Bdlnin
of iWafhemariru/
Biolo~,
Vol.
58, No. 3, pp. 471-492. Elsevier
0
1996 Society
for Mathematical 0092~8240/96
1996
Science $15.00
Inc.
Biology + 0.00
‘0092-8240(95hlO353-3
A MATHEMATICAL MODEL FOR THE POPULATION DYNAMICS OF ARMY ANTS ?? N. F. BRImON
Centre for Mathematical University of Bath, Bath BA2 7AY, UK (E.mail:
Biology and School of Mathematical
Sciences,
[email protected])
?? L. W. PARTRIDGE*
School of Biology and Biochemistry, University of Bath, Bath BA2 7AY, UK (Email:
mi405@mlun’.sari.ac.uk)
?? N. R. FRANKS
Centre for Mathematical Biochemistry, University Bath BA2 7AY, UK (Email:
Biology and School of Biology and of Bath,
bssntffbath.ac.uk)
A stochastic cellular automata model for the population dynamics of the army ant Eciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusionlike, although it is random. This has implications for the kind of model that is chosen.
1. Introduction. Army ants are among the most spectacular of social animals, living in organised colonies that may contain over half a million individuals. In the tropical rain forest their foraging raids sustain diversity among the litter floor fauna by creating a patchwork of areas in different stages of ecological succession (Franks, 1982c; Franks and Bossert, 1983; Otis et al., 1986). Many species of birds, lizards, and invertebrates also * Present address: Macaulay Land Use Research Institute, Craigiebuckler, Aberdeen AB9 2QJ, UK 471
472
N. F. BRI’ITON et al.
associate with them (Willis, 1967; Ray and Andrews, 1980). Army ants are therefore an important component of rain forest ecology. The data on which the paper is based are obtained from a well-studied (Rettenmeyer,l963; Willis, 1967; Schneirla, 1971; Franks 1982a, b, 1989) population of about 50 colonies of the neotropical army ant E&on @rchelZi that are confined to Barro Colorado Island (BCI) in Gatun Lake, Panama. These colonies have a single queen, reproduce by fission and the female reproductives are flightless. They are therefore descendants of those colonies marooned when the lake was formed in 1914 by the construction of the Panama Canal, and there has been no opportunity for further natural colonisation. The population is not necessarily genetically isolated, since the males can fly. It seems likely that the population is regulated by intraspecific competition between colonies for patches of social insect prey (not army ants) that take about 200 days to recover from army ant occupation on BCI (Franks, 1982a). E. burchelli also preys on large, non-social arthropods such as cockroaches, and the level of these prey recover within 10 days after a raid by army ants (Franks, 1982a). An early version of the model took into account both types of prey, but it was found that the rapidly recovering prey had very little effect on the competition between colonies. Such an effect would only be felt by a colony arriving in an area that had been exploited by another colony within the previous 10 days, and even then for only a few days. These times are short compared to the prevailing time scales in the problem. Moreover, the effect is small as well as short-lived. A colony that arrives in an area where non-social prey have recovered but social prey have not, decreases in size over the next time period, since non-social prey are not sufficient to sustain the population. This is the underlying biological reason that the social insect prey are more important than the non-social arthropods. For these reasons only social insect prey were considered in the present version of the model. Note that the faster recovery of the large non-social arthropods is not due to a faster growth rate but to immigration from surrounding areas, so that we are not claiming that the population is regulated by the prey population with the slower growth rate. Colonies of E. burchelli exhibit a 35 day activity cycle (Willis, 1967; Schneirla, 1971) consisting of a statary phase and a nomadic phase. During the statary phase of about three weeks the colony occupies a fixed bivouac from which raids are made into the surrounding forest on most days. A typical distance covered by such a raid is about 90 m (Franks, 1982a). During the nomadic phase raids take place every day, and the whole bivouac or nest is moved to a new position near the end of the raid. The direction taken each day is correlated with that taken on the previous day (Franks and Fletcher, 1983). By the end of the nomadic phase the colony
POPULATION DYNAMICS OF ARMY ANTS
473
has arrived at a new statary phase bivouac which is typically about 530 m from the old one (Willis, 1967). Patches currently occupied by another colony tend to be avoided (Franks and Bossert, 1983). This activity cycle is the basic time step in the model during which, for modelling purposes, colonies are assumed to exploit one patch of prey in the forest. Preliminary simulations modelled the colony movement on a day-to-day basis, but it was found that the important determinant of the population dynamics was where the colony spent the statary phase, when it was committed to a certain patch of forest for 21 days. During the nomadic phase some good and some bad areas tended to be exploited, so that the overall effect was similar for all colonies. Each patch represented an area of about 30,000 m2-roughly the area thought to be affected by the raids produced during the statary phase of the activity cycle. Note the implication that the probability density function for the position of the colony in statary phase t + 1 is not peaked at its position in statary phase t and is not monotonic decreasing away from this position. In this sense the motion is not diffusion-like, although it is random. Let us consider a habitat in which conditions vary over space in such a way that the variations have an effect on the population dynamics, and let the variations occur over a spatial scale of the same order as that corresponding to individuals in the population, or more generally to the units of population that are being considered. Then a continuum approximation to population density is not sufficient, as it averages out the effect of these variations. This is the situation we find ourselves in here, as it is crucially important to an army ant colony whether it is in a patch of forest that has recently been raided by another colony or not. In this case we must use a discrete variable-number of colonies-to describe the population size. The pattern of patches of forest that have and have not recently been exploited by army ants is also an essential determinant of the population dynamics, so we shall also use a discrete model for space. Finally, the colony activity cycle of five weeks-the time taken to raise a new brood-determines the movement of the colony and is an essential part of the biology, so that we shall work in discrete time steps. A partial differential equation model is therefore not appropriate for this biological system. We develop a discrete time stochastic cellular automata model (stochastic because the choice of which patch to move to at the end of each five week cycle is random) structured by taking into account the size of the colonies and including density-dependent mortality. There has recently been much discussion of the importance of such short-range spatial variations in ecology, and a realisation that stochastic cellular automata (also known as interacting particle systems or patch-occupancy models; Caswell and Etter, 1993; Czaran and Bartha, 1992; Durrett, 1992;
