Mathematical Model of Iteroparous and Semelparous Species Interaction


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(3) (2022), Pages 445-463 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: June 16, 2022 Reviewed: June 22, 2022 Accepted: June 27, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i3.16447        

Mathematical Model of Iteroparous and Semelparous Species 
Interaction 

Arjun Hasibuan*, Asep Kuswandi Supriatna, and Ema Carnia 

Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas 
Padjadjaran, Jl. Raya Bandung-Sumedang KM 21, Jatinangor, Sumedang, West Java 

Province, 45363, Indonesia 

 

Email: arjun17001@mail.unpad.ac.id  

ABSTRACT 

To study the survival of a species in an ecosystem it is very important to consider the dynamics of 
the species. A species can be categorized based on its reproductive strategy either semelparous or 
iteroparous. In this paper, we examine the dynamics involving both categories of species in an 
ecosystem. We focus on one semelparous and one iteroparous species influenced by density-
dependent and also by harvesting factors in which there are two age classes for each species. We 
study two different models, i.e competitive and non-competitive models. We also consider two 
type of competition, i.e intraspecific and interspecific competition. The approach that we use in 
this research is the multispecies Leslie matrix model. In addition, we use M-Matrix theory to obtain 
the locally stable asymptotically of the model. Our results show that the level of competition both 
intraspecific and interspecific competition affect the co-existence equilibrium point and the 
stability of the equilibrium point. We also present explicitly the conditions for all equilibrium 
points to exist and to be locally stable asymptotically. This theory can be applied to study the 
dynamics of natural resource models including the effects of different management to the growth 
of the resources, such as in fisheries. 

Keywords: density-dependent; harvesting; multispecies; Leslie matrix; age-structured model 

INTRODUCTION 

In an ecosystem, the survival of a species is an important thing to study. Species in 
the same ecosystem have reciprocal relationships between one species and other. The 
survival of each species can be affected by density-dependent, harvesting, competition, 
predator-prey, and so on. Of course, the important thing to do is to ensure the survival of 
these species to survive. The survival of a species can be studied with a system dynamics 
approach. In some studies, species in an ecosystem can be categorized based on their 
reproductive strategy, including species with semelparous and iteroparous strategy. 
Research on semelparous species can be seen in [1]–[3]. Then, research on iteroparous 
species can be seen in [4]–[6]. Semelparous species are species that reproduce only once 
in their lifetime shortly before dying. Then, iteroparous species are species that reproduce 
more than once in the lifetime of the species. Both species allow to live together and 
interact in the same ecosystem. In this research, we focus on studying the growth of 
multispecies cases consisting of one semelparous species and one iteroparous species 

http://dx.doi.org/10.18860/ca.v7i3.16447
mailto:arjun17001@mail.unpad.ac.id


Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 446 

using a dynamic system approach, especially using the Leslie matrix model. This model is 
a population growth model based on age class which was introduced in 1945 by Leslie in 
[7]. 

Research on studying the dynamics of population growth using the Leslie matrix 
model has been carried out by several researchers. These studies can be in the form of 
single species and multispecies cases. Several studies on single species cases include in 
[3], [8], [9], and many more. Then, several multispecies studies examine the effect of 
density-dependent on the Leslie matrix model which is one of the nonlinear models of the 
Leslie matrix model. In 1968, Pennycuick et al. [10] focused on simulating the case of 
single species and multispecies interacting in the form of competition and predator-prey 
using the Leslie matrix. In 1980, Travis et al. [11] reviewed two competing species and 
provided a case study on semelparous. In 2011, Kon [12] studied two semelparous species 
with one species containing two age classes while the other species amounting to one age 
class. In 2012, Kon [13] conducted a study on two semelparous species that have a 
predator-prey relationship and observed the effect of coprime traits from the number of 
age classes in both species. Coprime is a condition where two numbers have the greatest 
common factor of one, in which case the number is the number of age classes of each 
species. Then in 2017, Kon [1] examined the Leslie multispecies semelparous matrix 
model which has an arbitrary number of age classes. Then, there are also studies on 
multispecies but with other methods using the Rosenzweig-MacArthur model (See [14], 
[15]), the Leslie-Gower model (See [16], [17]), and the Lotka-Volterra model (See [18]–
[20]). 

Our aim in this paper is to study the growth dynamics of an ecosystem consisting 
of one semelparous species and an iteroparous species with two age classes in each 
species. In addition, we combined the density-dependent effect of the first age class for 
the two species. Then, we consider the effect of harvesting that occurs in the second age 
class in each species on the growth of each species. Next, we divide the case into two 
models consisting of without competition and with competition. Both models were 
analyzed and seen the influence of the level of intraspecific and interspecific competition 
on the equilibrium point and locally stable asymptotically for each equilibrium point. 

 

METHODS 

Leslie's Matrix Model with One Iteroparous Species and One Semelparous Species 
Without Competition 

In this section, we present one of the models that we studied, namely the multispecies 
Leslie matrix model with the case of one iteroparous species (𝑥) and one semelparous 
species (𝑦) in this case each species has two age classes. In this first model, we assume 
that the growth of both species is influenced by density-dependent occurrence in the first 
age class and harvesting is carried out in the second age class. In this case, the density-
dependent problem used in the model uses the classical Beverton-Holt function which is 
also used in Wikan's research [21]. This problem is presented in the following model and 
we refer to as Model 1: 

                𝑥1(𝑡 + 1) =
𝑓𝑥1

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡)  

𝑥2(𝑡 + 1) =
𝑠𝑥1(1 − ℎ𝑥2)

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑥1(𝑡)                  (1) 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 447 

𝑦1(𝑡 + 1) = 𝑓𝑦2𝑦2(𝑡)  

            𝑦2(𝑡 + 1) =
𝑠𝑦1(1 − ℎ𝑦2)

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑦1(𝑡)  

