and their homogenization: a higher-order principle of competitive exclusion. Ecol. Modelling ... higher-order "social" effects do appear in real data, what exactly is the proper role of ... Hutchinson proposed a "simple theoretical development" of the usual ... "It is well known that in the equations (1.1) the coefficients of competi-.
309
Ecological Modelling, 60 (1992).309-320 Elsevier Science Publishers B.V., Amsterdam
On Hutchinson's competition equations and their homogenization: a higher-order principle of competitive exclusion P. Antonelli a, X. Lin
a,,
and R.H. Bradbury b
a Department of Mathematics, University of Alberta, Edmonton, Alta. T6G 2G1, Canada, b National Resource Information Centre, Canberra, A.C.T. 2600, Australia (Accepted 23 October 1991)
ABSTRACT Antonelli, P., Lin, X. and Bradbury, R.H., 1992. On Hutchinson's competition equations and their homogenization: a higher-order principle of competitive exclusion. Ecol. Modelling, 60: 309-320. Replacement of cubic terms in Hutchinson's competition equations by simple homogeneous functions of the population densities results in interactions of higher-order (i.e. "social") which scale the same way as quadratic interactions and are hence, statistically, just as important as the usual quadratic terms. The resulting homogenized system, in addition, exhibits positive equilibria: either exactly one, exactly two or three equilibria, at most. In the latter case, two of these are certainly unstable, while the third may or may not be stable depending on coefficients. Thus, we arrive at a system whose competitive behaviour is strongly analogous to that of the classical competitive exclusion principle and yet one whose statistical behaviour reflects findings in real data on higher-order (i.e. "social") interactions, something which Hutchinson's original system does not do.
1. I N T R O D U C T I O N
Competition equations for two species involving third-order "social" interactions originate with a short article in 1947, by the great ecologist G.E. Hutchinson. In his famous treatise on population ecology, he diCorrespondence to: P. Antonelli, Department of Mathematics, University of Alberta, Edmonton Alta. T6G 2G1, Canada. * Partially supported by NSERC-A-7667 to P.L. Antonelli. 0304-3800/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
310
P. ANTONELLIET AL.
gresses at length on these equations (Hutchinson, 1978)• Admitting that higher-order "social" effects do appear in real data, what exactly is the proper role of such terms in the population dynamics of interactions, such as social interactions, vis-a-vis interactions such as predation or competition? This is just the question Hutchinson addressed in 1947. In the present article, we shall confine the discussion to the case of competition, as did Hutchinson, and refer the reader to Antonelli et al. (1991) for a discussion of higher-order interactions in predation. Hutchinson proposed a "simple theoretical development" of the usual G a u s e / W i t t (1935) model of a two species competitive interaction to cover what he called "incipient social p h e n o m e n a " by addition of terms - E ~ . N 1 • (N2) 2 and - E 2 . N 2 • (N1) 2 to
intraspecific interspecific
aN1 _
dt dN2 _
dt
biN1 (gl _N1 _o~N2) ] K1
biN2 ( g
J
(1.1)
2_N 2-]~N1)
K2
The result of this straight-forward generalization is the system
-blgl[gl-gl-aNe-Y(g2)2]}_ gl [ ~r [ ,] d N 2 b2N2 dt K2 lK2-1*2-13N'-6vNa12]
dNldl
(1.2)
Whereas (1.1) exhibits straight line isoclines, (1.2) exhibits parabolic isoclines and admits a generalization of the under a natural condition which ensures the equilibrium is unstable and hyperbolic. It is interesting that although there have been published papers on statistically significant higher-order interactions in real data, notably Vandermeer (1969), Hairston et al. (1968) and Wilbur (1971, 1972), Hutchinson's equations do not seem to be mentioned again in the literature until the appearance of the classic paper of Ayala et al. (1973). In this study, it is included along with 10 other models, in a data-fitting race. Coming in second, it is rejected for certain mathematical difficulties 1. One of these concerns the cooperative case which we do not consider here, while the
Principleof CompetitiveExclusion
1 In Ayala et al. there are typographical errors causing confusion. Equations 5 and 6 are mislabelled; plot isoclines to see which is being refered to in their text.