474
N. F. BRITTON et al,
Ermentrout and Edelstein-Keshet, 1993; Fisch et al., 1991; Hassell et al. 1991; Rand and Wilson, 1995) are appropriate in many situations. A survey of mathematical results on such systems is given by Durrett and Levin (1994). The cells in the model represent patches of land on BCI, each of which may be in one of a finite, but unusually large, number of states (if unoccupied, its state represents its stage of recovery from the last army ant occupation; if occupied, the size of the colony in occupation). An important parameter in the model is the length of time a patch of habitat takes to recover after army ant occupation. By varying this parameter the effect of habitats of differing quality is simulated, good quality habitat being defined to be that which recovers quickly. The model is simulated on a computer. It is then analysed using a mean field approximation, which gives reasonably good results. This allows a derivation of equilibrium population size. An improvement to the mean field approximation is derived that takes into account the difference between the mean field in a neighbourhood of any given colony and that in the island as a whole. This improvement gives slightly better agreement with the simulations, but its main advantage is in elucidating the role of the patch neighbourhood N. For an occupied patch the neighbourhood N is composed of the patches to which the occupying colony may move at the next time step, and for any patch is composed of the patches from which an occupying colony may arrive at the next time step. The advantage of moving a reasonably large distance between successive statary phase bivouacs is demonstrated. 2. The Cellular Automata Model and Computer Simulations. We set up a discrete time stochastic cellular automata (CA) model for the system. A CA model is a set of cells, each of which at any given time can be in one of a finite number of states, together with a set of rules governing how the state of each cell changes with time. Such stochastic cellular automata models have been notorious for their intractability at least since Onsager’s (1944) classic paper on the Ising model over 50 years ago. A good initial approach is to simulate the behavior of the system on a computer. The time step models the activity cycle time, which is about 35 days for E. burchdi. The set of cells is a lattice of K points on the computer, modelling K patches of forest on the island that can be exploited by the ants, and we shall refer to the cells as patches. Define the age of a patch to be the number of time steps since it was last exploited; it is of age zero if it is currently occupied by a colony of ants. Let n (typically six) be the number of time steps taken for a patch to recover from army ant exploitation. A patch that has not yet recovered nor is on the verge of recovery at a time
POPULATION DYNAMICS OF ARMY ANTS
475
step is referred to as young; all other patches are old. (A patch of age n is on the verge of recovery at the time step, so is old.) We shall divide the ant colonies into size classes, the smallest being of size 1 unit and the largest of size 2m - 1 (typically nine) units. The state of a patch is its age, if unoccupied, or the size of its occupying colony, if occupied. This would give an infinite number of states except that the age of an old patch is immaterial, so that all old patches are taken to be in the same state. The colony size referred to above is its size at the beginning of the time step. At the end of the time step, a colony that has been occupying a patch that was old on arrival increases in size by 1 unit; a colony that has been occupying a patch that was young decreases in size by 1 unit. A colony that by this rule would reach a size of 2m units divides into two colonies of size m units, a colony that would reach a size of zero units ceases to exist. Maximum colony sizes in nature are roughly 650,000 workers (Franks, 1985); therefore one size unit corresponds to about 65,000 workers (assuming m = 51, a not unreasonable net change in colony size over a single activity cycle. At the next time step the colonies move to any one of the patches a fixed distance away with equal probability, and the process is repeated. On BCI the diameter of each patch is about 180 m, whereas the distance between successive statary phase bivouac sites is about 530 m. There are various ways in which this can be approximated on a square lattice: We chose to allow each colony to move to any of the 24 patches at the edge of a 7 x 7 square whose centre is the current position. We shall refer to these 24 patches as the neighbourhood N of the current position. To investigate the consequences of moving such a large distance, we also ran simulations in which each colony was only allowed to move to one of its eight nearest neighbours. We shall refer to these two models as the jumping CA (JCA) and stepping CA (SCA) models. There are problems in interpreting the neighbourhood near edges of the island. We use periodic boundary conditions, i.e. any colony that disappears off one side of the island simply reappears at the other. This is for simplicity in coding only; more realistic boundary conditions would make colonies turn back in some random or determinate direction when they reach the coast, but we do not expect the particular choice of boundary conditions within this class to affect the results unduly for islands of reasonable size. Similarly, we do not expect the shape of the island to be crucial, as long as it is not very convoluted. Since the area of BCI is about 500 times the area of a patch we took a lattice of 20 X 25 points to represent the island. The simulation is allowed to run until the system reaches an asymptotic state, which might in general be a steady state, a periodic solution or chaotic behaviour. Since our system is stochastic, we can only expect this state (if it is non-trivial) to be asymptotic in distribution, for example, the
476
N.
