Note that this is part of a major chapter of this e-text: THE CONCEPT: Once bitten twice shy. If, on reading this section on models, you find something that you do not understand, then go to the beginning of the chapter and read from there.
Developing the Lotka Volterra (MELV)
necessary to better perceive the interactions involved in ecosystems
and to see how compatibility evolves naturally through the interactive
process. Note that competition is a component of the interactive
process and that the ritualisation of competition instead of crude
strife will form a part of the evolution of social systems. However,
instead of the term competition, I prefer to regard the various
intensities of interaction as a more objective term.
Smuts said, "To understand nature, we must take one of her own units, and not an abstract one of our own making. We must as it were, take a small sample section of nature which will include as one and indivisible both the element of activity or principle and the element of structure or concreteness in her." I quoted this statement in the chapter on Smuts. By now its import and significance should have increased in the mind of the reader, as it did for me while writing this book. When I first read this statement, it was like a vaguely discernible beacon in the mist while attempting to berth on a dark night in a harbour with all forms of traffic, official and unofficial, large and small. Later I found that Smuts was a reader of Kant , describing him as "one of the great kings and legislators of thought," and having just finished a preliminary study of Kant's teleology, the mist lifted and the beacon became clearly visible. Authors using their own abstract ideas find rational explanations but not the facts. These ideas can be entertaining, and if the primary purpose of the author is to entertain, this is fine. Richard Dawkins (1983) used such " thought experiment s" as he called them in his book, "The Extended Phenotype". Daniel Dennet (1995) in his book, "Darwin's Dangerous Idea", enjoyed introducing "thought experiments" that had no real substance or relation to nature. What are needed to understand the inherent mechansims of nature are nature experiments according to Smuts' principle .
Our conceptions of what Kant called "self-organising systems" needs to reflect or "explain nature by reference to herself and her own standards," thereby reducing the subjectivity of our perceptions. Nature's evolutionary processes and physical forms, represents a solution that has taken all of time to express. The evolution of the whole universe has culminated in the present life forms on earth. On other planets there is other life (1) (footnote: QsXLIIv29). We are earth's magnum opus and need to respect that responsibility. If we can recognise any principles in nature that will serve as a guide to human rationality and reason, then we will have a useful tool that enhances our objectivity when dealing with nature and so constraining the faults of our human nature.
An obvious necessity in nature is "survival". The law of survival
rules the great diversity of life. I introduce two indivisible elements
of survival, perpetuity and compatibility. To use the words of
Wynne-Edwards (1986): "Because of the fact that life is vested in
individuals and its transmission is hereditary, it is beneficial for
individuals to be tested by selection and the less fit phenotypes
eliminated from their ranks (termed simply perpetuity); and it is equally true that the fitness value
of an individual can be measured by the contribution it makes to the
viability of its group, and of the higher organisations of which the
group is part (termed simply compatibility)." Smut's element of
"activity" is perpetuity. A continuing output of life forms provides a
directional force similar in principle to the outpouring of light from
the sun or the force from a spring keeping a pendulum in motion. Smuts'
element of concreteness, inseparable from perpetuity, is compatibility
and it consists of the requirement for stability to enable perpetuity.
An animal is adapted to and depends upon the structure of the natural
habitat for its survival. Perpetuity , or the continual production
of a "reproductive surplus" (Mayr, 1993), as a principle needs
to include the stability and continuity (persistence) of the habitat
upon which the animal depends - thus compatibility .
The logistic equation (sometimes called the Verhulst model or
logistic growth curve) illustrates the main processes and
constraints to be found in nature and models population dynamics.
A population's rate of change (dN/dt), or the change in numbers of a population (N) with time (t) is expressed as follows:
This is a differential equation describing exponential growth.
where r is the intrinsic rate of increase of population numbers (N). Growth under such conditions is exponential and experiences no constraints. This form of growth has been illustrated in the exponential growth of the pheasant ( Phasianus colchicus ) over the years 1937 to 1942 (Putman, 1994).
In nature, we find limits , a major one being the carrying capacity of the environment. Resources are neither constant nor unlimited. Environmental carrying capacities for a species represent the maximum population size sustainable within a particular area. An animal's habitat is limited, so as the population grows, the increased density begins to inhibit growth in some way until it reaches an asymptotic level called the carrying capacity. This is the maximum number supportable in a given habitat (Smith, 1974). Generally, ecologists attribute these constraints to intraspecific competition (Putman, 1994). When we look at specific examples, we find the situation to be more complex.
Wildebeest or Gnu of Africa ( Connochaetes gnou , C. taurinus ) have to maintain an extremely high growth rate (r) to survive as a species. Of the calves born in late African summer (January and February), almost 50% are dead within a few weeks through predation and becoming separated from their mothers. At an age of six to seven months, a rinderpest epidemic afflicts them and lasts until they are about a year old. Those with immunity, imparted by the colostral in the first flow of the mother's milk, survive. By the end of this epidemic up to 80% of the calves born that generation has died! The remaining 20% of yearlings represent 8% of the whole population. As wildebeest populations are usually stable, this means that 8% of adults are lost each year and births to maintain this have to number 40% of the adult population! Effectively, almost every female has to give birth each year for the species to survive. Research has shown that 83% of yearling cows and 95% of other adult females breed each year (Purnell's Encl., G). Their survival strategy emphasises reproduction (the r-factor). The 8% of adult deaths annually are mostly due to lion predation (about 90%), but cheetah, leopard, hyena and wild dog are also predators. This is in the Serengeti reserve where there are no human "predators".
Without predation and disease controlling the Wildebeest population, we can imagine a situation similar to that described by Leopold when they eradicated the wolf to protect its natural prey, the deer: "I have lived to see state after state extirpate its wolves. I have watched the face of many a wolfless mountain, and seen the southern-facing slopes wrinkle with a maze of new deer trails. I have seen every edible bush and seedling browsed, first to anaemic desuetude, and then to death. I have seen every edible tree defoliated to the height of a saddlehorn. Such a mountain looks as if someone had given God a new pruning shears and forbidden Him all other exercise. In the end the starved bones of the hoped for deer herd, dead of its own too much, bleach with the bones of the dead sage, or molder under high-lined junipers" (Bratton, 1992). Actual limits to growth occur, but are quite complex.
To introduce this limit to growth, the carrying capacity of the
environment, we need the logistic equation. The model is the Verhulst-Pearl equation (Smith, 1974):
dN/dt = ------------
growth rate =
(average birthrate-average death rate)*number of individuals* (carrying capacity-number of individuals)/carrying capacity
As the species numbers (N) approaches the carrying capacity
(K), the top half of the equation approaches zero and the rate of
growth approaches zero. This is the logistic growth curve. The
population levels off to a plateau, a situation better reflecting the
real situation in nature. The interpretation of this behaviour is
that as population density increases, competition between members of a
species also increases (Smith, 1990). Population growth slows until
growth ceases due to increased mortality and decreased fecundity.
r is also called the Malthusian
Figure 1: Exponential and sigmoidal increase in population numbers (N) with time.
This logistic formula provides a general mechanism for the population growth process, so does not accommodate individual or specific characteristics. It reflects population growth inder idealsitic conditions. Populations respond as systems to positive and negative feedback from the environment. Over-crowding would be negative feedback, while food provides a positive feedback.
The Lotka_Volterra model, developed separately by Lotka and Volterra, modifies the logistic equation (curve) to take into account two interacting species sharing a common resource. They added a constant to the formula, reflecting the effect of one species upon the population growth of another. Putman (1994) observed, "The Lotka-Volterra equations greatly oversimplify the processes of population growth and interaction". It does not account for age structure within the population or that the carrying capacity is not a constant. They do not account for environmental factors, be it climate or terrain, but assume them constant and stable (Smith, 1990). It is useful in investigating the perpetuation of interacting species and organisms through time as it provides a summed or averaged response to these various factors. In statistical terms, it shows a type of probability of some condition developing. The equation allows for intraspecific interactions to be considered. A unit of measure does not necessarily reflect time, as days, but may reflect generations as an organism may have a life cycle of hours as with bacteria or years as in elephants.
