EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1674-1684
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Generalized Core Functions of Maximum Entropy
Theory of Ecology

Cristina B. Corcino1,2, Roberto B. Corcino1,2,5, Jay P. Picardal3,4

1 Research Institute for Computational Mathematics and Physics, Cebu Normal University,
6000 Cebu City, Philippines
2 Mathematics Department, Cebu Normal University, 6000 Cebu City, Philippines
3 Research Institute for Tropical Biology and Pharmacological Biotechnology,
Cebu Normal University, 6000 Cebu City, Philippines
4 Biology Department, Cebu Normal University, 6000 Cebu City, Philippines
5 Cebu Normal University-Balamban Campus, Balamban Cebu, Philippines

Abstract. Core distributions of Maximum Entropy Theory of Ecology (METE) are the Spatial
Structure Function (SSF) and the Ecosystem Structure Function (ESF). SSF is a by-species predic-
tion of the clustering of individuals over space. ESF is a kind of container function that describes
the probability space of how abundances are assigned to species and how metabolic energy is parti-
tioned over individuals in a community. In this study, these core functions of METE are generalized
by deriving the corresponding functions in the Tsallis q-entropy. Derivation used the method of
Lagrange multipliers. The generalized SSF and ESF are expressed in terms of the q-exponential
function. Numerical examples are provided to illustrate the generalized SSF.

2020 Mathematics Subject Classifications: 11B68, 42A16, 11M35

Key Words and Phrases: Probability Distribution, Ecology, Entropy, Tsallis Entropy, Spatial
Structure Function, Ecosystem Structure Function

1. Introduction

Maximum entropy framework has been utilized to construct ecological models, thus
providing support to emerging ecological relationships at a broader scale [4, 10]. Maximum
entropy models maximize information content from biological system while satisfying rel-
evant constraints which are primarily composed of bioclimatic and biophysical variables.
These models have extensive applications in biodiversity conservation and ecosystem man-
agement of a particular species [6, 8].

Mathematical rigor of MaxEnt models enhances the accuracy and reliability of predic-
tions, thereby supporting ecologists in making informed choices in designing conservation

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5137

Email addresses: corcinoc@cnu.edu.ph (C. Corcino),
rcorcino@yahoo.com (R. Corcino), picardalj@cnu.edu.ph (J. Picardal)

https://www.ejpam.com 1674 © 2024 EJPAM All rights reserved.



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1675

strategies for a particular species [12]. For instance, the model may provide inputs in
understanding species niche differentiation and species interactions [11], construction of
food web structures [2] and establishing trophic relationships [13].

Maximum Entropy Theory of Ecology (METE) is a theoretical framework of macroe-
cology that makes a variety of realistic ecological predictions about how species richness,
abundance of species, metabolic rate distributions, and spatial aggregation of species in-
terrelate in a given region. Underlying mathematics of METE relies on MaxEnt: the
maximization of information entropy. Primary equations that regularly occur in the max-
imization as presented in [1] are the given in (1.1)-(1.3).

The general expression for K constraints on the mean values of the variables fk(n),
where n follows the distribution p(n), is expressed as

n=N∑
n=1

fk(n)p(n) = ⟨fk⟩. (1)

Additional constraint provides for the normalization of the probability distributions, ex-
pressed as

n=N∑
n=1

p(n) = 1. (2)

To maximize Shannon information entropy subject to the above constraints, the tools
of variational calculus and the method of undetermined Langrange multipliers will be
employed. The function F to be maximized is an expression that incorporates the measure
of Shannon information entropy and the constraints. That is,

F = −
Nmax∑

n=Nmin

p(n) ln p(n)− λ0

 Nmax∑
n=Nmin

p(n)− 1

− λ1

 Nmax∑
n=Nmin

f(n)p(n)− ⟨f⟩

 . (3)

On the other hand, the q-logarithm function introduced in [14] is given by

lnq x =
x1−q − 1

1− q
, (q ∈ R, x > 0) (4)

and the q-exponential function is given by

expq x =

{
(1 + (1− q)x)

