A mathematical theory of the growth of populations of the flour beetle Tribolium confusum, Duval IX. The persistence of tunnels in the flour mass

J. Theoret. Biol. (1966) 13, 379-411

A Mathematical Theory of the Growth of Populations of the Flour Beetle Tribolium confusum, Duval IX. The Persistence of Tunnels in the Flour Mass JOHN

STANLEY

Department of Genetics, McGill University, Montreal, Quebec, Canada (Received 14 February

1966)

An attempt is made to formulate the total tunnel existing after a given time in flour containing the beetle Tribolium confusum. Tunnels in the flour mass are destroyed by transversal of the existing tunnels during boring of new ones. The half lives of tunnels are evaluated and solutions are sought in exponential equations describing the fate of various types of tunnel after a given interval. A numerical example is given,

l.Introdlwtion Stanley (1964) developed a mathematical theory descriptive of the growth of the plexus of tunnels which forms in a mass of flour of volume V, seeded with one beetle (Tribolium

confusum) at time T = 0. This was an improved

version of a previous theory (Stanley, 1949). It was pointed out that there are three kinds of tunnel: F = total tunnel, 9 = new tunnel (traversed only once) and JI = old tunnel, re-traversed at least once following the initial boring. All types of tunnel are subject to destruction by collapse, due to the disturbance of adjacent new boring (destructive new tunnelling). New tunnel may be converted to old tunnel by re-traversal (conversion tunnelling). At any arbitrarily selected time, T = x, there are definite amounts of the three kinds of tunnel in the flour, but new is being converted into old, and both are being destroyed. As time passes, thinking in terms of the “original” tunnel present at T = x, the relative composition of the tunnel-plexus will change and eventually all the original tunnel will be destroyed. It is replaced, of course, by other tunnel produced subsequent to T = x. Our present problem is to examine the fate of the original tunnel, present at T = x as T increases from x to approach infinity (see Fig. 1). 379

380

J.

STANLEY

2. Explanation

of Symbols

Stanley (1949) symbolized the three types of tunnel as P(T), &(T> and e(T), simplifying this notation in 1964 to F, 4 and $. In this paper we shall require more detailed identification of the various kinds of tunnel, and shall use a subscript notation, as shown below. These subscripts are appended to the principal symbols, F, 4 and @. x denotes an “original” tunnel volume, present at T = X, i.e. I;;, 4, and $,; d denotes a portion of tunnel destroyed by destructive new tunnelling in the interval x I T < 03, such as &,; L denotes a portion of tunnel which has not been destroyed in the above interval, and remains in the flour as a physically observable volume, such as $r; c denotes a volume of flour which has been converted from new to old tunnel in the above interval of time, such as +,a; T = time measured from the setting up of the experiment, when the single beetle is placed in the mass of un-tunnelled flour; T* = time measured onward from a selected time, x (see Fig. 1); t means “total”, and has no reference to “time”, for example #,,, (see Fig. 1). Double subscripts arise, thus c/I~,.means new tunnel, first converted to old tunnel and then left in the flour as such. Fed means total tunnel, first converted to old tunnel, and then destroyed as such. The subscript “1” will appear in such forms as 4,t = +=r + CJ!J~~, 4dt = $d + $Cd, and so on. Selecting any time, X, we have a quantity Fx of total tunnel and, as time increases, the situation will unfold as shown below. destroyed as new tunnel, without prior Total conversion = q& (equation (58)) destroyed = )bt to old tunnel and then (equation (157)) destroyed as such = &, (equation (67)) to old tunnel and then left Fs = total Total as such = &, (equation (45)) tunnel left = q& and left as new when (equation (78)) tunnel = q&, (equation (38)) T=x yl, = old tunnel destroyed as old tunnel = v/d h-wtion (la)) whenT=x = VL bwtion (11)) [ left as old tunnel

From the above, in view of certain obvious relations which exist, we can write some identities which are useful later for cross-checking, for the easy solution of some equations and for the calculations of a numerical example. Fx = h+ti,, 4, = 4i+hl+dJL+AL

(1) = kit+A.t,

ICI, = eci+k.9

(2) (3)

FIG.

T’

16-

20-

24-

28-

-

and

I

I

I 30 0

__rcTheory

, 40 10

,

, 50

begins

20

,

here

I 60 30

I,

Time

70 40

(

, 80 50

,

, 90 60

(

, 100

70

,

, 110

80

,

, 120

90

,

, 130

from Fig. 2 of Stanley (1964), showing the general growth of the tunnel-plexus, and the position

I

Build-up of $,,tix;. Fx in this period

1. Mod&d

-

ofT=x,T*=O.

5 xl 6 a ii

_

L

32%

382

J. FL,

= &fdbLf$L

STANLEY

(see equation (163))

(4)

4dt+tid,

(5)

= +“dt+h

Fcit =

4d+hd+$d

F,,

=

=

+‘ct

=

hd

*LA = h+ l(ldt

=

$d

+ ~cL.,

4CL9 + hd,

(6) (7) (8) (9)

3. Discussion (A) THE FATE

OF I,b,

The fate of the original old tunnel ($,) presents the simplest problem (see Fig. 6), as there are no complications due to possible conversion. From equation (7) of Stanley (1964):

where E = kkl(gWhence,

1) from equation (17) of Stanley (1964). $tL = $,e3x-T’,

(11)

and lim 1c/‘ = I(/,,

(12)

T-+X

lim $L = 0,

(13)

T-CO

lim $i=-+,
(14)

T-+X

lim I&, = 0.

(1%

T-tm

The limit of all higher derivatives of $r., is always zero at infinity, thus the approach to the value tiL = 0 at infinity is asymptotic. The function is plotted in Fig. 6, and the curve has no point of inflexion. From equations (3) and (1 I), (16)

whence (17) (18)

GROWTH

OF

383

POPULATIONS

lim JI; = 5*x > 0,

(20)

T+X

lim I++;= 0.

