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Interaction between hosts and parasites when some host individuals are more difficult to find than others
aT. Theoret. BioL (1962)
3, i-x8
Interaction b e t w e e n Hosts and Parasites w h e n s o m e H o s t I n d i v i d u a l s are m o r e D i f f i c u l t to find t h a n Others V. A. BAILEY,~ A.
J.
NICHOLSON++ AND E. J. WILLIAMS§
(Received 15 August
1961)
(I) The situation is examined in which some hosts are less likely to be attacked than others, either because they have superior intrinsic defensive properties, or because they occupy places where parasites experience relatively great difficulty in finding or attacking them; the object of the investigation is to determine whether, and under what conditions, a stable steady state may exist. (2) When some hosts are more difficult to find than others, or the parasites search some places more effectively than others, systems of damped or of growing oscillations are produced according to the circumstances, which are defined. (3) When the frequencies of the areas of discovery approach a normal or Gaussian distribution the populations are always unstable, oscillations in density growing in amplitude with time. (4) The likelihood of stability increases when the distribution of areas of discovery approaches a J-shape : that is, when hosts for which the parasites have the lowest area of discovery are the most numerous, higher categories progressively and rapidly becoming less numerous. Such frequency distributions seem improbable. (5) When more than the surplus fraction of the host offspring is inaccessible to the parasites a steady state is clearly impossible, the parasites being precluded from destroying the whole surplus of hosts. The host and parasite populations then tend to grow indefinitely. (6) When the fraction of hosts inaccessible to the parasites is less than this, host-parasite interaction leads to a system of growing oscillations, provided that the power of increase does not lie between certain limits associated with the power of increase and the fraction of hosts which is inaccessible (see Fig. 4). If the power of increase lies between these generally very narrow limits, the system is stable, interspecific oscillations being damped. x. I n t r o d u c t i o n
I n earlier papers (Bailey, i931 ; Nicholson, i933; Nicholson & Bailey, I935) dealing with the interaction of specific parasites and specific host species, the simplifying assumptions were m a d e that all parasite individuals have an equal efficiency in finding hosts, and that all host individuals are t School of Physics, University of Sydney, N.S.W. Division of Entomology, C.S.I.R.O., Canberra, A.C.T. § Division of Mathematical Statistics,C.S.I.R.O., Canberra, A.C.T. T.B,
I
2
V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
equally easy to find. It was concluded that the interaction of such animals leads to an essentially unstable system in which small displacements from the steady state lead to oscillations which progressively grow in amplitude with time, unless additional factors come into operation and check this growth. Varley (I947, page I8I) commented that "it can be shown with Nicholson and Bailey's theory that if a proportion of hosts is not available to parasitism, oscillations will be damped instead of increasing in amplitude", and in his summary that "these oscillations may not be of ever-increasing amplitude, as supposed by Nicholson and Bailey, but may in fact be damped if some hosts are less available to parasites than others". These conclusions need careful examination, for it commonly happens that some fraction of the hosts is unavailable to the parasites through occurring in inaccessible places, or at times when the parasites are not operating; and also that some hosts are more difficult to find or to attack than others. Nicholson (i954, page 4 z) examined these situations by constructing numerical models. He concluded that, in spite of the two factors mentioned, interspecific oscillations due to parasite-host interaction commonly grow in amplitude--although damped oscillations are possible, particularly if some of the hosts are much more difficult to find than the others, and if this is associated with a low power of increase of the host species. The investigations recorded in this article were undertaken to examine these situations more rigorously, and to define the limiting conditions between systems of damped and of growing oscillation. z. The Primary Postulates
It is assumed, as in the simpler situations already examined (Nicholson & Bailey, I935) , that a specific insect parasite attacks a specific insect host under constant environmental conditions. That is to say, the only variable that influences the population of either of the interacting animals is change in the density of the other. To simplify initial examination of such interaction it is also assumed : (I) that when a parasite contacts a host individual it lays one egg in this which gives rise to one parasite in the next generation, the attacked host individual being killed; (2) that the host has a constant power of increase, this being the multiplication rate of the hosts from generation to generation in the absence of the restrictive influence of the parasites; and (3) that both hosts and parasites occur in a succession of discrete generations, the interacting stages of the hosts and parasites coinciding precisely in the times of their production. The original model is modified in the present article by assuming that the area of discovery, instead of being constant, has a range of values representing the varying degrees of ease with which the parasites find
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
3
different host individuals. Some hosts are relatively difficult to find either because they possess superior intrinsic defensive properties, or because they occupy situations where they are less accessible to parasites than their fellows in other places. T h e area of discovery (Nicholson, i933; Nicholson & Bailey, I935) of a parasite for a given kind of host is a measure of the efficiency of the parasite in finding and attacking hosts of that kind under the given conditions. It may be defined as the area effectively explored by an average parasite individual. It should be noted that, although some parasite individuals may be intrinsically more effective than others, it is sufficient to refer to the average area of discovery of the individuals; for, as all the parasites remain in action throughout any particular generation, in spite of any variation in the fraction of hosts attacked, the average remains unchanged. This contrasts with the hosts, for if some are more difficult to find than others the survival of these tends to be disproportionately large in the population remaining after the parasites have operated for a while, and so the average area of discovery of the parasites for the hosts progressively falls as the fraction destroyed increases during any given generation. The notation and terms to be used are summarized here for convenience. a = the mean area of discovery of a parasite individual in the searching population for hosts with a given degree of accessibility.
(a)da = the relative frequency of a within the range a 4- ½da. F----- the power of increase of the host population, this being the potential degree of multiplication in the absence of parasites under the given conditions. Its value allows for any constant mortality due to factors other than attack by parasites. hn = the density of adult hosts, which give rise to Fhn offspring, this being the initial density of hosts in the (n 4- i)th generation which is subject to attack of the parasites Pn+lp,~ = the density of mature parasites which search in the nth generation. Steady state = that balance of host and parasite densities which is self-regenerating from generation to generation if undisturbed. Stable steady state = that balance of host and parasite densities which if disturbed will return eventually to the steady state. 1--2
4
V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
3. The Fundamental Equations Figure i illustrates the frequency function f(a) by means of two examples. The curve A represents a continuous distribution of areas of discovery and the curve B represents approximately a discontinuous distribution such as exists when, for example, the fraction o.6 of the
B
B
f(d
I
2
Area of discovery (a) Fro. I. Examples of a continuous u n i m o d a l distribution (A), and a discontinuous distribution concentrated at a ~- i and a = 2 (B).