There are several parameters in the Model 1. First, 𝑓𝑥1 > 0 and 𝑓𝑥2 > 0 are the birth rates 
of the 1st and 2nd age classes of species 𝑥, respectively. Second,  𝑓𝑦2 > 0 is the birth rate 

of the 2nd age classes of species 𝑦. Third, 0 < 𝑠𝑥1, 𝑠𝑦1 < 1 are the survival rates of the 1st 

age classes of species 𝑥 and 𝑦, respectively. Fourth, 0 < ℎ𝑥2, ℎ𝑦2 < 1 are the harvesting 

rates of the 2nd age classes of species 𝑥 and 𝑦, respectively. In addition, the variables 𝑥𝑖(𝑡), 
and 𝑦𝑖(𝑡) represent the total population of each species 𝑥 and 𝑦 for the age class 𝑖 = {1,2}. 
Simply put, equation 1 in (1) means that the population of the first age class of species 𝑥 
at time 𝑡 + 1 is obtained by adding the number of newborn from the first age class and the 
second class at time 𝑡. The newborn of first age class is affected by density-dependent 
while the newborn of the second age class is not affected by density-dependent. Then, 
equation 2 in (1) means that the number of population of the second age class of species 
𝑥 at time 𝑡 + 1 is obtained from the number of surviving populations which is influenced 
by density-dependent of the first age class at time 𝑡. Equations 3 and 4 in (1) have the 
same meaning as equations 1 and 2 in (1) but in species 𝑦 there is no birth in the first age 
class. Model 1 is constructed based on research conducted by Leslie [7], Travis et al. [11], 
and Wikan [21]. In addition, Model 1 is adjusted based on the assumptions and 
simplifications in this research. 
 

Leslie Matrix Model with One Iteroparous Species and One Semelparous Species 
with Competition Effect 

In this section, we present a model which is an extension of the previous model. The 
problems raised in this section involve the effect of competition between the same 
species, also known as intraspecific competition, and competition between different 
species, also known as interspecific competition. The following is an extension of the 
Model 1 and we refer to it as Model 2. 

                𝑥1(𝑡 + 1) =
𝑓𝑥1

1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡)
𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡)  

𝑥2(𝑡 + 1) =
𝑠𝑥1(1 − ℎ𝑥2)

1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡)
𝑥1(𝑡)                  (2) 

𝑦1(𝑡 + 1) = 𝑓𝑦2𝑦2(𝑡)  

            𝑦2(𝑡 + 1) =
𝑠𝑦1(1 − ℎ𝑦2)

1 + 𝑏𝑥1(𝑡) + 𝑎𝑦1(𝑡)
𝑦1(𝑡)  

The description of the parameters and variables in the Model 2 is the same as in Model 1 
where the difference is only in parameters 𝑎 > 0 and 𝑏 > 0. In Model 2, to simplify the 
problem, we assume that the level of competition between the first age class in species 𝑥 
and species 𝑦 has the same value, namely 𝑎. Then, the level of competition between the 
first age class in species 𝑦 against species 𝑥 and vice versa has the same value, namely 𝑏. 
The level of competition within the same species is referred to as intraspecific 
competition (𝑎) and the level of competition between different species is referred to as 
interspecific competition (𝑏). Details of Model 2 are the same as Model 1 but there are 
differences in the first age class birth rate, first age class survival rate and second age class 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 448 

harvesting rate in species 𝑥 which are influenced by density-dependent and competition. 
Then, species 𝑦 is only affected by density-dependent and competition on the survival rate 
of the first age class and the level of harvesting of the second age class. Model 2 is 
constructed based on research conducted by Leslie [7], Travis et al. [11], Wikan [21], and 
Cushing [22]. In addition, Model 2 is adjusted based on the assumptions and 
simplifications in this research. 
 

Local Stability Criteria Using M-Matrix 

In this section, we present the definition of M-matrix and the theorem that ensures a 
matrix has absolute eigenvalues less than one to determine the locally stable 
asymptotically of the model. 

 

Definition 1. (See [11] or [23]) (M-Matrix) 

A square matrix of size 𝑛, for example, 𝑀 = (𝑚𝑖𝑗) (1 ≤ 𝑖, 𝑗 ≤ 𝑛) is called an M-matrix 

if it is satisfied that 𝑚𝑖𝑗 ≤ 0 ∀𝑖 ≠ 𝑗 and if any of the following things are true: 

1. All minor principals of the 𝑀 matrix are positive 
2. All eigenvalues of the 𝑀 matrix have a positive real part 
3. The matrix 𝑀 is a non-singular matrix and 𝑀−1 is positive 
4. There is a vector 𝑣 > 0 so that it meets 𝑀𝑣 > 0 or 
5. There is a vector 𝑢 > 0 s so that it satisfies 𝑀𝑇𝑢 > 0 
 

Theorem 1. (See [11]) 

Suppose a matrix 𝐽 has the following form 

              𝐽 = [
𝐴𝑚×𝑚 𝐵𝑚×𝑛
𝐶𝑛×𝑚 𝐷𝑛×𝑛

]  

and the matrix 𝐺 = 𝐼 − 𝑆𝐽𝑆−1 is an M-matrix with 
              𝑆 = 𝐼 if 𝐵 and 𝐶 ≥ 0  

or  

              𝑆 = [
𝐼𝑚 0
0 −𝐼𝑛

] if 𝐵 and 𝐶 ≤ 0,  

where 𝐼, 𝐼𝑚, and 𝐼𝑛 are identity matrices with sizes 𝑚 + 𝑛, 𝑚, and 𝑛, respectively, then 
matrix 𝐽 has a spectral radius of less than one. The spectral radius is the largest modulus 
of all the eigenvalues. 

Theorem 1 and Definition 1 are used to determine the locally stable asymptotically of 
Model 1 and Model 2 in the Results and Discussion section. 

RESULTS AND DISCUSSION  

In the previous section, we have presented Model 1, Model 2, Definition 1 and Theorem 
1. In this section, the two models, definition, and theorem will then be used to analyze the 
equilibrium point and the locally stable asymptotically of each equilibrium point. 

 

Equilibrium Point of Model 1 

The equilibrium point of Model 1 can be obtained by expressing the variables 𝑥 and 𝑦 
to the left of the Model 1 depending on time 𝑡. The equilibrium Model 1 is obtained as 
follows: 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 449 

                𝑥1(𝑡) =
𝑓𝑥1

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡)  

𝑥2(𝑡) =
𝑠𝑥1(1 − ℎ𝑥2)

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑥1(𝑡)                  (3) 

𝑦1(𝑡) = 𝑓𝑦2𝑦2(𝑡)  

            𝑦2(𝑡) =
𝑠𝑦1(1 − ℎ𝑦2)

1 + 𝑥1(𝑡) + 𝑦1(𝑡)
𝑦1(𝑡)  

Then by determining the solution of equation (3), the equilibrium point of the Model 
1 is obtained as follows: 

1. The equilibrium point with both species going extinct is 

            𝐸0 = [

𝑥1
𝑥2
𝑦1
𝑦2

] = [

0
0
0
0

] .  

