Ecology: Niche, Basic Theories and Debates

-Terminology

The word “niche” derives from the Latin word “nidus”, which means “nest”. The

ecological derivation of this word borrows the concept from Hinduism. In India where

polytheism is a common practice, people have plenty of gods/ goddesses idols in their

house. Each of them is given a unique and proper place for worshiping, which are

called “niche”. Ecologists borrow such concept, and propose that each and every kind

of species in an ecological system plays a special and exclusive role, since each

species drill, prosper and reproduce only under a unique set of range of environmental

variables.

-Definition

Following the concept above, we can define ecological niche as below:

Suppose in an ecological system there are n kinds of environment parameters (ex:

temperature, solar insolation force etc.) of concern, say x1, x2, …, xn. For any

particular species, it thrives only under certain ranges of these parameters, say

𝜃 ∈ {𝑥1|𝐿1 ≤ 𝑥1 ≤ 𝑈1 , 𝑥2|𝐿2 ≤ 𝑥2 ≤ 𝑈2, … , 𝑥𝑛|𝐿𝑛 ≤ 𝑥𝑛 ≤ 𝑈𝑛}

Then we call this specific set of range of parameters θ the niche of the particular

species.

This is known as the Hutchinsonian niche. It is quite worthy to note that the definition

can be extended to populations, gene pools, groups or even individuals with similar

analogy.

-The Technique to Determine Niche of a Species

Usually we start by collecting the spatial distribution map of a species of concern. For

example, consider the dot plot below:

Now we name a list of environmental parameters we are interested in discussing. To

simplify things I will consider only temperature and precipitation here.

We then plot the temperature contour and precipitation contour of the area.

Next we combine these three maps and plot the distribution on the

temperature-precipitation plane.

We may find out that the species only appears in a certain area on the plane; this is the

niche of the species.

-Niche Overlapping and Competition, Abstract

From now on we consider the one dimensional case of food size for simplification.

We know that every species has its unique range of food size preference. But other

species might also have a range of preference which coincides with it at some value.

The phenomenon is called niche overlapping.

Intuitively, if the niche overlapping of the two species is greater, the interspecies

competition of these two species increases. This is because of a more similar foraging

pattern, causing the probability for two species to compete over the same food to

increase.

- Lotka-Volterra Competition Equation

We introduce here the Lotka-Volterra competition equation for a more complete and

deeper discussion on overlapping and competition later. It says that for two species N1

and N2 which only interacts as competitors, the amount of these two species is

governed by the following differential equation:

𝑑𝑁1

𝑑𝑡= 𝑟1𝑁1 (1 −

𝑁1

𝐾1− 𝑏12

𝑁2

𝐾1)

𝑑𝑁2

𝑑𝑡= 𝑟2𝑁2 (1 −

𝑁2

𝐾2− 𝑏21

𝑁1

𝐾2)

Where r, b, and k are some constants with their own ecological meanings. The first

term represents the reproduction rate, the second the intra-species competition, and

the third the inter-species competition.

Non-dimensionalization yields

𝑑𝑢1

𝑑𝑡= 𝑢1(1 − 𝑢1 − 𝛼12𝑢2)

𝑑𝑢2

𝑑𝑡= 𝜌𝑢2(1 − 𝑢2 − 𝛼21𝑢1)

At stationary point

[𝑢1

𝑢2] =

1

1 − 𝛼12𝛼21[1 − 𝛼12

1 − 𝛼21]

The differential field near the stationary points can be linearly approximated by the

Jordian matrix (the partial differential matrix):

𝑨 = [1 − 2𝑢1 − 𝛼12𝑢2 −𝛼12𝑢1

−𝜌𝛼21𝑢2 𝜌(1 − 2𝑢2 − 𝛼21𝑢1)] = 𝐒𝚲𝐒−𝟏

= [𝒙𝟏 𝒙𝟐] [𝑒𝜆1𝑡 00 𝑒𝜆2𝑡

] [𝒙𝟏 𝒙𝟐]−𝟏 𝑜𝑟 = 𝐌𝐉𝐌−𝟏

= [𝒙𝟏 𝒙𝟐] [𝑒𝜆𝑡 𝑡𝑒𝜆𝑡

0 𝑒𝜆𝑡] [𝒙𝟏 𝒙𝟐]−𝟏 (𝑖𝑓 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟 𝑟𝑜𝑜𝑡)

The outcome of the system relies on the eigenvalue(s) of the function. Only when all

the eigenvalues λi are negative will the system be stable. Otherwise one of the values

will go toward zero. Such phenomenon can be discussed further in another ecological

concept: competition exclusion.