hosts presents an area of discovery of i unit and the fraction 0. 4 presents an area of discovery of 2 units. The function f(a) is necessarily positive or zero for all a > o, and is necessarily zero for values of a exceeding the upper limit of areas of discovery. Clearly the area under any curve such as A or B must be equal to unity; that is, we must have CO
l" f(a) da = i
(I)
o
When the distributions are discontinuous, the integral should be replaced by a sum. Thus, for instance, if different groups of hosts presented areas of discovery of o, I, 2, . . . etc., units with respective probabilities to, ri, r2, ... etc., the integral in (I) should be replaced by rO ~- r l ~- r2 ~- . . . . i (ifl) For convenience in the discussion of the general problem we will introduce the function ~(p), which is the Laplace transform of f(a), defined by O0
¢(p) = ~ e-Paf(a)da o
(2)
INTERACTION
BETWEEN
HOSTS
AND PARASITES
5
where p is some number, representing in the present application the density of parasites. This function has an important physical interpretation, as will be seen below. For discontinuous distributions, as described above, the Laplace transform is q~(p) = ro + rle -~ + r~e-2~ + . . . (za) The Laplace transform takes the following values : (O) ~--- I ;
as p -+ oo ~(p) --~ o, for continuous distributions --~ ro for discontinuous distributionsj (3) Since ~(p) decreases as p increases, its derivative, ~'(p), is negative. We may now apply these results to the determination of the equations expressing the growth of the two populations. If at the nth generation there are h, hosts occupying unit area, the density of their offspring will be Fh,~. The expected number of these offspring for which the parasites have an area of discovery between a and a + da is F h J ( a ) da
Since the density of the parasites is p,,, the fraction of these host-offspring remaining undiscovered will be e-~"a (see Nicholson & Bailey, 1935, page 555). Hence, the final density of hosts that survive to make up the (n + i)th generation is h,+ 1 = f em"aFh,f(a) da 0
= Fh.4(p.) (4) The Laplace transform (2) is thus seen to represent the proportion of hosts surviving after attack by parasites of density p. Also the parasite density P.+t in the next generation is given by P,+t -~ Fhn -- h,~+t
(5) Equations (4) and (5) are thus the fundamental equations of the problem. They are a generalization of equations (8o) and (81) of Nicholson & Bailey (i935), where the problem is treated for constant area of discovery. In the present paper the exponential function of (8o) is replaced by a more general function $(p,), which is in fact an average of exponential functions, corresponding to the distribution of areas of discovery. 4. The Steady State The steady state is by definition the state in which the densities of hosts and of parasites remain constant from generation to generation. For the
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v.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
steady state we write
h.=h P. = P Substituting these values in equations (4) and (5) we are able to determine whether a steady state can exist, and to find the values of h and p for the steady state. The equations become ~=F$(p) (6) and p = h ( F - - ~) (7) Thus p is determined as the root of (6), and the corresponding value of h is then given by (7). Now since, as we saw above i > $(p) > o (for continuous distributions) > r o (for discontinuous distributions) it follows from (6) that, for a steady state to be possible when r o is non-zero, we must have I/r o > F > I (8) Continuous distributions impose no upper limit on F. Equation (8) shows that, when a fraction of the hosts is inaccessible (presenting zero area of discovery to the parasites), this fraction must be less than I/F. This result may be deduced on physical grounds. If r0 exceeded I/F, the inaccessible hosts alone would increase the population in each generation by a factor Fr o which exceeds unity, so that the population as a whole would increase indefinitely. It should be remarked that, when F = I/r0, it is apparently possible for the inaccessible hosts to maintain the population at a constant level; however, the actual steady densities of hosts and parasites in this case are indefinitely large, and so could not be supported by the resources of any environment. 5. Behaviour near the Steady State
(i) Derivation of criterion for stability In order to see how a developing population is likely to behave, it is not sufficient to establish that there exists a steady state, which will persist if undisturbed. Even small variations in the environment are likely to disturb the populations from their steady state, so it is important to know whether the steady state is stable or unstable. We consider the effect of small disturbances, which we denote by H n
I N T E R A C T I O N B E T W E E N HOSTS AND P A R A S I T E S
7
and P., from the steady-state values h and p respectively. T h u s we have h,~= h .-}- H , p. = p + v. (9) where H , / h and P,,]p are small quantities, so that only terms of the first degree in each need be considered. T h e n from (4) we have, keeping only linear terms H~ and P~,
h + H~+I = F.(h + H.)~(p + P~) = (h + H.)F[+@) + ~'~)P.] from (6) = (h + H.)[I + F~'(p)P.] = h + H . + FhqV(p)P. or Hn+l = H,~ - - G P , where G = - - Fhgb'(p ) and is therefore positive. Also, from (5), P,~+a = FH,~ -- Hn+ 1
(Io)
(II)
On eliminating H,, between (io) and (ii), we find that P., as a function of the generation n, satisfies the difference equation Pn+~. -- (G + I)P,,+I -[- FGP,, = o ; (i2) it is readily verified that H . satisfies the same equation. Now any solution of equation (i2) can be shown to be of the form Au'~ + Bu~ where ul and u2 are the roots of the quadratic equation uS - - ( G + i)u + F G = o
(i3)
T h e behaviour of the populations of hosts and parasites therefore depends on the nature of the roots of equation (i3). T h e populations are oscillatory if the roots of (I3) are complex, non-oscillatory if the roots are real. T h e condition for real roots is (G + I) 9">
4FG
(I4)
Of more importance is the stability of the populations, which requires that P,(or H~) diminish as n increases. T h e condition for this to hold is that the absolute values of the roots of (I3) are less than unity:
lu~l<~,
lu~l
(i5)
Now when ul and u2 are real, it is readily seen that they are both positive; for ut + u ~ = I + G and (16) ulu'a = F G
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V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
F r o m the second of these equations we see that it is necessary that FG < 1 ; moreover, since F > i we must also have G < L We now show that the quantities I -- ul and 1 -- u2 are both positive; their sum is 2--
Ul--
U2 =
I -- G > o
and their product is 1 - - u l - - u~ + u I u ~ = G ( F - - 1 )
> o
T h u s we have shown that, if the roots are real they are both less than unity if and only if FG < I (17) On the other hand, if the roots are complex, they are conjugate quantities and so have the same modulus; but since
ulu2 = F G it follows that
lull
---- [u~l = (FG)½ < I
Thus, when the condition (17) is satisfied, the roots are always less than unity in absolute value, and the steady state is stable. (ii) Graphical determination of nature of steady state W h e n ~b(p) is such that the calculations become complicated the following graphical process offers a simple method for determining the steady density p and the nature of the steady state. Y U
3
p
>"
~
X
FIG, 2. Showing graphical method for determining the steady parasite density p and the stability of t h e steady state; for details see text.
First the curve y = $(p) is drawn to some convenient scale as in Fig. 2. Next ON is set off equal to F -1 and the line through N parallel to OX is drawn to cut the curve at P. T h e tangent at P is now drawn to cut OY at T.
INTERACTION
BETWEEN
HOSTS AND PARASITES
Then p = N P and the steady state is stable if T U > N U ' ~ and unstable if T U < NU~J The proof of (18) is as follows : Since at P y = ~(p) = O N = F -~ therefore N P ----p. Also at P A p~b'(p) = N P tan N P T = - - N T Hence by (7) i --G=x_F
, -1 p $ ( p ) =
08)
OU N--u'NT
and so (17) is equivalent to OU NT >-NU
-- --
ON
that is --
OU(NU - - TU) > - - NU (OU - - NU)
or
TU > NU 2 This inequality therefore defines a stable steady state; the reverse inequality similarly defines an unstable steady state. 6. I n t e r a c t i o n w i t h D i s c o n t i n u o u s D i s t r i b u t i o n of Areas of Discovery
(i) The criterion f o r stability As an example in which the hosts are divided into distinct groups, we consider the binomial distribution. If A and B are positive numbers such that A+B=I k is a positive integer, and a o and as are positive numbers, with a o < ak, we here take a~ = a o + (as - - ao)i]k
and ri = (k)AS-~B~ (i = o, 1, . . . , k) s That is to say, r~ is the fraction of the population with area of discovery ae An example of this distribution is given by Fig. 3 for which A = 2]3 , B = I/3, k = IO. The ordinates r~ are given by the successive terms in the
V.A.
