Commentary
on the Malthus and Tetra models
The Malthus Model
No balance in nature
Perhaps the principle conclusion drawn from the Malthus model is
that there is no balance of nature. The vision of life that comes out
of this simulation model is one of cyclic change, of instability,
even chaos.
The orthodox account of population growth, the Logistic Equation,
has populations rising to a maximum and then levelling out. In the
Malthus model simulations, this only happens with populations growing
in a nutrient stream. It is the kind of population behaviour that can
be expected in laboratory chemostats which maintain a steady flow of
nutrients to a population of bacteria, or that can generally be
expected of plants whose power source is provided by a stream of
solar radiation. Where populations multiply in pools of energy, rather
than in streams, the tendency is for populations to oscillate.
The distinction between energy pools and streams is not one that
seems to be found in ecology. Energy is almost always shown moving
in streams. Thus plants are converting the stream of solar radiation
into a stream of nutrients on which a population of grazers feed,
and these multiplying grazers provide a stream of nutrients for
predators. While this view is certainly true over the long run, it
conceals the reality that plants and grazers (and predators) store
up energy, and create a pool of energy rather than a stream.
Its result is an implicit assumption that Logistic Equation population
growth applies to grazing animals and their predators, that plant,
grazer, and predator populations grow to the environment's 'carrying
capacity,' and remain stable.
The alternative view, which the Malthus model offers, is that the
natural world is always in a transient state, always in process of
change. If the face of nature appears constant, it is because the
pace of this change, relative to the span of human life, is usually
slow.
No reproductive imperative
Another conclusion from the Malthus model is that, in unpredated
populations, the selective pressure is for minimum rather
than maximum reproduction. Also, slow reproduction paradoxically
results in higher populations overall, because the slow reproducers,
using less energy, make lower demands on their environment, and
thus more of them can be supported.
Predation, in the model, reverses the selective pressure, and tends
to increase reproduction rates. If it is taken that predation is
a constant in natural life, like taxation in human life, then
maximum reproduction will result. But there does not seem to be any
natural law which states that as each new species appears, its
predator arrives beside it. Predation, in Idle Theory, is suggested
to result from grazers finding first a scavenging life, and then a predatory
life, resulting in increased idleness, and hence increased survival.
This is only likely to happen when plant populations, for one reason
or other, have fallen to relatively low levels, forcing grazers
to work longer to find food. When plant food is abundant, there is
no incentive for a grazer to become a predator.
Even if a population of predators does emerge, the Malthus model
suggests that, as predation drives up prey reproduction rates, it
becomes increasingly difficult for predators to restrict prey
population. Sooner or later, the numbers start rising, until
overgrazing produces a population crash, which predators
(and sometimes their grazer prey) may not survive. This suggests that
predation is itself a transient phenomenon, that an efficient
predator sooner or later, one way or other, creates the conditions
for its own demise. Seen this way, the only long-term predator
strategy to ensure predator survival is to keep predator populations
low, and tap off a small numbers of grazers. Such a strategy would
result in selection pressure on the prey population switching to
low reproduction.
The conclusion that slow reproduction has selective adavntages over
fast reproduction raises the question of how a population can restrict
its reproduction rate. In the a asexually reproducing Malthus model
populations, slow reproducers simply expend less energy on reproduction
than fast reproducers. Since, in the natural world, a great deal
of reproduction is sexual, one possibility for the evolution of
sexual reproduction is that the complication it introduces itself
acts as a brake on reproduction, and that, in principle, sexual
reproduction allows reproductive flexibility by varying the ratio of
sexes. If, under predation, sexually reproducing populations can
increase the relative proportion of females to males, overall
reproduction increases. And if, in the absence of predation, the
proportion of females is reduced, overall reproduction falls. Thus
sexual reproduction, quite apart from the genetic diversity it
allows, also offers a means of varying the reproduction rate of a
species.
The suggestion that the creatures serve their long-term interests
best by minimizing reproduction runs entirely counter to an orthodox
view that the creatures are all trying to reproduce as rapidly as
possible, and that any adaptation which increases reproduction rates
is beneficial.
Any change in a replicating nucleic acid molecule which speeded up
its replication would be favoured by selection.
(Maynard Smith. The Theory of Evolution. Ch.6)
If replicator molecules of type A make copies of themselves on average
once a week while those of type B make copies of themselves once an
hour, it is not difficult to see that pretty soon type A molecules
are going to be far outnumbered, even if they live much longer than
B molecules. There would therefore probably have been an 'evolutionary
trend' towards higher 'fecundity' of molecules...