One purpose of models is to create a common terminology that scientists can use to compare, describe and understand a diverse range of results. Through models they can discover trends and patterns in apparently unrelated data. A value of such a formal model is that we can better understand some principles of the system than without the model (Axelrod, in Poundstone, 1992). Models cannot display or include all the variables of the real system. Nevertheless, simplification allows one to discover or confirm large-scale patterns (Dennett, 1995). As explained the model is merely another "tool of analysis" and as such has inherent failings and characteristics . An idea such as "community ecology", "depends upon, but is simultaneously constrained by the identification of the appropriate macrodescriptors (models) for rationalisation" (Putman, 1994). For example, in describing trophic (feeding) levels within an ecosystem, the multitude of individual interactions across the trophic levels reduces to a unitary relation between one trophic block and the other. The rationalisation of dynamics derived from such models is subject to compounding "errors of precision or incorrect assumptions" especially as the system becomes more complex (Putman, 1994). Often defining biological systems in quantitative or measurable terms is difficult. This may lead to:
 An inability to apply "statistical significance" to data,
 Difficulty establishing a measurable dimension or component in the field,
 Problems verifying assumptions in the field,
 Problems establishing the dependence or independence of assumed variables (Putman 1994).
There are even cases where "two different distributions, derived from conflicting initial premises, both adequately fit an observed set of data such that even if a hypothesized distribution does fit the observed species-abundance distribution, that fit neither proves nor disproves the postulates of the model" (Putman, 1994). I present these problems, not to provide solutions to them but to reiterate the fact that models, as mere tools of analysis, have inherent limitations. Biological models, to be as real as possible need to incorporate biologically meaningful assumptions. This is why "thought experiments" have limited practical value and may further confuse the issue.
Ecologists have developed the logistic equation further as the Lotka-Volterra model (Begin, 1986) to investigate interspecific interactions. Traditionally, this equation models competition (Smith, 1990), while here interactions of any form between associated species are considered. Putman (1994) recognises this possibility (2) .
[i] Population sizes are denoted as N1, N2, N3, etc., where N1 is the number of a species being considered in the interaction. They relate it to the real numbers within the population.
[ii] K1, K2, K3, etc. represent carrying capacities, and reflect the size of the population supportable by its environment (Souls & Piper, 1990). This capacity represents density-dependent constraints (factors that change with population size) acting as a form of negative feedback, and tending to stabilise the population. Many density-related factors cause this effect, such as shortages of food, nesting sites or space, increased competition or disease, fewer resources available to devote to the production of offspring, self-regulation and so forth. As the carrying capacity often relates to available food resources, it may change from year to year, season to season or with the change from one food source to another. A winter carrying capacity should generally be lower than a summer carrying capacity. Such an effect was found with the coal tit ( Parus ater ) foraging largely on a moth ( Operophtera brumata ) in summer and dormant insects and spiders in winter in Scots pine ( Pinus sylvestris ) and Corsican pine ( P.nigra ). This food stock varied in quantity from year to year and diminished through the winter (Wynne-Edwards, 1986). This bird interacted with other tits and the goldcrest ( Regulus regulus ). Coal tit densities declined and were significantly correlated with the disappearance of the food stock through winter.
[iii] Rates of increase of each population as r1, r2, r3, etc. represent the rate of population increase over time, usually annually to standardise and compare periods as generation periods vary from species to species.
[iv] Later, I introduce separate mortality rates as m1, m2, m3 etc. which represents the death rate over time, so that if necessary, r and m can be separated. Usually if m is not considered, r is the net effect after m has been deducted.
[v] The most important variable to this topic is the competition coefficient , alpha, which I shall call the interactive coefficient ( i-factor ) to accord with the compatibility concept and allow the consideration of all interactions, including competition and cooperation.
The competition coefficient (alpha) represents the strength of interaction ( i-factor ), showing the extent to which the growth rate of species one influences that of associated species two (Putman, 1994). i12 is the interactive effect of species two on species one . INTRASPECIFIC interactive effects represented by the i-factor , or the "cost" of the interactive effect of two individuals of the same species is given a value of one. All other interactions with other species are measured RELATIVE to this value. In other words, we allocate the average interactive effect of two individuals of the same species a "cost" of one. There are many ways of expressing this. Brewer (1994) explains it by saying that "the inhibitory effect of one or more individuals of species one on itself is 1/K1 ." This is a valid assumption as animal behaviour has evolved and is predictable so that it has a measurable average cost for the species concerned. Colinvaux (1973) explains that "if an individual of species one treats an individual of species two as one of its own kind, i21 = K2/K1 = 1."
Behavioural ecologists studying interactions between animals recognise communication as occurring between individuals, where "specially designed signals or displays" influence the behaviour of others. Ecological constraints and the effect upon an interactor influence these signals. These behavioural signals or patterns evolve through natural selection. In this process they become more effective through becoming stereotyped, repetitive and exaggerated, a process called ritualisation. Ritualisation results from the coevolution of interactors (Krebs & Davies, 1987). Ethologists have undertaken comparative studies of displays in closely related species to show evidence for the ritualisation of behaviour. Traditionally, such displays evolve to reduce the possibility of ambiguous signals. To reinforce this, the displays of closely related species are clearly different, reducing the chance of confusion between interactors.
We can measure the i-factor (interactive cost) as the proportion of the animal's daily energy budget devoted to intraspecific interactions. Sometimes, simple behaviour of some form, such as territorial markers, has evolved so the interactive cost, the i-factor , is low. A study of the blue wildebeest, (Connochaetes taurinus ), of the Etosha National Park in Namibia, established that they spend 53% of their time standing, lying or resting in the shade, 33% grazing, 12% moving, 0.5% drinking and 1-15% in social interactions. Territorial bulls economise on interactions through ritual displays. If a bull approaches another's territory, the territory holder adopts a threat display. He holds his head horizontally and his neck erect while he approaches the intruder with a "rocking horse" canter (Skinner & Smithers, 1990). This behaviour is usually sufficient, but there may be some horn sparring and head butting. These encounters are threats rather than actual fighting and seldom lead to injuries to either interactor.
Black wildebeest Connochaetes gnou has a social organisation made up of:
 territorial males
 bachelor herd males
 female herds
Territorial males of the black wildebeest, defend or maintain their territories through ritualisation , which eliminates costly or dangerous interactions. They often display by simply standing with a head-up posture, or stiff-legged cantering. Adjacent bachelor herds occupy marginal habitat with lower quality grazing. As territorial males may have to defend their territory at any time, they spend more time standing than members of bachelor herds or females. Serious fighting between territorial males is rare (due to evolved, ritualised behaviour) and within the bachelor herds there is a remarkable lack of aggression between members. They mark territories with faeces, urine and various glands on the body. They vocalise with a two-syllable call and adopt advertising stances. At their most energetic, they gallop at high speed, with heads up and swishing their tails, sometimes kicking out with their back legs. If they approach another territory, they become submissive, with head low or in a grazing attitude (Skinner & Smithers, 1990).
If, in the interspecific interaction, i12> 1 then individuals of species 2, perhaps a lion, have on average a greater inhibitory effect on individuals of species one, the wildebeest, than do the individuals of species 1 upon each other. Seeing that in wildebeest is easy. This value has to represent an average and probability for the population as male-male interactions may be more aggressive than female-male interactions and each individual interaction may be different. Further, the formula requires that each consideration in determining the i-factor be strictly between the two species involved so that i-factors relate to that interaction and cannot be directly compared. In practise, such an i-factor will be difficult to measure. It must reflect the effect of one species upon another, including the cost of interaction and perhaps the loss of niche space due to the presence of the other species. This is clearly difficult to measure, but it is a real factor in nature, whether or not it is measurable. In the model, the "inhibitory effect" is an energy value reflecting the cost of the interaction compared with the average cost of intraspecific interactions.
As interactions diminish, they reach a point of no interaction. Where two species do not interact, the i-factor becomes zero (Colinvaux, 1973). At this point the two populations do not affect each other, so their carrying capacities are independent of each other. A weed plant in a forest, which can only colonise open areas, has specialised in a dynamic niche, so reducing the interactive effect of other plant species. Its role reflects an adaptation to its habitat, including other species.
[vi] t is time and the notation dN/dt means, the change of numbers
The equation then becomes:
----- = r1N1 ---------------------- [a]
Dealing with two (or more) interacting species, we get:
Multiplication of N2 by i12 (the effect of species two on species
one in [a] above) brings the population N2 into relation with
population N1 as
concerns their relative interactive effects. If the presence of N2
individuals has the same average effect upon N1 as does other N1
individuals, i12 is one.
By definition, each species competition coefficient upon itself is 1, i.e., = 1, a11 = 1. The coefficient a12 is the competition coefficient of N2 on N1, and a22 is the interactive (competition) coefficient of N1 on N2. If a12 equals zero, then the first equation results in logistic growth for species 1, i.e., species 2 has no impact on its population growth. Likewise, if a21 equals zero, then the second equation results in logistic growth for species 2, with no impact of species 1 on its growth (White & Winkelman, 2005).