1
1−q , if 1 + (1− q)x > 0,

0, otherwise
(5)

The functions expq x and lnq x converge to expx and log x as q → 1, respectively and the
following relations are true,

expq(x+ y + (1− q)xy) = expq x expq y, (6)

lnq xy = lnq x+ lnq y + (1− q) lnq x lnq y. (7)



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1676

A new approach of handling the Lagrange multipliers involved in the extremization
process leading to Tsallis’ statistical operator was presented in [9]. To understand this
new approach, the following discussion taken from [9] is provided.

The Tsallis’ generalized entropy will be defined as follows,

Sq

k
= −

ω∑
i=1

pi lnq pi, (8)

where k ≡ k(q) tends to the Boltzmann constant kB in the limit q → 1 [14] subject to the
constraints

w∑
i=1

pi = 1, (9)

∑w
i=1 p

q
iO

(i)
j∑w

i=1 p
q
i

= ⟨⟨Oj⟩⟩q, (10)

where the pi is the probability assigned to the microscopic configuration i (i = 1, 2, . . . , w)

and theO
(i)
j denote the n relevant observables whose generalized expectation values ⟨⟨Oj⟩⟩q

are a priori known.

Following the generalization in [3], the second generalized Tsallis’ entropy that will be
used is defined by

Sq

k
= −

ω∑
i=1

pqi lnq pi. (11)

The method of Lagrange multipliers requires to maximize the function

F =
Sq

k
− λ0

(
w∑
i=1

pi − 1

)
−

s∑
i=1

λj

(∑w
i=0 p

q
iO

(i)
j∑w

i=1 p
q
i

− ⟨⟨Oj⟩⟩q

)
. (12)

The new approach replaces (12) by

F =
Sq

k
− α0

(
w∑
i=1

pi − 1

)
−

s∑
j=1

αj

w∑
i=1

pqi (O
(i)
j − ⟨⟨Oj⟩⟩q), (13)

which will yield the same pi and the relation of the Lagrange multipliers λj and αj is given
by

λj = αj

w∑
i=1

pqi . (14)

The core distributions of METE are the Spatial Structure Function (SSF) and the
Ecosystem Structure Function (ESF). The SSF is a by-species prediction of the clustering
of individuals over space. The ESF is a kind of “container function” that describes the
probability space of how abundances are assigned to species and how metabolic energy



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1677

is partitioned over individuals in a community.These core functions of the MaxEnt were
derived in [1] using the Shannon entropy. A phase space entropy model of ecosystems was
developed in [7] and ecosystem equilibria was specified by conservation of biomass and
total metabolic energy. A new proof of the theorems on the maximum entropy principle
in Tsallis statistics without using the Lagrange multipliers method was presented in [3].
Analysis of the climate fluctuations in past deuterium records corresponding to the last
glacial period was done in [5] using nonadditive entropy on which nonextensive statistical
mechanics is based.

In this paper, core functions of METE are generalized by deriving the corresponding
functions in the Tsallis q-entropy given in (8) and (11). The ecological state variables
introduced in [1] namely, A, H, N , and E, representing respectively, total area, total
number of species, total abundance and total metabolic energy of an ecological system
will be used in the discussion below.

2. Generalization of Spatial Structure Function

Spatial Structure Function (SSF) also called ”Pi Distribution” [1], is defined as the
probability that n individuals of a species are found in a cell area A if it has n0 individuals
in the total area A0 under consideration. The variables involved in the function are A and
n which is the abundance of a single species at the total spatial scale.

