficient temperature, the ants begin their swarm. It is the beginning of the swarm, then, which leads naturally to an initial value and boundary value problem. A nest may be considered an initial distribution. For this paper, the newly excited nest will be modeled by a bell curve distribution of ants in space. One may view this distribution as occurring shortly after the ants are excited, but immediately before the swarm begins. Then, with an initial distribution of ants on a strip or a plane, can a reasonable model describe Eciton burchelli propagation using PDE techniques? Only the most basic ant intelligence will be assumed ~V in the preliminary models presented, ants are assumed to behave completely randomly. More complex models assume simple internal preference for outside influences which affect turning rates exclusively. The Model
Initially, it was the goal of this project to describe ant propagation in a plane. After all, army ants do function, normally, within a modified two dimensional world. That is, they don~Rt fly, and the biggest obstacles are hills, trees, and rocks. Initial research in this direction lead to work by C. S. Patlak from 1953 [4,5]. It became quickly apparent that, in two dimensions, without significant simplifications which would trivialize it, the model would be of unusable complexity. Any model in two dimensions would necessarily involve a system of first-order partial integral-differential equations. Based on work by Keshet and Watmough, however, this paper presents models which may be sufficient to describe Eciton burchelli raids while remaining simple enough to acquire analytical results. Let us begin with a single one-dimensional strip with some arbitrary length and centered at the point zero. Ants may travel in one of two directions ~V in the positive direction or in the negative direction, and do so with a constant speed V. The density of ants traveling in the positive direction may be called Uplus(x,t) and, similarly, the density traveling in the opposite direction may be called Uminus(x,t). As suggested by Keshet and Watmough~Rs work, a telegraph-style system assumes that ants switch directions at some constant rates Lplus and Lminus. Then: partial(Uplus,t) + V*partial(Uplus,x) = -Lplus*Uplus + Lminus*Uminus partial(Uminus,t) + V*partial(Uminus,x) = Lplus*Uplus ~V Lminus*Uminus describes this basic scenario in a one dimensional strip. The initial condition would be sufficiently described by Uplus(x,0) + Uminus(x,0) = exp(-A*x^2) for various values of the positive real constant A ~V analysis of this system focuses on the case where Uplus(x,0) = Uminus(x,0). Reflecting boundaries at either end of the strip are also assumed, but analysis focuses on cases where the ~Send of the ants~R world~T is not reached. While this system of equations is relatively simple, it~Rs irritatingly complex to solve, or even analyze, by hand. Separation of Variables, Laplace Transformation methods, and computer-based techniques proved incapable of finding an exact solution sufficient to meet initial and reflective boundary conditions. This system is easily tackled numerically by Maple 8, however, with its numerical PDE module. The required code to do so is included as Appendix A. Interestingly, this system contains more information about Eciton burchelli behavior than was anticipated, or even intended. With sufficiently large values of V, and reasonable