(21)

T-rm

Again, all higher derivatives approach zero as T approaches zero, so that the approach is asymptotic. The function is shown in Fig. 6, and the curve has no points of inflexion. (B) THE

FATE

OF

f-,

By exactly similar reasoning, the functions describing the fate of F, may be obtained, thus: lim I;, = F,,

(23)

T-+X

lim FL( = 0,

(24)

T-ra,

limFt,=-$F,
lim F;, = 0,

(26) (27)

T-r-

and again the approach is asymptotic. A cross-check using equations (22), (163) and (11) will show that the identity of equation (4) is satisfied. From equations (1) and (22), F dl = F,[l-e~‘x-T’],

Gw

lim Ed, = 0,

(2%

lim Fdt = Fx,

(30)

T-r%

T+a,

E

F;, = -vF&’

+x-T)

lim Fit = EF ‘v x >0,

,

(31) (32)

T-rX

lim Fi, = 0. T-r@2

(33)

384

J.

STANLEY

Again, a cross-check : using equations (28), (157) and (16) will show that the identity of equation (5) is satisfied. The function is plotted in Fig. 2, using the values of the numerical example of Stanley (1964). 251,

T’

I

I *.*. ,

0

IO

I

/

I

,

I

30

20

I

I

40

I

r

50

I

I

I

60

70

,

I

00

I

I

,

90

Time

FIG.

(C)

2. The fate of F, subsequent

THE

HALF-LIVES

OF

Fx

to T*

= 0.

AND

4,

The curves of FL, and Fd cross only once, as seen from Fig. 2 and we may determine the time, T(F,, = Fdt) at which FL, = (j)Fx. For such determinations it is better to measure time as T*, setting T* = (T-x). It will then be apparent, comparing equations (11) and (22) and also equations (16) and (28) that T*(k, = t41) = T*@L, = Ftit). (34) From equations (11) and (16), setting T*(tjL = +,J as TT (for ease of typesetting) $se+1* = $x[l-&:], T: =;log,2

as shown in Fig. 6.

= TWL

(35)

= tidh

(36)

GROWTH

OF

POPULATIONS

385

From Stanley (1964), where = relative preference of the beetle for new tunnelling as opposed to re-traversal; $1 = mean speed of the beetle while boring a new tunnel; Q = cross-sectional area of a tunnel; k i- - a parameter measuring the friability of the flour; Q = mean radius of a tunnel plus the thickness of a “destruction sheath” around the tunnel. (This is a cylindrical zone around the tunnel, such that, if the centre of an adjacent tunnel, being bored, comes to lie in it, the tunnel under discussion will collapse (see Stanley, 1964, Fig. l).); r = mean radius of a tunnel, and we have E = kk,(g- I), where g = Qz/r2, k = plsla. Pl

Clearly then, from equation (36), T*($,, = $J occurs earlier the larger pi, s1 or a, and earlier the greater the ratio between the cross-sectional area of the destruction sheath and that of the tunnel. This is equivalent to saying that the half-life of the original total or old tunnel is briefer the more intensive the destructive effect of adjacent tunnelling. It will be noted that the effect of conversion tunnelling does not enter here because in the case of FLt, it makes no difference whether or not a portion of tunnel be old or new, and in the case of eL we are thinking in terms of the original tunnel, and thus do not include additional old tunnel provided by conversion tunnelling. Conversion tumelling would enter into any formulation describing a “half-life” derived from I,& (see equation (133)). (D)

THE

FATE

OF 4,

The fate of the original new tunnel presents a much more complex picture (see Figs 3, 4 and 5) because we must deal with the types +,,1, composed of A and AA, and &, composed of +eL and &. These are discussed below in that order which makes the reasoning simplest. From equations (6) and (9) of Stanley (1964), (37)

(38) and

T.I).

386

J.

T’

0

IO

20

30

STANLEY

40

50

70

60

00

90

Ttme FIG. 3. The general fate of the original new tunnel, &, subsequent to T* = 0. This figure should be examined together with Figs 4 and 5.

IO ....

.............

I I . .................

............

9 l--Y /

/

1

/

I

/

I

I1

I

t

11

II

. ... . .....

’

.?. ... 7; I-

40

60

50 10

20

00

70 30

100

90 50

40

60

120

110 70

00

130 90

Time FIG. 4. The etkt of destructive new tunnelling on & without regard to conversion [email protected].

GROWTH

<

6-

g B

5-

4

43-

OF POPULATIONS

387

2loT T*

0

IO

20

30

40

50

60

70

80

SO

Time

FIO. 5. The effect of conversion tumelling on & without regard to destructive new tumelling. Seealso Figs 3 and 4.

ELI-- (k,+E) xe~(x-T) , v 4 lim & -qQJ~
(41) (42)

T-+X

lim &=O.

(43)

T-+CQ

Thus, in the end, all the original new tunnel is destroyed, with an asymptotic approach to zero as T (or T*) approaches infinity (see Fig. 4). Some of the original new tunnel is, however, converted to old tunnel (see Fig. 5), and may then remain as such (c/I,L) or be destroyed after conversion ($3. From equation (4) 4 ~L=J'L,-+L-#L. WI Whence, via equations (ll), (22) and (38) #cL = ~,e+-n[l

-$:(X-T)I,

lim AL = 0,

(45) (46)

~~,=O.

(47)

Tda,

From equation (45), 4;L = 9*[(k,=E)~~~(x-T)-~e~(x-“],

W)

388

J.

STANLEY

whence

lim fjiL = 7

> 0,

(49)

T-rX

(50)

lim (pCL= 0, T+a,

and the approach as T (or T*) approaches infinity is asymptotic. Equation (48) implies a maximum value of ~$=r at

T(4&J = x+;1og. 2

(51)

LL,, = )I(&E)q&).

(52)

From equation (48), (53) and this implies a point of inflexion at

T*(&) = $ log,(y),

(54)

whence, oia equation (CA),

T*(AL,) = 2T*b4~m.J This is somewhat curious, as it is independent From equations (45) and (54)

(55) of all the biotic parameters.

