Contenidos

Witteman

et al.

(1990)

provide evidence which appears to support Turchin's

(1993)

concerns regarding the utility of the single dimension model used by Hassell

et al. (1976).

The authors used auto-correlation analysis and lag­ plot reconstruction technique s on

.71

long term population studies and demonstrate that complex dynamics are far more prevalent in natural population behaviour than is suggested by the earlier publication of Hassell

et al.

(1976).

2 . 5. 3 Empirical Testing The Lotka- Volterra Equation

Lotka

(1932)

and Volterra

(1931),

used the logistic equation as the basis of their competition equations. Important theoretical developments and applications of these equations are reviewed by Wangersky

(1978).

For some time after the initial formulation of the Lotka-Volterra competition models, it was assumed that the outcomes of these interacting equations was independent of changes in the specific growth rate parameters of the two equations. Dorschner

et al.

(1987)

present the results of computer simulation studies along with analytical arguments to show that

this earlier

interpretation of the effect of r on the outcome of competition in the

indeterminate case is in error.

The authors demonstrate that when bi�tic potentials are similar, it is possible to reverse the competitive outcome of an initial set of densities based on competition coefficients and equilibrium values as per the Lotka-Volterra indeterminate case (Dorschner

et al. 1987).

The theta ( e ) form of the logistic equation was developed by Gilpin and Ayala

(1973)

to allow for different forms of density-dependence to be described. If

theta is small, per capita birth or death rates decrease rapidly with increasing population size, even at small population densities. If theta is large, the probabilities of birth and death do not change much until the prey population

reaches its carrying capacity. The competition form of the theta logistic model (27

a, b) appears to give a better fit to experimental data with

D ro s oph ila

pseudoobscura

and

D. serrata

(Ayala 1969). With the exception of the theta term, the parameters and coefficients of this equation are identical to those specified in equation (22 a, b).

(27a, b)

= rN2

(

1 -

(

N

!xJ

8

_ aN!xJ

Initially, Ayala (1969) interpreted his results as disproof of the principle of competitive exclusion. Gilpin and Justice (1972), pointed out that Ayala's results with

Drosophila pseudoobscura

and D.

serrata

could more usefully be thought of as illustrating the inadequacy of the logistic competition equation formulation

(22 a, b) to analyse all cases of competition.

� 1 -l r �· I

�- 1:2(--

_ ! O r­ ! > 8 � i 6 � 2 Succhurom.vces alnnt.' - - - -:>- - - � - - - - e - o ... - - - 0" Schi:.usaccharomyces· 3lone ..,. ... "' o-- - 0 ..,."() 20 40 80 100 1 20 uo Time (hrj

Fig 2.29The two experimental yeast populations of Gause (1932) fitted to the logistic model (from Krebs 1985).

Istock (1977) studied the behaviour of two species of waterboatmen

(Corixidae)

and fitted seasonal data to the Lotka-Volterra model. Both species were found to eo-dominate in their pond environment and were present at all times. One species

(Hesperocorixa lobata)

was found to be numerically dominant in the spring and early summer. while the other species

Sigara macropala)

dominated in the latter half of the summer. !stock concludes that

S. macropala

is able to inhibit the development of

H. lobata,

but the interaction is not necessarily reciprocal. The behaviour of these two populations appears to be equilibrium centred and yet quite dynamic on an annual basis.

Much of the early research and theoretical developments that have come from the Lotka-Volterra equations can be traced back to the research efforts of the

104

competition between two species of yeast

Saccharomyces cervisiae

and

S chizosaccharomyces kephir.

To begin with, Gause (1932) grew the two yeast species separately and found that he could obtain a good fit between their growth

data and the logistic model (fig 2.29).

Before the research efforts of Gause in 1932, Richards (1928), had discovered

that the action of alcohol produced by the breakdown of sugar for energy under anaerobic conditions would slow down the rate of growth of yeast populations. Richards discovery suggested that yeast were principally limited by ethyl alcohol accumulation and this association could be illustrated by plotting alcohol

concentration alongside population growth (fig 2.30).

Z!. 60 � " ;)i) f-·

..

. �t! t-" 30 i- 1 I f) �0 ;r - - - � - - - �

I ��

/

,.

-= / � ,d • ₆ 1 /

.

� cf �t Aknh,)l ' cont•entration

/

I I Cl I I I 60 tlO -i l L_j___L.] (i lOO l�O . Tinw fhr;

Fig

2.30 Experimental data for the growth of yeast fitted to the logistic model (solid line). Accumulation of ethyl alcohol (Richards 1928), (from Krebs 1985).

In a nice example of lateral thinking, Gause (1932) reasoned that if alcohol

accumulation was indeed a critical limiting factor then it should be possible to determine the coefficients of competition in the Lotka-Volterra equations by measuring the alcohol production rates of the two yeasts. To test this hypothesis, Gause grew the two yeast species in a single culture, fitted the data to the Lotka­ Volterra model and compared the competition coefficients with those determined by direct measurements of alcohol production rates. While the two sets of coefficients were not identical, they were very close. Gause attributed differences to the presence of other waste matter.

Birch (1953) experimented with interacting populations of grain beetles

Calandra oryzae

and

Rhizopertha dominica.

Birch found that the outcome of interaction between the two beetles in a mixed culture was dependent on

temperature. At 290C,

Calandra

would usually eliminate

Rhizopertha,

while at

32oc,

Rhizopertha

would eliminate

Calandra.

Once again, it is interesting to

note that in the experimental results of Birch plotted in figure 2.31, the species

which wins out in the competitive encounter does not reach a stable

equilibrium point as predicted by the Lotka-Volterra model.

..., ' fi00[- 1

r

4oot­ ; i �

I

:!OOf-

i

r

0�

�(I 80 1 :!0 1 60 Tim..- ( wt"t"k.• J

Fig 2.31 A plot showing the outcome of a competitive interaction between the two species of grain beetle grown in a mixed culture by Birch (1953). In this example from the work of Birch, both species of grain beetle !are raised in wheat at 14% moisture content and 29.1 °C constant temperature (from Krebs 1985).

Park (1948) explored interspecific competition by experimenting with

mixed cultures of two flour beetles

Tribolium confusum

and

Trib o l i u m

castaneum.

Park experimented with a range o f environmental variables including: space, initial densities, food quantities, climate, genetic variability and the addition of sporozoan parasites. Park found that space did not greatly affect the outcome of competition between these two species.

Park (1948), also noted that it was not possible to predict outcomes with any certainty. The results appeared to be more probabilistic in nature. This was especially the case at intermediate climates when sometimes

T. con

fu

sum

would win and sometimes

T. castaneum

would win. Furthermore, Park found that different genetic strains of

Tribolium

differed greatly in their competitive ability.

The work of Park was instrumental in demonstrating that the outcome of competition is not always invariant and may depend on abiotic factors like

weather and internal biotic factors like genetic · composition. In the examples of

interacting populations considered so far, the outcome of the interaction has also meant the loss of one species.

Crombie (1945) raised two species of grain beetle in wheat and found that they could coexist indefinitely. The mechanism of coexistence appeared to be difference in food preference. The larvae of

Rhizopertha

live and feed on the inside of the wheat

gr

ain while the larvae of

Oryzaephilus

live and feed on the

106

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