To study the model for each population under consideration, the equation runs repeatedly (iterated) to decide the outcome of the interaction over long periods. These iterated formulae answer the question of what happens to two interacting populations over time and how different i-factor s or coefficients will affect the outcome of interspecific interactions. If two species interact, one needs to find i12 and i21; for three species, i12, i21, i31, i13, i23, and i32; for four species, i12, i21, i31, i14, i24, i34, i41, i42, i43, i13, i23, and i32, or N*(N-1).
Under such complex situations as in natural systems, the
recognition of universal trends becomes difficult as each time we
change a variable, the outcome changes. The more alike that two species
are, the more likely they are to have similar requirements for space,
food nesting sites and the like. Such interactions between species that
occupy similar niches needs to be investigated to further understand
natural ecosystems. This was the original purpose of my use of this
model: to observe the dynamics of population growth and interactions to
better understand the mechanism that leads to the
perpetuity-compatibility complex in nature. The surprise came when the
model illustrated so clearly how interaction affects two species, but
first we need to deviate for a short while.
Wynne-Edwards (1986), a proponent of group selection, details this
idea in his book, "Evolution through group selection." Finding out how
this idea relates to the perpetuity-compatibility idea is necessary
(pc-concept). We need to investigate this topic, as an interaction of
the two ideas will reveal details of each.
Standard evolutionary teaching promotes that all forms of natural selection acts on individuals, whereas Wynne-Edwards believes that such a mechanism cannot account for all attributes that organisms possess. Wynne-Edwards' problem with traditional natural selection upon organisms is that "some of their attributes are wholly dependent on mutual cooperation for the achievement of beneficial effects, and require that individuals conform to rule in order to promote the common good." The main example he gives is the universal precaution of animals in not overexploiting their food resources - "that co-operators, bequeathing productive resources as well as genes to posterity, have developed an unsurpassed strategy for survival." He admits that this is largely unrecognised by ecologists, in turn identifying the area of controversy within traditional ecology. According to Wynne-Edwards (the group selection camp), proponents of individual selection (the traditional selectionist camp) claim, "advantage depends solely on the number of progeny that an individual contributes to the breeding stock of the next generation, the fittest contributing the most." They believe any good for the group or the species "to arise incidentally from the production of fit individuals." Group selection on the other hand, "is much concerned with the cooperation of individuals in local populations to protect the welfare and survival of their own stock; and that may depend on their having a social organisation, imposing a way of life with which they have to conform. For co-operative systems like this to evolve it seems necessary to invoke a selection process acting between group and group , with the groups persisting as semipermanent units, giving time for the better integrated ones to prosper and supplant those that are less vigorous."
Natural selection in the holistic model acts upon the individual . Compatible (co-operative) behaviour results through natural selection at the level of the individual, because compatible behaviour in its many forms is the fittest possible mechanism in terms of survival. In the interactive process, the evolution of compatible interactive mechanisms (the evolution of reduced interactive effects) between associated organisms leads to the most stable organisation. Maximised reproductive output leads through many generations to overpopulation of the reproducer, destruction of its habitat and the extinction of associated species. Ecological instability results, negatively affecting the reproducer. In this pc-concept, rather than protecting the interests of their own stock in a group selectionist fashion, the animal acts purely in self-interest in the traditional selectionist fashion. However, natural selection has defined this "self-interest" or perpetuity within the animal's interactive realm, which is its niche within the whole ecosystem. An organism's niche requires the environment in which the animal is found for full definition. In this holistic ecological context, compatibility becomes a component of "self-interest" (perpetuity), in deciding the fittest through the process of natural selection. Units evolve naturally within the complex organisation of the system, but with the individual still acting to best maximise the perpetuity of its own kind. That the behaviour selected for is co-operative is a consequence of its superiority over the long term as displayed by the Lotka-Volterra model. Adaptation, when between organisms that are both evolving, leads to what appears like group selection, but is coadaptation (in its many guises).
In any ecosystem, the members of a single species "compete" (interact) and the MELV model can illustrate the same mechanism. Reduced interactive costs (reduced "competition"), improves survival potential (in a teleonomic sense), through increased ecological stability. Within a species, selection for reduced interactive costs leads to the evolution of social systems. The ultimate objects of competition often become ritualised or replaced by artificial substitutes through this process. Social behaviour is therefore explainable as a consequence of the same mechanism. The conclusion drawn from the MELV model is that behaviour that reduces the interactive effect or "cost" between individual organisms of the same or different species, is selected for through natural selection, as such behaviour leads to individual fitness and ecological stability.
Ritualised and instinctive social behaviour is even found in humans and can be traced to behaviour in our most ancient ancestors. We practise visual communication without much thought. Of 12 facial expressions found in chimpanzees at least 11 are found in humans and eight in the more primitive lemurs. The smile ("silent bared-teeth face") is found in bushbabies, lemurs, cebus and spider monkeys, baboons, Guenons, chimpanzees and humans.
Insectivores show the most primitive form of the smile, drawing back the corners of their mouths and shaking their heads in response to unpleasant food or a strong smell. This behaviour, a primitive rejection or protective, nonsocial response evolved into a ritualised fear grin in monkeys and apes. A dominant conspecific elicits this behaviour in the subordinate. The stump tailed macaque expresses this fear grin and a related submissive greeting, both of which a nervous human pupil may display in front of his teacher. In humans and chimpanzees this has evolved further from a signal of fear and submission to appeasing greetings (Jolly, 1972).
Wynne-Edwards (1986) recognised the social qualities of cooperation
at the group level. Unable to postulate a mechanism for their evolution
through natural selection at the individual level, he reasons that
natural selection must be acting upon groups to achieve the observed
social qualities: "adaptations for the common good could not evolve
through individual selection, pure and simple, but only through
selection acting between groups." This was his error!
An evolutionary process recognised is that animals specialise, often by becoming specialist feeders due to interactions between competing species. One mechanism to reduce the interactive effect between competing species is to SPECIALISE, through improved means of finding, handling and digesting a specific food. Increased nutritional efficiency will be a competitive advantage over some generalist while also segregating consumer species. This reduces competition by reducing the number that shares the same resource (Wynne Edwards, 1986).
Specialisation implies improved efficiency, but it starts with competitive exclusion. Two species of flatworm, for example distribute across a wider range of temperature extremes in a river when present alone (allopatry) than when together (sympatry). If together in the same stream, each species excludes the other from its preferred temperature zone. Planaria montenegrina dominates lower temperature zones and P. gonocephala occupies warmer temperature zones within the same river system (Putman, 1994). This enhances adaptation to their "preferred" zones. Natural selection leads to a decreasing i-factor through the DIVERSIFICATION OF SPECIES. Divergence through evolution has led to the present diversity of nature (Beck, et al, 1991). Natural selection decreases the intensity of interaction between two associated species through their adaptation and diversification, as this is an economic advantage. Specialization and diversification are an adaptive advantage, improving survival potential.
The mechanism of such specialisation and diversification will be similar to that which adapts some invertebrates to the annual change and logistics of its food resource, so that they have evolved a period of growth and a period of dormancy . Similar too is the stimulus to migrate found in some birds. Both are an evolved behavioural response to a predictable phenomenon of their environment. To see dormancy or migration as an adaptive response to food supply, which is in turn coupled to the climate is easy to visualise. To see specialisation and diversification as reducing the interactive cost is just as logical but a little more abstract. The one is adaptation to abiotic environmental factors, the other, adaptation to biotic factors.
Examples of the results of evolution through natural selection provided in this book illustrate various perspectives of this evolutionary process. Ant insect castes evolve without the mother being subject to the selective pressures of each caste, garter snakes evolve behavioural differences within a species, newborn children in a hospital are still subject to the forces of natural selection, the American Mojave squirrel competes by not competing, the dung fly evolves an optimal behavioural strategy through the selection of decreased competition etc.
We can show that reducing the interactive cost enhances ecosystem stability through the iterated MELV model (see below). Animals within their natural environment often display some mechanism of reducing what they traditionally call competition. In the 1930's a Russian, G.F. Gause tested the Lotka-Volterra model in the laboratory. Placing protozoas in an artificial environment - a test tube - he found that often, one species drove the other to extinction. Developed from this observation, the Gause principle or Principle of competitive exclusion states, "Stable populations of two or more species cannot continuously occupy the same niche" (Colinvaux, 1973). This principle became a cornerstone of ecological theory and so competition came to dominate the development of this science.
In the standard analysis of the Lotka-Volterra Model, ecologists recognise the possibility of coexistence through decreased interactive effects between species (Smith, 1990). When intraspecific competition inhibits population growth more than interspecific interactions do, coexistence is possible. Smith (1990) notes that coexistence occurs when "Each species inhibits its own growth through density-dependent mechanisms more than it inhibits the growth of the other species." Colinvaux (1973) notes, "For weakly competing populations , then, the Lotka-Volterra equations predict that both should persist indefinitely, their populations fluctuating only gently about equilibrium levels." Unfortunately, Smith and other ecologists did not realise the evolutionary implications of this "WINDOW OF OPPORTUNITY" that opens between two interactors. The principle of competitive exclusion dominated ecological science.