IO
BAILEY,
A. J . N I C H O L S O N
AND
E. J . W I L L I A M S
expansion of (A + B) ~, and lie equally spaced between a 0 and ak. The maximum ordinate corresponds to the value of i nearest to kB (in the figure, hB = 3½, i = 3), so that a~ is close to (a0A + akB). For this distribution, ¢(p) = (Ae-r,~o/,~+ Be-P=~/~)k so that, from (6), the steady parasite density p is the positive root of Ae -ra°/k + Be-~'~/k = F-1/k (19)
A
"6
N
,
....
O0
I ......N...
Area of discovery (o)
r-
--
ek
Fro. 3. An example of a binomial distribution of areas of discovery, with k = IO.
We also find that
G= p ~
F1/k
(Aaoe-r~°/k + Bake -p~k/~)
so that, from (I7), the criterion for stability is Fl+l/k
p ~
(Aaoe-p~°/k + Bake -pak/k) < I
When a0 = o, there is a finite proportion A k of inaccessible hosts, so that there is a limit to the values of F that can lead to steady states. This case is, however, simpler than that when a0 > o, and will be considered first. (ii) Some hosts inaccessible (a o = o) Here (19), the equation for the steady density, becomes A + Be -~ak/k -----F -1/k A steady state will be possible only if F < I/A ~
(20)
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
The condition for stability becomes FZ+Z/k B F -----~ pake-~'ak/k < x
IZ
(2x)
The steady states are stable for all values of F between this lower limit and the upper limit defined by (20). In Fig. 4, Curve I gives the upper limit of F for all positive integral values of k, and Curve II gives the lower limit for stability defined by (2t), with k = I. Each is plotted against A, the proportion of inaccessible hosts.
"1
10
IO0
-|.0(~
Power of Increase of Hosts (F)
Fzo. 4. Boundaries to regions of stable steady states when a fraction of the hosts is inaccessible to the parasites. I. General upper bound for steady states (AF = i), above which the hosts and parasites multiply without check. I I . - V . Lower bounds for stable steady states when: II. the area of discovery for accessible hosts is constant (k = I). III. the range of areas of discovery is very great (k --~ oo). IV. the areas of discovery for accessible hosts are distributed according to the gamma distribution with t = 2. V. the areas of discovery for accessible hosts are distributed in a triangular frequency distribution with mode at zero. Below the curves I I - V , steady states are unstable, under the conditions of each distribution, interactions leading to oscillations that grow in amplitude with time.
I2
V.A.
BAILEY,
A.
J.
NICHOLSON
A N D E. J . W I L L I A M S
When k is large the binomial distribution approaches the Poisson distribution, for which the relative frequencies are ri :
e-kB(kB)i/i[
As necessarily kB remains finite, we write kB = A, so that r i = e-~A~[i!
and the inaccessible proportion becomes e -~. We also put ak/h = b. A steady state will be possible only if F < e: (22) The equation for the steady density is log F = A(I -- e - # ) so the condition for stability becomes ()~ - - l o g F ) log
~ _ log F
< ~
(23)
As for the binomial distribution, conditions (22) and (23) set upper and lower bounds for values of F that can give rise to stable steady states. Curve I in Fig. 4 gives the upper bound, and Curve III the lower bound, corresponding to (23). For values of y , the inaccessible fraction, and F defining points above the upper bound of Fig. 4, no steady states are possible. For values defining points below the lower bound for a given distribution there are steady states but they are all unstable. The region of stable steady states is narrow; that is, for given y, the range of F values giving stable states is small. It seems likely that the chance of a value of F and a proportion inaccessible occurring so as to give rise to a stable steady state is quite small. (iii) A l l hosts accessible (a o > o) Without loss of generality we take a 0 ---=-x, ak > i. Since there are no inaccessible hosts, a steady state is always possible, whatever the value of F. With h = i, it can be shown that a k ( : a l ) must exceed 5.8 3 for a stable steady state to exist. As a 1 increases without limit, the lower bound for stability approaches that for a 0 = o, since the condition depends only on the ratio of a 0 to a k. In this limiting ease the upper bound to the region of stable steady states is given by A e - V = F-:/k and Fl+l/k
h P y-_- I A e - r
=
or P--1
A = F-r/he Fk
i
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B E T W E E N H O S T S AND P A R A S I T E S
13
T h e proportion in the least accessible class is y=A
e(F-1)/F ~ -- - F
T h e boundaries to the region for stable steady states in terms of F and
\
",',o.o
\.
\
\
\
\
I'0
\
\
\ 0'11
I0
100
~L000
Power of Increase of Hosts (FI
FIG. 5. Boundary regions of stable steady states when all hosts are accessible, and are distributed into two groups, in one of which the area of discovery for the parasites is b times that of the other. I. b = 6
II. b = io
III. b = oo
Within the curves I, II, and III, the steady states are stable for the given ratio of areas of discovery in each distribution. Outside these curves, the steady states are unstable, and interactions lead to oscillations that increase in amplitude with time.
the proportion in the least accessible class are given in Fig. 5 for k = x and a I = 6, :co and co. For all points outside the region of stable states, steady states without stability are possible.
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V. A. BAILEY, A. J. N I C H O L S O N AND E. J. WILLIAMS 7- I n t e r a c t i o n w i t h C o n t i n u o u s D i s t r i b u t i o n F u n c t i o n s
(i) A l l hosts accessible Discontinuous frequency distributions of areas of discovery, though theoretically possible, are not likely to occur in nature. One would expect that f ( a ) would more usually be a continuous function with a single maximum, such as the unimodal curve A in Fig. i, possibly associated with a finite proportion at zero accessibility. A family of unimodal distributions which adequately represents many observed continuous frequency distributions is the family of gamma distributions, for which f(a) =
e - a at-1 F(t) ' with t > °
In general the distribution will contain a scale factor, but since the scale of the distribution does not affect the nature of the steady states, it has been omitted. When o < t < i the distribution has its maximum at zero and is described as "J-shaped". W h e n t > I the distribution has a maxim u m at t -- I, and becomes more nearly symmetrical about this m a x i m u m as t increases. T h u s t is a parameter defining the " s h a p e " of the distribution. Figure 6 shows the curves of the frequency distributions for t = ½, I, 2, and 2o. For distributions of this family we have ¢(p) -
+
i
--t -
+p),+1
T h e conditions for a stable steady state are (i + p ) e = F F~pt (F --
+
< i
For t > I there is no value of F > I which will satisfy these relations. For 0 < t < I, all values of F will satisfy these relations. At the transition value, t = i, f ( a ) = e -~ ;
further investigation is needed before the nature of the steady states can be established for this particular distribution. For the triangular distribution, f ( a ) = 2(1 -- a) from a = 0 to a = i, at the steady state, •(p) = 2(e - ~ - - i + p)/p2
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B E T W E E N H O S T S AND P A R A S I T E S
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o
>*
o
I
2
3
4 $ Area of discovery
6
7
$
FIG. 6. Curves showing the " s h a p e " of the gamma distributions for various values of t (with scale of a for each reduced by a factor ~/t). I. t = ½ II. t = I III. t = 2 IV. t = 2 o Of these, Curve I admits stable steady states for all values of the power of increase, because it increases indefinitely near zero accessibility; Curves III and IV never admit stable steady states. For Curve II, describing the exponential distribution, the nature of the steady state will vary, depending on the initial host and parasite densities.
and ~'(p) = - -
21i0(1 + e - y ) - - 2(1 - - e-l~)][p 3
The condition for stability, 2F ~ [ I + e -p F--I ~ p
2(I
y))
S not satisfied for any value of F. (ii) Some hosts inaccessible; illustrative case We here consider the case where a finite proportion A of hosts is inaccessible, the remainder being distributed in a unimodal continuous distribution. For this purpose we conveniently use the gamma distributions already discussed.