F: BRI’ITON et al.
numbers of colonies of various sizes averaged over many simulations may reach a steady state. For an island the size of BCI, our simulations resulted in an oscillation of small amplitude in the numbers of colonies and other variables of interest. The mean of the oscillation for various variables is given in Table 1 and compared with the data that are available for BCI and with the mean field approximations that are to be discussed in the next two sections. The data are not good enough to decide whether the oscillation found in the simulations also occurs on the island or whether it is an artefact of the model. The oscillation is weakened if a third type of patch is introduced which is intermediate in effect between young and old patches. This may be a more realistic model of the biological situation.
3. The Mean Field Approximation. In this section we derive an approximation to the CA model of section 2 that is amenable to mathematical analysis, which we shall refer to as the mean field (MF) model. The variables of interest are the numbers of ant colonies of various sizes and the numbers of patches of forest of various ages. At each time step, the rules of colony death and size change as given in section 2 are followed. The colonies then move to any patch on the island with equal probability, subject to the constraint that no two colonies end up on the same patch, and the process is repeated. This constraint was not applied in the simulations, but was felt to be biologically realistic and simpler to analyse. In the real situation and in the simulation of section 2 the colonies may of course only move to patches in the neighbourhood N of the given patch. The MF approximation is not intended to be a realistic model of such a process, but it may nevertheless approximate its behaviour very closely. It will isolate the temporal features of the population dynamics from the spatial features and give some clues as to what behaviour to expect from the real system. It is akin to the mean field approximation of statistical mechanics, and its success depends on the typical distances that are moved at each time step; for distances that are biologically realistic in this application we expect to get reasonably good results (Durrett and Levin, 1994). It is crucial that the
Table 1. Summary of results of the various models compared with data from BCI
ki*
KY* Ei Ej Turnover
BCI
JCA
SCA
MF
JMF
SMF
45-55 ? ? ? 1.6
54.5 233 5 9.17 1.22
40.8 174 5 15.45 1.16
54.5 250 5 8.17 1.09
52.9 240 5 8.76 1.06
48.7 218 5 10.4 0.97
POPULATION DYNAMICS OF ARMY ANTS
477
distance moved between successive statary phase bivouacs is much greater than the diameter of the patches. Let the proportion of patches that are of age j be yi; thus C~ZOy~= 1. Let the proportion of young patches be y’ = Cy,,‘yi. Let the proportion of patches that are occupied at time t by an ant colony of size i units be xf. Then the proportion of patches that are occupied by an ant colony of any size is given by x’ = C?Z; ‘xf. Clearly, x’=y&
(3.1)
At any time step, the chance of a colony finding itself in a favourable environment (an old patch) is 1 - yt, and the chance of an unfavourable environment is y’. Therefore the equations satisfied by the ant (n) variables are x;+l =x;yt, x:+l =x:(1 -y’)
X m-l t+l
=xL_z(l
+x5yf,
-y’)
x’m +l =x&-*(1 -y’) xt+l m+l
t+1
X2m-2
t+1
_ -x,(1
t
-Y’)
+x:,+lyf
(1
+ 2x&-,(1
-y’),
+x;+2yt,
-y’)
=.&-j(1
XZm-l=4,-2
+x:,y’,
+x;,_lyt, (3.2)
-Y’).
At time t + 1, the number of patches of age j 2 1 is equal to the number of patches of age j - 1 at time t that are not occupied by an ant colony at time t f 1, so that the equations satisfied by the patch (y) variables are y;+*
=yi'_,(l -x’+*)
(3.3)
for each j 2 1. The equations (3.11, (3.2) and (3.3) comprise the system we must analyse, and we first derive some consequences that will be useful later. Adding together the equations (3.2), x t+l = (x’+x;,_J(l
-y’)
+ (x’-x;)yt.
(3.4)
418
N. F. BRITTON
et al.
(3.2) by i, for i = 1,2,. . . ,2m - 1, and
Multiplying the ith of equations adding gives z r+l
= (z’+x’)(l
-y’)
+ ( z’ -x’)y’ =z’+x’(l
- 2y’),
(3.5)
where we have defined z’ = Cfr; ‘ix:. (Although this observation is not relevant to the analysis, we may note that 65,OOOKz’is a measure of the total number of individual ants on the island.) Let us now find the steady state of this system. We represent the values of the variables at the steady state by asterisks. From (3.5), z* =z* +x*(1-
2y"),
so that for a steady state with x* # 0, y* = L
(3.6)
2’
Then from (3.41,
x* = (x*
+&_,)(1
-y*> + (x* -x;>y*,
so that x~~._r(l -y*) -xTy* = 0, or with y* = 3, x&,_~ =xT. Then, using the steady state versions of equations (3.2),
=iqrnmi = ix;
x’
(3.7)
for i = 1,2,. . . , m and 2m-1
x *=
c
i=
x*=j&;.