2. The equilibrium point with species 𝑥 exists while species 𝑦 is extinct, i.e 

            𝐸𝑥 = [

𝑥1
𝑥2
𝑦1
𝑦2

] =

[
 
 
 
 

𝑅𝑥 − 1
(𝑅𝑥 − 1)𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑥
0
0 ]

 
 
 
 

.  

The condition 𝐸𝑥 exists if it is fulfilled 𝑅𝑥 = 𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 > 1. 𝑅𝑥 is referred 
to as the expected number of offspring per individual per lifetime when density-
dependent effects are neglected on harvest-influenced growth of species 𝑥. 

3. The equilibrium point with species 𝑦 exists while species 𝑥 is extinct, i.e 

            𝐸𝑦 = [

𝑥1
𝑥2
𝑦1
𝑦2

] =

[
 
 
 
 
 

0
0

𝑅𝑦 − 1

𝑅𝑦 − 1

𝑓𝑦2 ]
 
 
 
 
 

.  

The condition 𝐸𝑦 exists if it is fulfilled 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) > 1. 𝑅𝑦 is referred to as 

the expected number of offspring per individual per lifetime when density-
dependent effects are neglected on harvest-influenced growth of species 𝑦. 

It can be seen that in the model (1) there is no equilibrium point where the two species 
survive or co-existence equilibrium point. 
 

Locally Stable Asymptotically at the Equilibrium Point of Model 1 

In this section, we perform a locally stable asymptotically analysis of the Model 1. The 
equilibrium point is said to be asymptotically stable if it is satisfied that the spectral radius 
of the Jacobian matrix at the equilibrium point is less than one. The locally stable 
asymptotically of each equilibrium point of the Model 1 is stated in the following theorem. 

 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 450 

Theorem 2. (Locally Stable Asymptotically at the Equilibrium Point Model 1) 

For the Leslie multispecies matrix model with the case of one iteroparous species and one 
semelparous species in which there are two classes each whose growth is influenced by 
density-dependent, harvesting and without the influence of competition described in the 
Model 1, among others: 

1. The equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1 and 𝑅𝑦 < 1. 

2. The equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. 

3. The equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 𝑅𝑥 and 𝑅𝑦 > 1. 

Proof : 
In determining the stability of all equilibrium points of the Model 1, it can be obtained by 
determining the linearization of the Model 1, namely  

         𝐽(𝐸) = 𝐽 ([

𝑥1
𝑥2
𝑦1
𝑦2

]) =

[
 
 
 
 
 

𝑓𝑥1(1 + 𝑦1)

(1 + 𝑥1 + 𝑦1)
2

𝑓𝑥2 −
𝑓𝑥1𝑥1

(1 + 𝑥1 + 𝑦1)
2

0

𝑃𝑥(1 + 𝑦1) 0 −𝑃𝑥𝑥1 0
0 0 0 𝑓𝑦2

−𝑃𝑦𝑦1 0 𝑃𝑦(1 + 𝑥1) 0 ]
 
 
 
 
 

(4) 

where 

            𝑃𝑥 =
𝑠𝑥1(1 − ℎ𝑥2)

(1 + 𝑥1 + 𝑦1)
2
 and 𝑃𝑦 =

𝑠𝑦1(1 − ℎ𝑦2)

(1 + 𝑥1 + 𝑦1)
2
.  

The reason for using the M-Matrix theory in this study is due to the complexity of 
determining the spectral radius matrix 𝐽(𝐸) at the corresponding equilibrium point 𝐸. The 
use of Definition 1 and Theorem 1 guarantee that the spectral radius value of the 𝐽(𝐸) 
matrix at the corresponding equilibrium is less than one. 

Based on the 𝐽(𝐸) matrix in (4) the elements of rows 1-2 columns 3-4 and rows 3-4 
columns 1-2 are non-positive because the elements of the equilibrium point are 
guaranteed to be zero or positive. Theorem 1 says matrix 𝑆 for matrix 𝐽(𝐸) is 

            𝑆 = [

1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1

] .  

The next step is to substitute all the equilibrium points in (4) and analyze their stability 
one by one. 

1. The Jacobian matrix for the equilibrium point 𝐸0 is  

            𝐽(𝐸0) =

[
 
 
 

𝑓𝑥1 𝑓𝑥2 0 0
𝑠𝑥1(1 − ℎ𝑥2) 0 0 0

0 0 0 𝑓𝑦2
0 0 𝑠𝑦1(1 − ℎ𝑦2) 0 ]

 
 
 

.  

Next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸0))𝑆
−1 and examine the element 𝑔𝑖𝑗 for  

𝑖 ≠ 𝑗 that is non-positive and that all minor principals of 𝐺 are positive. If these 
conditions are met, then the 𝐺 matrix is an M-Matrix so that the spectral radius 
𝐽(𝐸0) is less than one. As a result, the equilibrium point 𝐸0 is locally stable 
asymptotically. Here we present the obtained matrix 

            𝐺 =

[
 
 
 

1 − 𝑓𝑥1 −𝑓𝑥2 0 0
−𝑠𝑥1(1 − ℎ𝑥2) 1 0 0

0 0 1 −𝑓𝑦2
0 0 −𝑠𝑦1(1 − ℎ𝑦2) 1 ]

 
 
 

 

and all minor principals of 𝐺 obtained are 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 451 

           𝑃𝑀1 = |𝑔11| = 1 − 𝑓𝑥1, 𝑃𝑀2 = |
𝑔11 𝑔12
𝑔21 𝑔22

| = 1 − 𝑅𝑥,  

          𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| = 1 − 𝑅𝑥, and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑥)(1 − 𝑅𝑦).  

In this paper, we define 𝑃𝑀𝑖 (𝑖 = 1,2,3,4) as the 𝑖-th minor principal of the matrix 
𝐺 for each equilibrium point under consideration. It is clear that the element 𝑔𝑖𝑗 <

0 for 𝑖 ≠ 𝑗 in the 𝐺 matrix is nonpositive by recalling the previously defined 
parameters. Then, 𝑃𝑀2 and 𝑃𝑀4 will be positive if met 𝑅𝑥 < 1. In addition, 𝑅𝑥 < 1 
implicitly results in 𝑓𝑥1 < 1 so that 𝑃𝑀1 > 0. Furthermore, 𝑃𝑀4 is positive if 𝑅𝑦 <

1 because it must be fulfilled that 𝑅𝑥 < 1. Therefore, 𝐺 is an M-Matrix, that is if it is 
filled with 𝑅𝑥 < 1 and 𝑅𝑦 < 1. Hence, according to Theorem 1, the equilibrium 

point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1 and 𝑅𝑦 < 1. 