(Richard Dawkins. The Selfish Gene. Ch.2)
Underlying this genetically-centred view is a simple
presumption that those creatures which reproduce fastest will in
time outnumber slower reproducers, and drive them into extinction.
Since most genetically-centred texts
concern themselves with the details of genetic processes, and seldom
consider the energetics of reproduction or ecosystems, or the dynamics
of population growth, this account of competitive exclusion seems to
be based on a simple extrapolation of basic population trends, without
actually carefully considering what is actually likely to happen.
In the energy-centred Malthus population model, the opposite happens,
because gas-guzzling fast reproducers are at a disadvantage next to
more economical slow reproducers.
Model limitations
The Malthus simulation model is a very simple model. It has only
one kind of plant, which grows at a constant rate to a maximum size.
One improvement would be to introduce varieties of plants, some
multiplying faster than others, some having defences against grazers.
The populations in the Malthus model are small. The average
population of grazers is usually not much more than 30, and there
are seldom more than 4 or 5 predators. With these low numbers, the
probability of chance events (predators finding themselves in places
where there are no grazers, for example) dictating the model's
performance is high. In a larger model, with 3000 grazers and 100
predators, the chance element would be reduced. But because the
Malthus model works out how each plant will grow, and how
each grazer and predator will behave, much greater computing power
than a Pentium PC is needed.
The Malthus predators (unlike the Tetra model omnivores) are always
in danger of extinction if prey levels fall too low. Allowing dead
prey carcases to slowly rot after a mass death is really a fix to
ensure that predators have a chance of surviving mass grazer deaths.
Equally, having predator offspring flee from their mothers after
birth is a fix to stop mothers eating them (or them eating their
mothers), as well as an attempt to disperse predators across the
field.
Because the model works out what each creature will do, one after
the other, direct competition is effectively disallowed, because
no two creatures will head for the same food source simultaneously,
because it will have already been eaten by A by the the time B or
C or Z start searching. In order to avoid a 'pecking order', the model
alternately flips the sequence, so that it goes A-Z, Z-A, A-Z, and so
on. Little things like this can have a large effect.
The Tetra Model
The Tetra simulation model deals with whole populations of 'tetrapods' rather
than with individuals. The Tetra model works out how whole populations,
of any number, behave - not how individuals behave.
Another advantage over the Malthus model
is that an attempt is made to 'design' the tetras, and calculate
their energy requirements. The tetras come in a range of sizes, each
one an order of magnitude different, the largest tetra being the size of
a matchbox. Their size determines their basic metabolic rate, and
the length of their legs, which in turn determines their speed of
movement.
One in a million tetra offspring are mutants an order of magnitude
larger or smaller.
One criticism of the Tetra model is that the tetras feed on
a photosynthetic soup whose energy density is unlimited (it just
keeps storing solar energy), and everywhere equal.
Another criticism of the tetra model is that when one 'species' of
evenly-distributed tetras feeding from the homogeneous soup reaches
zero idleness, the entire population dies simultaneuosly. If the
soup was more heterogeneous, with some parts more energy-rich than
others, this would not happen.
The tetra predators are also omnivores. When the soup is energy-rich,
they feed off that. When the soup thins out, they switch to predation
of that evenly-distributed population of slower-moving tetras which offers the
highest idleness life. In this, the tetras are like fund managers who
keep shifting assets into investments yielding the best returns.
One result of this, it seems, is that tetra predation acts to hold
predated populations at near-constant levels for long periods.
But then, each size or 'species' of tetra comes with its own inbuilt
reproduction rate, which is invariant - so predated tetras don't
respond like the Malthus model grazers, upping their reproduction
rate when subject to predator attack.
The long periods of population stability in the tetra model end
with falling soup energy density claiming first the smallest
and slowest-moving tetras, whose slow speed disallows them from
catching larger, faster ones, and enforces feeding from the soup only.
This triggers off a chain reaction, as the next largest tetra is forced
to become a grazer, until it too can no longer survive. After a
series of extinctions, the soup energy density starts to climb back
up, and the remaining tetras begin to rediversify, and gradually
re-establish predation stability. But since the fastest tetra, the top
predator, itself has no predators, its numbers inexorably rise, and
it is this which precipitates the final extinction.
Nevertheless, the tetra model again shows the same overall cyclical
behaviour as the Malthus model. No stable state is found. The main
difference is that the tetra omnivore system does actually work to
stabilize populations for long periods, unlike the predators of
the Malthus model, which aren't omnivores, and can easily wipe out
their prey, or be wiped out when the prey numbers fall too low.