(3.8)
1
A testable prediction of the model is therefore that the size distribution of colonies is triangular. From the steady state versions of (3.1) and (3.3), yj* =x*(1
-x*)’
(3.9)
for j = 0, 1,. . . and n-l y*
=
c
j=O
n-l
y;
= c x*(1 -x”)j= j=O
1 - (1 -x*Y
POPULATION DYNAMICS
OF ARMY ANTS
419
so that x* = 1 - (1 -y*)l/”
= 1 - (5)“”
(3.10)
Equations (3.6) to (3.10) give the values of the variables at the steady state. The system (3.1), (3.2) and (3.3) is an infinite system of difference equations. There is a decoupled subsystem of order 2m + II - 1 consisting of (3.1), (3.2) and the first II - 1 equations of (3.3). With m = 5 and n = 6 this is a system of order 15, which can be analysed numerically. The numerical results indicate that the system is stable. Numerical values of the steady state when m = 5 and IZ= 6 and the total number of patches K = 500 are given by fi* = K( 1 - (i)““)
(3.11)
= 54.6,
Ky” = +K = 250.
(3.12)
The mean colony size is given by
cy=,ix;
E(i) = c”
(3.13)
nc” =m=5
r-l
1
and the variance of this size is given by V(i) =Hi2)
Cy! li2xi* - (E(i)>2 = c” xT I-1 1
m2-1
=~
6
= 4. (3.14)
By defining the generating function y(s)
=
i
yi*sj =
j=O
2 x*(1
X”
-x*)‘sj = 1 -
j=O
(1 -x*)S
’
it is easy to show that the mean patch age is given by 1 -x* E(j) = t jy; = Y’(1) = ~
= 8.17
(3.15)
X*
j=O
and the variance of this age is given by 2 V(j)
=E(j2)
-
(E(j))2
=
jcoi2Y;
i
jgoi,:
i
= Y’(1) + Y’(1) - (Y’(1))2 = (1 -x*)/(x*~)
= 74.8.
(3.16)
480
N. F. BRI’ITON
et al.
The standard deviation is therefore 8.65, which is greater than the mean. This suggests that most patches are younger than eight units (in agreement with the result that half the patches are young), but a few are much older. The mean number of colony extinctions per unit time is equal to the mean number of fissions per unit time, since we are in a dynamic steady state, and is given by
Ky”xT =K(l
-y”).&_,
ICC* = -2m2 = 1.09.
(3.17)
4. An Improved Mean Field Approximation. The model above takes no account at all of spatial effects. It therefore does not permit one to explore the consequences of moving longer or shorter distances in the nomadic phase, since the approximation is’independent of these distances. The fact that a colony is more likely to return to a patch that it has recently exploited than one, on the other side of the island, simply because the former patch is nearer, is not considered; we shall remedy this deficiency here. Consider the situation at the beginning of the time step t, when colonies move to the patch that they will occupy for that time step. We have so far proposed that they move at random to any patch on the island (including the patch they currently occupy) with equal probability, so that the chance that any given colony has of landing on a good patch is equal to the proportion of good patches on the island, and the chance that any given patch has of being occupied is equal to the proportion of occupied patches at the next time step. That is, these chances depend on the two mean fields on the island: those of good patches and colonies. In fact, though, a colony occupying a patch A lands somewhere in its neighbourhood N(A) at the next time step, and a patch A can only be occupied by a colony which was somewhere in N(A) at the previous time step. Thus the mean fields in N(A) depend, among other things, on the state of the patch A. The idea of this section is to take into account this dependence of the mean fields in N(A) on the state of A through the first order effects described above. Some higher order effects were considered in preliminary work on this system, but made very little difference to the results, so are neglected here. The order of the effect is the power of l/M in the approximation, where A4 is the number of elements in N. Zeroth order effects, which explicitly take account of the fact that a colony cannot occupy the same patch twice in succession, are smaller than first order effects, because although they are O(1) in an expansion in powers of l/M, they turn out to be 0(1/k> in an
POPULATION DYNAMICS OF ARMY ANTS
481
expansion in powers of I/K, and we shall neglect them. The analysis is thus appropriate for a large island. Consistent with this and with the rest of the paper, we shall also neglect boundary effects. The patch equation (3.1) is unaffected by this change, so we have t+1
Yo
_ -xl+‘ *
(4.1)
We next derive the ant equations. At each time step, a colony occupying patch A is taken to move from A to any point in its neighbourhood iV with equal probability, again subject to the constraint that no two colonies end up on the same patch. We have to calculate the chance of a colony finding itself in a favourable patch, that is, the mean field of good patches in the neighbourhood N. Define y(S) to be the mean field proportion of young patches in a set S. Let the mean field y&N(A)) of patches of age j in the neighbourhood of the patch A satisfy yj(N(A))
if A is occupied,
;,
=
ii, i
if A is unoccupied,
I’
where yj and yy are independent of A. Then, by definition of the mean field, the mean of the neighbourhood mean fields is equal to the total mean field, g$ + (1 -x>y;
=yj
for each j 2 0. Now, the mean field of patches of age 1 in the neighbourhood of an occupied patch is greater than that in the neighbourhood of an unoccupied patch, y; > y’;, since we know that in the first case there is at least one patch B of age 1, that from which the occupying colony arrived. There is a corresponding diminution in the mean fields of patches of any other age in the neighbourhood of an occupied patch. Using the set equality N = N\B U B, with y,(B) = 1, y,(B) =y,(B) =y,(B) = e-e = 0, we have y;=y;’
12 + l i M M’
I
yb=y{
1-k i
y;=y;
, i
1-L i M’i (4.2)
482
N. F. BRIlTON
Eliminating
et al.