Smith explains the phenomenon of compatibility from the purely competitive perspective: "Neither species reaches a high enough density to cause any serious competition between them and the population growth of each is not strongly controlled by the same limiting condition. Since a different resource limits each species and both are only weakly competitive , then the two species will continue to coexist." He does not consider the fact that coevolution leads to weak competitive interactions. With the MELV model, I have shown how such coexistence has evolved. However, I have interpreted and illustrate the relationship where, with interacting species, intraspecific competition inhibits the growth of a population more than interspecific interactions as an association of interdependence . I have recognised the evolution of interdependence and coadaptation. In doing so, I have followed the advice of Edward O. Wilson (1992), "What we understand best about evolution is mostly genetic, and what we understand least is mostly ecological. I will go further and suggest that the major remaining questions of evolutionary biology are ecological rather than genetic in content. They have to do with selection pressures from the environment as revealed by the histories of particular lineages, not with genetic mechanisms of the most general nature." . . . "I think the greatest advances in evolutionary biology will be made in ecology, explaining more fully in time why the diversity of life is of such and such a nature and not some other." (Wilson, 1992).
What Gause found, occurs in the confines of a test tube, but not necessarily in nature. Through the process of natural selection, a decrease in the intensity of interactions evolves. We see the effect as specialisation and diversification. Gause took two species belonging to two different niches and placed them together in an artificial environment and the inevitable occurred - the competitive extinction of one. Nature finds many ways of reducing the interactive effect between two species, achieved through evolution by natural selection. In the tropical forests of Costa Rica, two species of stingless bees coexist on the same resource, the pollen of a single species of flowering shrub (Brewer, 1994). Their coexistence results from an economic solution. One species has specialised in clumped resources that it defends against the other. Scouts locate these flowering clumps each day. The other species has specialised in scattered, low-density small clumps and individual flowers. These required less energy to find than the large clumps, but also gave less return or benefit. Large clumps of flowers required more energy to locate and defend, but gave a higher energy return. Both species of stingless bee flourish in the same environment and upon the same food resource as they had become specialists and evolved a mechanism of reducing the intensity of interaction between them. They had used one aspect of their complex niches as a basis for reducing interactive costs. Had Gause placed these two species inside a bottle with some flowers, he would have found the one drive the other to extinction and so devised his principle of competitive exclusion! In the ecosystem, however, what has coevolved is the reduction in the cost of the interaction through diversification - two compatible species. Many examples of this are to be found in scientific texts. Four different antbird species ( Myrmotherula sp .) separate their niches by foraging at different heights (Putman, 1994) - again reducing interactive costs.
So, after Gause, competition dominated the perception of ecologists dealing with the Lotka-Volterra model and ecosystems. They modified the Lotka-Volterra to explain other forms of interaction such as mutualism. Biological mutualism is a beneficial, reciprocal relationship between two species. The interaction favours survival, growth or fitness of both species (Smith, 1990). To model this, "The general approach has been a modification of the terms of the Lotka-Volterra equations for competition in which the negative alphas of competition become positive." They term this positive coefficient the coefficient of mutualism. By the time Putman published his book, "Community Ecology" in 1994, they had established this positiveness, negativeness or neutralness of biotic relationships: "Indeed all the various relationships through which the various members of a community might interact, may be represented as ++ (mutually beneficial); +0 (beneficial to one participant, not affecting the other); and +- (beneficial to one, damaging to the other)." To this he adds two more in a table, -- (for competition) and 0- (for incidental damage) (3) .
All of this makes sense and is quite reasonable until one asks if this is reflecting something from nature or merely another "thought experiment". Colinvaux (1973) showed how an i-factor of zero implies no interaction, while a value of one is the average intraspecific value of the i-factor. If you ask what this intraspecific value of one represents, the conclusion must be that it represents the average cost of intraspecific interactions, represented as a statistical probability of occurrence. The intensity of the intraspecific interactions, measured in energy terms is allocated a value of 1. They compare all other interactions with this.
The small addition that I make to this model, which has profound
evolutionary implications, is that between 0 (no interaction), and 1
(interactions equal to intraspecific interactions), exists the realm of
coadaptation, if both species have an i-factor of less than
1 for the other species. This is what I model and is the source of
compatibility. It is the window of opportunity that leads to holism.
Let us further explore "nature experiments" to better understand this
Already the Lotka-Volterra formula has become a bit complex, but it has a certain beauty. It includes the primary factors involved in the growth and interaction of species within an ecosystem. The complexity of natural interactions will become evident from slight changes to the variables of the model.
A hypothetical field mouse and a rabbit may have interactive effects upon each another. As the mouse is small compared with the rabbit, it may take many mice to have the same interactive effect as one rabbit upon another rabbit. As for the carrying capacity, the habitat would probably support far more field mice than rabbits, so their K-values will differ, unless K is expressed as a biomass instead of numerically. However, a biomass relation of K will be inaccurate, as the energetics of mouse biomass will not relate directly to that of rabbit biomass. As such K, the CARRYING CAPACITY, should be a little more abstract and represent actual BIOMASS MULTIPLIED BY THE METABOLIC RATE of the species represented by that biomass. Both are measurable and quantifiable.
Further, mouse and rabbit diets are likely to differ to some extent, so the K-values for the rabbit and mouse are different as defined by the terms K1 and K2. The habitat can support a higher biomass of field mice and rabbits than either field mice or rabbits alone. This niche differentiation effect is not found in the formula, but is a real factor in nature. Therefore,  the formula restricts one strictly to a consideration of INTERACTIVE EFFECTS or the interactive regime between the two species. The intersection of two circles can represent a shared diet, where each circle represents the total dietary complement of the species. The red area of intersection represents the interactive regime in the below figure. Only resources shared by the two species are involved, whereas in nature the system is a complex whole. INDIVIDUAL CARRYING CAPACITIES FUNCTION WITHIN THE CONTEXT OF THE WHOLE ECOSYSTEM, but the model reflects only the area of interaction .
Species B may be forced out of the red area of intersection by
Species A. It is forced by Species A to feed upon the resources
represented by the green area. Natural selection acts upon the species
tied into this association, so that Species A forces Species B to
specialise upon the food resources in the green area. Coevolution
between closely associated species is in this way inevitable.
In the standard model ecologists define K1 as "the effects of both intraspecific and interspecific competition on resources used by species 1" (Putman, 1994). I restrict this even further to the area of niche-overlap as this is strictly the area of interaction to which the i-factor s apply.
Niche separation may be to such an extent that the two species depend upon completely different food resources, nesting sites etc. There would then be no interaction, the i-factor being 0 and the two K-values would be mutually exclusive values or independent variables. If the i-factor is above 0 the two species interact and share some resource and their TOTAL K-values must intersect to an extent. This interaction affects the carrying capacity of interacting species. As the model considers the interaction of the species, those areas outside the niche-overlap do not form part of the k-values of the model. Carrying capacities, in the model, must therefore be a constant. The area of interaction is a single "area" or resource and therefore must be the same for each species involved.
An important point to note is that when allocating i12 for the effect of the rabbit (2) on the mouse (1), one also has to consider i21 or the effect of the mouse on the rabbit.
THERE IS A TWO-WAY EFFECT IN INTERACTIONS, with both interactors affected in some way by the other, but the effects working each way are different. Computers allow one to run iterations of each side of the interaction simultaneously, to see the effects of the interaction. Relations are definable in terms of their interactive values ( i-factor s). In the field mouse interaction with the rabbit, the hypothetical rabbit may consume the feed equivalent to that consumed by five field mice. Its mere presence may exclude five field mice. The i-factor effect of the rabbit upon the mouse would then be five (ignoring other possible interactive effects), while the corresponding field mouse effect upon the rabbit would be 0.2. In other words, where the two species' feeding component of their niches intersects, the presence of a single rabbit has the same interactive cost as five field mice upon the mouse.
From a reductionist perspective, where they reduce the whole to smaller parts and they study the smaller parts, one may note and object that 0.2 implies a compatible relationship. This is not so, due to the mutual interaction. Interactions are all compared with intraspecific interactions and the interaction is a two-way affair. To be a compatible relationship the i12, for the effect of the rabbit (2) on the mouse (1), needs to also be less than one. Pairs of i-factors are needed. This is because the i-factor is an energy value converted into a relative factor . The i-factor is the interactive effect of one species upon another, compared with the effect the affected species has upon itself. If the rabbit and mouse diets were so different that the rabbit consumed so little of the mouse's natural food, which it had an interactive impact of less than one mouse, then we would have (by definition) a compatible relationship (while the rest of their niches do not interact and add to the i-factor). Note that there is a cumulative density effect here, for each additional rabbit interacting with the mouse must contribute to the overall effect of the rabbit on the mouse.