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V.A. BAILEY, A. J. NICHOLSON AND E. J. WILLIAMS
If the inaccessible proportion be A, the relative frequency elsewhere will be e-aat-1
f(a)
= (I - - A) ~
(a > o)
Then I--A $(p) = A ~ (i + p)-----q (i - - A)t
¢'(P) =
(i +p),+l
The conditions for stability are I--A A + (I +p-----)*-- I/F F2pt(i -- A)
( F - i)(i +p),+l
The upper bound for F is given as usual by I/A. The lower bound is given when the limit of the above inequality is attained. It is found that admissible lower limits are given only when t > I. For t < I, all admissible values of F ( < I/A) lead to stable steady states. As t increases without limit, the distribution approximates a concentration at the single point a = t -- i, corresponding to a binomial distribution with k = I, so that the lower boundary for stability is given by Curve II of Fig. 4. The lower boundary for t = I is given by Curve IV of Fig. 4. For comparison, the lower bound for the triangular distribution with mode at zero accessibility and a finite proportion at zero is given by Curve V of Fig. 4. 8. D i s c u s s i o n
Field observation strongly suggests that the interaction of predators and their prey has a profound influence upon population numbers, but so many other influences affect populations in nature that it is difficult to assess the part played by such interaction. Because of this, simple theoretical models have been constructed which represent predator-prey interaction alone, in the simplest conceivable situations. This is only a first step. Later, variations in the primary postulates need to be made to take into account the additional influences that sometimes affect populations. Many distinctive models are required for, with different species and in different places, the interaction of predators and their prey is influenced by a wide variety of different factors. Lotka (1925) and Volterra (i926), with their formulation of simple interaction between predators and prey, concluded that this leads to a
INTERACTION
BETWEEN
HOSTS AND PARASITES
17
stable condition in which the populations of the two species oscillate continuously about their steady densities, which are determined by the properties of the two species. Bailey (193 I, i933a, b) and Nicholson (1933) made essentially the same assumptions for simple interaction between such animals, with the exception that they allowed time for both kinds of animals to develop to the stage at which they can interact; whereas Lotka and Volterra assumed that the interacting animals are fully mature at birth. It was found that this change in the formulation caused the system to be unstable, corrective reactions induced by displacement from the steady densities being over-violent, so leading characteristically to a system of oscillations that progressively grow in amplitude--although, if the numbers are not large, over-violent reactions may lead to the extinction of one or of both animals in a single pulse. Bartlett (196o) found that the addition of a term representing developmental lag into the Lotka-Volterra equations greatly increases the likelihood of rapid extinction of the predator, or of both the interacting species, when the system is subject to stochastic fluctuations. With the possible exception of some simple protozoa, all animals take some time to develop. Consequently the models suggest that instability is an underlying characteristic of predator-prey interaction, but this tendency must be countered in some way, for predators and their prey succeed in living together indefinitely in nature. It should be noted that such instability is the resultant, not only of developmental lag, but also of the postulate that the number of prey destroyed and the number of predators born at any moment are proportional to the product of the numbers of the interacting animals at that moment. This postulate is used in the simple models of all the investigators mentioned: but, as Nicholson (1933, page 148) has pointed out, it is tenable even as a rough first approximation only when considering those predators usually referred to as insect parasites--and only to these when near their steady densities. Using verbal arguments, he concluded that with other predators, such as birds and mammals, interaction should in general be less violent than with parasites and their hosts, so tending to maintain them at, or fluctuating about, their steady densities (Nicholson, 1933, page 167). It was also concluded that many kinds of influence can counter the tendency to instability which results from parasite-host interaction, so permitting parasites and their hosts to live together indefinitely in the same region. According to their characteristics, these influences lead to systems of sustained oscillation, damped oscillation, progressive approach to the steady densities of the animals, or to the fragmentation of the populations of the interacting animals, with an ever-changing pattern of T.Bo
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I8
V . A . BAILEY, A. J. NICHOLSON AND E. J. WILLIAMS
local distribution (Nicholson, I933, Conclusions 4 o, 4 I, 42, 43, 63, 64 and 65; Nicholson & Bailey, i935, Conclusions 5 I, 52, 53 and 54). In the absence of such influences, parasite-host interaction should lead to extinction, of the parasite at least. It is probable that this has happened frequently in isolated areas that are not very large. It has long been thought that modification of some of the other primary postulates could reveal circumstances under which the inherent tendency to instability of host-parasite interaction would be countered. In this article the postulate that the parasites can find all host individuals with equal ease is abandoned, and it is shown that when some host individuals are more difficult to find than others this may lead to stability, but often does not do so. When a fraction of the hosts is inaccessible (the area of discovery for these being zero) this also may lead to stability, but only if the magnitude of the inaccessible fraction lies within narrow limits, related to the power of increase of the hosts--otherwise the system remains unstable, or both hosts and parasites tend to multiply indefinitely. These results give some support to Varley's (i947) conjectures, but disprove their generality. Leslie & Gower (I96o), like Lotka and Voherra, assume that predators and prey are born mature, and so interaction does not lead to instability in their model. Their deterministic model of simple predator-prey interaction differs from those of Lotka and Volterra in that a damping term is introduced. They conclude that this leads to a greater degree of stability when the system is subject to stochastic fluctuations, although random extinction of one or of both animals may occur in time. They then examine the situation in which a fraction of the prey is inaccessible, the remainder being uniformly subject to attack by the predators, and conclude that the likelihood of chance extinction of the predators is then reduced. Thus, when population numbers are small, the inaccessibility of some hosts may exercise a stabilizing influence additional to that deduced in the present article. REFERENCES BAILEY, V. A. (193I). Q. ft. Math. 2, 68-77. BAILEY, V. A. (1933a). Proc. Roy. Soc. A, I43, 75-88. BAILEY, V. A. (x933b)- ft. and Proc. Roy. Soc. N . S . W . 66, 387-393 . BARTLZrT, M. S. (I96o). "Stochastic Population Models in Ecology and Epidemiology", pp. 4o-4 z. Methuen, London. LESLIE, P. H. & GOV~ER,J. C. (x96o). Biometrika, 47, 219-234. LOTKA, A. J. (I925). "Elements of Physical Biology". Williams and Wilkins, Baltimore, NICHOLSON, A. J. (i933). ft. Anim. Ecol. 2, I32-I78. NICHOLSON, A. J. (I954). Australian ft. Zool. 2, 9-65. NXCHOLSON, A. J. & BAILEY,V. A. (I935). Proc. Zool. Soc. London, 551-598. VARLEY, G. C. (1947). ft. Anita. Ecol..,x6, I39-187. VOLTmmA, V. (x9z6). Memorie della R. Accademia Nazionale dei Lincei, 2, 3I-rx~.