2. The Jacobian matrix for the equilibrium point 𝐸𝑥 is 

            𝐽(𝐸𝑥) =

[
 
 
 
 
 
 
 

𝑓𝑥1
𝑅𝑥

2
𝑓𝑥2

𝑓1(1 − 𝑅𝑥)

𝑅𝑥
2 

0

𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑥
2

0
𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥)

𝑅𝑥
2

0

0 0 0 𝑓𝑦2

0 0
𝑠𝑦1(1 − ℎ𝑦2)

𝑅𝑥
0

]
 
 
 
 
 
 
 

 

Next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸𝑥))𝑆
−1 and make sure the matrix 𝐺 is an 

M-Matrix. Here we present the obtained matrix 

            𝐺 =

[
 
 
 
 
 
 

𝑅𝑥
2−𝑓𝑥1

𝑅𝑥
2 

−𝑓𝑥2
𝑓1(1−𝑅𝑥)

𝑅𝑥
2 

0

−𝑠𝑥1(1−ℎ𝑥2)

𝑅𝑥
2 1

𝑠𝑥1(1−ℎ𝑥2)(1−𝑅𝑥)

𝑅𝑥
2 0

0 0 1 −𝑓𝑦2

0 0 −
𝑠𝑦1(1−ℎ𝑦2)

𝑅𝑥
1 ]

 
 
 
 
 
 

 

and all minor principals of 𝐺 obtained are 

           𝑃𝑀1 = |𝑔11| =
𝑅𝑥

2 − 𝑓𝑥1
𝑅𝑥

2 
, 𝑃𝑀2 = |

𝑔11 𝑔12
𝑔21 𝑔22

| = −
1 − 𝑅𝑥

𝑅𝑥
,  

           𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| = −
1 − 𝑅𝑥

𝑅𝑥
, and 𝑃𝑀4 = |𝐺| = −

(1 − 𝑅𝑥)(𝑅𝑥 − 𝑅𝑦)

𝑅𝑥
.  

Based on the previously defined parameters, the element 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 will be non-

positive if 𝑅𝑥 > 1 is satisfied. Because of 𝑅𝑥 > 1, consequently 𝑃𝑀2 and 𝑃𝑀3 are 
positive. Besides that, 𝑃𝑀4 is also positive but with the additional condition that is 
𝑅𝑥 > 𝑅𝑦. Then, it is clear that 𝑓𝑥1 < 𝑅𝑥

2 = (𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1)
2 so that 𝑃𝑀1 >

0. Therefore, 𝐺 is an M-Matrix, if it is fulfilled 𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. Hence, 

according to Theorem 1, the equilibrium point 𝐸𝑥 is locally stable asymptotically 
if  𝑅𝑥 > 𝑅𝑦 and 𝑅𝑥 > 1. 

  



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 452 

3. The Jacobian matrix for the equilibrium point 𝐸𝑦 is 

            𝐽(𝐸𝑦) =

[
 
 
 
 
 
 
 
 

𝑓𝑥1
𝑅𝑦

𝑓𝑥2 0 0

𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑦
0 0 0

0 0 0 𝑓𝑦2

−
𝑠𝑦1(1 − ℎ𝑦2)(𝑅𝑦 − 1)

𝑅𝑦
2

0
1

𝑓4𝑅𝑦
0

]
 
 
 
 
 
 
 
 

 

Next, determine the matrix 𝐺 = 𝐼 − 𝑆 (𝐽(𝐸𝑦)) 𝑆
−1 and make sure the matrix 𝐺 is 

an M-Matrix. Here we present the obtained matrix 

            𝐺 =

[
 
 
 
 
 
 
 

𝑅𝑦 − 𝑓𝑥1

𝑅𝑦 
−𝑓𝑥2 0 0

−𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑦
1 0 0

0 0 1 −𝑓𝑦2
𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦)

𝑅𝑦
2

0 −
1

𝑓4𝑅𝑦
1

]
 
 
 
 
 
 
 

 

and all minor principals of 𝐺 obtained are 

𝑃𝑀1 = |𝑔11| =
𝑅𝑦 − 𝑓𝑥1

𝑅𝑦
, 𝑃𝑀2 = |

𝑔11 𝑔12
𝑔21 𝑔22

| =
𝑅𝑦 − 𝑅𝑥

𝑅𝑦
,  

𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| =
𝑅𝑦 − 𝑅𝑥

𝑅𝑦
, and 𝑃𝑀4 = |𝐺| =

(1 − 𝑅𝑦)(𝑅𝑥 − 𝑅𝑦)

𝑅𝑦
.  

Note that all elements 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are nonpositive except for 𝑔41. Then, 𝑔41 will be 

negative if 𝑅𝑦 > 1. Next, focus on the minor principal terms of the 𝐺 matrix. 𝑃𝑀2 

and 𝑃𝑀3 are positive if 𝑅𝑦 > 𝑅𝑥. Because of 𝑅𝑦 > 1 and 𝑅𝑦 > 𝑅𝑥, consequently 

𝑃𝑀4 are positive. Then, since 𝑅𝑦 > 𝑅𝑥 where 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) and 𝑅𝑥 =

𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 it follows that 𝑃𝑀1 is positive because 𝑅𝑦 > 𝑓𝑥1. Hence, 𝐺 is 

an 𝑀-Matrix and the equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 >

𝑅𝑥 and 𝑅𝑦 > 1. 

∎ 

 
Figure 1. Population growth graph for each age class of each species 𝑥 and 𝑦 in case i Model 1 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 453 

Equilibrium Point of Model 2 

With the same treatment as determining the equilibrium point in the Model 1, the 
equilibrium Model 2 is obtained as follows: 

                𝑥1(𝑡) =
𝑓𝑥1

1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡)
𝑥1(𝑡) + 𝑓𝑥2𝑥2(𝑡)  

𝑥2(𝑡) =
𝑠𝑥1(1 − ℎ𝑥2)

1 + 𝑎𝑥1(𝑡) + 𝑏𝑦1(𝑡)
𝑥1(𝑡)                  (5) 

𝑦1(𝑡) = 𝑓𝑦2𝑦2(𝑡)  

            𝑦2(𝑡) =
𝑠𝑦1(1 − ℎ𝑦2)

1 + 𝑏𝑥1(𝑡) + 𝑎𝑦1(𝑡)
𝑦1(𝑡)  

Then, there are four equilibrium points from the Model 2, namely  
1. The equilibrium point with both species going extinct is 

            𝐸0 = [

𝑥1
𝑥2
𝑦1
𝑦2

] = [

0
0
0
0

] .  