In a predator prey relationship the predator may have a very heavy effect upon its prey compared with prey interactions upon each other (conspecific) and we need to quantify this effect in some way. Biotic relationships under the conventional scheme of the Lotka-Volterra model are represented by a competition coefficient that is positive (+) if the interaction effects an increase to population growth and negative (-) if it effects a decrease to population growth or 0 if no effect is felt (Putman, 1994). As such, this method allocates ++ as the effect of the bee-flower interaction or mutualistic relationships, with both competition coefficients being positive and a predatory or parasitic relationship as +- (Putman, 1994). The conventional model uses the interactive coefficient as a measure of the effect of one species upon POPULATION GROWTH of another, which is difficult to quantify. My adaptation of this model defines the i-factor as a relative energetic cost. How does one measure and quantify a positive or negative relationship? In nature one can measure energy, but the interpretation of a positive relationship is very subjective. Energy is measurable in nature, but positiveness is not. By completely removing one interactor of an association, we may measure the effect, but this is not always practical in the field.
Predator-prey relationships have unusual character. A predator's numbers depend to an extent upon the prey population. Prey numbers affect the predator birth rate (Brewer, 1994). It will also affect the predator carrying capacity. Prey numbers also depend upon the predator population, as more predators increase the mortality rate (Brewer, 1994). Again, the prey carrying capacity may be lowered in the presence of many predators. These two factors need to be added to the standard Lotka-Volterra model.
Because of the interaction, the predator has on average a net gain in energy, while the prey on average, has to expend energy in escaping from its predator. A predator's energy gain from the interaction has to be considered, or else the energy expended in this interspecific interaction will be greater than the cost of intraspecific interactions and the association will appear detrimental to the predator. Only those prey that escape survive to perpetuate, but their total escape activities are what makes up their i-factor, which may be greater than one (the intraspecific interactive cost). Perhaps, they expend on average, three times as much energy avoiding predators as they do in interactions with their own species. The i-factor for the predator upon the prey will then be three. For the predator, there is on average a net gain from the interaction, or else the creature would fail as a predator. As such, although they may expend much energy in hunting for prey, the value of the i-factor of the prey upon the predator is less than one. For the predator, the interaction, resulting in a net gain of energy, does not fall between zero and one and is not neutral (0). Effects of the prey upon the predator must be a negative value. (In this model energy costs in an interaction are positive, so energy gains through an interaction must be negative.) The i-factor may be -0.1, transforming the negative sign of the conventional Lotka-Volterra model to a positive value.
If the predator is dependent upon this prey species, its r-factor should in some way be linked to the relation between prey and predator numbers. The standard formula does not provide for this situation, but the i-factor for the effect of the prey upon the predator cannot be a constant. It must change as the ratio of predator numbers to prey numbers changes. As the prey becomes scarce; its energetic return to the predator becomes less, eventually zero and perhaps finally positive. If the predator can turn to other prey species, it will give up the effort of seeking scarce prey. If it is dependent upon the prey, it will spend ever more energy in search of its required prey. With plentiful prey, it may specialise and concentrate upon this food through a change in behaviour. Such a change in behaviour has been found mostly in vertebrates and termed a functional response. Another form of response, common in insects is a rapid increase in predator numbers, termed a numerical response.
As the prey numbers available determine the predator carrying capacity, the prey numbers determine both the r-factor and the k-factor. Such a relationship is dynamic.
In summary, we see that the holistic (biotic (living) and abiotic (nonliving)) environments affect the r-factor, the i-factor and the k-factor of the predator. This principle applies to most animals. Many species may avoid such limits by feeding on more than one species, so if one declines, they turn to another. We encounter two major types of association. First, populations feed upon and are therefore getting negative feedback from prey/food populations that directly determine their numbers. Changes in the prey population affect the r-factor of the predator directly. The same may apply to grazers and grazing, bees and flowers etc. Usually, an animal/organism will have evolved behavioural mechanisms that involve the i-factor, to prevent heavy negative feedback affecting their r-factor.
This i-factor relationship is the second major form of association. It promotes environmental stability. Lions do it through territoriality, their mating system, and their social structure. Their i-factor determines their r-factor and their k-factor, and their i-factor has evolved as an adaptation to their environment.
Now comes the difficulty of modelling this. An animal's r-factor can adjust by direct negative feedback. How does the lion "determine" its i-factor through evolution? Territory sizes vary in relation to the availability of prey. With less prey, lions defend larger territories. The lion defines its intraspecific relationship through its behaviour. In our MELV model, we allocate this average intraspecific relationship (i-factor) a cost of one. Naturally, the lion has to sustain itself, so when food is scarce, it ranges further in search of food, extending its territorial aggression and thus its territory. They require more energy for maintenance and less is available for reproduction, so the r-factor declines as the i-factor increases. An interactor has to be able to get enough food to maintain this intraspecific interaction and its other activities, or else it loses the race, so there are constraints, optimal and suboptimal habitats. The result is that intraspecific behaviour decides the reproductive output of the species. The i-factor determines the r-factor . Evolution through natural selection has defined intraspecific behaviour that is an adaptation to the environment and determines the reproductive output (r-factor) of the species. As natural selection selects for the fittest, and those that perpetuate are the fittest, what is found is fitness defined by the i-factor. As a territorial species maintains a certain area for its maintenance, the i-factor often determines the k-factor.
With the i-factor, the question being asked is whether intraspecific interactions, allocated a value of one are more or less beneficial, or costs more or less energy, than the interspecific interaction under consideration. With this scheme a bee-bee interaction has a value of one. A bee-flower interaction is beneficial to each and so has an i-factor of less than one for the bee and the flower. For the bee, a nectivore, a bee-flower interaction has a greater energetic return than a bee-bee interaction. An i-factor of zero in this scheme implies that there is no interaction or no benefit. For the i-factor to be zero, niches cannot in any way interact or affect each other. In complex systems this may be difficult to figure out. An i-factor of less than one suggests that the flower is "more valuable" to the bee than another bee. Still, for the flower, there is no obvious energy return from the activities of the bee. Activities of the bee enhance the plant's perpetuity . This cannot be objectively quantified in energy terms. It is the other part of the holistic "perpetuity-compatibility complex". A bee gains energy, as pollen and nectar, and the flower has an enhanced perpetuity. This return of energy from the flower means that the i-factor of the pollinator must be negative in this model, so providing a positive return of energy to the pollinator from the interaction.
The interdependence between the bee and the flower has advanced to the stage where the flower is designed to attract the bee or other pollinators (called nectivores). Individual flowers do not interact directly, but do so in attracting bees. There is no obvious energy required for flower-flower interactions. A plant expends energy in creating the flower and providing the pollen and nectar for the pollinator or nectivore. As other plant's flowers are also attracting bees, there may be some form of coevolution between flowers. Some plants reduce the potential competitive effect of flower coevolution by having a single pollinator species. There are tropical flowers that only open at night and attract a species of bat as their pollinator. Others can only be "serviced" by a single bird species. Coevolution between this flower and the bird has left the bird with an enormously long beak to reach into the depths of its specific flowers. Darwin saw the Madagascar star orchid ( Angraecum sesquipedale ) with 30-centimetre long tubes containing nectar. He predicted that a specialised pollinator would be found and 40 years later a hawkmoth pollinator was found (Begon, et al, 1986)
Through specialisation, birds and flowers avoid competition and become interdependent. Flowering of a single species is usually seasonal, so reducing competition with other flowers that are abundant at other times. We see this in flowers of the Heliconia species that flower in sequential, non-overlapping peaks. This group has some species adapted to shady and some to sunny habitats and others suiting behavioural differences between hummingbird species. In natural environments one finds flowers with short seasonal flowering periods and nectivores following a progression of flowering plants through the season (Smith, 1990).
In the flower's association with pollinator's, the flowers adapt to achieve cross-fertilisation. Different pollinators, such as beetles, carrion flies, bees, butterflies, moths, birds and bats lead to differing flower shapes, colours, scents and the provision of nectar or other products. The carrion flower smells of decaying flesh, while orchid flowers of the genus Ophrys look and smell like the females of bees and wasps! In these orchids, flowering coincides with the emergence of male bees or wasps. Adaptation is so intricate that it is miraculous. Bat and moth flowers that open at night are often white, while for insects and other pollinators with colour vision that operate in the day, flowers are often red or another bright colour (Brewer, 1988). This expenditure only makes sense holistically as the r-factor takes predominance over the i-factor here.