(56) Turning now to the question of the amounts of new tunnel which are destroyed, the amount destroyed without prior conversions to old tunnel will depend solely on &, and in no way directly on $CL or 4=*. Therefore, as per equation (6) of Stanley (1964), 44 - $bL. Substituting integrating,

(57)

the value of #I~ from equation (38), separating the variables and 4a _ Wx

[l-;@$%-T)],

k,fE

(58)

whence lim & = 0, T-rX

(59)

GROWTH

OF

389

POPULATIONS

(61)

WX lim 4; =y>o, T-W lim 4; T-CO

(62)

(63)

= 0,

and the approach to rj+, = 0 as T approaches x is asymptotic. There is no maximum, other than that of equation (61) and no point of inflexion (see Fig. 4). Some of the original new tunnel is first converted to old tunnel and may then be left as such, as per equation (45), but other portions are subsequently destroyed. The total quantity? converted (whether or not it is subsequently destroyed) is the original amount of 4, minus that left as new tunnel, minus that destroyed as new tunnel, i.e. Substituting

43 = A-4L-dJd. the values of & and 4,, from equations (38) and (58),

(64)

but of course, At = 4,a+ +cL. (66) Substituting the values of 4CL and 4,t from equations (45) and (64), and solving for 4Cd, 4ca = ~,[{l-e$(x-T’j-jl-~~~‘x-T’~],

(67)

2

whence lim f/&j = 0,

w-0 (69)

1, lim t#& = 0, T-r%

lim & = 0,

T-rCO t see equatkms

(149)

to (153).

(70) (71) (72)

390

J.

STANLEY

and the approach to the value of (pnl as T approaches itity (Fig. 4).

is asymptotic

From equation (70),

whence, there exists a point of inflexion at

(74) Substituting

the value of T@,,)

from equation (74) in equation (67)

~=*,=)I[(~E)(l-~~)(~E)~}].

(75)

As a cross-check, it should be noted that as T approaches infinity, t$,, and &, = 0 and

~m[),~+k+~~l=C(~)+~+~l=F,

[Q.E.W.

$CL,

(76)

Be itioted that the time T(c$L = c#+,)has no real meaning, owing to the losses of new tunnel by conversion. See, however, equations (171) and (172).

4. Some Additional Functions We are now in a position to determine certain additional functions having the subscript “t” meaning “total”, other than FL1 of equation (22), Fd, of equation (28) and 4,, of equation (65) which were, of necessity, dealt with en route in previous discussion. (A)

TOTAL

OLD

TUNNEL,

LEFT

AS OLD

TUNNEL

This does actually exist in the flour (see Fig. 5), consisting as it does of that portion of $, which has escaped destruction, plus that old tunnel converted from $J, and also escaping destruction. We have then that $I+, = A. + d%L, whence, from equations (11) and (45)

(77) (78)

Equation

(78) could have been obtained from equations (22) and (38) as

ti = FL, - 41.. From equations (12) and (46)Ldr from equations (24) and (40), lim kt = ti*s T-X

(79) 030)

GROWTH

OF POPULATIONS

lim Jlt* = 0.

391 (81)

Tea,

and equations (80) and (81) hold regardless of the particular selected. From equation (78)

value of x

(82)

whence (83) (84) From equations (32) and (35) of Stanley (1964), limp -=- kz+E T-mf#’

E

*

(85)

Whence, since F = $+4,

From equations (45A) and (37) of Stanley (1964), limM=O.

(87)

TdO

We have dM ==

W-4’JI q)2

P

033)

and from equations (25) and (26) of Stanley (1964), it can be shown that dM/dT is zero when Be-BT+(A+B)e-(A”B)T-Ae-AT = 0, (89) where A = (k, + E)/V and B = (k+E)(Y (see Stanley, 1964, p. 206). The only solutions of equation (89) are T = 0 and T = 00. The former is the minimum, where M = 0 and the latter is the maximum, where M = k,/E. In the interval 0 I; T 5 00, dM/dT > 0. Therefore, for all realistic Tribolium tunnel-plexi, lim &,t * 0. (90) T-W

For a brief period, then, the total quantity of old tunnel increases subsequent to T* = 0 (see Fig. 6). This is so because in any naturally produced

392

J. STANLEY

T T*

40 0

50 IO

60 20

00

70 30

40

90 50

100 60

110

120

70

00

130 90

Time

FIG. 6. The fate of the original old tunnel (vz). Note in particular above the original value v/r.

the increase of vLt

plexus during this period t+kLtand $r are always so related that the increase in the quantity of old tunnel (due to conversion from new tunnel) is greater than the loss of old tunnel by destruction. The effect occurs regardless of the value of x selected. This is somewhat difficult to appreciate. One might suggest that if kz = E (presuming we could fIind a strain of Tribolium for which this can be true), then if x be selected at a value greater than T(4 = $), we would have M > kz/E and there would be no such initial increase in tiL1. This situation, however, would not arise, as can be seen by reference to equation (51) of Stanley (1964). In expanded form this shows that T(c$ = 1,4)occurs when (k,+E)(k,+k)e-BT-2k,(k+E)e)e-AT

Substituting

= (k,+E)(k,+k)-2k,(k+E).

(91)

the value kz = E, except in the indices (A and II), we have e -BTEe-AT

(92)

GROWTH

OF

POPULATIONS

393

This can be true only when T = 0 (when 4 naturally equals 9) and at T = co. That is to say, when k2 = E, the moment T(+ = $) occurs only at infinity. The quantity k2 = pzs,a is a measure of the conversion-tunnelling effect as per equations (9) and (10) of Stanley (1964). E is the parameter controlling the destruction effect as per equations (5) to (7) of Stanley (1964). If the two are equal, re-traversal is so uncommon as compared to destruction that old tunnel builds up so slowly that it equals new tunnel only at inlinity. ThusMel = kz/E and the initial increase of GLt occurs (0 < x < co). From equation (82) lim #it = 0, (93) T-CO

indicating a maximum value of $Lt in the interval x c T < co or 0 < T* < co. From equation (82), $Lt is zero at

and that this is a maximum

can be shown thus. From equation (82)

Noting that T* = -(x-T) equation (94)

and substituting

the value of T*(I,&J

from

(96) showing the presence of a maximum. Substituting the value of T*($ rt,J

from equation (94) in equation (78)

Inasmuch as, from equation (81), $rl becomes zero at infinity, it must, having increased above JI, at T*(J/ LtJ decrease again to equal I,+, at T*(ht = klFrom equation (78) this will occur when F,e-7

EP

-4xe

&+JOTI V = $, = F,-4,.

This is an implicit equation similar to equation (51) of Stanley (1964) and no wholly general solution exists for all possible values of E/V and (k, + E)/ V. However, a solution is possible in some cases. Setting k2 = nE and - !!T. ev = 2 and noting that F, = Cp,+ J/, we substitute these values in equation (98) and divide by (- 4) Z~+l-(M+l)Z+M = 0. WI

394

J.