y;(N) from these equations for each j, we obtain Cl- l/M)y,
r; =
+ l/Ml
-x)
l-x/M
Cl- l/M)Yi JG=
)
l-q/j
(4.3)
for j # 1. Hence, summing from j = 0 to j = IZ- 1, assuming n 2 2, Cl- l/My
+ l/it4(1 -x)
y'=
1 -X/M
=P+Qy
or 1 -y’ = Q(1 -y), where l-l/M
Q= 1 -X/M
=l--+
1 A4
r+o M
1 ( M2 1 ’
(4.4)
Hence the chance of a favourable P = 1 - Q = (l/M)(l -x)/(1 -x/M). patch is Q and that of an unfavourable one is P + Qy. The equations (3.2) become
xi+’= (P'
+ Qfyt)x;, + (P’+ Qfyt)x;,
x:+’ = Q’(1 -y’)x; .
x;:r
+ (P’+ Q’y’)x:,,
= Q’(1 -y’)x;_,
xz,” = Q’(1 -y’)x&_, X ;,fl
=
Q’(1 -y’)x:,
+ (P’+ Qfyt)~;+l
+ 2&‘(1 -yt)&_,,
+ (P’+ Q’~‘)x;+~,
x;;!, = Q'(1 -y’)~;~-~ x;;!_, = Q’(1 -Y’)x;~-~.
+ (P’+ Qtyt)~;,_l, (4.5)
Now let us consider the patch equations. Each of the equations (3.3) must be modified to take account of the adjusted mean fields of colonies in their neighbourhoods. It is important to note that these adjusted mean fields are different depending on the age of the patch. Let the patch A be i units old, let the mean field of colonies in its neighbourhood be x(‘)(N(A)), and let x(‘)(N(A)) = x@ be independent of the particular patch of age i
483
POPULATION DYNAMICS OF ARMY ANTS
units that is being considered. We also assume that x(j) is independent of i for i # 1, i.e. x(‘) =x(O) for all i 2 2, since the difference arises from second and higher order effects. Then the mean of the neighbourhood mean fields is equal to the total mean field, yox@) +y,x(‘) +y#
+ *+*= (1 -y,)x’O’ +y#)
=x.
Also, the mean field of colonies in the neighbourhood of patches of age 1 is greater than that in the neighbourhood of patches of any other age, x(l) >x(‘) for i # 1, since we know that in the first case there is at least one occupied patch B, occupied by the colony that left A on the previous time step. The rest of the neighbourhood N\B is indistinguishable from any subset of N of M - 1 patches of a patch of age i # 1. Using the set equality N = N\B U B, we have 1 1 x(1) = 1 - M ,-@ + M)
( i
using
~(0 =x(O)
for all i 2 2. Solving the last two equations for x(O)and x(l),
we obtain x(o) =
x -Y,/M 1 - (l/M)y,
’
+ l/MO -yJ
Cl- l/Mb
x(1) _
(4.6)
1 - (l/M)y,
-
or (1 - l/M)0
l-x
1 -x(o) = I-
(l/M)y,
’ -‘(‘)
’
=
1 _ (l/M)y,
-x) ’
(4.7)
Equations (3.3) become r+l
Yl
_
yg1 -xr+9
- 1-
(l/M)y;
’
(1 - l/M)y;(l
-x’+l)
y+ I-
y;+l =
(l/M)y;
’
yi’_,(l -x’+l) l-
(l/M)y;
(4.8)
’
the last of these holding for all j 2 3. Adding together equations (4.21, X f+l =
Q’(1 -y’)(x’ +x;,,,_~) + (I”+
Q’y’)(x’-xi).
484
N. F. BRITTON et al.
Multiplying the ith of equations (4.2) by I and adding gives z r+l =
@(I
-y’)(z’+x’)
+
(P+
Q'y')(z'--x').
(4.9)
Equation (4.9) at the steady state gives Q*(l
-y*)
=f’*
+ Q*y*
= +
or, explicitly, 2(M-
l)y* --x* =M-
2.