Theorem 2.1. The generalized Spatial Structure Function (SSF), denoted by Π(n, q), is
given by

Π(n, q) =
expq(α1(n− µq))∑N0
n=1 expq(α1(n− µq))

. (15)

Proof. The generalization of the Spatial Structure Function is obtained using the
Tsallis’ q-entropy (11) subject to the normalization constraint which is given by

N∑
n=1

Π(n) = 1, (16)

and the second constraint comes from the measurement of the average value of the per-
species abundance which is given by∑N

n=1 nΠ(n)
q∑N

n=1Π(n)
q

= µq. (17)

The function to be maximized is

F =
Sq

k
+ α0

(
N∑

n=0

Π(n)− 1

)
+ α1

(
N∑

n=0

Π(n)q(n− µq)

)
, (18)



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1678

where α0 and α1 are the Lagrange multipliers. Taking the derivative of F with respect to
Π(n) and setting it equal to 0,

0 = −1− qΠ(n)q−1 lnq Π(n) + α0 + α1(n− µq)qΠ(n)
q−1

0 = −1− q

1− q
+ α0 +

(
q

1− q
+ α1(n− µq)q

)
Π(n)q−1.

Solving for Π(n),

Π(n) =

(
q

1− (1− q)α0

) 1
1−q

{1 + (1− q)α1(n− µq)}
1

1−q

=

(
q

1− (1− q)α0

) 1
1−q

expq(α1(n− µq)). (19)

Applying the normalization constraint will give(
q

1− (1− q)α0

) 1
1−q

N0∑
n=1

{1 + (1− q)α1(n− µq)}
1

1−q = 1. (20)

Let

Zq =

N0∑
n=1

expq(α1(n− µq)). (21)

Now, with Π(n) being a function of q, we can write Π(n) = Π(n, q). Hence,

Π(n, q) =
expq(α1(n− µq))

Zq
.

3. Generalization of Ecosystem Structure Function

Ecosystem Structure Function (ESF) denoted by R(n, ϵ) is a joint probability distri-
bution, with R(n, ϵ)dϵ by definition being the probability that a randomly selected species
has abundance n, and that a randomly selected individual from any species with abun-
dance n has metabolic requirement in the interval ϵ, ϵ+ dϵ. The normalization constraint
is given by

N∑
n=1

∫ E

ϵ=1
R(n, ϵ)dϵ = 1. (22)

The additional constraints are aggregated measures of variables n and nϵ. With f1(n) = n,
and f2(n)ϵ = nϵ, the corresponding means of the data sets are respectively,〈

f1(n)

〉
= N/H ,

〈
f2(nϵ)

〉
= E/H. (23)



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1679

These give the pair of constraints,

N∑
n=1

∫ E

ϵ=1
nR(n, ϵ)dϵ =

N

H
, (24)

N∑
n=1

∫ E

ϵ=1
nϵR(n, ϵ)dϵ =

E

H
. (25)

Theorem 3.1. The generalized Ecosystem Structure Function (ESF) R(n, ϵ, q) is given
by

R(n, ϵ, q) = α2(2− q)
expq (α1(n− µq,1) + α2(nϵ− µq,2))∑N
n=1

1
n

[
(expq uE)

2−q − (expq u1)
2−q
] , (26)

where u = α1(n− µq,1) + α2(nϵ− µq,2), uE = u(ϵ = E), u1 = u(ϵ = 1).

Proof. The generalization for the Ecosystem Structure Function is obtained using (11).
Following [9], the normalization constraint (22) will be kept while the constraints (24) and
(25) will be replaced respectively, by∑N

n=1

∫ E
ϵ=1 nR

q(n, ϵ)dϵ∑N
n=1R

q(n, ϵ)
= µq,1, (27)

∑N
n=1

∫ E
ϵ=1 nϵR

q(n, ϵ)dϵ∑N
n=1R

q(n, ϵ)
= µq,2, (28)

where Rq(n, ϵ) = [R(n, ϵ)]q. The function to be maximized is

F =
Sq

k
+ α0

[
N∑

n=1

∫ E

ϵ=1
R(n, ϵ)dϵ− 1

]
+ α1

[
N∑

n=1

∫ E

ϵ=1
Rq(n, ϵ)dϵ(n− µq,1)

]