STANLEY

This equation can be solved for 2 in any case in which it can be transformed into an equation of the form Z”-(M+ l)Zb+M = 0, uw where a and b are integers less than or equal to 5. This transformation is possible for n = 0, 1, 2, 3, 4 and for the non-integral values n = 0.25, O-333, 0.50, l-50 and O-666. Of these cases, when n = 0 there is no re-traversal, and no true plexus of tunnels. When n = 1, old tunnel equals new tunnel only at infinity (as discussed above). When n = 2, a tunnel plexus does develop, and this is the simplest case relating to a real situation. When n = 3 we have the case used in the numerical example to follow, where k2 = 6 and E = 2. When n = 4, we must solve a quintic equation, but this is possible because one root is known (2 = 1) and the equation can be reduced to a quartic. When n is non-integral, as say n = 1.5, it is possible to transform either to a quartic, or to a quintic which can be reduced to a quartic. These cases are discussed below. Case (i), n = 0 Equation (99) becomes Z-(l+M)Z+M = 0 whence 2 = 1 and T = 0. This is an unrealistic situation, re-traversal.

(101) there being no

Case (ii), n = I

This case has already been discussed in connection with equations (91) and (92), but it is of interest to see how it develops via equation (99). Equation (99) becomes z2-(1 +M)Z+M = 0. (102) There are two roots, 2 = 1 and 2 = M. For the root 2 = 1, T(t+hkt = +.J = x and T*(t+hk, = $3 = 0. For the root 2 = M, T*(&,

= $3 = -:

log, M > 0

OeM<
(103)

But for n = 1, k,/E = 1, and this is the limiting value of M as T approaches infinity, so that in the range 0 c T < cc, M 4 1, log, M 4 0, and there is always an initial increase in $rt. In point of fact, the time x, being also at infmity, if one attempts to find a time at which M > k,/E; at infinity, both roots, 2 = 1 and 2 = M, yield the same value of T*&, = ex). This case is unrealistic for Tribolium, but it would apply in the case of an insect which rarely traversed its own tunnels after the initial boring.

GROWTH

OF

395

POPULATIONS

Case (iii), n = 2 In this case, equation (99) becomes 2%(l+M)Z+M = 0, andZ = 1 being a root, this reduces to Z2+Z-M = 0, of which there are two roots = -1+J1+4M Z 2 ’

ww (105)

(106)

-l-&+4M (107) 2 From equation (106), in order to yield a real value of T*(@r, = $3, 2 must be in the range 0 < Z s 1 and M will be in the range 0 < M s 2 because k2/2, the limiting value of M is 2. The root Z = 1 yields T* = 0, T = X, while equation (106) yields

z=

T*h

= $3 = ilog.

[

-1+&+4M 2

] > 0.

(108)

Equation (107) does not lead to a root of equation (98) because this equation has no roots other than those already determined. Case (iv), n = 3

This is the case which would be used for the numerical example to follow, where k2 = 6 and E = 2. Equation (99) becomes Z4+(1+M)Z+M = 0, (1W and2 = 1 being a root, this reduces to Z3+Z2+Z-M = 0, uw a cubic equation in standard form: Z3 $ bZ2 +cZ+ d = 0, which can be solved by standard methods. This done, it is found that z=

jM

M2

J z + 0.129630 + J 4 + 0.129630M + 0.027778 M M2 -;i-+O*129630M+OG27778 + J 2 + 0.129630 J -0.333333. (111) Certain problems which may arise in the solution of cubic equations do not arise here, and one can always obtain one real root 0 < 2 5 1. The remaining roots are either imaginary, or show 2 as less than 0 or greater than 1, and

396

J.

STANLEY

yield thus no real value for T*(& = @J. Knowing 2 from equation (111) the value of T*&, = $.J is easily calculated (see Fig. 6),

=$J=-~log,Z>O. T*WL,

uw

Case (v), n = 4 In this case equation (99) becomes Z5-(1+M)Z-M and2

= 0,

(113)

= 1 being a root, this reduces to

Z4+Z3+Z2+Z-M=O,

(114) which can be solved for a further root 0 -C Z I 1, the remaining roots again being of no use. Case (vi), n = 1.5 This is a practicable case for an insect which re-traverses its own tunnels, and is interesting because of the non-integral value of n. Other cases with non-integral values of n can be solved by similar methods. Equation (99) becomes Z2”-(l+M)Z+M = 0, (115) and setting Y = Z”‘, we have

Y'+(l+M)Y'+M=O.

(116)

Now Z = 1 is a root of equation (115) and thus Y = 1 is a root of equation (116) and this equation then reduces to Y4+Y3+Y2-MY-M=O, (117) which can be solved as a standard quartic equation. Returning now to further discussion of the plotted curve of $r(, from equation (95), $& = 0 when

T* = T*(&)=;

2 log

(118)

but this point of inflexion can never be at T* = 0, (T = x), because this would require that FJ&, = (E/(k,+E))’ and, from equations (85) and (86) this can never be the case in the interval 0 5 T < 03. Noting that T* = -(x-T) and substituting the value of T*(&J from equation (118) in equation (78),

(k2+E)2

(119)

GROWTH

(B) TOTAL

OLD

OF

TUNNBL,

397

POPULATIONS

DESTROYED

AS SUCH

This is old tunnel, remaining from the original JI, and then destroyed (see Fig. 6), plus additional amounts provided by conversion, and then destroyed. Thus Jlat = $d + (Pcd, whence, from equations (16) and (67)

(120)

We could also write elf* = Felt- 4&o and obtain equation (121) through equations (28) and (58). It is seen that lim Jldt = 0,

(122)

(123)

T+X

and from equations (18) and (69)

while from equations (30) and (60) (125) Noting equations (120) and (122) it is easy to show that the limits of equations (124) and (125) are the same. In point of fact, the results stemming from either equations (120) or (122) are always the same, and we shall work through equations (121) to (122) in the subsequent discussion bearing on qat. From equation (121), uiu equations (19) and (70)

lim & = E+ = lim $/; > 0. T-x

Td%

Equation (127) is to be expected because, until the lirst iota of conversion tunnelling can take place in the interval 0 5 T* < (T*+ AT*) as AT* approaches zero, I&~ = $A. Furtha, lim #& = 0, W-9 T-b.20

and the approach as T approaches infinity is asymptotic.

398

J.