(Adapted from Mathematical Biology and Ecology, Dr. Ruth E. Baker, 2011,

p.32-p.33)

-Niche Overlapping and Competition, Deeper

Above we demonstrated how to determine the ultimate outcome of a system. We

know that in the long term, for two species to co-exist we must have a stable system;

or else one of the species should have already evolved or adapted another range of

niche, or would have already become extinct.

What this implies is that you cannot have two species having a niche overlapping too

great, or else the competition is too fierce, referring to an eigenvalue too large and

thus the system cannot achieve dynamic equilibrium.

Then it would be natural for one to ask the relationship between the amount of niche

overlapping and the intensity of competition so that we can determine the maximum

niche overlapping for a particular species.

The logic is discussed as below: suppose for each species the preference of the food

size is a normal distribution and, just for simplification, having the same value of

variation w and a spacing of mean d.

The coefficients b, r and k we mentioned earlier represent the probability for which

the two species compete for the same food. For any value of food size x0, the

probability density of such competition to occur would be

𝑓(𝑥0, 𝑑) =1

2𝜋𝑤2𝑒

−𝑥0

2

2𝑤2𝑒−

(𝑑−𝑥0)2

2𝑤2

A convolution appears again in front of us, just as we have experienced in solving

heat equations wave equations and hydrology curves. Try find yourself a satisfying

physical meaning of the convolution in these example, as I had done in the heat

equation case.

We realize that the equation above can be represented on a bi-variable probability

joint density function. To find the probability density for any distance d we can of

course do a line integral, since we have the constrain of x+y=d. But there is another

more quicker yet harder to observe way to evaluate its value. The heart of this is the

symmetry of this joint density function. Transforming the coordinate system into

𝑢 =1

√2(𝑥 + 𝑦), 𝑣 =

1

√2(𝑥 − 𝑦)

We find out that in this new coordinate system, the form of the probability density

function does not change. So for the line x+y=d, its probability density is equivalent

to the one variable probability density which has a distance of d/sqrt(2) from the

origin.

So its probability density should be simply

f(d) =1

√2𝜋𝑤𝑒

−(𝑑

√2)2

2𝑤2 =1

√2𝜋𝑤𝑒

−𝑑2

4𝑤2

Hence we have found the competition coefficient(rb/k). If we go further and diagonize

the matrix, we find that the eigenvalues will always be negative except for d=0 where

the matrix becomes singular. This means that in a deterministic case, the distance d

can be arbitrary close to zero and thus the niche overlapping has no constrain. But in a

real environment where parameters are variable, the reproduction rate should also be

stochastic. It can be shown that the minimum distance d has a constrain for which

𝑑 ≥𝜎2

𝑟

Where sigma is the standard division of the reproduction constant and r is the

reproduction constant at deterministic case.

Now for sigma so small and for large amount of interacting species, the constrain can

be approximated as d≥w.

This is a very important result, because it will lead to the next important theory.

(Adapted from Niche Overlap as a Function of Environmental Variability, Robert M.

May and Robert H. Mac Arthur,1972)

-Important Theory: Niche Variation Hypothesis

In the real environment, resources are limited. Say the food size spectrum spans only

for a length of L. Then for a homogeneous spacing distance of d among n species, we

have an approximate constrain of

d ≤𝐿

𝑛

Then there is also a constrain over the standard deviation w:

w ≤ d ≤𝐿

𝑛

So we see that for a limited range of resources, the more species existing the narrower

each species’ niche width should be. The niche variation hypothesis states that

1. The more species exist in a same place, the more competition, and thus a narrower

niche width for each species in order to minimize the niche overlapping.

2. The niche width can be quantified by the phenotypes, morphology or behavior of

the species to confirm such hypothesis.

-Debate over Whether the Theory is Valid

The mechanism of the NVH seems to be intuitively plausible, but since the niche

variation hypothesis is introduced in 1965 by Van Valen, debates have taken place to

confirm or reject its validity. The topic is still controversial today.

In a 2005 study, a research team examines the theory. Since the NVH theory states

that more competition leads to narrower niche width (and should therefore result in a

narrower morphology variation), it predicts that for a same species, its island

populations should have wider variation than its continental counterparts, for

populations in the islands should face less competition. Their research rejected such

prediction, and they claimed that genetic interchange is still the main cause of

variation, which continental populations have more access to.

A study in 2007 however mentions that morphology variation is not the only way to

examine the NVH. Behavior variations can also reduce competition. The team

examined the behavior of snails, frogs and lizards (their diet variation to be specific)

and did find a positive correlation between niche width and foraging variation.

Reference:

VARIABILITY AND SEXUAL SIZE DIMORPHISM IN CARNIVORES: TESTING

THE NICHE VARIATION HYPOTHESIS, SHAI MEIRI et al. , 2005

Comparative support for the niche variation hypothesis that more generalized

populations also are more heterogeneous, Daniel I. Bolnick et al. ,2007