ON HUTCHINSON'S COMPETITION EQUATIONS AND T H E I R H O M O G E N I Z A T I O N
311
other is related to the scaling properties of the various terms in (1.2). In order to discuss this we first quote directly from Hutchinson (1947, p. 320). "It is well known that in the equations (1.1) the coefficients of competition a and /3 can change as a result of environmental changes. Since the number of individuals in a closed system is a factor in the environment of any one of the individuals, the values of the coefficients may in some cases change continuously throughout the experiment. Considering the simplest case first, let it be supposed that the effect of one individual of one species or the other is determined not by a single constant coefficient (a or/3), but by a factor (Y" N2 or 8 • N 1) proportional to the total number of competing species. The equations of competition now become (1.2)." He goes on the remark, "when one individual of the second species ( N 2 = 1) is introduced into the population of the first species, its effect will be Y" (1) 2, while in equations (1.1) the same effect will be a" 1. It is therefore evident that in general the new coefficients (% 6) can be expected to have values of the same order of magnitude as those (a, /3) derived from previous experiments" [our italics]. Thus, Hutchinson seems to have though that the K~ and K 2 of (1.2) are statistically similar to the usual carrying capacities of (1.1). Yet, Ayala et al. (1973) disagree. They are correct and Hutchinson has made a mathematical mistake, unfortunately. The choice of one individual is the choice of a special case. More generally, if, say 10 or 100 individuals of N 2 are chosen, then the effects on N 1 through the y - N 2 term in (1.2) relative to the effects on N 1 through the a . N 2 term in (1.1) are not of the same order of magnitude. Instead, the effects in the case of 100 individuals of the term in (1.2) compared to its counterpart in (1.1) is ten times the effect in the case of 10 individuals. Clearly, the effect & not scalable in the same way in the two models. We believe this is the reason that the K's are not ordinary carrying capacities in Hutchinson's equations. However, it goes deeper than this and has everything to do with the relative importance (i.e. scaling) of within- and between-species interactions in population dynamics, generally. Ultimately, Hutchinson's equations can be "adjusted" so that his quoted arguments come out right. The procedure is what we call Homogenization of (1.2). Before describing this formal process let us recall that in the experimental studies on communities of bacteria and protozoa, Hairston et al. (1968) found "significant second-order effects," while Wilbur (1972) makes a very strong case for statistically significant higher-order interactions in his experimental work with field enclosure communities of Ambystoma salamanders. He states, "Frequently the higher-order interactions are as important as the main effects". Wilbur means that, according to rigorously applied analysis of variance, the "social" interactions are of equal importance with the usual quadratic interactions.
312
P. A N T O N E L L I
E T AL.
This is brought out more clearly when he states (p. 18) "Mathematically, the additivity assumption neglects higher-order interactions, such as the/3's in the following equations. A failure of the assumption means that the drag terms must be modified to the form (K 1
- N 1 - a12N2 - a13N 3 - ~I23N2N3) /KI
(K 2 -
N 2 -
a21N
1 -
(K 3 - N 3 - t~31N1 - -
a23N
3 -
i~213N1N3)/K2
a32N
2 -
~312N1N2)/K3
which is a nonlinear system of equations when it is set to zero to solve for the equilibrium densities. Such a nonlinear system does not have the analytical power of the first-order system derived from the Lotka-Volterra model (Vandermeer, 1969)." Further, "Nonadditivity was certainly the rule in the salamander community, both within and between species". In the V a n d e r m e e r paper quoted, the interspecific competition equations are extended formally in Taylor series fashion for a more general case and V a n d e r m e e r notes that such equations are mathematically intractable. Indeed, this is a common lament by all who have considered higher-order interactions [e.g. see Pianka's article "Competition and niche theory" (1981)]. Yet, it could be that virtually any statistically significant higherorder interaction must be "as important as the main effects" simply because of the nature of the linear theory of analysis of variance. In any case, we are suggesting that when this is the case, higher-order terms of a Taylor series could be replaced by simple algebraic functions which scale as if they are quadratic. This replacement process results in what mathematicians call second-degree positively homogeneous functions. The results can be biologically meaningful and mathematically tractable. Now let us give the formal definitions. Let Q(NI,..., N n) be a function of positive variables N~,..., Nn with the property that, under the transformation N~ ~ / ~ .N,., i = 1 . . . n where ~ > 0 is any positive real number, Q ~ ~2. Q. In this case, Q is said to be second-degree (p)-homogeneous. For example, if we replace
Y" bKli N 1N 2 and 6"-~22N2 N2 in Hutchinson's equations (1.2) by
b, (N, N2)2/3 and 6" ~2(N2N2)2/3
313
ON HUTCHINSON'SCOMPETITION EQUATIONS AND THEIR HOMOGENIZATION
respectively, we obtain the Homogenized Hutchinson equations dN1 blNl( dt - K 1 dN:
b2N2(
dt
K2
( N1N2) 2/3 ) K1
N1
-
')/1
- °~12N2 -
N~
(N2N2)2/3) K2 - N2 - a21N1 - Y2
(1.3)
N2
In Section 2 we give the equilibria and isocline structure of the system (1.3), and thereby demonstrate a generalization of the Principle of Competitive Exclusion. 2. ISOCLINE ANALYSIS OF THE HOMOGENIZED HUTCHINSON EQUATION A number of facts can be readily established. The isocline
(N1.N2) 2/3 K1 - N1 -
°~12N2
--
Yl
-0
NI
(2.1)
can be solved for N 2 in terms of N~. This follows from the non-vanishing, everywhere relevance, of d N 2 / d N 1, dN 2 dN1
(K1-NI-al2N2)2"[(K1-NI-al2N2)-3N1] y13[4N23 +3a12 T
NI( gl
- - N1 - -
(2.2)
°q2 N2)2]
Furthermore, the isocline N 2 = F1(N 1) has the property lim F[(N1)= +o~
(2.3)
N I --~0
where the prime indicates the relevant derivative. Also, FI(O ) --- FI(K1)
=
0
(2.4)
holds. Likewise, the dual relations hold so that for the dual isocline
72( N2N2) 2/3 K 2 - N 2 - Cez1N1 -
N2
= 0
(2.5)
We have then lim F~(N2)= + ~ N2-~0
(2.6)
314
v. ANTONELLIlETAL.