How to incorporate this pollinator-flower association into an energy-based model is a bit of a problem. We need to explore a " nature experiment " to see where it leads us. Insects or other animals do not pollinate all plants. Wind pollination is common and possibly preceded the evolution of living pollinators. Conifers and most grasses are examples of wind-pollinated plants with small greenish flowers with no nectar or scent (Brewer, 1988). Pollen is full of carbohydrates and so potentially nutritious. It would not take long for an insect to find and exploit this food resource. An insect that can survive by eating pollen initiates a coevolutionary process between itself and the pollen producing plant. This becomes a cost to the plant, which requires pollen for fertilisation, but very easily becomes a benefit when the insect visits and accidentally fertilises another plant.
As a measure of i-factors, the wind-pollinated plant has a value of one. The insect may initially be a costly pest and so have an impact upon the plant of greater than one. Once the insect achieves accidental cross-fertilisation, coevolution of the flower's reproductive mechanism and the insects feeding behaviour is possible. Pollination by the insect may require less pollen, especially if the pollen can become stickier or in some way be more easily transferred to another plant. Coevolution leads to a more efficient pollination method that is more effective and requires less pollen and less energy than wind borne methods. After some time (far greater than our short life times), a measure of the energy expended by the coevolved plant would show a decrease in the energy expended in reproduction.
As in the horse-human example, benefit is estimated compared with the initial condition before the association. The insect, as the vector for the plant's improved reproductive efficiency, becomes tied into a coevolutionary association of interdependence. Although initially costing more than the i-factor of one for the plant, it becomes a benefit to the plant and so has an i-factor of less than one compared with the condition of wind pollination. An ecologist measuring energy expenditure may find that the plant expends more energy in interacting with the insect pollinator than with its own species. By definition, the average energy expenditure within a species has a value of one. In this measure, the apparently beneficial interaction with the insect should, by definition, be less than one, but the ecologist measures an energy expenditure of greater than one. He is unable to go back in time and measure the sequential decrease in the i-factor. Although the ecologist can measure the various complexities of ecosystems, he cannot measure changes in efficiency along a very real dimension - time. The expenditure also involves another holistic parameter - perpetuity.
What we are encountering here, is a holistic interaction between the r-factor and the i-factor. This is a perpetuity-compatibility or yin-yang interaction. Holism requires a comprehension of the change across time.
If we return to the original consideration of two competitors, we must relate the interactive effect in some way to the energy expenditure required to compete with another species compared with competition with one's own species, but it is still simply an interaction. Interaction with one's own species, we allocate a value of one, but this represents the average energy expended through such interactions under normal conditions. This is a one to one interaction. WITHIN THE TOTAL SYSTEM THERE ARE MANY SUCH INTERACTIONS, THE NUMBER RELATED TO THE POPULATION DENSITY. This may give a clue as to the mechanism of self-regulation of the population density. An interaction is an energetic cost while the habitat provides an energetic return. They will exclude weaker interactors when the effort to maintain intraspecific interactions uses more energy than the animal gains from its habitat. The adjustment can thus be a natural mechanism dictated by inherited behaviour determining interaction rates and intensities, the average energy expended to maintain daily activities and the return in energy as food. Without the intraspecific interactions the animal may have survived and had time to gather more food. Interactions may therefore lead to animals maintaining a population density below the carrying capacity of the environment. This is a very significant principle resulting from the model.
However the formula deals only with a simple one on one interaction and represents the average energy expenditure under normal conditions. There is no energy return for this expenditure (unless they exchange some food). THE I-FACTOR THUS REPRESENTS THE AVERAGE INTERSPECIFIC INTERACTIVE ENERGY COST compared with A VALUE OF 1 ALLOCATED TO THE AVERAGE INTRASPECIFIC INTERACTIVE ENERGY COST. As an ecosystem evolves, natural selection favours lower i-factors for each species. This results through the adaptation of associated (coevolving) species. Lower i-factors mean greater energetic efficiency for the whole ecosystem and therefore greater stability. A comparison of ancient i-factors, such as associations of lemurs from Madagascar, or marsupials from Australia, with animals in Africa or America, may show that the more highly evolved creatures have lower i-factors. In history, when the life forms that have evolved separately on two continents or islands suddenly interact, the competitive exclusion that occurs may be due to the lower i-factor of the more successful species.
Scientists have achieved a decrease in the i-factor of an organism experimentally. Viruses and bacteria evolve very rapidly, so allow one to test some evolutionary theories. A classic experiment carried out in the mid-1960's by the biochemist, Sol Spiegelman provides an important part of scientific practise, which is the experimental verification of the i-factor hypothesis. He took a primitive QB virus and supplied it with a replicase enzyme and free nucleotides that it needs for replication and survival. This was done in a test tube with a flow through system continually adding the nutrients (Casti, 1989).
With the provision of these materials, the virus was freed from dependence upon a cell to continue its life cycle. Starting with about 4500 nucleotides, the virus competed against itself, natural selection promoting a reduction in the i-factor. In this case the virus achieved a reduced i-factor, improving the efficiency of replication through the evolution and survival of viruses that did not produce the supplied (now unnecessary) materials. From a teleonomic perspective, it had inherited a protective protein coat and produced the replicase enzyme, as both were necessary for survival in the host cell. Mutant strains evolved, so that within seventy generations, rapidly reproducing strands with about 220 nucleotides had replaced the other variants. They had neither the protective coat, nor the ability to produce the replicase enzyme. (Here is a possible cure for AIDS.) Adaptation to the new environment had followed the course, as predicted by the compatibility theory, of a decrease in the i-factor, leading to greater efficiency.
One should immediately argue that this is not a decrease in the i-factor, as the remaining interactors are no more compatible than before. Still, they are. The outcome of the evolutionary process was a massively increased r-factor, with a cheaper cost of reproduction. By definition, a reduced i-factor leads to an improved efficiency of interaction. The virus, as it evolves a simpler and more specialised form, is reducing the cost of interactions within its environment. As it discards unnecessary processes, interaction becomes cheaper. Through adaptation, it achieves greater efficiency and a decreased i-factor.
This experiment created the Spiegelman Monster, a specialised creature adapted to its whole environment, including the biotic and abiotic component. If this were an AIDS virus that had shed its protective coat and virulence, it would not be able to infect humans. So, its inoculation would not lead to it out competing its virulent relative. However, if one treated an AIDS patient as the system (holistically) and provided some essential metabolic component that the virus normally produced, it would probably evolve along the lines of a decreased i-factor . Those viruses that no longer produce the metabolic products would be more efficient and out compete their robust relatives. In such a disease that progresses over so many years this is a feasible treatment. Once laboratory tests have verified that only the mutant form is present in the body, doctors suddenly remove the treatment. Without this, now essential metabolite, the AIDS virus perishes. This is a principle for holistic medicine. It may again evolve the lost mechanism, but with "starvation" and the body's natural defences, it does not have the time to mutate and do so. Losing nonessential components is far easier than evolve new structures through mutation.
This principle of the i-factor has broad explanatory power. It explains why so many cave dwelling creatures are albinos. There is no way that an albino creature (frog, shrimp, millipede) living in complete darkness can have a selective advantage over a black relative based on colour. However, producing the black colour costs energy, so the albino is more efficient and has a lower i-factor than its black relative. Here it is not efficiency in interaction, but holistic efficiency, for in the same environment it is more efficient. A similar explanation goes for the loss of eyes by many cave dwellers. As described in detail in the next chapter, the decrease in the i-factor through natural selection explains the persistence of a species once evolved.
Returning to the effect of the prey upon the predator, the predator expends energy in catching the prey as a part of the ENERGY BUDGET of the predator's niche, but there is an energy return from the prey. A leopard catching its prey may expend a lot more energy in this activity than intraspecific interactions, but there is a payoff in the energy value of the prey as food. As such, the energy value of the prey has to be included in the calculation of the predator's (e.g.. cheetah's) i-factor. The idea of an energy budget for the total niche of the creature is a sound idea, as the animal will have a minimum maintenance level merely to remain alive (like a car idling) and an average operational level representing energy consumption during normal activity. Most activities do not have any energy return, and not all activities are interactive. In comparing the influence (i-factor) of a prey item to an intraspecific i-factor, it seems that we have to include the energy of the prey in the calculation of this i-factor. The result is that the i-factor becomes negative (energy expended in interaction minus energy gained). As the predator and prey exist on different trophic levels, not all of the prey's energy is utilised (roughly about 10%).