3, i-x8
Interaction b e t w e e n Hosts and Parasites w h e n s o m e H o s t I n d i v i d u a l s are m o r e D i f f i c u l t to find t h a n Others V. A. BAILEY,~ A.
J.
NICHOLSON++ AND E. J. WILLIAMS§
(Received 15 August
1961)
(I) The situation is examined in which some hosts are less likely to be attacked than others, either because they have superior intrinsic defensive properties, or because they occupy places where parasites experience relatively great difficulty in finding or attacking them; the object of the investigation is to determine whether, and under what conditions, a stable steady state may exist. (2) When some hosts are more difficult to find than others, or the parasites search some places more effectively than others, systems of damped or of growing oscillations are produced according to the circumstances, which are defined. (3) When the frequencies of the areas of discovery approach a normal or Gaussian distribution the populations are always unstable, oscillations in density growing in amplitude with time. (4) The likelihood of stability increases when the distribution of areas of discovery approaches a J-shape : that is, when hosts for which the parasites have the lowest area of discovery are the most numerous, higher categories progressively and rapidly becoming less numerous. Such frequency distributions seem improbable. (5) When more than the surplus fraction of the host offspring is inaccessible to the parasites a steady state is clearly impossible, the parasites being precluded from destroying the whole surplus of hosts. The host and parasite populations then tend to grow indefinitely. (6) When the fraction of hosts inaccessible to the parasites is less than this, host-parasite interaction leads to a system of growing oscillations, provided that the power of increase does not lie between certain limits associated with the power of increase and the fraction of hosts which is inaccessible (see Fig. 4). If the power of increase lies between these generally very narrow limits, the system is stable, interspecific oscillations being damped. x. I n t r o d u c t i o n
I n earlier papers (Bailey, i931 ; Nicholson, i933; Nicholson & Bailey, I935) dealing with the interaction of specific parasites and specific host species, the simplifying assumptions were m a d e that all parasite individuals have an equal efficiency in finding hosts, and that all host individuals are t School of Physics, University of Sydney, N.S.W. Division of Entomology, C.S.I.R.O., Canberra, A.C.T. § Division of Mathematical Statistics,C.S.I.R.O., Canberra, A.C.T. T.B,
I
2
V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
equally easy to find. It was concluded that the interaction of such animals leads to an essentially unstable system in which small displacements from the steady state lead to oscillations which progressively grow in amplitude with time, unless additional factors come into operation and check this growth. Varley (I947, page I8I) commented that "it can be shown with Nicholson and Bailey's theory that if a proportion of hosts is not available to parasitism, oscillations will be damped instead of increasing in amplitude", and in his summary that "these oscillations may not be of ever-increasing amplitude, as supposed by Nicholson and Bailey, but may in fact be damped if some hosts are less available to parasites than others". These conclusions need careful examination, for it commonly happens that some fraction of the hosts is unavailable to the parasites through occurring in inaccessible places, or at times when the parasites are not operating; and also that some hosts are more difficult to find or to attack than others. Nicholson (i954, page 4 z) examined these situations by constructing numerical models. He concluded that, in spite of the two factors mentioned, interspecific oscillations due to parasite-host interaction commonly grow in amplitude--although damped oscillations are possible, particularly if some of the hosts are much more difficult to find than the others, and if this is associated with a low power of increase of the host species. The investigations recorded in this article were undertaken to examine these situations more rigorously, and to define the limiting conditions between systems of damped and of growing oscillation. z. The Primary Postulates
It is assumed, as in the simpler situations already examined (Nicholson & Bailey, I935) , that a specific insect parasite attacks a specific insect host under constant environmental conditions. That is to say, the only variable that influences the population of either of the interacting animals is change in the density of the other. To simplify initial examination of such interaction it is also assumed : (I) that when a parasite contacts a host individual it lays one egg in this which gives rise to one parasite in the next generation, the attacked host individual being killed; (2) that the host has a constant power of increase, this being the multiplication rate of the hosts from generation to generation in the absence of the restrictive influence of the parasites; and (3) that both hosts and parasites occur in a succession of discrete generations, the interacting stages of the hosts and parasites coinciding precisely in the times of their production. The original model is modified in the present article by assuming that the area of discovery, instead of being constant, has a range of values representing the varying degrees of ease with which the parasites find
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
3
different host individuals. Some hosts are relatively difficult to find either because they possess superior intrinsic defensive properties, or because they occupy situations where they are less accessible to parasites than their fellows in other places. T h e area of discovery (Nicholson, i933; Nicholson & Bailey, I935) of a parasite for a given kind of host is a measure of the efficiency of the parasite in finding and attacking hosts of that kind under the given conditions. It may be defined as the area effectively explored by an average parasite individual. It should be noted that, although some parasite individuals may be intrinsically more effective than others, it is sufficient to refer to the average area of discovery of the individuals; for, as all the parasites remain in action throughout any particular generation, in spite of any variation in the fraction of hosts attacked, the average remains unchanged. This contrasts with the hosts, for if some are more difficult to find than others the survival of these tends to be disproportionately large in the population remaining after the parasites have operated for a while, and so the average area of discovery of the parasites for the hosts progressively falls as the fraction destroyed increases during any given generation. The notation and terms to be used are summarized here for convenience. a = the mean area of discovery of a parasite individual in the searching population for hosts with a given degree of accessibility.
(a)da = the relative frequency of a within the range a 4- ½da. F----- the power of increase of the host population, this being the potential degree of multiplication in the absence of parasites under the given conditions. Its value allows for any constant mortality due to factors other than attack by parasites. hn = the density of adult hosts, which give rise to Fhn offspring, this being the initial density of hosts in the (n 4- i)th generation which is subject to attack of the parasites Pn+lp,~ = the density of mature parasites which search in the nth generation. Steady state = that balance of host and parasite densities which is self-regenerating from generation to generation if undisturbed. Stable steady state = that balance of host and parasite densities which if disturbed will return eventually to the steady state. 1--2
4
V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
3. The Fundamental Equations Figure i illustrates the frequency function f(a) by means of two examples. The curve A represents a continuous distribution of areas of discovery and the curve B represents approximately a discontinuous distribution such as exists when, for example, the fraction o.6 of the
B
B
f(d
I
2
Area of discovery (a) Fro. I. Examples of a continuous u n i m o d a l distribution (A), and a discontinuous distribution concentrated at a ~- i and a = 2 (B).