2. The equilibrium point with species 𝑥 exists while species 𝑦 is extinct, i.e  

            𝐸𝑥 = [

𝑥1
𝑥2
𝑦1
𝑦2

] =

[
 
 
 
 
 

𝑅𝑥 − 1

𝑎
(𝑅𝑥 − 1)𝑠𝑥1(1 − ℎ𝑥2)

𝑎𝑅𝑥
0
0 ]

 
 
 
 
 

.  

The condition 𝐸𝑥 exists if it is fulfilled 𝑅𝑥 = 𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1 > 1. 

3. The equilibrium point with species 𝑦 exists while species 𝑥 is extinct, i.e 

            𝐸𝑦 = [

𝑥1
𝑥2
𝑦1
𝑦2

] =

[
 
 
 
 
 

0
0

𝑅𝑦 − 1

𝑎
𝑅𝑦 − 1

𝑎𝑓𝑦2 ]
 
 
 
 
 

.  

The condition 𝐸𝑦 exists if it is fulfilled 𝑅𝑦 = 𝑓𝑦2𝑠𝑦1(1 − ℎ𝑦2) > 1. 

4. The equilibrium point with species 𝑥 and 𝑦 exists if one of them is satisfied, namely 
𝑎2 > 𝑏2 or 𝑎 > 𝑏, 𝑎(𝑅𝑦 − 1) > 𝑏(𝑅𝑥 − 1), and 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), or 𝑎

2 < 𝑏2 or 

𝑎 < 𝑏,  𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1), and 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1) with  



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 454 

            𝐸𝑥𝑦 = [

𝑥1
𝑥2
𝑦1
𝑦2

] =

[
 
 
 
 
 
 
 
 
 

𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)

(𝑎2 − 𝑏2)

𝑠𝑥1(1 − ℎ𝑥2) (𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1))

(𝑎2 − 𝑏2)𝑅𝑥
 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1)

(𝑎2 − 𝑏2)

 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1)

(𝑎2 − 𝑏2)𝑓𝑦2 ]
 
 
 
 
 
 
 
 
 

.  

In this second model, we obtain four equilibrium points where an equilibrium point 
appears with all species existing or a co-existence equilibrium point. The level of 
competition in both species affects the existence of a co-existence equilibrium point. 

 

 
Figure 2. Population growth graph for each age class of each species 𝑥 and 𝑦 in case ii Model 1 

 

Locally Stable Asymptotically at the Equilibrium Point of Model 2 

This section discusses the locally stable asymptotically of Model 2 which is presented 
in Theorem 3 below.  

 

Theorem 3. (Locally Stable Asymptotically at the Equilibrium Point of Model 2) 

For the system in the case of one iteroparous species and one semelparous species, each 
of which consists of two classes whose growth is affected by density-dependent, 
harvesting and competition which is specifically described in the Model 2, among others: 

1. The equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 < 1, and 𝑅𝑦 < 1. 

2. The equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 1 and 𝑎(𝑅𝑦 − 1) <

𝑏(𝑅𝑥 − 1). 
3. The equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 1 and 𝑎(𝑅𝑥 − 1) <

𝑏(𝑅𝑦 − 1). 

4. The equilibrium point 𝐸𝑥𝑦 is locally stable asymptotically if 𝑎 > 𝑏, 𝑎(𝑅𝑦 − 1) >



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 455 

𝑏(𝑅𝑥 − 1), 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), and 𝑓𝑥1(𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑥 − 𝑎𝑅𝑦)) < (𝑎
2 −

𝑏2)𝑅𝑥
2. 

Proof: 
The steps to determine the stability of all equilibrium points of the Model 2 can be carried 
out as in Model 1. Linearization of the Model 2, namely  

         𝐽(𝐸) = 𝐽 ([

𝑥1
𝑥2
𝑦1
𝑦2

]) =

[
 
 
 
 
 

𝑓𝑥1(1 + 𝑏𝑦1)

(1 + 𝑎𝑥1 + 𝑏𝑦1)
2

𝑓𝑥2 −
𝑓𝑥1𝑥1𝑏

(1 + 𝑎𝑥1 + 𝑏𝑦1)
2

0

𝑃𝑥(1 + 𝑏𝑦1) 0 −𝑃𝑥𝑏𝑥1 0
0 0 0 𝑓𝑦2

−𝑃𝑦𝑏𝑦1 0 𝑃𝑦(1 + 𝑏𝑥1) 0 ]
 
 
 
 
 

(6) 

with 

            𝑃𝑥 =
𝑠𝑥1(1 − ℎ𝑥2)

(1 + 𝑎𝑥1 + 𝑏𝑦1)
2
 and 𝑃𝑦 =

𝑠𝑦1(1 − ℎ𝑦2)

(1 + 𝑏𝑥1 + 𝑎𝑦1)
2
.  

The elements of rows 1-2 columns 3-4 and rows 3-4 columns 1-2 in (6) are non-positive 
because the elements of the equilibrium point are guaranteed to be zero or positive, so 
Theorem 1 says matrix 𝑆 for matrix 𝐽(𝐸) is 

            𝑆 = [

1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 −1

] .  

The next step is to substitute all the equilibrium points in (6) and analyze its stability one 
by one. 

1. The Jacobian matrix for the equilibrium point 𝐸0 is  

            𝐽(𝐸0) =

[
 
 
 

𝑓𝑥1 𝑓𝑥2 0 0
𝑠𝑥1(1 − ℎ𝑥2) 0 0 0

0 0 0 𝑓𝑦2
0 0 𝑠𝑦1(1 − ℎ𝑦2) 0 ]

 
 
 

.  

