IdleTheory 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.

Idle Theory

Author: Chris Davis
Last edited: 7 April 1998