From equation maximum when

STANLEY

(126) casual observation F x e+T)

would indicate

an ordinary

= s,;~~~-T’+

(129) But, as seen from equations (22) and (38) this can never occur, except as T approaches infinity, From equation (126)

(130) whence there is a point of inflexion when

Substituting

the value of T*($,,J

from equation (131) in equation (121),

9 cumbersome expression which cannot be put in any simpler or more useful form. (C)

HALF-LIVES

ASSOCIATED

WITH

vL,

vLc

We have previously determined the half-lives of I;, and tiX as in equation (34), without taking into consideration any additional old tunnel provided by conversion. A similar time, T*(fiL1 = $dJ can be found (see Fig. 6) but it is not a half-life in the true sense because of the additional old tunnel provided by conversion, changing JIL to eL1, see Fig. 6. From equations (77) and (120), via equations (ll), (16), (45) and (67), it can be shown that T*&, = tji&) occurs when

2(k,+E)FxeVPrx-T)-~k2+2~)~,~~'X-J'=(k,+E)F,-E~,.

(133)

This equation, similar to equation (51) of Stanley (1964) and to equation (98) of the present paper has no general explicit solution but, where the conditions noted under equation (100) obtain, it can be so solved. In point of fact, in the practical sense, given a calculating machine, one can solve it as quickly by trial and error. From Fig. 6 it will be noted that T*(&, = @,,J > T*($r, = $,,) but the question arises as to whether this is a peculiarity of the particular numerical example chosen, or whether it is always the case. If it is the case, then at T+WL = $d) We should observe that lC’Lt > $dt. Substituting the value of T*(& = @,J from equation (34) in equations (78) and (121), we have

GROWTH

399

OF POPULATIONS

whence ;%yy.

(?)

(135)

From equation (139, setting (k, + E)/E = Y, this is equivalent to showing that

Now in any real Tribolium We have also that

L>l (136) Y+l * tunnel-plexus, k2 % 0, E 9 0 and (k, + E)/E % 1, * 2p ky+l=’ 2r lim y+jTl=

1

(137)

*.

(138)

From equation (136) cl 2’ --= dY ( Y+l > and this is greater than zero if

2r[(Y+l)log,2-1] (Y+1)2

’

(139)

1 1 = 0.4427. h?, 2 Therefore, in any real Tribolium tunnel-plexus, the left-hand side of equation (136) is monotonically increasing for any realistic value of Y, and equation (134) is proven correct. Thus T*(&, = e,,J always occur later than T*&+ 1+4k),see Fig. 6. There are, in fact, many such interesting sequential relations between the various special times, such as shown in equations (55) and (172). If certain mathematical difhculties can be overcome, it is hoped to deal with this matter in extenso in a later publication.t Y>--

(D)

THE

TOTAL

NON-CONVERTED

TUNNEL

Inasmuch as only new tunnel CM be converted, we can write 4 net = 4x- ht = F.tis

(141)

t Added in the proof. This study is progressing, but is proving to be of monumental complexity. A FORTRAN IV programme is being deai@cd to solve the several hundred equations involved.

400

J.

STANLEY

where the subscript “n” means “not”. Whence, from equation (65) (142)

We may also write Al,, = hi + 4LY and obtain equation (142) via equations (58) and (38). Then :

(143)

(144)

lim c$,,, = - WX T-rCO 4,

=

kz+E

kz&ve(~(x-r)

“Et

= lim 4a & 0, T-+ca

GO,

V

(XI

lim & i T-.X

T-+X

T-cm)

* 0, I

lim $J& = 0.

(145) (146) (147) (148)

T-tCO

There is no maximum, no point of inflexion, and the approach as T* approaches infinity is asymptotic. The function is plotted in Fig. 5. It will be noted from equation (145) that the value of $,,, is not zero as T approaches infinity, and this may seem odd, inasmuch as all the original tunnel (4,) is eventually destroyed (equation (40)). One must remember that 4,,, is not a quantity of tunnel physically existent in the flour. It is merely that portion of c$, which escapes conversion. Some will remain for a time as c#J~,which could be symbolized as c#J,,,“*. This decreases to zero as T approaches infinity. (E)

THE

TOTAL

CONVERTED

NEW

TUNNEL

This is that portion of 4, which undergoes conversion, without regard to whether or not it is then left as old tunnel, or is destroyed. Only a portion of it (#eL) actually remains in the flour at any time. This function was determined en route as equation (65), from which lim fj,, = 0, (149) T-+X

(151)

GROWTH

OF

401

POPULATIONS

052)

(153) The approach as Tapproaches infinity is asymptotic, and there is no maximum or point of inflexion. The function is plotted in Fig. 5. (F)

THE

HALF-LIFE

OF

Q%, IN

REGARD

TERMS

OF

CONVERSION,

WITHOUT

TO DESTRUCTION

This determines the moment at which half of $, shall have been converted to old tunnel, without regard to its subsequent fate (see Fig. 5). From equations (65) and (142) it can be shown that T*(d,, = c$,,J occurs when -(x-T)

k,-2kze

= E

V

(154)

,

whence (155) (G)

THE

TOTAL

NEW

TUNNEL DESTROYED, CONVERSION

WITHOUT

REGARD

TO

This is that portion of (b, which is destroyed by time T, regardless of whether or not it has been converted to old tunnel prior to destruction (see Fig. 4). We have that hit = A+ ki, (156) whence, from equations (58) and (67) fjnt = fj,[l-e+-T)],

(157)

lim f& = 0,

(158)

Eh*

(159)

T-r%? &=-feV

EC#J

= rb,, !((x-T)

E&c lim &it =y>o,

,

MO 061)

T-rX

lim C#&= 0.

(162)

T-03

There is no maximum or minimum and no point of inflexion, and the approach as T approaches infinity is asymptotic. The function is plotted in Fig. 4. 26 T.B.

402 (H)

J. TOTAL

NEW

TUNNEL,

STANLEY

REMAINING

IN

THE

FLOUR,

AND

NOT

DESTROYED

This is that portion of 4, which escapes destruction up to time T, though it may meanwhile have been converted to old tunnel. It could be symbolized as & but is more correctly denoted as +ndl. We have that 9 odt

=

4,

-

(163)

+dc,

+ndt = f$, Qcx- T).

(164)

&dt

(165)

By an alternate approach, = 4‘

+ 6cL)

whence, via equations (38) and (45) we again obtain equation (164). From equation (164) lim q$,dt = 4,.