values for the turning rates, this telegraph model does create propagation with a significant spike of density leading away from zero in either direction. That is, a mass of ants are at the front of the swarm (in both the positive and negative directions), primarily moving away form zero, followed by a mixture of ants moving forward and backward respectively, but with lesser density. As expected, however, this result diminishes with time and, eventually, the colony assumes a flattened shape like a broad pile of dirt. The above model contains only one dimension, which is a particularly alien environment for army ants. Therefore, as interesting as the results may be, they are hardly applicable to ants in the wild. While avoiding the complexity of a Patlak-style model, it is still reasonable to expand the above result into a simplified two-dimensional geometry by considering a system of strips ~V an array of several parallel one-dimensional telegraph systems which allow for ants to leave and join adjacent trails at constant rates. A system of [n] trails may look as follows: partial(Uplus[i],t) + V*partial(Uplus[i],x) = -Lplus*Uplus[i] + Lminus*Uminus[i] + T[i] ~V Aplus[i]*Uplus[i] for the positively oriented portion of the system. Here, Aplus[i] would be a constant rate at which ants leave a given trail. T[i] is a set of terms describing ant arrival from adjacent trails. T[i] depends on the track~Rs position in the grid for its specific terms. T[i] = 1/2 *Aplus[i-1]*Uplus[i-1] + 1/2 *Aplus[i+1]*Uplus[i+1] for i between 1 and n. T[1] = 1/2 *Aplus[2]*Uplus[2], and T[n] = 1/2 *Aplus[n-1]*Uplus[n-1] This assumes ants are equally likely to join an adjacent track in either direction. The negatively oriented track and initial conditions would be similar to the single strip model, but the initial distribution may be isolated to one track, or a small number of them. Again this system is simplistic yet irritatingly complex, if not impossible, to solve explicitly. Numerical methods were tested for a 5-track setup and, similar to the previous model, swarms with higher density of ants were characteristically found at the front of propagation provided sufficiently large speed V. Also similar to the previous model, the ants eventually form a flattened pile. The Maple 8 code to numerically solve and graph this 5-track system is provided as Appendix B. Further Research
The models presented in this paper contain surprisingly accurate information about ant movement, but completely ignore what is believed to be an integral part of army ant communication: pheromone. Keshet, Watmough, and Ermentrout only found trail following behavior in systems which involved ant attraction to pheromone which the ants themselves excrete. This is similar to Keller and Segel~Rs work with slime molds where they found that attraction to chemical depositions was important for the complex behavior of the organisms [6]. It would be reasonable to consider a more complex extension of the models presented in this paper for future research. As a guideline for future work on this problem, one may modify the turning rates to describe chemical attraction. By providing an extra equation for each strip describing the density of chemical attractant, the Lplus, Lminus, Aplus[i], and Aminus[i] terms may be described in terms of that new density. That is, perhaps, the turning rates would be small when ants are at high concentrations