+ α2

[
N∑

n=1

∫ E

ϵ=1
Rq(n, ϵ)dϵ(nϵ− µq,2)

]
. (29)

Taking the derivative of F with respect to R := R(n, ϵ),

δF

δR
= −

N∑
n=1

∫ E

ϵ=1

(
1 + qRq−1R

1−q − 1

1− q

)
dϵ+ α0

N∑
n=1

∫ E

ϵ=1
dϵ

+ α1

[
N∑

n=1

∫ E

ϵ=1
q(n− µq,1)R

q−1(n, ϵ)dϵ

]
+ α2

[
N∑

n=1

∫ E

ϵ=1
q(nϵ− µq,2)R

q−1(n, ϵ)dϵ

]
. (30)

Setting (30) equal to zero and solve for R(n, ϵ),

0 = −
(
1 +

q

1− q
− q

1− q
Rq−1

)
+ α0 + α1q(n− µq,1)R

q−1 + α2q(nϵ− µq,2)R
q−1



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1680

1 +
q

1− q
− α0 =

(
q

1− q
+ α1q(n− µq,1) + α2q(nϵ− µq,2)

)
Rq−1

1− α)(1− q)

q
R1−q = 1 + (1− q) {α1(n− µq,1) + α2(nϵ− µq,2)} .

Solving for R = R(n, ϵ),

R(n, ϵ) =

(
q

1− α0(1− q)

) 1
1−q

(1 + (1− q)(α1(n− µq,1) + α2(nϵ− µq,2))
1

1−q . (31)

Substitution to (22),

N∑
n=1

∫ E

ϵ=1

(
q

1− α0(1− q)

) 1
1−q

(1 + (1− q)(α1(n− µq,1) + α2(nϵ− µq,2)))
1

1−q dϵ = 1,

which will give(
q

1− α0(1− q)

) 1
1−q

=
1∑N

n=1

∫ E
ϵ=1 expq(α1(n− µq,1) + α2(nϵ− µq,2))dϵ

. (32)

Let

Zq =
N∑

n=1

∫ E

ϵ=1
expq(α1(n− µq,1) + α2(nϵ− µq,2))dϵ. (33)

Now, with R(n, ϵ) being a function of q, we can write R(n, ϵ) = R(n, ϵ, q). Hence,

R(n, ϵ, q) =
expq(α1(n− µq,1) + α2(nϵ− µq,2))

Zq
. (34)

The expression in (34) will be called the q-ESF denoted by R(n, ϵ, q) and Zq given in
(33) is the corresponding partition function. Solving the integral involved in the partition
function,∫ E

ϵ=1
expq(α1(n− µq,1)+α2(nϵ− µq,2))dϵ

=

∫ E

ϵ=1
(1 + (1− q){α1(n− µq,1) + α2(nϵ− µq,2)})dϵ

=
1

α2n(2− q)

(
exp2−q

q (uE)− exp2−q
q (u1)

)
,

where u = α1(n− µq,1) + α2(nϵ− µq,2), uE = u(ϵ = E), u1 = u(ϵ = 1). Then

Zq =
1

α2(2− q)

N∑
n=1

1

n

[
(expq uE)

2−q − (expq u1)
2−q
]
, (35)

and

R(n, ϵ, q) = α2(2− q)
expq (α1(n− µq,1) + α2(nϵ− µq,2))∑N
n=1

1
n

[
(expq uE)

2−q − (expq u1)
2−q
] .



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1681

4. Examples

In this section, examples are provided to illustrate the generalized SSF.
Example 1. To be able to give an example for (15), the values of q, µq, and n must be

specified. Taking q = 1
2 , µ 1

2
= 2 and n = 1, 2, 3, the partition function (21) is

Zq =
3∑

n=1

exp1/2(α1(n− µ1/2)) =
3∑

n=1

(
1 +

1

2
α1(n− 2)

)2

= 3 +
(α1)

2

2
.

The probability function (15) becomes

Π(n, 1/2) =
exp1/2(α1(n− 2))

3 +
α2
1
2

=

(
1 + 1

2α1(n− 2)
)2

3 +
α2
1
2

.