hosts presents an area of discovery of i unit and the fraction 0. 4 presents an area of discovery of 2 units. The function f(a) is necessarily positive or zero for all a > o, and is necessarily zero for values of a exceeding the upper limit of areas of discovery. Clearly the area under any curve such as A or B must be equal to unity; that is, we must have CO
l" f(a) da = i
(I)
o
When the distributions are discontinuous, the integral should be replaced by a sum. Thus, for instance, if different groups of hosts presented areas of discovery of o, I, 2, . . . etc., units with respective probabilities to, ri, r2, ... etc., the integral in (I) should be replaced by rO ~- r l ~- r2 ~- . . . . i (ifl) For convenience in the discussion of the general problem we will introduce the function ~(p), which is the Laplace transform of f(a), defined by O0
¢(p) = ~ e-Paf(a)da o
(2)
INTERACTION
BETWEEN
HOSTS
AND PARASITES
5
where p is some number, representing in the present application the density of parasites. This function has an important physical interpretation, as will be seen below. For discontinuous distributions, as described above, the Laplace transform is q~(p) = ro + rle -~ + r~e-2~ + . . . (za) The Laplace transform takes the following values : (O) ~--- I ;
as p -+ oo ~(p) --~ o, for continuous distributions --~ ro for discontinuous distributionsj (3) Since ~(p) decreases as p increases, its derivative, ~'(p), is negative. We may now apply these results to the determination of the equations expressing the growth of the two populations. If at the nth generation there are h, hosts occupying unit area, the density of their offspring will be Fh,~. The expected number of these offspring for which the parasites have an area of discovery between a and a + da is F h J ( a ) da
Since the density of the parasites is p,,, the fraction of these host-offspring remaining undiscovered will be e-~"a (see Nicholson & Bailey, 1935, page 555). Hence, the final density of hosts that survive to make up the (n + i)th generation is h,+ 1 = f em"aFh,f(a) da 0
= Fh.4(p.) (4) The Laplace transform (2) is thus seen to represent the proportion of hosts surviving after attack by parasites of density p. Also the parasite density P.+t in the next generation is given by P,+t -~ Fhn -- h,~+t
(5) Equations (4) and (5) are thus the fundamental equations of the problem. They are a generalization of equations (8o) and (81) of Nicholson & Bailey (i935), where the problem is treated for constant area of discovery. In the present paper the exponential function of (8o) is replaced by a more general function $(p,), which is in fact an average of exponential functions, corresponding to the distribution of areas of discovery. 4. The Steady State The steady state is by definition the state in which the densities of hosts and of parasites remain constant from generation to generation. For the
6
v.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
steady state we write
h.=h P. = P Substituting these values in equations (4) and (5) we are able to determine whether a steady state can exist, and to find the values of h and p for the steady state. The equations become ~=F$(p) (6) and p = h ( F - - ~) (7) Thus p is determined as the root of (6), and the corresponding value of h is then given by (7). Now since, as we saw above i > $(p) > o (for continuous distributions) > r o (for discontinuous distributions) it follows from (6) that, for a steady state to be possible when r o is non-zero, we must have I/r o > F > I (8) Continuous distributions impose no upper limit on F. Equation (8) shows that, when a fraction of the hosts is inaccessible (presenting zero area of discovery to the parasites), this fraction must be less than I/F. This result may be deduced on physical grounds. If r0 exceeded I/F, the inaccessible hosts alone would increase the population in each generation by a factor Fr o which exceeds unity, so that the population as a whole would increase indefinitely. It should be remarked that, when F = I/r0, it is apparently possible for the inaccessible hosts to maintain the population at a constant level; however, the actual steady densities of hosts and parasites in this case are indefinitely large, and so could not be supported by the resources of any environment. 5. Behaviour near the Steady State
(i) Derivation of criterion for stability In order to see how a developing population is likely to behave, it is not sufficient to establish that there exists a steady state, which will persist if undisturbed. Even small variations in the environment are likely to disturb the populations from their steady state, so it is important to know whether the steady state is stable or unstable. We consider the effect of small disturbances, which we denote by H n
I N T E R A C T I O N B E T W E E N HOSTS AND P A R A S I T E S
7
and P., from the steady-state values h and p respectively. T h u s we have h,~= h .-}- H , p. = p + v. (9) where H , / h and P,,]p are small quantities, so that only terms of the first degree in each need be considered. T h e n from (4) we have, keeping only linear terms H~ and P~,
h + H~+I = F.(h + H.)~(p + P~) = (h + H.)F[+@) + ~'~)P.] from (6) = (h + H.)[I + F~'(p)P.] = h + H . + FhqV(p)P. or Hn+l = H,~ - - G P , where G = - - Fhgb'(p ) and is therefore positive. Also, from (5), P,~+a = FH,~ -- Hn+ 1
(Io)
(II)
On eliminating H,, between (io) and (ii), we find that P., as a function of the generation n, satisfies the difference equation Pn+~. -- (G + I)P,,+I -[- FGP,, = o ; (i2) it is readily verified that H . satisfies the same equation. Now any solution of equation (i2) can be shown to be of the form Au'~ + Bu~ where ul and u2 are the roots of the quadratic equation uS - - ( G + i)u + F G = o
(i3)
T h e behaviour of the populations of hosts and parasites therefore depends on the nature of the roots of equation (i3). T h e populations are oscillatory if the roots of (I3) are complex, non-oscillatory if the roots are real. T h e condition for real roots is (G + I) 9">
4FG
(I4)
Of more importance is the stability of the populations, which requires that P,(or H~) diminish as n increases. T h e condition for this to hold is that the absolute values of the roots of (I3) are less than unity:
lu~l<~,
lu~l
(i5)
Now when ul and u2 are real, it is readily seen that they are both positive; for ut + u ~ = I + G and (16) ulu'a = F G
8
V.A.
BAILEY,
A. J . N I C H O L S O N
A N D E. J . W I L L I A M S
F r o m the second of these equations we see that it is necessary that FG < 1 ; moreover, since F > i we must also have G < L We now show that the quantities I -- ul and 1 -- u2 are both positive; their sum is 2--
Ul--
U2 =
I -- G > o
and their product is 1 - - u l - - u~ + u I u ~ = G ( F - - 1 )
> o
T h u s we have shown that, if the roots are real they are both less than unity if and only if FG < I (17) On the other hand, if the roots are complex, they are conjugate quantities and so have the same modulus; but since
ulu2 = F G it follows that
lull
---- [u~l = (FG)½ < I
Thus, when the condition (17) is satisfied, the roots are always less than unity in absolute value, and the steady state is stable. (ii) Graphical determination of nature of steady state W h e n ~b(p) is such that the calculations become complicated the following graphical process offers a simple method for determining the steady density p and the nature of the steady state. Y U
3
p
>"
~
X
FIG, 2. Showing graphical method for determining the steady parasite density p and the stability of t h e steady state; for details see text.
First the curve y = $(p) is drawn to some convenient scale as in Fig. 2. Next ON is set off equal to F -1 and the line through N parallel to OX is drawn to cut the curve at P. T h e tangent at P is now drawn to cut OY at T.
INTERACTION
BETWEEN
HOSTS AND PARASITES
Then p = N P and the steady state is stable if T U > N U ' ~ and unstable if T U < NU~J The proof of (18) is as follows : Since at P y = ~(p) = O N = F -~ therefore N P ----p. Also at P A p~b'(p) = N P tan N P T = - - N T Hence by (7) i --G=x_F
, -1 p $ ( p ) =
08)
OU N--u'NT
and so (17) is equivalent to OU NT >-NU
-- --
ON
that is --
OU(NU - - TU) > - - NU (OU - - NU)
or
TU > NU 2 This inequality therefore defines a stable steady state; the reverse inequality similarly defines an unstable steady state. 6. I n t e r a c t i o n w i t h D i s c o n t i n u o u s D i s t r i b u t i o n of Areas of Discovery
(i) The criterion f o r stability As an example in which the hosts are divided into distinct groups, we consider the binomial distribution. If A and B are positive numbers such that A+B=I k is a positive integer, and a o and as are positive numbers, with a o < ak, we here take a~ = a o + (as - - ao)i]k
and ri = (k)AS-~B~ (i = o, 1, . . . , k) s That is to say, r~ is the fraction of the population with area of discovery ae An example of this distribution is given by Fig. 3 for which A = 2]3 , B = I/3, k = IO. The ordinates r~ are given by the successive terms in the
V.A.