(166)

T-.X

lim fjndt = 0,

(167)

T-rCC

lim$$dl=-~=-‘lim$;,


IT-+X lim &,a, = 0.

T-+X

(169)

I

(170)

T-+a)

There is no maximum and no point of inflexion, and the approach as T approaches infinity is asymptotic. The function is plotted in Fig. 4. (I)

THE

HALF-LIFE

OF

4,

WITHOUT

REGARD

TO CONVERSION

This is the moment at which half of the original 4x has been destroyed (see Fig. 4), regardless of whether or not it has been converted to old tunnel prior to destruction. From equations (157) and (164), T(+ndt = +dt) occurs when

whence T*(t#Jnd, = &) = f log, 2 =

T*($L

=

$,,)

=

T*(f’u

=

Fdt).

072)

5. The Problem of “Tunnel-guidance”

Superficially, a Tribolium tunnel-plexus bears some resemblance to an ant-hill. There are tunnels in which insects move; tunnels are broken down and new tunnels are made. It differs from the ant-hill, however, in the matter

GROWTH

OF POPULATIONS

403

of the extent to which the inhabitants are guided in their movements by the tunnels. In the ant-hill, the walls of the tunnels are strong and hard, and an ant will follow a tunnel, even though this may force it to make quite sharp turns. It seems unlikely that even a “non-conformist” ant would, on reaching a corner, simply plough ahead and drill a new tunnel collinear with the former line of march. To some extent, Tribolium beetles are “tunnelconformist”. They do follow tunnels, but they are quite likely to branch off and make a new tunnel at any moment, and are physically free to do so. Thus, the “tunnel-guidance” in a Triboliwn tunnel-plexus is relatively slight, but not negligible. There is also the fact that, as T* comes to exceed 0, parts of the original 4, and 4, become closed off by blockage of the ends, like parts of an abandoned mine. If there is much guidance, such isolated regions are seldom or never traversed. In terms of the theory thus far developed in this present paper and in Stanley (1964), where a completely random disposition of the tunnels, and random movement in them are assumed, the above effects play no part. How important they are is at present quite unknown. If they do have a sign&ant effect, the limiting values of the functions at T* = 0, or as T* approaches infinity, will not be changed. There will, however, be a tendency to delay the sequence of events to an ever greater and greater degree as T* increases. At the moment, the writer can see no intellectual approach to theory which would cover this matter. 6. A Numerical Example Stanley (1964) displayed the plots of #J, $ and F as well as those of other special functions, by the use of a numerical example employing arbitrary values of the parameters. This was necessary because actual determinations of the true values have defied all attempts over many years (Stanley, 1938, 1949, 1964). For the purposes of this example, we require only some values from Stanley (1964), i.e. k, = 6, E = 2 and V = 100. The starting time, x, is chosen as x = 35 (see Fig. 1). This choice has no particular virtue except that it avoids the special time T(4 = JI) and so presents a more general picture. It also permits the plots to be drawn within reasonable abscissa1 values in terms of page-size and scale. When T = 35 (T* = 0), F, = 21.6687417, rjfX = 11-9146410 and 4, = 9-7541007. On the above bases, Table 1 shows ordinates of the various functions from T* = 0 to T* = 95 (T = 35 to T = 130). Table 2 shows the special times such as T(tiL1 5: JIdJ and their ordinates, while Table 3 shows the limiting values as T* = 0 and approaches infinity. As in Stanley (1964) the

t With sometimes throughout

!ii 95 100 105 110 115 120 125 130 co

!i 75 80

ii

45 50

T

11.915 13.069 13.359 13.115 12.578 11.823 11-008 10.168 9-339 8.543 7.792 7.093 6.446 5.851 5.307 4.811 4.359 3.948 3.575 3.236 OGOO

11.915 10.781 9.755 8.821 7-987 7.227 6-539 5.917 5.354 4.844 4.383 3.966 3.589 3,247 2.938 2.659 2406 2.177 l-969 1.782 O-000

l-134 2.160 3.088 3.928 4.688 5.376 5.998 6.561 7.071 7.532 7.949 8.326 8,668 8.977 9.256 9.509 9.738 9.945 10.133 11.915

O-000

ld

6, 7

16

1.259 2.585 3.912 5.198 6.417 7.560 8.618 9.594 10.487 11.304 12+47 12.725 13.339 13.897 14QtOl 14.860 15.214 15.650 15.991 19.230

GUm 0400

6

121

9.754 6.538 4-382 2-938 1.969 l-320 O-884 0.593 0.398 0.267 0.179 0.120 0.080 0.054 0.036 o-024 O-016 o-01 1 O-007 0.005 OWO

4‘

3, 7

38

3604 4.288 4.569 4.596 4.468 4.251 3.985 3.699 3409 3.127 2.857 2.604 2.369 2.152 1.953 1 .I11 1606 1.454 0400

4CL

3, 5, 7

4, 7

nos 4, I

157

nos

4

164

O-803 1.343 1,704 1.946 2.109 2.217 2.290 2.339 2.312 2.394 2409 2.418 2.425 2.429 2.432 2.434 2.436 2.437 2.438 2.439

0.124 0.425 0.824 1 a270 1.729 2.184 2.620 3.033 3.416 3.772 4.098 4-399 4.671 4.920 5.145 5.351 5.536 5,705 5.858 7.316

0.928 1.768 2.528 3.216 3.838 4401 4.910 5.371 5.788 6.166 6.507 6.817 7.096 I.349 7.518 7.785 7.972 8.141 8.295 9.754

8.826 7.986 7.226 6.538 5.916 5.352 4-844 4.383 3.966 3-588 3.247 2.937 2.658 2405 2.176 1.969 1 .I82 l-613 1.459 OQOO

4d 4Cd 4*t 4m OGOO OGOO OGOO 9.754

Function

7

to figure

67

to equation 58

3,4,

Reference

45

Reference

5

142

2.412 4.029 5.112 5.839 6.325 6.652 6-871 7.017 7.115 7.181 7-226 7.256 7.276 7.288 7.297 7.303 7.307 7.311 7.312 7.316

7.342 5.726 4649 3.916 3.429 3.102 2.883 2.737 2.638 2.573 2.528 2.499 2.479 2.466 2.457 2.451 2446 2444 2442 2.438