A broad distinction of activities would be interactive and non-interactive. Preening would usually be a non-interactive activity. In the mandarin duck ( Aix galericulata ), some of its wing feathers form a bright orange sail that is erected during courtship preening. This bird has ritualised preening behaviour, turning its head quickly so that the bill points to the orange sail (Krebs & Davies, 1987). Preening here has become interactive.
This problem with the general model as concerns predator-prey interactions, is that the predator carrying capacity should depend on prey numbers but the formula errs and cannot deal with a low K for the predator and large prey numbers. A solution is that K must not be numeric but represent an energy value. There is an indication here that NOT ONLY METABOLIC RATES AND BIOMASS needs TO BE CONSIDERED, IN DETERMINING THE CARRYING CAPACITY, BUT THE ENERGY RELATION BETWEEN THE TWO TROPHIC LEVELS ACROSS WHICH THE INTERACTION TAKES PLACE, NEEDS TO BE CONSIDERED. We need to multiply the biomass of a higher trophic level by some factor to represent the energy lost in the conversion of ENERGY from one trophic level to the other.
By changing the variables of the model, we reflect the real situation as best as possible. This type of theoretical development is a necessary consequence of using the model, but is not necessary for the completion of this book.
Field mice, being smaller than rabbits, should have a higher sustainable carrying capacity (say 50000 and 10000 respectively) within the same habitat and perhaps a higher reproductive output (say 10 and 8). We can equate mortalities at one. Under such conditions, the field mouse out competes the rabbit and the rabbit is driven to extinction. However, this "extinction" only takes place in the area defined by the mathematical formula, and that is the area of niche-overlap, here shared food sources. The mouse will eventually dominate the shared food resource and the rabbit must subsist upon its alternate food resources. This is a demonstration of the competitive exclusion principle (Stable populations of two or more species cannot continuously occupy the same niche. (Colinvaux, 1973)). Eventually, the mouse excludes the rabbit from food resources. It has to specialise upon a narrower range of food, or find new food resources. Forces of natural selection will then shift because of the interactions between species that share niche space . Coadaptation and divergence of form then occurs, as each interactor will be adapting to the other's presence through the process of natural selection. Holism emerges from the complex system because of interactions and natural selection.
As an example, the bearded vulture or Lammergeier ( Gypaetus barbatus ) of mountainous areas of Africa is the last animal to scavenge a carcass. It has adapted to eat bones, coming in to feed after all the other scavengers have taken their share. It has learned to drop bones from a height onto rocks to break the bones and get access to the marrow inside (Perrins & Harrison, 1979). Interactions with other scavengers and natural selection have pushed this creature into this specialised niche. Had other scavenging interactors been absent, it could have fed on the whole carcass. With the changes to the biota of Africa over the last 200 years, this large bird (2.7m wingspan) has been driven to the edge of extinction. A change in the structure of the ecosystem has led to a change in its food supply.
Brewer (1988) recognises the principle described above, but the tools of his analysis, his terminology, reduces the significance of his observation: "For every species involved in an interaction that is harmful to it, there will be an evolutionary prize to any individual that can do either of two things: (1) disaffiliate - escape the interaction - or (2) capitalize on the interaction - put it to its own benefit. Genetic factors allowing either outcome should spread rapidly in a population." Both are simply ways of reducing the cost of the interaction and broad roads to greater efficiency. He goes on to conclude: " . . .evolution of host and prey tends to break up parasitic or predatory interactions (disaffiliation) or to pull a profit from them (capitalisation). Coevolution on the part of the predator or parasite tends to oppose disaffiliation but usually to favour capitalisation. Thus, the overall trend is for the development of mutualistic coactions."
Robert Smith tabulates various standard population interactions in his book, Ecology and Field Biology (1990). He lists neutral (0,0), mutualism (+,+), commensalism (+,0), amensalism (-,0), parasitism (+-), predation (+-) and competition (--). I illustrate these below. You can run simulations online as well.
Random changes that the computer has generated are interesting, but do not illustrate the real situation. The random variables included do not properly model nature, where changes are seasonal and extremes of the random variable follow the normal or Gaussian distribution of statistics. This is the standard model for plotting variation and is a bell-shaped curve. Most data clusters around an average forming the hump of the bell shape (Gleick, 1987). Such a distribution may be skewed or symmetric. Temperature for example, will be seasonal, seldom reaches the very extremes and is often skewed in one direction or the other. All these factors can influence the interaction between organisms creating further complexity to the real situation in nature.
Under complex situations as found in natural systems, the recognition of universal trends becomes difficult. As each variable changes, the outcome changes. This is evident in GRAPH 2.12 below where only one variable is randomly generated within narrow constraints. The more alike that two species are, the more likely they are to have similar requirements for space, food nesting sites and the like and so drive each other to extinction locally as in GRAPH 2.3. A solution to this, in order to survive, is to disperse, differentiate and utilise different food resources. Success in this depends upon space and time. If the organism can occupy a different space for enough time it can evolve enough differences so that it no longer competes.
It is important to investigate interactions between species that occupy similar niches to understand natural ecosystems. This was my original purpose for using this model, to observe the dynamics of population growth and species interactions to better understand the mechanism that leads to the observed perpetuity-compatibility complex (holism) in nature. The surprise came when the model illustrated so clearly how interactions affect two species. We observe compatibility in nature and from the model we see interactions and natural selection have caused this.
Mutualism, as defined in standard texts seems incomplete. By definition, both species benefits from the interaction, so they denote it as ++. What is happening when both species benefits? The presence of and interaction with the other species results in a benefit for both interactors. This has to be defined in terms of the parameters of the Lotka-Volterra model. The i-factor quantifies interactive effects. A value of one implies an interactive cost equal to that of intraspecific interactions. A value of above one for both species implies that the interaction is costly. If each species beneficially changes some parameter of the other interactor, such as mortality or reproductive rates, this is still translated into lower i-factor s. If there is no interaction, it is neutral in that they derive no benefit. The area of interaction where mutualism can occur is where both species i-factors lie between 0 and 1, the compatible association.
Smith (1990) defines mutualism as "a positive, reciprocal relationship at the individual or population level between two different species." He explains that because of the association at the individual level, both species improve their survival, growth, or fitness. He provides an expression of the Lotka-Volterra equation to express mutualism, where alpha multiplied by N2 is added to the K-value of N1. We redefine the carrying capacity as the equilibrium density of the one species in the absence of the other species. As he notes, this leads to runaway positive feedback where the numbers of both populations in the association increase unrealistically. Even with i-factors below one there is a runaway population explosion. This modification of the Lotka-Volterra model is therefore an unrealistic representation of mutualism or compatibility. The model that I provide above, with the i-factors for both species lying between zero and one better represents the real situation.
Examples of mutualism are numerous. As they have energetically based interactions, there needs to be some type of reciprocal exchange for both species to benefit. Standard textbooks include:
 Fungal-algal associations that we call lichens: algae gain protection and moisture and the fungi gain metabolites;
 Plant roots with the mycelia of fungi, plants gain:
(i) An extended root system and better access to soil nutrients such as phosphorus,
(ii) More nutrients through a more rapid soil organic matter decomposition process,
(iii) Protection from some pathogens;
(iv) Fungi gain a constant supply of carbohydrates;
 algae with anthozoans that form coral reefs: in nutrient poor tropical waters, the algae gains nutrients from their hosts, access to light and protection, the corals gain about 86% of their energy from the algal fixation of carbon and nitrogen;
 the yucca plant and its moth pollinator: they pollinate the plant and the larvae of the moth feed on yucca seeds;
 ants and the Acacia plants: the plant gains protection from herbivores and the ants gain shelter and food from the plant;
 the Clark nutcracker ( Nucifraga columbiana ) and the white barked pine ( Pinus albicaulis ): the pine achieves seed dispersal while the nutcracker gets food from the seeds;
 Many associations between fruiting plants and seed dispersing frugivores (Smith, 1990);
All the above interactions represent highly evolved expressions of
compatibility. Smith (1990) notes that the study of mutualism is "still
embryonic, lacking in empirical studies in spite of the relatively
large literature available." I hope that this book has contributed to
the empirical and theoretical development of this important field.
Again quoting Smith (1990), "Mutualism, long overshadowed by
interspecific competition, is now receiving the attention it deserves."
"What Darwin described in the Origin of Species was the steady
background kind of evolution. But there also seems to be a
non-Darwinian kind of evolution that functions over extremely short
time periods - and that's where all the action is" (Narbonne, Queen's
Univ. paleontologist) (in Nash, 1995).
"Many species once formed never undergo any further change . . . ;
and the periods during which species have undergone modification,
though long as measured by years, have probably been short in
comparison with the periods during which they retain the same form"
(Darwin quoted from Dennett, 1995).