IO
BAILEY,
A. J . N I C H O L S O N
AND
E. J . W I L L I A M S
expansion of (A + B) ~, and lie equally spaced between a 0 and ak. The maximum ordinate corresponds to the value of i nearest to kB (in the figure, hB = 3½, i = 3), so that a~ is close to (a0A + akB). For this distribution, ¢(p) = (Ae-r,~o/,~+ Be-P=~/~)k so that, from (6), the steady parasite density p is the positive root of Ae -ra°/k + Be-~'~/k = F-1/k (19)
A
"6
N
,
....
O0
I ......N...
Area of discovery (o)
r-
--
ek
Fro. 3. An example of a binomial distribution of areas of discovery, with k = IO.
We also find that
G= p ~
F1/k
(Aaoe-r~°/k + Bake -p~k/~)
so that, from (I7), the criterion for stability is Fl+l/k
p ~
(Aaoe-p~°/k + Bake -pak/k) < I
When a0 = o, there is a finite proportion A k of inaccessible hosts, so that there is a limit to the values of F that can lead to steady states. This case is, however, simpler than that when a0 > o, and will be considered first. (ii) Some hosts inaccessible (a o = o) Here (19), the equation for the steady density, becomes A + Be -~ak/k -----F -1/k A steady state will be possible only if F < I/A ~
(20)
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
The condition for stability becomes FZ+Z/k B F -----~ pake-~'ak/k < x
IZ
(2x)
The steady states are stable for all values of F between this lower limit and the upper limit defined by (20). In Fig. 4, Curve I gives the upper limit of F for all positive integral values of k, and Curve II gives the lower limit for stability defined by (2t), with k = I. Each is plotted against A, the proportion of inaccessible hosts.
"1
10
IO0
-|.0(~
Power of Increase of Hosts (F)
Fzo. 4. Boundaries to regions of stable steady states when a fraction of the hosts is inaccessible to the parasites. I. General upper bound for steady states (AF = i), above which the hosts and parasites multiply without check. I I . - V . Lower bounds for stable steady states when: II. the area of discovery for accessible hosts is constant (k = I). III. the range of areas of discovery is very great (k --~ oo). IV. the areas of discovery for accessible hosts are distributed according to the gamma distribution with t = 2. V. the areas of discovery for accessible hosts are distributed in a triangular frequency distribution with mode at zero. Below the curves I I - V , steady states are unstable, under the conditions of each distribution, interactions leading to oscillations that grow in amplitude with time.
I2
V.A.
BAILEY,
A.
J.
NICHOLSON
A N D E. J . W I L L I A M S
When k is large the binomial distribution approaches the Poisson distribution, for which the relative frequencies are ri :
e-kB(kB)i/i[
As necessarily kB remains finite, we write kB = A, so that r i = e-~A~[i!
and the inaccessible proportion becomes e -~. We also put ak/h = b. A steady state will be possible only if F < e: (22) The equation for the steady density is log F = A(I -- e - # ) so the condition for stability becomes ()~ - - l o g F ) log
~ _ log F
< ~
(23)
As for the binomial distribution, conditions (22) and (23) set upper and lower bounds for values of F that can give rise to stable steady states. Curve I in Fig. 4 gives the upper bound, and Curve III the lower bound, corresponding to (23). For values of y , the inaccessible fraction, and F defining points above the upper bound of Fig. 4, no steady states are possible. For values defining points below the lower bound for a given distribution there are steady states but they are all unstable. The region of stable steady states is narrow; that is, for given y, the range of F values giving stable states is small. It seems likely that the chance of a value of F and a proportion inaccessible occurring so as to give rise to a stable steady state is quite small. (iii) A l l hosts accessible (a o > o) Without loss of generality we take a 0 ---=-x, ak > i. Since there are no inaccessible hosts, a steady state is always possible, whatever the value of F. With h = i, it can be shown that a k ( : a l ) must exceed 5.8 3 for a stable steady state to exist. As a 1 increases without limit, the lower bound for stability approaches that for a 0 = o, since the condition depends only on the ratio of a 0 to a k. In this limiting ease the upper bound to the region of stable steady states is given by A e - V = F-:/k and Fl+l/k
h P y-_- I A e - r
=
or P--1
A = F-r/he Fk
i
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
13
T h e proportion in the least accessible class is y=A
e(F-1)/F ~ -- - F
T h e boundaries to the region for stable steady states in terms of F and
\
",',o.o
\.
\
\
\
\
I'0
\
\
\ 0'11
I0
100
~L000
Power of Increase of Hosts (FI
FIG. 5. Boundary regions of stable steady states when all hosts are accessible, and are distributed into two groups, in one of which the area of discovery for the parasites is b times that of the other. I. b = 6
II. b = io
III. b = oo
Within the curves I, II, and III, the steady states are stable for the given ratio of areas of discovery in each distribution. Outside these curves, the steady states are unstable, and interactions lead to oscillations that increase in amplitude with time.
the proportion in the least accessible class are given in Fig. 5 for k = x and a I = 6, :co and co. For all points outside the region of stable states, steady states without stability are possible.
14
V. A. BAILEY, A. J. N I C H O L S O N AND E. J. WILLIAMS 7- I n t e r a c t i o n w i t h C o n t i n u o u s D i s t r i b u t i o n F u n c t i o n s
(i) A l l hosts accessible Discontinuous frequency distributions of areas of discovery, though theoretically possible, are not likely to occur in nature. One would expect that f ( a ) would more usually be a continuous function with a single maximum, such as the unimodal curve A in Fig. i, possibly associated with a finite proportion at zero accessibility. A family of unimodal distributions which adequately represents many observed continuous frequency distributions is the family of gamma distributions, for which f(a) =
e - a at-1 F(t) ' with t > °
In general the distribution will contain a scale factor, but since the scale of the distribution does not affect the nature of the steady states, it has been omitted. When o < t < i the distribution has its maximum at zero and is described as "J-shaped". W h e n t > I the distribution has a maxim u m at t -- I, and becomes more nearly symmetrical about this m a x i m u m as t increases. T h u s t is a parameter defining the " s h a p e " of the distribution. Figure 6 shows the curves of the frequency distributions for t = ½, I, 2, and 2o. For distributions of this family we have ¢(p) -
+
i
--t -
+p),+1
T h e conditions for a stable steady state are (i + p ) e = F F~pt (F --
+
< i
For t > I there is no value of F > I which will satisfy these relations. For 0 < t < I, all values of F will satisfy these relations. At the transition value, t = i, f ( a ) = e -~ ;
further investigation is needed before the nature of the steady states can be established for this particular distribution. For the triangular distribution, f ( a ) = 2(1 -- a) from a = 0 to a = i, at the steady state, •(p) = 2(e - ~ - - i + p)/p2
INTERACTION
B E T W E E N H O S T S AND P A R A S I T E S
15
o
>*
o
I
2
3
4 $ Area of discovery
6
7
$
FIG. 6. Curves showing the " s h a p e " of the gamma distributions for various values of t (with scale of a for each reduced by a factor ~/t). I. t = ½ II. t = I III. t = 2 IV. t = 2 o Of these, Curve I admits stable steady states for all values of the power of increase, because it increases indefinitely near zero accessibility; Curves III and IV never admit stable steady states. For Curve II, describing the exponential distribution, the nature of the steady state will vary, depending on the initial host and parasite densities.
and ~'(p) = - -
21i0(1 + e - y ) - - 2(1 - - e-l~)][p 3
The condition for stability, 2F ~ [ I + e -p F--I ~ p
2(I
y))
S not satisfied for any value of F. (ii) Some hosts inaccessible; illustrative case We here consider the case where a finite proportion A of hosts is inaccessible, the remainder being distributed in a unimodal continuous distribution. For this purpose we conveniently use the gamma distributions already discussed.