(4.10)
The equations (4.1), (4.5) and the first n - 1 of (4.8) comprise a system of order 2m + n - 1 that we must analyse. Numerical results show that the steady state is stable for all biologically reasonable values of M. The quantities of primary interest are therefore the steady state values of the number of colonies and their size distribution, the number of young patches and the age distribution of patches, and the rates of colony fission and extinction. These of course depend on the number M of points in the neighbourhood, which depend on the model for the nomadic phase. As in the computer simulations we consider two possibilities: the jumping mean field (JMF) model in which each colony is assumed to move to one of the 24 patches a distance of three patch lengths away at the end of each time step, and the stepping mean field (SMF) model in which each colony is assumed to move to one of the eight patches adjacent to its current patch. The JMF model is more biologically realistic. It is more difficult to obtain numerical values for the quantities of interest in the revised model of this section than in that of the previous section, since some of the equations whose couhterparts gave explicit formulae in the last section no longer do so in this section. In particular (4.10) and the steady state version of (4.8) can no longer be solved to give explicit formulas for X” and y*. However a good approximation can be obtained as a perturbation of the solution in the last section or the equations may be used to calculate x* and y* by an iterative procedure. The difficulties do not apply to the mean of the colony size, where the calculations go through exactly as before, so that equations (3.13) and (3.14) still hold. Results for some other variables are given in section 5. Expressions for the mean and variance of the patch age may be found by a generating function approach, as in section 3. 5. Comparison of the Models. The results of sections 2 to 4 are summarised in Table 1. All the models except the stepping models give results in reasonable agreement with biological data, although they underestimate
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DYNAMICS
OF ARMY
ANTS
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the true value of the turnover rate by about 35%. The turnover rate is the rate of fission (or rate of death, since in a dynamic steady state the two quantities are equal) measured in colonies per time step. This may be improved by reducing the number of size classes, which is probably not biologically realistic, or by including senescence or some other form of density-independent mortality in the model. The jumping simulation (JCA) model is assumed to reflect biological reality most closely, the mean field (MF) model is a simple approximation of it, and the jumping mean field (MJF) model is an improved approximation. The stepping (SCA and SMF) models represent the effect of shorter distances being covered in the nomadic phase. It is interesting to note how close the MF model is to the JCA model, which vindicates the use of the mean field approximation in this case and allows very little room for improvement by the JMF model. However, the MF is not nearly such a good approximation to the SCA model, as expected from the general theory (Durrett and Levin, 19941, and the improvement given by the SMF model is much more necessary in this case. This is one reason for the development of the improved mean field approximations; another is that it allows us to compare the jumping and stepping strategies. We shall do this in the next section.
6. Competition between Nomadic Phase Strategies. In this section we consider the advantage of moving a reasonably long distance in the nomadic phase. Let us consider a population with a particular nomadic phase strategy I? (represented by the neighbourhood to which it moves) invaded by a mutant population with a strategy fi. We shall denote those variables and parameters associated with the wild type population by tildes and those associated with the mutant population by carets. In particular we will label patches with a tilde or a caret according to whether they were last occupied by the wild type or the mutants; for example yj is the proportion of patches that were both (a) last occupied j time units ago and (b) last occupied by a colony of wild type. We define Y= CT=,yj (the proportion of patches, young and old, that are occupied 0; were last occupied by a colony of wild type) and Y=C~X=o~, so that Y+Y=l. We also define x=2+2, y=y+j. The patch equation (3.1) still holds for the total population and for each population separately, so we have
(6.1)
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For the wild type ant equations, we have to calculate the chance of a colony finding itself in an unfavomable patch, that is, the mean field of bad patches in the neighbourhood N. The mean field of bad patches last occupied by a colony of wild type is calculated as in section 4 and the mean field of bad patches last occupied by a mutant colony is simply j, so that we obtain -t+l Xl
_ -
(Fr + @jr+j+:,
q!”
=
((jr - Qr,t
_@;
+
(p + (jtyt +jY)f:,
where 0 = (1 - l/&/(1 -i/A?) and p = 1 - &. Similar equations hold for the mutant ant population with the tildes and carets transposed. Now let us consider the patch equations. We need to calculate the mean fields of both kinds of colonies in the neighbourhood of each kind of patch. These adjusted mean fields are again different depending on the age of the patch, but also on the kind of colony that last occupied it. If a patch A is occupied or was last occupied by colony of wild type, then we know nothing about the mean field of mutant colonies in its mutant neighbourhood to distinguish it from the rest of the island, so its mean field of mutants is R. If the patch is occupied or was last occupied by a colony of wild type, then its mean field of colonies of wild type may be found by a similar argument to that in section 4. Let the patch be i units old, let the mean field of colonies of wild type in its wild type neighbourhood be f(‘)(N( A)), and let $‘)(N( A)) = ici) be independent of A. Also assume that Zci)= Z(O)for all
POPULATION
i L 2. Then the mean of the neighbourhood field, fog(O) +ylp
+y2~F’2)
+ ... =
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mean fields is the total mean
(I;_j+‘O’
+jp)
=
&.
AlSO
_w= (1 -
l/A?)f’O’ + l/G.