of chemical, but large when at low concentrations. The track switching rates may be similarly affected. With pheromone concentration W(x,t) and decreasing positive functions F( ) and G( ), a simple model may have: Lplus(x,t) = LMinus(x,t) = F(W(x,t)) Aplus[i](x,t) = Aminus[i](x,t) = G(W(x,t)) This approach is simple in that it does not require information about adjacent concentrations, while it still provides reasonable descriptions of real ant behavior. It also avoids the complexity of a Patlak-style model, allowing for easier use in predictive applications. Unfortunately this approach leads naturally to a non-linear system, the complexity of which is beyond the methods used for the simple models in this paper. For the case where ant movement is affected by density at a given location, turning rates may again be modified. This scenario assumes that ants prefer to keep some distance from their neighbors. For positive functions B( ) and C( ), the turning rates in this scenario may look like: Lplus(x,t) = LMinus(x,t) = B(Uplus(x,t)+Uminus(x,t)) Aplus[i](x,t) = Aminus[i](x,t) = C(Uplus(x,t)+Uminus(x,t)) where we may assume that large values of Uplus and Uminus lead to large turning rates, and low values to low rates accordingly. More complex cases could account for turning rates dependent on densities of ants in specific directions individually ~V allowing for the possibility that ants prefer to, at least, move in the same direction as large concentrations of their sisters. Again this leads to a nonlinear system which is beyond the methods of analysis in this paper. Both systems, however, are worth further research as they contain significantly more information about actual ants. Citations:
[1] Sumpter and Pratt, A modeling framework for understanding social insect foraging, Behav Ecol Sociobiol (2003) 53:131-144 [2] T.C. Schneirla, Further studies on the army-ant behavior pattern: mass-organization in the swarm-raiders, Journal of Comparative Psychology 29: 401-460 [3] L. Edelstein-Keshet, J. Watmough, and G. Ermentrout, Trail following in ants: individual properties determine population behaviour, Behav Ecol Sociobiol (1995) 36:119-133 [4] C.S. Patlak, A mathematical contribution to the study of orientation of organisms, Bulletin of Mathematical Biophysics (1953) 15:431-475 [5] C.S. Patlak, Random walk with persistence and external bias, Bulletin of Mathematical Biophysics (1953) 15:311-338 [6] E. Keller and L. Segel, Initiation of Slime Mold Aggregation Viewed as an Instability, J. Theor. Biol. (1970) 26:399-415 [7] L. Edelstein-Keshet and J. Watmough, A one-dimensional model of trail propagation by army ants, J. Math. Biol. (1995) 33:459-476 Appendix A (Maple code)
restart; V:=5;Lplus:=.5;Lminus:=.5; sys:={diff(Uplus(x,t),t) + V*diff(Uplus(x,t),x) = - Lplus*Uplus(x,t) + Lminus*Uminus(x,t), diff(Uminus(x,t),t) - V*diff(Uminus(x,t),x) = Lplus*Uplus(x,t) - Lminus*Uminus(x,t)}; Init:=exp(-x^2);IBC:={Uplus(x,0)=1/2*Init, Uminus(x,0)=1/2*Init, D[1](Uplus)(20,t)=0,D[1](Uminus)(-20,t)=0}; pds := pdsolve(sys,IBC,numeric,range=-20..20,timestep=.01,spacestep=.01): p1:=pds:-plot(Uplus + Uminus,t=0,numpoints=1000): p11:=pds:-plot(Uplus + Uminus,t=2,numpoints=1000,color=blue): plots[display]({p1,p11}); Appendix B (Maple