16
V.A. BAILEY, A. J. NICHOLSON AND E. J. WILLIAMS
If the inaccessible proportion be A, the relative frequency elsewhere will be e-aat-1
f(a)
= (I - - A) ~
(a > o)
Then I--A $(p) = A ~ (i + p)-----q (i - - A)t
¢'(P) =
(i +p),+l
The conditions for stability are I--A A + (I +p-----)*-- I/F F2pt(i -- A)
( F - i)(i +p),+l
The upper bound for F is given as usual by I/A. The lower bound is given when the limit of the above inequality is attained. It is found that admissible lower limits are given only when t > I. For t < I, all admissible values of F ( < I/A) lead to stable steady states. As t increases without limit, the distribution approximates a concentration at the single point a = t -- i, corresponding to a binomial distribution with k = I, so that the lower boundary for stability is given by Curve II of Fig. 4. The lower boundary for t = I is given by Curve IV of Fig. 4. For comparison, the lower bound for the triangular distribution with mode at zero accessibility and a finite proportion at zero is given by Curve V of Fig. 4. 8. D i s c u s s i o n
Field observation strongly suggests that the interaction of predators and their prey has a profound influence upon population numbers, but so many other influences affect populations in nature that it is difficult to assess the part played by such interaction. Because of this, simple theoretical models have been constructed which represent predator-prey interaction alone, in the simplest conceivable situations. This is only a first step. Later, variations in the primary postulates need to be made to take into account the additional influences that sometimes affect populations. Many distinctive models are required for, with different species and in different places, the interaction of predators and their prey is influenced by a wide variety of different factors. Lotka (1925) and Volterra (i926), with their formulation of simple interaction between predators and prey, concluded that this leads to a
INTERACTION
BETWEEN
HOSTS AND PARASITES
17
stable condition in which the populations of the two species oscillate continuously about their steady densities, which are determined by the properties of the two species. Bailey (193 I, i933a, b) and Nicholson (1933) made essentially the same assumptions for simple interaction between such animals, with the exception that they allowed time for both kinds of animals to develop to the stage at which they can interact; whereas Lotka and Volterra assumed that the interacting animals are fully mature at birth. It was found that this change in the formulation caused the system to be unstable, corrective reactions induced by displacement from the steady densities being over-violent, so leading characteristically to a system of oscillations that progressively grow in amplitude--although, if the numbers are not large, over-violent reactions may lead to the extinction of one or of both animals in a single pulse. Bartlett (196o) found that the addition of a term representing developmental lag into the Lotka-Volterra equations greatly increases the likelihood of rapid extinction of the predator, or of both the interacting species, when the system is subject to stochastic fluctuations. With the possible exception of some simple protozoa, all animals take some time to develop. Consequently the models suggest that instability is an underlying characteristic of predator-prey interaction, but this tendency must be countered in some way, for predators and their prey succeed in living together indefinitely in nature. It should be noted that such instability is the resultant, not only of developmental lag, but also of the postulate that the number of prey destroyed and the number of predators born at any moment are proportional to the product of the numbers of the interacting animals at that moment. This postulate is used in the simple models of all the investigators mentioned: but, as Nicholson (1933, page 148) has pointed out, it is tenable even as a rough first approximation only when considering those predators usually referred to as insect parasites--and only to these when near their steady densities. Using verbal arguments, he concluded that with other predators, such as birds and mammals, interaction should in general be less violent than with parasites and their hosts, so tending to maintain them at, or fluctuating about, their steady densities (Nicholson, 1933, page 167). It was also concluded that many kinds of influence can counter the tendency to instability which results from parasite-host interaction, so permitting parasites and their hosts to live together indefinitely in the same region. According to their characteristics, these influences lead to systems of sustained oscillation, damped oscillation, progressive approach to the steady densities of the animals, or to the fragmentation of the populations of the interacting animals, with an ever-changing pattern of T.Bo
2
I8
V . A . BAILEY, A. J. NICHOLSON AND E. J. WILLIAMS
local distribution (Nicholson, I933, Conclusions 4 o, 4 I, 42, 43, 63, 64 and 65; Nicholson & Bailey, i935, Conclusions 5 I, 52, 53 and 54). In the absence of such influences, parasite-host interaction should lead to extinction, of the parasite at least. It is probable that this has happened frequently in isolated areas that are not very large. It has long been thought that modification of some of the other primary postulates could reveal circumstances under which the inherent tendency to instability of host-parasite interaction would be countered. In this article the postulate that the parasites can find all host individuals with equal ease is abandoned, and it is shown that when some host individuals are more difficult to find than others this may lead to stability, but often does not do so. When a fraction of the hosts is inaccessible (the area of discovery for these being zero) this also may lead to stability, but only if the magnitude of the inaccessible fraction lies within narrow limits, related to the power of increase of the hosts--otherwise the system remains unstable, or both hosts and parasites tend to multiply indefinitely. These results give some support to Varley's (i947) conjectures, but disprove their generality. Leslie & Gower (I96o), like Lotka and Voherra, assume that predators and prey are born mature, and so interaction does not lead to instability in their model. Their deterministic model of simple predator-prey interaction differs from those of Lotka and Volterra in that a damping term is introduced. They conclude that this leads to a greater degree of stability when the system is subject to stochastic fluctuations, although random extinction of one or of both animals may occur in time. They then examine the situation in which a fraction of the prey is inaccessible, the remainder being uniformly subject to attack by the predators, and conclude that the likelihood of chance extinction of the predators is then reduced. Thus, when population numbers are small, the inaccessibility of some hosts may exercise a stabilizing influence additional to that deduced in the present article. REFERENCES BAILEY, V. A. (193I). Q. ft. Math. 2, 68-77. BAILEY, V. A. (1933a). Proc. Roy. Soc. A, I43, 75-88. BAILEY, V. A. (x933b)- ft. and Proc. Roy. Soc. N . S . W . 66, 387-393 . BARTLZrT, M. S. (I96o). "Stochastic Population Models in Ecology and Epidemiology", pp. 4o-4 z. Methuen, London. LESLIE, P. H. & GOV~ER,J. C. (x96o). Biometrika, 47, 219-234. LOTKA, A. J. (I925). "Elements of Physical Biology". Williams and Wilkins, Baltimore, NICHOLSON, A. J. (i933). ft. Anim. Ecol. 2, I32-I78. NICHOLSON, A. J. (I954). Australian ft. Zool. 2, 9-65. NXCHOLSON, A. J. & BAILEY,V. A. (I935). Proc. Zool. Soc. London, 551-598. VARLEY, G. C. (1947). ft. Anita. Ecol..,x6, I39-187. VOLTmmA, V. (x9z6). Memorie della R. Accademia Nazionale dei Lincei, 2, 3I-rx~.