Hence
or
Equations (4.8) become -c+l
Y2
=
-t
Yl
1 (1-
l/tip+1 P
-?+l)
--it+1
j$/iti
) 1
(1 _i”I)f:r _21+1
j7;+1 =j;_l (
Ft -y;/ti
)
(6.3)
I
the second of these holding for j = 1 and all j 2 3, and with similar equations holding with the tilde and the caret transposed. The equations of this section reduce to those of section 4 if there is initially only one nomadic phase strategy, so the steady state analysis of that section leads to two steady states here. To determine the stability of these steady states we solved the equations numerically, taking initial conditions to be a small number of J strategists and a large number of 5’strategists. The results presented in Fig. 1 show that the J strategists successfully invade the S strategists, finally driving them to extinction. 7. Conclusions. Some aspects of the population dynamics of army ants may be successfully modelled by a cellular automata approach and simulated on a computer. The mean field approximation represents a tractable
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I
-0
50
,
100
150
200
250
300
,
I
350
400
450
500
time in cycles
Figure 1. The improved mean field approximation was used to simulate the effect of introducing a single colony of mutant jumpers, i.e. colonies that move three patch diameters in the nomadic phase, into a population of wild type steppers, i.e. colonies that move one patch diameter. The mutants drive the wild type to extinction within 500 cycles (less than 50 years).
alternative to simulation in modelling such populations whenever the distances moved in the nomadic phase are large compared to the statary phase patch diameter, as they are in Eciton burcheZZi.However it is much less successful when movements are small, that is, when interactions are more local. An improved approximation can then be made by treating the mean fields in the relevant neighbourhoods differently from those in the rest of the island. It will be important to have the mean field approximation available as a tractable alternative to simulation when we come to estimate expected times to extinction (Britton et al., 1996) since on large islands these times are very long and therefore impracticable to obtain by simulation. Equilibrium population sizes given by the model were in good agreement with data, even using the standard mean field approximation. The equilibrium mean field of young patches is 0.5, which means that at equilibrium a
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colony has a 50% chance of hitting a good patch at each step, as would be expected. The equilibrium mean field of colonies is given by (3.10), so the equilibrium number of colonies is K( 1 - (i)““). This is independent of one of the parameters of the problem-the number of size classes 2m - 1. The equilibrium number of colonies is proportional to island size, as would be expected. The dependence on n is more interesting and is shown in Fig. 2. The parameter IZ is the time in cycles taken for the habitat to recover, and may be thought of as an inverse measure of habitat quality. In conservation biology, there is often a need to maintain a given minimum population size in order to ensure a reasonable probability of population survival. Clearly one way to do this is to ensure a sufficiently large habitat island, but if this is not possible, changes in habitat quality may make an appreciable difference. To fix ideas, a habitat of quality n = 6 would have to be 45% larger than a habitat of quality II = 4 to maintain the same population. Quality is therefore an important determinant of population size and one that has perhaps been too often neglected in much research focussing on the size and number of reserves in conservation (Simberloff, 1988; Caughley, 1994). Model turnover rates underestimated the data by about 35%. This suggests that density-dependent mortality is not the only cause of turnover
10
area of island in hectares
0
2
patch recovery time n in cycles
Figure 2. The equilibrium population is shown as a function of patch recovery time n in cycles, an inverse measure of the quality of the habitat, and the area of the island in hectares. The number of patches on the island is one-third of its area in hectares.
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and that some kind of density-independent mortality may be important, A variant of the model including density-independent mortality does give higher turnover rates. For example, a model in which queens live on average six years, colonies die when their queen dies and with m = 5 and n = 6 as usual, gives with the mean field approximation an equilibrium population size of XV* = 48.6 and a turnover rate of about 1.3, an underestimate of only about 20%. The alternative is that spending the statary phase in a bad patch has an even harsher effect and/or spending it in a good patch has an even more beneficial effect, than it does in our model. A value of m = 4, equivalent to a net loss or gain of 25% more ants in these situations, and with n = 6, gives an equilibrium population size of KY* = 54.6 and a turnover of about 1.7, which exceeds the field data by about 5%. We have also considered the importance of the large distances covered between successive statary phase bivouacs from a biological point of view. The analysis here suggests that the most important parameter in determining the effect that the nomadic phase strategy has on growth rate is M, the number of patches in N. In other words, the more patches there are in the neighbourhood N that is accessible in the nomadic phase, the higher is the growth rate of the individual colony concerned. We suggest that this effect is driven by the probability that a colony, having moved from A to B during one nomadic phase, immediately moves back to A during the next. Since B E N(A), then A E N(B) and is one of the M patches in N(B), any of which the colony is equally likely to raid, so this re-raid probability is l/M. Because of this, the factor 1 - l/M +x/M in equation (4.4) would naively be expected to be simply 1 - l/M. The extra term n/M arises because the mean field in the re-raid patch is worse than would be expected from the uniform distribution assumption, so the mean field in the rest of the neighbourhood must be better to obtain the correct overall field. We have not taken into account any costs associated with moving large distances and we suggest that were these to be included there would be an optimum distance to move in the nomadic phase. It is interesting to note that if the colony does not always move the full distance possible, then more patches are available to it. A strategy that has some random variation in raid length and in the directions moved on successive days in the nomadic phase may therefore be better than one in which these quantities are always very similar (see Franks and Fletcher, 1983). The behaviour of colonies near the coast was not modelled in this paper and the simplest possible (periodic) boundary conditions were chosen. These were assumed to have effects similar to more realistic conditions, as long as the island was reasonably large and not too irregular. This conclusion is supported by simulations carried out by one of us (Partridge, 1994) on a detailed map of Barro Colorado Island with realistic boundary
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conditions, which gave very similar results. However, for small or irregular islands it might be necessary to model the effect of the behaviour of the colonies near the coastline more carefully. In the cellular automata approach, some kind of reflection might be adopted. In the standard mean field approach, boundary conditions are not required, but the method of improved mean field approximation might be adapted to take account of these effects. As mentioned above, this paper suggests that the number of neighbouring patches that can be reached from any given patch is the most important feature of the spatial organisation. This could be estimated for any given size and shape of island, and the corresponding equilibrium population size could be estimated. We would predict that an island with a very convoluted coastline would not be able to support as large a population as a rounder island of the same area.