"Soon after a major innovation, discovery of profoundly different
variations is easy. Later innovation is limited to modest improvements
on increasingly optimised designs" (Stuart Kauffman, At Home in the
Universe, quoted in Nash, 1995).
As mentioned early in this chapter, the paleontologist Niles
Eldredge finds patterns of evolution in the fossil record that
geneticists cannot explain by standard evolutionary models. He sees
energy flow within ecosystems as the "actual organising ingredients of
ecosystems." Whereas geneticists believe that species show a gradual
change in response to environmental change, the naturalist Eldredge
claims that as the environment changes, the species will track its
required habitat and become extinct with the loss of that habitat. He
therefore sees much less flux in a species form over time.
Evolutionary stasis is evident from the fossil record: species
may remain unchanged for millions of years once formed (Avers, 1989).
With this pattern, termed punctuated
equilibrium , he found the rapid evolution of new species before
the period of stasis. Because of this process, there is a glaring
absence of transitional or intermediate forms in the fossil record.
Geneticists cannot explain this with their models. Scientists need to
explain this pattern in the fossil record if a sound evolutionary model
is to be found.
Research has shown that punctuated equilibrium is quite common. Hundreds of studies have been made, with the general finding that species remain stable for millions of years and then change so rapidly that no fossil record of the transition is to be found. Even under the influence of environmental change, species remained static over millions of years (Prothero, 1992). Upon this observation, Prothero (1992) notes, "Mayr (1992) argues that it is merely the integration of species as complex wholes, so that small-scale changes are insufficient to upset the complex balance of integrated genes. Others suggest that fundamental developmental constraints play an important role in restricting the possible avenues of change (Gould and Lewontin, 1979; Kauffman, 1983). Still others suggest that there might be properties of species that may not have been discovered yet by geneticists and evolutionary biologists, properties which operate on scales of millions of generations and years (Vrba and Eldredge, 1984)."
In contradiction to this are the textbook examples of the gradual evolution of species. A classic example is the herring gull of the Northern Hemisphere (Dennett, 1995). This species forms a broad ring around the North Pole. Along its distribution its form changes gradually, but it is one species. Where the geographically extreme forms of the ring around the North Pole meets in Europe, the effect of this gradual change over the continuum is evident. They form two species, the herring ( Larus argentatus ) and the lesser black-backed gull ( Larus fuscus ) that do not interbreed. Their young from the first winter are indistinguishable (Tuck & Heinzel, 1979).
They term this graded spectrum of forms (phenotypes) of a species over its geographical range a cline (Avers, 1989). Neighbouring populations can and often do interbreed, while those at the extremes cannot usually interbreed. Was an intermediary population that linked the distribution to become extinct, we could call the two extreme limits of distribution separate species!
How does the MELV model account for these observations? Let me explain. A part of the adaptive process resulting from evolution through natural selection of long-associated species is a decrease in the i-factor (cost of interaction). The MELV model shows this. A lowered i-factor results simply as a product of the improved survival potential or fitness of compatible or even interdependent interactors. We must apply the same principle to intraspecific interactions that are subject to natural selection. A major difference here is that the gene pool is common to both interactors. As such, a decrease in the i-factor through natural selection is economically based and does not lead to divergence. Individuals have higher fitness if they can be more efficient while still reproducing. With time this leads to greater efficiency and fitness for the species as a whole.
Improved economy is reflected in the fossil record. Mammals are generally less robust than dinosaurs and dinosaurs generally have a more efficient mode of locomotion than the earlier reptiles. Ecologists can find the result of this reflected in behavioural repertoires that reduce the energy expenditure or other damage resulting from such intraspecific interactions. The result of this process upon the whole species is that once speciation has taken place, subsequent evolutionary rates will slow. Economic factors come into play under the force of natural selection acting to decrease the i-factor. Variations from some economic norm are less fit and are therefore eliminated through natural selection. The species becomes a real entity maintained in a state of stasis because of natural selection, intraspecific interactions and reproduction.
Now we must enter the possibility of the genetic isolation of part of the population. An immediate assumption has to be that this founding population occupies a niche at least partly isolated reproductively through some mechanism. Geographical isolation is the most easily visualised example, but the possible ways this separation could occur are as diverse as nature herself.
Once such an isolated subpopulation is founded, that genetic population is subject to natural selection decreasing the i-factor of that population. Involved too, are mutation , genetic drift , the founder effect and the adaptation to the new environment through natural selection. The individuals of such an isolated genetic unit may encounter conditions that they are unable to cope with and become extinct. If a species has already evolved, this will fall under the category of species sorting . Heavy forces of natural selection, shaping the necessary adaptation of the new or potential species, leads to the rapid rate of evolution at this early phase of speciation. Adaptation through natural selection brings the population into accord with its new environment. At this early phase of speciation, the population is occupying a new niche. Initially the struggle for survival and adaptation enables a capable variant to persist despite its economic efficiency. At this stage survival needs override social competition for economic efficiency. As time proceeds, increased intraspecific interactions lead to a decrease in the i-factor for the incipient species. As the animal becomes adapted to its new environment, the effect of intraspecific interactions increases in importance, so that after an initial, very rapid evolution the form of the species stabilises.
Are there any patterns in nature to support this result of the model? The above model is very similar to the founder effect and peripatric speciation proposed by Ernst Mayr in 1954 (Avers, 1989). He requires the same conditions for his model:
 Small isolated populations at the periphery of the main population;
 Genetic drift and founder individuals;
 No gene flow between the isolated and main population;
 Subjection to different selective pressures in the new habitat;
 Changes in allele frequencies at many sites, even of previously fixed features (Avers, 1989);
The result predicted is a genetic revolution. Evolution through natural selection leads to reduced intraspecific and interspecific interactive costs. If two newly formed, but similar species meet, but are reproductively incompatible for some reason, natural selection will select mechanisms of divergence to reduce the interactor's i-factors and improve their relative fitness. Intraspecifically the same selective force leads to the stasis of the species, as variants are generally less efficient. The driving force of this stasis is intraspecific interactions. Where two interactors are very different, such as the bee and the flower, the same process of natural selection upon the interactive cost of the association may lead to interdependence through coadaptation . Where associated species do not interact there can be no coadaptation, but there can be holistic adaptation . This occurs where an organism is adapting to biotic and abiotic factors within its environment (e.g.. the shady conditions under a forest canopy). Smith (1990) terms holistic adaptation indirect mutualism, stating, "it could influence community organisation." All that we are really identifying here is an unrecognised aspect of natural selection and that is selection for and the evolution of economic efficiency to improve relative fitness .
In support of peripatric speciation and by coincidence the predicted effect of the i-factor upon speciation, Mayr found:
 The main population of species shows little obvious variation in major features;
 Small, isolated populations showed greater variation in features.
The MELV model does not require total reproductive isolation but simply that:
 Hybrids of the two interacting populations have lower fitness than either population;
 Each variety has a higher relative fitness in its own habitat that in its neighbours.
As Darwin explained, individuals from either population will then
continually colonise the interactive zone, but neither population can
penetrate the other's zone. Natural selection favours all means of
divergence that reduce the interactive costs. The MELV model explains the rapid evolution
followed by species stasis, while genetic models predict enhanced
variance in new populations (Avers, 1989). The MELV model supports the
punctuated equilibria model of Niles Eldredge and Steven Jay Gould
(1972) that is based on the fossil record. It shows an ecologically
based mechanism that overrides and interacts with genetic processes and
in the genetics-ecology debate, restores the organism as the unit of
selection. Perpetuity and compatibility represent an evolutionary and
ecological process resulting the diversity of forms and associations
found in nature. Its principle is so simple that it must be true.
This book ends on a very positive note. Our potential as the human race and beings of earth is far greater than even we can imagine. Our future does not lie in the mechanical hell portrayed in so many modern films. That is a choice that we do not need. True advance lies in a holistic blend of technology and nature. I doubt that this means that they will genetically engineer all future life. There are limits to what we can or should do. As a species that thrives on communication, technology should open immense possibilities here. As a biological creature, a love for nature should bring us to repair the damage we have done and again let life flourish. With such short lives, we will not witness the change, but each generation will find itself a little closer to the ideal. How painful or effortless the travail is, depends on how violently we oppose the God Ordained path. Transformation has two routes, the one that we try to determine (with force), with no success and the one that we learn, through intelligence, to find. As Henry Thoreau said, "Would it not be well to consult with nature in the outset?" (Worster, 1994).
1. (footnote: QsXLIIv29: And among His Signs is the Creation of the Heavens and the Earth, and the living creatures that he has scattered through them: and He has power to gather them together when He wills)
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