Next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸0))𝑆
−1 and make sure the matrix 𝐺 is an 

M-Matrix. Here we present the obtained matrix 

            𝐺 =

[
 
 
 

1 − 𝑓𝑥1 −𝑓𝑥2 0 0
−𝑠𝑥1(1 − ℎ𝑥2) 1 0 0

0 0 1 −𝑓𝑦2
0 0 −𝑠𝑦1(1 − ℎ𝑦2) 1 ]

 
 
 

 

and all minor principals of 𝐺 obtained are 

             𝑃𝑀1 = |𝑔11| = 1 − 𝑓𝑥1, 𝑃𝑀2 = |
𝑔11 𝑔12
𝑔21 𝑔22

| = 1 − 𝑅𝑥,  

           𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| = 1 − 𝑅𝑥, and 𝑃𝑀4 = |𝐺| = (1 − 𝑅𝑥)(1 − 𝑅𝑦).  

By considering the matrix 𝐺, it is clear that the values of all 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are 

nonpositive. Next is focus on determining the conditions for 𝑃𝑀𝑖 > 0 (𝑖 = 1,2,3,4). 
𝑃𝑀2 and 𝑃𝑀3 are positive if 𝑅𝑥 < 1 is satisfied. Because of 𝑅𝑥 < 1 so that 𝑃𝑀1 is 
positive and an additional condition for 𝑃𝑀4 to be positive is 𝑅𝑦 < 1. Therefore, 𝐺 

is an M-matrix, and the equilibrium point 𝐸0 is locally stable asymptotically if 𝑅𝑥 <
1 and 𝑅𝑦 < 1. 

  



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 456 

2. The Jacobian matrix for the equilibrium point 𝐸𝑥 is 

            𝐽(𝐸𝑥) =

[
 
 
 
 
 
 
 

𝑓𝑥1
𝑅𝑥

2
𝑓𝑥2

𝑏𝑓1(1 − 𝑅𝑥)

𝑎𝑅𝑥
2 

0

𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑥
2

0
𝑏𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥)

𝑎𝑅𝑥
2

0

0 0 0 𝑓𝑦2

0 0
𝑎𝑠𝑦1(1 − ℎ𝑦2)

𝑎 + 𝑏(𝑅𝑥 − 1)
0

]
 
 
 
 
 
 
 

.  

Next, determine the matrix 𝐺 = 𝐼 − 𝑆(𝐽(𝐸𝑥))𝑆
−1 and make sure the matrix 𝐺 is an 

M-Matrix. Here we present the obtained matrix 

            𝐺 =

[
 
 
 
 
 
 
 

𝑅𝑥
2 − 𝑓𝑥1
𝑅𝑥

2 
−𝑓𝑥2

𝑏𝑓1(1 − 𝑅𝑥)

𝑎𝑅𝑥
2 

0

−𝑠𝑥1(1 − ℎ𝑥2)

𝑅𝑥
2

1
𝑏𝑠𝑥1(1 − ℎ𝑥2)(1 − 𝑅𝑥)

𝑎𝑅𝑥
2

0

0 0 1 −𝑓𝑦2

0 0 −
𝑎𝑠𝑦1(1 − ℎ𝑦2)

𝑎 + 𝑏(𝑅𝑥 − 1)
1

]
 
 
 
 
 
 
 

 

and all minor principals of 𝐺 obtained are 

𝑃𝑀1 =  |𝑔11| =
𝑅𝑥

2 − 𝑓𝑥1
𝑅𝑥

2 
, 𝑃𝑀2 = |

𝑔11 𝑔12
𝑔21 𝑔22

| = −
1 − 𝑅𝑥

𝑅𝑥
,  

𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| = −
1 − 𝑅𝑥

𝑅𝑥
, and 𝑃𝑀4 = |𝐺| =

(1 − 𝑅𝑥) (𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1))

𝑎 + 𝑏(𝑅𝑥 − 1)
. 

In the matrix 𝐺, it can be seen that all 𝑔𝑖𝑗 for 𝑖 ≠ 𝑗 are non-positive because 𝑅𝑥 > 1 

which is a condition for 𝐸𝑥 to exist. Therefore, the next step is to focus on 
determining the positive terms of the minor principal of the 𝐺 matrix. It is clear 
that 𝑃𝑀2 and 𝑃𝑀3 are positive because 𝑅𝑥 > 1. Then, it is clear that 𝑃𝑀1 is positive 
because in fact 𝑓𝑥1 < 𝑅𝑥

2 = (𝑓𝑥2𝑠𝑥1(1 − ℎ𝑥2) + 𝑓𝑥1)
2. Since 𝑅𝑥 > 1, 𝑃𝑀4 is positive 

if 𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1) < 0 or 𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1). Therefore, 𝐺 is an M-

matrix, and the equilibrium point 𝐸𝑥 is locally stable asymptotically if 𝑅𝑥 > 1 and 
𝑎(𝑅𝑦 − 1) < 𝑏(𝑅𝑥 − 1). 

3. The Jacobian matrix for the equilibrium point 𝐸𝑦 is 

            𝐽(𝐸𝑦) =

[
 
 
 
 
 
 
 
 

𝑎𝑓𝑥1

𝑎 + 𝑏(𝑅𝑦 − 1)
𝑓𝑥2 0 0

𝑎𝑠𝑥1(1 − ℎ𝑥2)

𝑎 + 𝑏(𝑅𝑦 − 1)
0 0 0

0 0 0 𝑓𝑦2

𝑏𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦)

𝑎𝑅𝑦
2

0
1

𝑓4𝑅𝑦
0

]
 
 
 
 
 
 
 
 

.  

Next, determine the matrix 𝐺 = 𝐼 − 𝑆 (𝐽(𝐸𝑦)) 𝑆
−1 and make sure the matrix 𝐺 is 

an M-Matrix. Here we present the obtained matrix 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 457 

            𝐺 =

[
 
 
 
 
 
 
 
 −

𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1)

𝑎 + 𝑏(𝑅𝑦 − 1) 
−𝑓𝑥2 0 0

−𝑎𝑠𝑥1(1 − ℎ𝑥2)

𝑎 + 𝑏(𝑅𝑦 − 1)
1 0 0

0 0 1 −𝑓𝑦2
𝑏𝑠𝑦1(1 − ℎ𝑦2)(1 − 𝑅𝑦)

𝑎𝑅𝑦
2

0 −
1

𝑓4𝑅𝑦
1

]
 
 
 
 
 
 
 
 

.  

and all minor principals of 𝐺 obtained are 

           𝑃𝑀1 = |𝑔11| = −
𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1)

𝑎 + 𝑏(𝑅𝑦 − 1)
,  

          𝑃𝑀2 = |
𝑔11 𝑔12
𝑔21 𝑔22

| = −
𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)

𝑎 + 𝑏(𝑅𝑦 − 1)
,  

          𝑃𝑀3 =  |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| = −
𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)

𝑎 + 𝑏(𝑅𝑦 − 1)
,  

and 

           𝑃𝑀4 = |𝐺| =
(𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)) (1 − 𝑅𝑦)

𝑎 + 𝑏(𝑅𝑦 − 1) 
.  