REFERENCES Britton, N. F., L. W. Partridge, and N. R. Franks. 1996. A model of survival times for population of army ants, Submitted to Bull. Math. Biol. Caswell, H. and R. J. Etter. 1993. Ecological interactions in patchy environments: from patch-occupancy models to cellular automata. In Patch Dynamics. Lecture Notes in Biomathematics, S. A. Levin, T. M. Powell, and J. H. Steele (Ed& Vol. 96, pp. 93-109. New York: Springer. Caughley, G. 1994. Directions in conservation biology. J. Anim. Ecol. 63, 215-244. Czaran, T. and S. Bartha. 1992. Spatiotemporal dynamic models of plant populations and communities. Trends Ecol. Evol. 7, 38-42. Durrett, R. 1992. Stochastic models of growth and competition. In Patch Dynamics. Lecture Notes in Biomathematics, S. A. Levin, T. Powell, and J. Steele (Eds), Vol. 96, pp. 176-183. New York: Springer. Durrett, R. and S. A. Levin. 1994. Stochastic spatial models: A user’s guide to ecoIogica1 applications. Phil. Trans. Roy. Sot. London Ser. B 343, 329-350. Ermentrout, G. B. and L. Edelstein-Keshet. 1993. Cellular automata approaches to biological modelling. J. Theor. Biol. 160, 97-133. Fisch, R., J. Gravner, and D. Griffeath. 1991. Threshold range scaling of excitable cellular automata. Statist. Comp. 1, 23-29. Franks, N. R. 1992a. Ecology and population regulation in the army ant Eciton burchelli. In The Ecology of a Tropical Forest: Seasonal Rhythms and Long-Term Changes, E. G. Leigh, A. S. Rand, and D. M. Windsor (eds.), pp. 389 -395. Washington, DC: Smithsonian Institution Press. Franks, N. R. 1982b. A new method for censusing animal populations: the number of Eciton burchelli army ant colonies on Barro Colorado Island, Panama. Oecologia Berlin 52, 266-268.
Franks, N. R. 1982~. Social insects in the aftermath of swarm raids of the army ant Eciton burchelli. In Biology of Social Insects. Proceedings of the 9th International Congress of the International Congress of the International Union for the Study of Social Insects, M. D. Breed, C. D. Michener, and H. E. Evans (Eds.), pp. 275-279. Boulder, CO: Westview Press. Franks, N. R. 1985. Reproduction, foraging efficiency and worker polymorphism in army ants. In Experimental Behavioral Ecology. Fortschritte der Zoologie, M. Lindauer and B. Hiilldobler (Eds.), Vol. 31, pp. 91-107. Stuttgart: G. Fischer.
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Franks, N. R. 1989. Army ants: a collective intelligence Am. Scientist 77, 139-145. Franks, N. R., and W. H. Bossert. 1983. The influence of swarm raiding army ants on the patchiness and diversity of a tropical leaf litter ant community. In E. L. Sutton, The Tropical Rain Forest: Ecology and Management, E. L. Sutton, A. C. Chadwick, and T. C. Whitmore (eds), pp. 151-163. Oxford: Blackwell. Franks, N. R. and C. R. Fletcher. 1983. Spatial patterns in army ant foraging and migration: Eciton burchelli on Barre Colorado Island, Panama. Behav. Ecol. Sociobiol. 12, 261-270. Hassell, M. P., H. N. Comins, and R. M. May. 1991. Spatial structure and chaos in insect population dynamics. Nature London 353,255-258. Onsager, L. 1944. Crystal statistics. I. A two-dimensional model with order-disorder transitions. Phys. Rev. 65, 117. Otis, G. W., C. E. Santana, D. L. Crawford, and M. L. Higgins. 1986. The effect of foraging army ants on leaf-litter arthropods. Biotropica 18, 56-61. Partridge, L. W., 1994. Facets of the ecology, behaviour and evolution of ants. PhD thesis, University of Bath. Rand, D. S. and H. B. Wilson. 1995. Using spatio-temporal chaos and intermediate-scale determinism to quantify spatially extended ecosystems. Proc. Roy. Sot. London Ser. B 259,111-117. Ray, T. S. and C. C. Andrews. 1980. Antbutterflies: butterflies that follow army ants to feed on antbird droppings. Science 210, 1147-1148. Rettenmeyer, C. W. 1963. Behavioral studies of army ants. Kansas Univ. Sci. Bull. 44, 281-465. Schneirla, T. C. 1971. Army ants. In A Study in Social Organization,H. R. Topoff (Ed.). San Francisco: W. H. Freeman. Simberloff, D. 1988. The contribution of population and community biology to conservation science. Ann. Rev. Ecol. Systematics19, 473-511. Willis, E. 0. 1967. The behavior of bicolored antbirds. Univ. Calif. Publ. Zool. 79, 1-127.
Received 3 December 1994