code)restart; V:=5;Lplus:=.5;Lminus:=.5;A1minus:=.5;a1minus:=.5;A2minus:=.5;a2minus:=.5;A3minus:=.5;a3minus:=.5;A4minus:=.5;a4minus:=.5;A5minus:=.5;a5minus:= .5;A1plus:=.5;a1plus:=.5;A2plus:=.5;a2plus:=.5;A3plus:=.5;a3plus:=.5;A4plus:=.5;a4plus:=.5;A5plus:=.5;a5plus:=.5; sys:={diff(Uplus_1(x,t),t) + V*diff(Uplus_1(x,t),x) = - Lplus*Uplus_1(x,t) + Lminus*Uminus_1(x,t) + a1plus*A2plus*Uplus_2(x,t) - A1plus*Uplus_1(x,t), diff(Uminus_1(x,t),t) - V*diff(Uminus_1(x,t),x) = Lplus*Uplus_1(x,t) - Lminus*Uminus_1(x,t) + a1minus*A2minus*Uminus_2(x,t) - A1minus*Uminus_1(x,t), diff(Uplus_2(x,t),t) + V*diff(Uplus_2(x,t),x) = - Lplus*Uplus_2(x,t) + Lminus*Uminus_2(x,t) + A1plus*Uplus_1(x,t) + a2plus*A3plus*Uplus_3(x,t) - A2plus*Uplus_2(x,t), diff(Uminus_2(x,t),t) - V*diff(Uminus_2(x,t),x) = Lplus*Uplus_2(x,t) - Lminus*Uminus_2(x,t) + A1minus*Uminus_1(x,t) + a2minus*A3minus*Uminus_3(x,t) - A2minus*Uminus_2(x,t), diff(Uplus_3(x,t),t) + V*diff(Uplus_3(x,t),x) = - Lplus*Uplus_3(x,t) + Lminus*Uminus_3(x,t) + a3plus * (A2plus*Uplus_2(x,t) + A4plus*Uplus_4(x,t)) - A3plus*Uplus_3(x,t), diff(Uminus_3(x,t),t) - V*diff(Uminus_3(x,t),x) = Lplus*Uplus_3(x,t) - Lminus*Uminus_3(x,t) + a2minus * (A2minus*Uminus_2(x,t) + A4minus*Uminus_4(x,t)) - A3minus*Uminus_3(x,t), diff(Uplus_4(x,t),t) + V*diff(Uplus_4(x,t),x) = - Lplus*Uplus_4(x,t) + Lminus*Uminus_4(x,t) + a4plus*A3plus*Uplus_3(x,t) + A5plus*Uplus_5(x,t) - A4plus*Uplus_4(x,t), diff(Uminus_4(x,t),t) - V*diff(Uminus_4(x,t),x) = Lplus*Uplus_4(x,t) - Lminus*Uminus_4(x,t) + a4minus*A3minus*Uminus_3(x,t) + A5minus*Uminus_5(x,t) - A4minus*Uminus_4(x,t), diff(Uplus_5(x,t),t) + V*diff(Uplus_5(x,t),x) = - Lplus*Uplus_5(x,t) + Lminus*Uminus_5(x,t) + a5plus*A4plus*Uplus_4(x,t) - A5plus*Uplus_5(x,t), diff(Uminus_5(x,t),t) - V*diff(Uminus_5(x,t),x) = Lplus*Uplus_5(x,t) - Lminus*Uminus_5(x,t) + a5minus*A4minus*Uminus_4(x,t) - A5minus*Uminus_5(x,t)}; Init:=exp(-x^2); IBC:={Uplus_3(x,0)=1/2*Init, Uminus_3(x,0)=1/2*Init, D[1](Uplus_3)(20,t)=0, D[1](Uminus_3)(-20,t)=0,Uplus_2(x,0)=0, Uminus_2(x,0)=0, D[1](Uplus_2)(20,t)=0,D[1](Uminus_2)(-20,t)=0, Uplus_1(x,0)=0, Uminus_1(x,0)=0, D[1](Uplus_1)(20,t)=0,D[1](Uminus_1)(-20,t)=0, Uplus_4(x,0)=0, Uminus_4(x,0)=0, D[1](Uplus_4)(20,t)=0,D[1](Uminus_4)(-20,t)=0, Uplus_5(x,0)=0, Uminus_5(x,0)=0, D[1](Uplus_5)(20,t)=0,D[1](Uminus_5)(-20,t)=0}; pds := pdsolve(sys,IBC,numeric,range=-20..20,timestep=.05,spacestep=.05): p3_0:=pds:-plot(Uplus_3+Uminus_3,t=0,numpoints=5000): p2_0:=pds:-plot(Uplus_2+Uminus_2,t=0,numpoints=5000): p1_0:=pds:-plot(Uplus_1+Uminus_1,t=0,numpoints=5000): p4_0:=pds:-plot(Uplus_4+Uminus_4,t=0,numpoints=5000): p5_0:=pds:-plot(Uplus_5+Uminus_5,t=0,numpoints=5000): p1_1:=pds:-plot(Uplus_1+Uminus_1,t=1,numpoints=5000): p2_1:=pds:-plot(Uplus_2+Uminus_2,t=1,numpoints=5000): p3_1:=pds:-plot(Uplus_3+Uminus_3,t=1,numpoints=5000): p4_1:=pds:-plot(Uplus_4+Uminus_4,t=1,numpoints=5000): p5_1:=pds:-plot(Uplus_5+Uminus_5,t=1,numpoints=5000): p1_2:=pds:-plot(Uplus_1+Uminus_1,t=2,numpoints=5000): p2_2:=pds:-plot(Uplus_2+Uminus_2,t=2,numpoints=5000): p3_2:=pds:-plot(Uplus_3+Uminus_3,t=2,numpoints=5000): p4_2:=pds:-plot(Uplus_4+Uminus_4,t=2,numpoints=5000): p5_2:=pds:-plot(Uplus_5+Uminus_5,t=2,numpoints=5000): p1_3:=pds:-plot(Uplus_1+Uminus_1,t=3,numpoints=5000): p2_3:=pds:-plot(Uplus_2+Uminus_2,t=3,numpoints=5000): p3_3:=pds:-plot(Uplus_3+Uminus_3,t=3,numpoints=5000): p4_3:=pds:-plot(Uplus_4+Uminus_4,t=3,numpoints=5000): p5_3:=pds:-plot(Uplus_5+Uminus_5,t=3,numpoints=5000): plots[display]({any combination of p1_0 to p5_3}); |
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