4,t 4ncL OGOO 9.154

5

65

21.669 19607 17.741 16.053 14-525 13.143 11.892 10.761 9.736 8.810 7.971 7.213 6.526 5905 5.343 4.836 4.375 3.959 3.582 3.241 0-000

FL0

2, 7

22

2.062 3.928 5.616 7.144 8.526 9.177 10.908 11.933 12.859 13.698 14.456 15.143 15.764 16.326 16.833 17.294 17.710 18.087 18.428 21.669

FdL 0900

2

28

E 95 al

iz 45 50 55 60 65 70 75 80

5 10 15 20 25 30

0

T*

only three places of decimals, it is not always possible to obtain exactly correct cross-checks to three places, there being a discrepancy of +OT)Ol between two cross-checks leading in theory to the same result. The second decimal place is correct the table, as far as careful cross-checking will determine.

ht

6, I

6, 7

VL

18

11

(T* = O&T* = 95) Ordinates? of the various functions from T = x = 35 to T = 130, where k, = 6, E = 2 and V = ZOO (see text). Exact values of eX, 4, and F, are 1c/,= 11-9146410, (6, = 9-7541007, F, = 21.6687417

TABLE 1

GROWTH

OF

POPULATIONS

405

TABLE 2

Special values from the various functions when x = 35, kz = 6, E = 2 and V = 100. T* = T-x. All values are calculated to the nearest three places from explicit equations except T*(JIL, = $3, T(tiLt = $3 T*(eL, = e,,,), T(h = Tht and tiLt = Gdt which are from implicit equations, by trial and error Special times and associated ordinates SYmbOl

or ordinate

Time

34657 69.657 5.957 9.802 44.802 13.358 24.410 59.410 32907 67907 10.519 39.275 74.275 9.457 9.802 44.802 2.533 23,105 58.105 4608 46.210 81.210 3.629 23.105 58.105 1.555 34.657 69.657 4.877 13.733 48.733 4.877 34.657 69.657 10.834

Reference to equation nos

i: ;; 111 111 118 118 119 78, 133 78,133 131 131 132 51 5”: 544 56 E 75 172 172 164 155 155 65 36 36 22

406

J.

STANLEY

TABLE 3 Limiting ordinates of the variousfunctions as T* approaches 0 and T approaches x and when both T* and T approach injinity, using the same parameters as In Table 2. Note that exact values of I),, 4, and F, are: $, = 11*9146410, 0, = 9-7.541007 and F, = 21.6687417

Symbol

T* approaches 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0 w 0

T approaches X w X w X w X w X w X w X w X to n w x w x w x w x w X W X W X W x W X W X W X w X W X

Limiting ordinates = 11.915 0 -0.238 0 y* = 11,915 0 0.347

Reference equation

to 00s

yr

:: V/r = 11.915 $0.238 0 0 19.230 $0.238 $4 =“9.754 -0y780 z 0 0585 0 0 2.439 0.195 0 0 7.316 0 0 d&4

(* =09.754 0 -0.195 0 0 7.316 0.585

:: 14 iii i: 93 17 18 20 21 123 124, 125 127 128 39 40 42 43 46 47 49 50 2 62 63 69 69 71 72 158 159 161 162 166 167 169 170 149 150 152

GROWTH

OF

407

POPULATIONS

TABIJZ 3-continued T Limitiqg Symbol approaches T* approaches ordinates 03

co

f%.

0

x co

2::

T co 0

al x

2:

FI& FL,,

co 0

00 0 co 0 al

2

F: Fit Fit

X

w X

03 X

aJ x co-

$&&l54 0.585 0 F, = 21.669

0

-0.433 0 0 F* = 21669 0.433 0

Reference to equation nos 153 144 145 147 148

it E 29 30 32 33

values were calculated initially to nine places and are then rounded to three places, this being the minimum which permits accurate cross-checks. It is not, of course, implied that in an actual experiment, anything like this accuracy could be attained. The values from these Tables are plotted in Figs 2 to 6, arranged in suitable combinations of curves. Fx This is shown in Fig. 2. It is seen that FL1 decreases with decreasing rapidity (equation (22) and Table 1) and reaches zero as T* approaches infinity (equation (24)). The total tunnel destroyed (F& increases concomitantly (equation (28)) and finally equals F, = 21.669 (equation (30)). The halt-life of I;, is at T(FL, = F,,,) = 34.657 where FL1 = F,/2 = 10.834 (equations (34) and (36) and Table 2). There are no points of inilexion. It must be realized of course that while the above is going on, additional total tunnel has been produced as F, as per equation (18) of Stanley (1964). (A)

THE

GENERAL

FATE

OF

(B) THE

GENERAL

FATE

OF

4,

The fate of the original new tunnel (4.J requires discussion from three points of view, the general fate (discussed here), the matter of destruction, and the matter of conversion. The general fate of 4, is shown in Fig. 3. The new tunnel remaining as such (until destroyed eventually), decreases rapidly, but at a decreasing rate (equation (38) and Table 1) and soon nearly disappears due to the heavy losses by destruction (Fig. 3) and conversion to old tunnel (Fig. 5). Eventually, as T* approaches infinity, all is lost (equation (40) and Table 3). The total quantity of $, destroyed increases at a decreasing rate (equation (157)) and reaches 4, as T* approaches infinity (equation (159)).

408

J.

STANLEY

Apart from the new tunnel left as such (&) and that destroyed both before and after conversion (c$~J, only that portion converted and not as yet destroyed still remains. This is $J,.~ (equation (45)). This is of course zero when T* = 0, and then increases to a maximum T*($I~~,J = 23.105 when 4 CLlIl.x = 4.608 (equations (51) and (52) and Table 2). Thereafter, r#~~= decreases to zero as T* approaches infinity (equation (47)), passing a point of inflexion at T*(dcL,) = 46.210, where c#J=~,= 3.629 (equations (55) and (56) and Table 2). The initial rise to a maximum (Fig. 5) is due to the initially larger amount of new tunnel, followed by a decline concomitant with reduced values of &., resulting from destruction and conversion. The three crossing points of the curves of Fig. 3 are artifacts of plotting, and have no biological meaning. (C)

