From: Alan McGowen <alanm@igc.org>
Newsgroups: sci.environment
Date: 13 Apr 92 21:49 PDT
Subject: ECO CENTRAL 11
Nf-ID: #N:cdp:1466601332:000:12006
Nf-From: cdp.UUCP!alanm    Apr 13 21:49:00 1992


ECO CENTRAL 11

                Tolerance, Limits, and the Niche


Tolerance 


Living systems are homeostatic dynamical systems which maintain 
themselves in the face of a stochastic universe. As such, they 
have regions of stability, within which they survive and outside 
of which they perish: so long as the environment does not stray 
beyond the range of conditions which the system can tolerate, 
homeostasis can be maintained.

We have seen that this is true for a multispecies community of 
populations with a codynamics governed by a community matrix A. If 

lambda = -max [eigenvalues of A] 

and if sigma^2 measures the stochasticity of the environment, 
then so long as lambda >> sigma^2 the community will be quite 
stable: lambda measures the amount of environmental fluctuation 
which this system can tolerate.

The physiology of an organism is also a homeostatic system, with 
a range of conditions which it can tolerate. For any measure of 
the performance of an organism -- i.e. fitness, foraging 
efficiency, survivorship etc. -- and a relevant environmental 
variable, there is a range of values of the variable outside 
which death or nonperformance ensues (the measure of performance 
is zero), but within which performance is possible (the measure 
of performance is positive.) When such a measure is graphed as a 
function of the variable, it is found to be a bell-shaped curve 
within the region of tolerated values of the variable: there is 
some optimum value at which performance is at its peak, and 
performance falls off as we move away from the optimum, becoming 
zero at the limits of the range of tolerance. The *principle of 
tolerance* is the observation that organismic responses -- 
indeed, the responses of all living systems from cells to 
ecosystems -- are typified by such tolerance curves. As we have 
seen, this very general principle actually arises from the fact 
that living systems are homeostatic dynamical systems with 
compact regions of stability. 


The niche


This leads to a definition of the niche as the hypervolume 
enclosed by all an organism's tolerance curves for a complete set 
of independent environmental variables: the complete "region of 
stability" of the organism in the space of environmental 
variables. This is known as the hypervolume, or Hutchinson, 
definition of the niche. The niches of individuals forming a 
population can be averaged, as can those of an entire species; 
often "the niche" under discussion is such an average. In 
evolutionary models, mean fitness, averaged over an interbreeding 
population, is the measure of performance of greatest interest, 
and the niche is defined as the hypervolume of mean fitness 
curves over environmental variables. 


Limits to growth: density dependence


A resource is something such that when one organism uses it 
another organism cannot. By this simple definition, many things 
are resources: any oxygen I consume cannot be consumed by you, so 
according to this definition, oxygen is a resource. However, 
oxygen is usually so plentiful that it is not a scarce, or 
*limiting* resource. But it can be limiting, under the right 
circumstances, as anyone who has ever maintained an aquarium is 
aware. 

Limits arise from tolerances coupled with scarcity. Growth often 
proceeds until enough of a resource is in use that the remainder 
has become scarce: the physiological cost of obtaining an 
additional increment of the resource becomes an increasing 
function of the number of organisms exploiting it. This is the 
condition that brings about density dependence. This can happen 
either because the organisms are potentially able to exploit more 
of the resource than exists (exploitative competition) or because 
they interfere with one another's access to the resource 
(interference competition). In either case, as the number of 
organisms increases, so does either the physiological cost of 
obtaining each increment of the resource, or the mortality rate 
(as the probability of obtaining the minimum required increment 
of the finite resource decreases). In effect, the resource 
increment has an increasing marginal cost per organism added to 
the population -- that is the effect of competition.

When the number of organisms is small compared to the total 
quantity of each resource they need, the increment of each 
resource per organism added to the population has no such 
increasing marginal cost, and the rate of intrinsic growth of the 
population does not decrease with increasing density; then we 
have exponential growth:

dN/dt = r(N)N 

r(N) = r0 = constant  (exponential growth)
r(N) is a decreasing function of N   (density dependence).

As we have seen in earlier ECO CENTRALs, the simplest model of 
density dependent growth is the logistic equation, in which r(N) 
decreases linearly with N:

r(N) = r0(1 - N/K)   (logistic growth)

Logistic growth amounts to assuming a linear increase in the 
physiological cost of some limiting resource as the number of 
organisms exploiting it rises, but more general functions than 
the linear could be used, for example a function r(N) which first 
rises with N up to some maximum and then falls: this sort of 
growth would typify many social species, which derive fitness 
from one another's company up to -- but not beyond -- the point 
of overcrowding. 


An example of more complex density dependence


Cooperative hunters, e.g. wolves, derive significant advantages 
from hunting in groups as the physiological costs of obtaining, 
say, one caribou, are shared by the pack (also, the chances of 
bringing down a caribou will increase with N). Thus the cost per 
pack member per hunt declines (and probability of success 
increases) with increasing N up to some optimal pack size, but 
net payback decreases above it. 