using the dual d N 1 / d N 2 to (2.2). Also Fz(O)=Fz(K2)=O
(2.7)
holds. It is easy to see that N 2 = FI(N 1) and N 1 = F2(N 2) are both concave-down and that the maximum point of F 1 is
K1 ( 3 )/ 4 _ ( 3 ] 3/4' ~1
.
4 + 0/12 Z ]
K1 ( 3 )/ 4 _ 4 -t--19/12 ")/1
while that for F 2 is K2
( 3 )/ 4 _
4 + 0/21
.
K2 4 + 0/21 - -
Y2} From (2.1) through (2.7) we conclude that THEOREM A. The system (1.3) has at least one positive equilibrium. Indeed, straightforward geometric analysis (see figures below) yields THEOREM B. The system (1.3) has more than one positive equilibrium if and only if the following four inequalities hold 4 -Yl }
K2 q 0/12
-~ ~2 -I--- - -q-0/21
(2.8)
4 --
K1 '] 0/21
--~]/1 "]- - - "~ 0/12 'Y2
(2.9)
'Y2 ]
Yl
K1 ( 3 ) 3/4 K~ ( 3 ) 3/4 ~ -')/2 (~) __3 3/4 4 + 0/12 4 -l- 0/21 "}/2 K2
[ 3
3/4 < 1 - - t 3/4 (~2) "Yl]
4 + O/21
K1
( 3 ) 3/4
4 + a~2 y~-
Also, (2.8) and (2.9) cannot be equalities simultaneously.
(2.10)
(2.11)
ON HUTCHINSON'S COMPETITION EQUATIONS AND THEIR HOMOGENIZATION
N2~ (b)
N2'I (a) I , ~ S L22
/-L22
N1
N2,
315
(c)
N1
N21 (d)
/ N1
N1
Fig. 1. One equilibrium.
THEOREM C. The system (1.3) has at most three positive equilibria. If only one of (2.8) and (2.9) becomes an equality, then (1.3) has exactly two positive equilibria, otherwise there are three positive equilibria. If both (2.8) and (2.9) are equalities, there is exactly one positive equilibrium. All possible intersections of the isoclines are given in Figs. 1-3. See the text following for the definitions of Lij. Now we state THE INSTABILITY THEOREM. If (1.3) has a unique positive equilibrium, then it is unstable. If (1.3) has two positive equilibria, both are unstable. If the model
N2
(a)
N2.
(b) ~L11
~
L
L22 !-12
V-1.,2 N1 Fig. 2. Two equilibria.
It
N1
316
P. A N T O N E L L I
ET AL.
N2 , Lll
2
L21
N1
Fig. 3. Three equilibria. has three positiue equilibria, at least two are unstable, and the remaining one can be stable or unstable. We now proceed with the proof. Assume that N 2 = FI(N 1) for the isocline (N1NZ) 2/3 K1
- N1 - alzN2
-
T1
= 0, from (2.1)
N1
and N 1 = F2(N 2) for the isocline ( N z N 2 ) 2/3 K2 - N2 - -
°/21N1 -
"Y2
0,
from (2.5).
F 1 and F 2 a r e both concave-down. The maximal point of F 1 is
Then,
Ka NI= 4 +a12\~-] The maximal value of F~ is 3
g2'max = F l ( / ~ l ) =
Z)
~3/4 /~1
Similary, the maximal point of
U2=
K2
(3] 3/4
4 + or2] Y---~21
F 2
is
ON HUTCHINSON'SCOMPETITIONEQUATIONSANDTHEIRHOMOGENIZATION
and the maximal value
Nl,ma x =
Let
317
3 F2(-~2) = (--)3/4/~ 2 Y2
Lt, = {(N 1, F,(N1)); 0 < N 1 ( T71) 3/4, replacement of them in (2.12) gives N1 > (_~)3/4 ' U
det J