It can be verified that
∑3

n=1Π(n, 1/2) = 1. To determine α1, impose the second constraint
(17) to yield

3∑
n=1

nΠ(n, 1/2)
1
2 = 2

3∑
n=1

Π(n, 1/2)
1
2

−Π(n, 1/2)
1
2 +Π(3, 1/2)

1
2 = 0,

from which α1 = 0. Thus, the desired probability function is the uniform distribution,

Π(n, 1/2) =
1

3
, n = 1, 2, 3.

Example 2. As a second example for (15), take q = 2
3 , µ2/3 = 2, n = 1, 2, 3, 4, 5. The

partition function (21) is

Z2/3 =
5∑

n=1

exp2/3(α1(n− 2))

=

5∑
n=1

(
1 +

1

3
α1(n− 2)

)3

= 5 + 5α1 + 5α2
1 +

35

27
α3
1.

To determine α1, impose the second constraint (17)

5∑
n=1

nΠ(n, 2/3)
2
3 = 2

5∑
n=1

Π(n, 2/3)
2
3 ,

which will yield the equation

45 + 90α1 + 35α2
1 = 0.



C. Corcino, R. Corcino, J. Picardal / Eur. J. Pure Appl. Math, 17 (3) (2024), 1674-1684 1682

The two solutions to the preceding equation obtained using Wolfram alpha equation solver
are α1 = −3

7 (3 +
√
2), and α1 = 3

7(
√
2− 3). The second value for α1 will give the desired

probability function. In particular, for n = 1, 2, 3 , 4, 5 the values of the probability
function are:

Π(1, 2/3) = 0.527, Π(1, 2/3) = 0.28571, Π(3, 2/3) = 0.132,

Π(4, 2/3) = 0.048, Π(5, 2/3) = 0.0094.

Remark 4.1. Owing to the complexity of (26) no example will be provided for the gener-
alized ecosystem structure function.

5. Conclusion and Recommendation

This paper has delved into the theoretical framework of METE, specifically focusing on
the development of q-generalizations for its two core functions. METE, or the Maximum
Entropy Theory of Ecology, is a fundamental framework used to understand the structure
and dynamics of ecological systems. By introducing q-generalizations, the paper extends
the applicability of METE to systems exhibiting non-trivial behavior, potentially offering
deeper insights into ecological patterns and processes.

In the context of this study, the q-generalizations were elucidated with regards to
the Spatial structure function, which plays a crucial role in characterizing the spatial
distribution of species within an ecosystem. Through illustrative examples, the paper
provided sample probability functions for the q-Spatial structure function, showcasing
how these generalized formulations can be applied in practical scenarios.

However, a notable gap exists in the paper’s treatment of the q-ecosystem structure
function. Despite the detailed exploration of the q-Spatial structure function, no corre-
sponding example was provided for the q-ecosystem structure function. This omission
leaves a significant aspect of METE unaddressed, limiting the comprehensiveness of the
study. Furthermore, the paper did not extend its analysis to encompass physical appli-
cations of the derived generalized functions. While theoretical developments are valu-
able, their utility often lies in their practical applicability. By demonstrating how these
q-generalizations can be applied to real-world ecological data or modeling scenarios, re-
searchers can validate their effectiveness and enhance their relevance to ecological studies.

In light of these considerations, it is recommended that future research endeavors
focus on bridging these gaps. Specifically, efforts should be directed towards exploring
sample probability functions for the q-ecosystem structure function, thereby completing
the theoretical framework. Additionally, researchers should actively seek out opportunities
to apply these generalized functions in physical contexts, such as ecological modeling or
data analysis. By doing so, we can advance our understanding of METE and its relevance
to the study and management of ecological systems.



REFERENCES 1683

Acknowledgements

The authors would also like to thank Cebu Normal University (CNU) for funding this
research project through the Office for Research, Development and Publication (RDP).

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