The decrease of net payback is due to inability to coordinate 
larger groups, so that a pack of 2N wolves will not in general 
kill twice as many caribou as one of N wolves will. Increasing N 
then reduces the average physiological cost of the kill somewhat 
but also reduces the average payback in quantity of food, and the 
balance of the two dictates that there must be an optimal pack 
size: the food caloric payback must exceed the caloric cost of 
the hunt. The N which maximizes net payback, (payback - cost), is 
the optimal pack size. For cooperative hunters, we would expect 
r(N) to increase up to this N and decrease above it, to 0 at the 
point where (payback - cost) = 0. [Of course, for real 
cooperative hunters such as wolves the computation will be 
complicated by other factors, e.g. in the social structure of a 
wolf pack, food is not shared equally. Also, many smaller prey 
are caught and eaten by individuals, so that the food obtained 
from hunting in packs is not the whole diet.]


Saturation


The tolerance principle allows us to say that there is some 
minimum increment of the limiting resource which is necessary to 
the survival of any organism added to the population. There is 
also some maximum physiological outlay which can be expended for 
the increment: when the physiological cost of the minimum 
tolerable increment of resource is itself too great for 
tolerance, the limits to growth have been reached, no more 
organisms can be added, and the environment is *saturated* with 
the organisms. In the logistic population growth model, the N at 
which saturation occurs is the carrying capacity K, and moreover 
this is a stable equilibrium of the system (if environmental 
fluctuation is not too strong). But in other models, the 
saturation point need not be a stable equilibrium -- it may be a 
maximum from which a decline, or *dieback* commences. Thus in 
general a long-term sustainable K need not be the same as the 
maximum saturation which is possible for some period of time, 
unless we can be sure that the growth is logistic. 

The tolerance principle, together with the fact that any organism 
has only a finite amount of physiological capital to expend while 
obtaining resources, allows us to understand why all growth must 
eventually become density dependent -- all growth must eventually 
saturate if it proceeds far enough. Indeed, even if no external 
resources are limiting, the waste products of the organisms 
themselves eventually increase the physiological cost of survival 
to the point where saturation occurs. Such "waste limited" 
communities occur in wine vats, for example.

More common types of limitation are: food or water, predation, 
and space (e.g. nesting sites, space for attachment to the 
substrate in littoral communities, territorial limitations, space 
providing access to light for plants). Populations can be limited 
by more than one factor at the same time, e.g. by both food and 
predation. 


Some extremes: r and K selection


Some organisms (r-selected organisms) rarely or never reach the 
point of saturation before their populations have become extinct. 
Such species survive by propagating whole new populations of 
themselves with an adequate probability of success even though the 
fate of any particular population is rapid extinction. These 
species live in an adaptive regime where growth is density-
independent, but mortality so high, and life so short, predators 
so numerous and the environment so prone to fatal fluctuations, 
that they never come close to saturation. For such species, 
success means high r and they are selected for the maximum 
reproductive potential.

Long-lived species whose populations persist for long times must 
evolve mechanisms for dealing with saturation -- for dealing with 
limits. They become more efficient at the use of limiting 
resources, for example. This is K-selection: such species are 
selected for their ability to cope with scarcity, for their 
ability to survive a broad range of adversities, for lower 
mortality of offspring. They invest more in survival, and in the 
survival of each offspring. Their r is small, so they must use it 
carefully. Such species become choosy about mates, and sexual 
selection can become important.


Niche overlap, limiting difference, character displacement


If two organisms' niches overlap, the organisms are potential 
competitors for limiting resources. In the case where the niches 
of two species overlap, if the overlap is large enough, one of 
the species may exclude the other.

We must distinguish between the *fundamental niche*, which is set 
only by an species' tolerances, and its *realized niche* which is 
constrained by competition for scarce resources, predation, 
sociality, different ecological roles at different life stages, 
or of different sexes, etc. The realized niche is often smaller 
than the fundamental niche. This phenomenon, that a niche shrinks 
in the presence of competition, is known as *niche compression*. 
The converse process, of a niche expanding when competitors or 
other limiting factors are removed, is called *ecological 
release*.

The minimum distance apart of the centers of two niches which are 
able to coexist is called the *limiting difference* of the two 
species: if they resembled each other more closely than this, one 
would exclude the other. Competitive interactions tend to evolve 
so that characters are displaced from each other by at least the 
limiting difference: i.e. the competing niches are "pushed apart" 
enough to reduce competition to the point where coexistence is 
possible. For example, in food limited *guilds* (groups of 
species with similar niches that live together) the size of the 
food item is frequently the character which is displaced: guilds 
of seed-eating birds are commonly marked by a fairly regular 
spacing of bill sizes and preferred seed size when the species are 
ordered by these characters.  In principle, the limiting 
separation controls the density with which species can be packed 
into the environment, i.e. it controls the maximum possible 
species diversity.


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Alan McGowen




Newsgroups: sci.environment