THE

DESTRUCTION

OF

4,

IN DETAIL

This is set forth in Fig. 4. New tunnel may be destroyed as such, or it may be converted to old tunnel and then destroyed. The total destroyed is the sum of the above, i.e. $dt = CJ&+ #ea. New tunnel destroyed as such (&J increases from zero when T* = 0 at a decreasing rate, and the quantity destroyed soon becomes almost constant (equation (58)), due to the fact that & has almost disappeared (Fig. 3). The new tunnel which is converted prior to destruction (~$3 is zero when T* = 0 (equations (67) and (68)) and passes a point of inflexion at T*($& = 23.105 when +Cd, = l-555 (equations (74) and (75)). It increases to a final value of 7.316 (equation (69)). The total amount destroyed (+dt) as per equation (157) is zero when T* = 0, and reaches a final value equal to A = 9.754 as T* approaches infinity (equation (159)). Note that as T* approaches intiity, 4, = 9.754 = (4Cd+ r#~,,)= 7.316+2-439 = 9.755 (accurate to two places and exact in theory). Thinking in terms of destruction only, there is a pseudo half-life of #X at T*(dndt = 4,J = 34.657 when 4at = &/2 = 4.877 (equation (172)). The other three crossing points of the plots of Fig. 4 are of no importance. (D)

THE

CONVERSION

OF

4,

IN

DETAIL

This is displayed in Fig. 5. The total quantity of new tunnel converted (c#J,,)increases from zero when T* = 0 (equation (149)) to reach a final value of 7.316 (equation (150)) as T* approaches infinity. Now (pCtconsists of 4ed (the new tunnel first converted and then destroyed) plus +EL (the new tunnel converted and then left as old tunnel, pending eventual destruction). The former is discussed in connection with Fig. 4 and the latter in connection with Fig. 3.

GROWTH

OF

POPULATIONS

409

The total new tunnel not converted commences as &, when T* = 0 (equation (144)), decreasing to 2.439 as T* approaches infinity (equation (145)). Thinking then in terms of conversion only, there is a pseudo half-life of 4, at T*bL = 4,,) = 13.733 when C& = &/Z = 4.877 (equation (155)). The remaining crossing points of the plots of Fig. 5 are of no importance.

(E)

THE

GENERAL

FATE

OF

y/,

This somewhat complex situation is shown in Fig. 6. Disregarding for the moment additional quantities of old tunnel arising from conversion (4=r), the old tunnel still left in the flour from the original $, decreases at a decreasing rate to approach zero as T* approaches infinity (equations (12) and (13)). Meanwhile, the original old tunnel destroyed (equation (16)) increases at a decreasing rate from zero when T* = 0 (equation (17)) to equal eX = 11.915 as T* approaches infinity (equation (18)). There is thus, in terms of the original $, a half-life at T*(&, = I,+~) = 34.657, where JIL = JlX/2 = 5.957 (equations (34), (35) and (36)). This has the same value as T*(&, = F,,J and T*(4,,a, = &,), as shown by equations (36) and (172). If we take into consideration the additional old tunnel brought into being by conversion, we have I,& of equation (78). Initially, for a period, eL, increases to become greater than $, (due to the additions from conversion exceeding the losses by destruction). JILt reaches a maximum at T*(JILt,3 = 9.802 where I,&,,,.~ = 13.358 > I,+, (equations (94) and (97)). It is shown (equations (98) to (117)) that this initial increase always occurs in any realistic Tribolium tunnel-plexus. Subsequent to T*($ LtmJ, the total old tunnel decreases again to equal 9, for the second and last time at T*(eL, = eX) = 24.410. In the general case, this value is determinable only by the trial and error solution of equation (98) but equation (98) can be solved explicitly in a few special cases, in particular that in which kz = 3E as in this numerical example (see equations (109) to (111)). There is then a point of inflexion at T*(&) = 32.907 where = 10.519 (equations (118), (119)). tiLli The total old tunnel destroyed is Jla, = $,,+ c#,, as per equation (120), and this increases from zero when T* = 0 (equation (123)) to reach 19.230 > eX as T* approaches infinity (equation (124)). There is a point of inilexion at T*($& = g-802, where J/,,,, = 2.533 (equations (131), (132)). There is also a type of pseudo halt-life at T*(tiL, = I,+~,) = 39.275 where $Lt = 9.457 # J/J2 and it is shown in equations (134) to (140) that this must always occur later than T*(tiL = $,,). The remaining two crossing points of the plots of Fig. 6 are of no importance.

410

J. STANLEY

In Fig. 7, an attempt is made to show the whole general picture of the gradual destruction of Fx and the interplay of conversion from new to old tunnel. The shaded portion, of vertical depth F, when T* = 0, is eroded away by destruction of both new and old tunnel, while the original new tunnel (C&J, subjected to a double onslaught by both destruction and conversion, rapidly declines to a minute amount. Be it noted that the two lines bounding & which appear to join, actually do not do so until T* approaches infinity. It was not possible to keep them apart due to the thickness of the lines in the drawing.

FIG. 7. This figure attempts to portray the distribution of the original tunnel (F;I into the various kinds, subsequent to T* = 0. This arrangement of the data does not permit showing t,u,,$but it brings all the still existent tunnel together as the shaded portion. The vertical depth of this portion becomes zero as T* approaches infinity.

There are many ways to arrange an illustration such as Fig. 7. By no arrangement can everything be shown. The arrangement used cannot show the total old tunnel destroyed (1,6,&),It consists of t+Vd + (Pcdand these two areas are separated on the figure by the areas of & (bCLand &. It will now be realized that the tunnel-plexus of Tribolium is in a sort of “moving steady state”. It changes with time, but is continually being destroyed and renewed. This fact is easily demonstrated if one makes up a flour of alternate coloured and white layers. After some time these will be found stirred into each other, even in regions which contain no tunnels at the moment of examination. They once contained tunnels, but they have been

GROWTH

OF

411

POPULATIONS

destroyed, and the flour re-consolidated. the stirring of the soil by earthw0rms.t

The same phenomenon

occurs with

The writer is grateful to McGill University for generous financial assistance with this and related work on Tribolium, and he is indebted to Miss Beatrix Wanke who carried forward experimental work with the re-tunnelling problem.

STANLEY, STANLEY. STANLSY,

REFERENCES J. (1949). Ecology, 30,209. J. (1964). Can. J. Zoo/. 42,201. J. & hALLMAN, B. N. (1938). Can. J.fRes. D.

16,221.

t Note to fishermen:To obtain earthwormsanywhere,find a worm-hole.Squirt in 2 cm3of a dilutesuspension of mustardpowder.Standback!