The equilibrium point of 𝐸𝑦 is exist if 𝑅𝑦 > 1 consequently all elements of 𝑔𝑖𝑗 for 

𝑖 ≠ 𝑗 are non-positive. Next is the focus on determining the conditions so that all 
the principal minor matrices 𝐺 are positive. Since 𝑅𝑦 > 1, so we have 𝑃𝑀2, 𝑃𝑀3, 

and 𝑃𝑀4 are positive if  𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1) < 0 or 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1). In 

addition, 𝑎(𝑅𝑥 − 1) < 𝑏(𝑅𝑦 − 1) the result is satisfied 𝑎(𝑓𝑥1 − 1) − 𝑏(𝑅𝑦 − 1) < 0 

or 𝑎(𝑓𝑥1 − 1) < 𝑏(𝑅𝑦 − 1). Then, because of 𝑅𝑦 > 1 and  𝑎(𝑓𝑥1 − 1) < 𝑏(𝑅𝑦 −

1) so that 𝑃𝑀1 is fulfilled with a positive value. Therefore, 𝐺 is an M-matrix and the 

equilibrium point 𝐸𝑦 is locally stable asymptotically if 𝑅𝑦 > 1 and 𝑎(𝑅𝑥 − 1) <

𝑏(𝑅𝑦 − 1). 

4. The Jacobian matrix for the equilibrium point 𝐸𝑥𝑦 is 

            𝐽(𝐸𝑥𝑦) =

[
 
 
 
 
 
 
 

𝐴𝑥𝑓𝑥1

𝐶𝑅𝑥
2

𝑓𝑥2 −
𝐵𝑥𝑏𝑓𝑥1

𝐶𝑅𝑥
2

0

𝑠𝑥1(1 − ℎ𝑥2)𝐴𝑥

𝐶𝑅𝑥
2

0 −
𝑏𝑠𝑥1(1 − ℎ𝑥2)𝐵𝑥

𝐶𝑅𝑥
2

0

0 0 0 𝑓𝑦2

−
𝑏𝐵𝑦 

𝐶𝑅𝑦𝑓𝑦2
0

𝐴𝑦

𝐶𝑓𝑦2𝑅𝑦
0

]
 
 
 
 
 
 
 

.  

Next is the 𝐺 matrix for the equilibrium point 𝐸𝑥𝑦 is 

            𝐺 =

[
 
 
 
 
 
 
 1 −

𝐴𝑥𝑓𝑥1

𝐶𝑅𝑥
2

−𝑓𝑥2 −
𝐵𝑥𝑏𝑓𝑥1

𝐶𝑅𝑥
2

0

𝐴𝑥𝑠𝑥1(ℎ𝑥2 − 1)

𝐶𝑅𝑥
2

1 −
𝑏𝑠𝑥1(1 − ℎ𝑥2)𝐵𝑥

𝐶𝑅𝑥
2

0

0 0 1 −𝑓𝑦2

−
𝑏𝐵𝑦

𝐶𝑅𝑦𝑓𝑦2
0 −

𝐴𝑦

𝐶𝑅𝑦𝑓𝑦2
1

]
 
 
 
 
 
 
 

 

and the principal minor of the matrix 𝐺 are 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 458 

𝑃𝑀1 =  |𝑔11| = 1 −
𝐴𝑥𝑓𝑥1
𝐶𝑅𝑥

2
, 𝑃𝑀2 =  |

𝑔11 𝑔12
𝑔21 𝑔22

| =
𝑎𝐵𝑥
𝐶𝑅𝑥 

 

𝑃𝑀3 = |

𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33

| =
𝑎𝐵𝑥
𝐶𝑅𝑥

, and      𝑃𝑀4 = |𝐺| =
𝐵𝑥𝐵𝑦

𝐶𝑅𝑥𝑅𝑦 
 

where 

          𝐴𝑥 = (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑥 − 𝑎𝑅𝑦)), 𝐴𝑦 = (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑦 − 𝑎𝑅𝑥))  

      𝐵𝑥 = (𝑎(𝑅𝑥 − 1) − 𝑏(𝑅𝑦 − 1)) , 𝐵𝑦 = (𝑎(𝑅𝑦 − 1) − 𝑏(𝑅𝑥 − 1)) , and 𝐶 = (𝑎
2 − 𝑏

2).  

In this case, the conditions that meet the requirements will be determined so that 
𝐺 is called the M-Matrix. First focus on 𝑃𝑀4 is positive. Because of 𝐸𝑥𝑦 exists if 

𝐵𝑥, 𝐵𝑦, 𝐶 > 0 or 𝐵𝑥, 𝐵𝑦, 𝐶 < 0. However, 𝑃𝑀4 is positive if it is fulfilled 𝐵𝑥, 𝐵𝑦, 𝐶 > 0. 

Because  𝐵𝑥, 𝐵𝑦, 𝐶 > 0 consequently fulfilled 𝑔13, 𝑔23, 𝑔31 < 0, and 𝑃𝑀2, 𝑃𝑀3 > 0. 

Then, 𝑃𝑀1 is positive if 𝑓𝑥1𝐴𝑥 < 𝐶𝑅𝑥
2. Finally, all the conditions for 𝐺 to be called 

an M-matrix have been fulfilled. Therefore, 𝐺 is an M-matrix, and the equilibrium 

point 𝐸𝑥𝑦 is locally stable asymptotically if 𝑓𝑥1 (𝑎(𝑎 − 𝑏) − 𝑏(𝑏𝑅𝑦 − 𝑎𝑅𝑥)) <

(𝑎2 − 𝑏2)𝑅𝑥
2, 𝑎(𝑅𝑦 − 1) > 𝑏(𝑅𝑥 − 1), 𝑎(𝑅𝑥 − 1) > 𝑏(𝑅𝑦 − 1), and 𝑎

2 > 𝑏2 or 𝑎 >

𝑏. 
∎ 

Numerical Simulations of Model 1 and Model 2 

In the previous subsection, an analysis of the existing condition and local stability 
asymptotically from each equilibrium point has been carried out on Model 1 and Model 2. 
In this section, we perform a numerical simulation of the results from the analysis of 
Model 1 and Model 2. In this case, we divide the two models into two cases and each case 
is divided into as many subcases as the asymptotically local stability conditions of 
Theorem 2 and Theorem 3. 

In the simulation of Model 1, we assume for all subcases of the Model 1 case, including: 
1. 𝑠𝑥1 = 0.6 and 𝑠𝑦1 = 0.4 respectively that if there are 10 individuals in the first age 

class of species 𝑥 and 𝑦 then only 6 individuals and 4 individuals are able to survive 
from species 𝑥 and 𝑦. 

2. ℎ𝑥2 = 0.5 and ℎ𝑦2 = 0.3 respectively that if there are 10 individuals in the first age 

class of species 𝑥 and 𝑦 then there are only 5 individuals and 3 individuals 
harvested from species 𝑥 and 𝑦. 

 
Figure 3. Population growth graph for each age class of each species 𝑥 and 𝑦 in case iii Model 1 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 459 

Then, the birth rate for the simulation in the Model 1 case is divided into 3 subcases based 
on Theorem 1, including: 

i. 𝑓𝑥1 = 0.7, 𝑓𝑥2 = 0.9, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 0.97 < 1 and 𝑅𝑦 = 0.84 < 1. 

ii. 𝑓𝑥1 = 1, 𝑓𝑥2 = 5, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 2.5 > 1 and 𝑅𝑦 = 0.84 < 𝑅𝑥. 

iii. 𝑓𝑥1 = 0.7, 𝑓𝑥2 = 0.9, and 𝑓𝑦2 = 10 consequently 𝑅𝑥 = 0.97 < 1 and 𝑅𝑦 = 2.8 > 𝑅𝑥. 

The results of the Model 1 simulation for each subcase i-iii are presented in Figure 1-
3. Figures 1-3 respectively for the parameters given in each subcase i-iii of the Model 1 
simulation show that the locally stable asymptotically towards the equilibrium point 𝐸0 =
[0,0,0,0]𝑇, 𝐸𝑥 = [1.5,0.18,0,0]

𝑇, and 𝐸𝑦 = [0,0,1.8,0.18]
𝑇. 

In the simulation of Model 2, we assume for all subcases of Model 2 for survival and 
harvesting rates are equal to Model 1. Then, the levels of intraspecific and interspecific 
competition are 𝑎 = 0.2 and 𝑏 = 0.1, respectively. Then, the birth rate for the simulation 
in the Model 1 case is divided into 4 subcases based on Theorem 2, including: 

i. 𝑓𝑥1 = 0.5, 𝑓𝑥2 = 1, and 𝑓𝑦2 = 3 consequently 𝑅𝑥 = 0.8 and 𝑅𝑦 = 0.84. 

ii. 𝑓𝑥1 = 30, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 60 consequently 𝑅𝑥 = 36 and 𝑅𝑦 = 16.8. 

iii. 𝑓𝑥1 = 5, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 100 consequently 𝑅𝑥 = 11 and 𝑅𝑦 = 28. 

iv. 𝑓𝑥1 = 20, 𝑓𝑥2 = 20, and 𝑓𝑦2 = 60 consequently 𝑅𝑥 = 26 and 𝑅𝑦 = 16.8. 

 

 
Figure 4. Population growth graph for each age class of each species 𝑥 and 𝑦 in case i Model 2 

 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 460 

 
Figure 5. Population growth graph for each age class of each species 𝑥 and 𝑦 in case ii Model 2 

 
Figure 4-7 is the simulation result of Model 2 for each subcase i-iv. Figure 4-7 for each 

parameter that satisfies Theorem 2 conditions in subcases i-iv of the Model 2 simulation 
that sequentially locally stable asymptotically towards the equilibrium point 𝐸0 =
[0,0,0,0]𝑇, 𝐸𝑥 = [175,1.46,0,0]

𝑇, 𝐸𝑦 = [0,0,135,1.35]
𝑇 dan 𝐸𝑥𝑦 = [114,1.32,22,0.36]

𝑇. 

 

 
Figure 6. Population growth graph for each age class of each species 𝑥 and 𝑦 in case iii Model 2 

 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 461 

 
Figure 7. Population growth graph for each age class of each species 𝑥 and 𝑦 in case iv Model 2 

 

CONCLUSIONS 

In this paper, we compare two different models: Model 1 and Model 2. Our focus is to 
compare the presence and absence of the influence of intraspesific and interspecific 
competition in the equilibrium point and its local stability of Model 1 and Model 2. 
Mathematically the conditions under which the positive/non-trivial equilibrium point 
exists and the local stability of this equilibrium of the model is easy to interpret. However, 
biologically only some conditions can be interpreted because of the complexity of 
conditions. Simply put, the results of our study show that the level of competition has a 
role in the equilibrium point and its local stability of the Model 1 and Model 2. Model 1 
shows that there is no coexistence equilibrium point so model 1 is never locally stable at 
the point where both species exist. In Model 2, one of the conditions that is easily 
interpreted is that the coexistence equilibrium point occurs when 𝑎 > 𝑏 which means the 
intensity of the intraspecific competition level is greater than the intensity of the 
interspecific level competition. The inequality of 𝑎 > 𝑏 is one of the locally stable 
asymptotically conditions of the co-existence equilibrium point in Model 2. The results of 
this study can be applied to problems that have similarities mathematical structure to this 
case. There still some limitation in this model to fit in a realistic real case, and hence we 
think that this research should be further developed, for example by increasing. the 
number of species, the number of classes, and so on, which is mathematically interesting 
and realistically important. This theory can be applied to study the dynamics of natural 
resource models including the effects of different management to the growth of the 
resources, such as in fisheries.     
 



Mathematical Model of Iteroparous and Semelparous Species Interaction 

Arjun Hasibuan 462 

ACKNOWLEDGMENTS  

The work is part of the research done by first author in the Master of Mathematics Study 
Program at Padjadjaran University.  The first author would like to thank for the tuition fee 
during the candidature covered by ALG-Unpad Research Grant with contract number 
1959/UN6.3.1/PT.00/2021. The preparation and the APC (Article Processing Charge) for 
manuscript publication is funded by “Penelitian Tesis Magister (PTM)” from the 
Indonesian Ministry of Education, Culture, Research and Technology in 2022 with 
contract number 1318/UN6.3.1/PT.00/2022.  

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