STABILITY ANALYSIS OF THE POPULATION MATRIX MODEL WITH TWO ITEROPAROUS SPECIES USING THE M-MATRIX

,


INTRODUCTION
Different species may live in a community, which is a portion of an ecosystem. Species within a community can be divided into semelparous species and iteroparous species. Female individuals of semelparous species can only give birth once in the last age group, shortly before death. Cicadas [1]- [4], beetles [1], [5], [6], giant Australian cuttlefish [7]- [9], and salmons [10]- [13] are a few examples of semelparous species. An iteroparous species, on the other hand, has several reproductions throughout its entire lifespan. This species is abundant in a variety of environments.
In particular for species that are a source of food for humans, population dynamics must be studied in order to ascertain the state of the population in the future. Harvesting by humans to meet food demands is a factor that has an impact on growth in population [14], [15]. For this reason both researches those focus on the dynamics of the resources and those focus on the optimal harvesting of the resources are equally important [16]- [20]. In addition, several other factors can influence the number of species in that community. First, a species may be density-dependent because of the scarcity of natural resources like food and suitable habitat [21]- [23]. Second, competition between species is conceivable and may result from lacking natural resources [24]- [26].
For a very long time, population dynamics have been studied using mathematical models. In ecological studies on plants and animals, the population matrix model has grown in popularity 3 STABILITY ANALYSIS OF THE POPULATION MATRIX MODEL recently [27], [28]. According to their characteristics, populations are categorized in the population matrix model. Age, developmental stage, body size (for instance, in small and big animals), and other traits might be considered as population characteristics. Leslie [29] first proposed an age-based discrete time population growth model in 1945. The Leslie matrix model is the name of this one. On sometimes, we are unable to determine the population's chronological age. According to their developmental stage, the population in this instance is divided into several categories [30]- [32]. The Lefkovitch matrix is a type of population matrix organized by developmental stage that was first presented in 1965 [32].
Research on population growth dynamics using the Leslie matrix model in the special situation of the multispecies scenario started in 1968. Pennycuick et al. [33] investigated the multispecies model using the Leslie matrix model and computer simulations. In that study, Pennycuick et al. [33] separated the instances into two categories of species interactions: predator-prey and competitive. The Leslie matrix model was then studied in 1980, with species divided into semelparous and iteroparous species. Travis et al.'s [34] stability requirements for the Leslie matrix model were created for two semelparous species that compete or are mutually exclusive. Kon [35] investigated the Leslie matrix model of two species, one of which has two age classes and the other of which has one. The next year, Kon [36] investigated how the coprime number of numbers affected the age classes of two semelparous species. When two numbers have the largest common factor of 1, they are called coprime. Later, Kon [37] expanded the earlier study to include an arbitrary number of semelparous species and an arbitrary number of age classes for each species. Furthermore, Hasibuan et al. [38] expanded on Kon [37]'s research by including the harvesting factor, although only for two semelparous species with two age classes. Next, Hasibuan et al. [39] established models of [38] for one semelparous species and one iteroparous species in the same community.
As an extension of the models investigated in [39], we study two iteroparous species, each with two age classes. It is possible to use the Leslie matrix model with two age classes for animals with life spans of two years, two months, two weeks, or two-time units. In this study, the population dynamics of the two iteroparous species are assumed to be affected by human harvesting and are density dependent between the two species. In both iteroparous species, harvesting occurs in the second age class. Then, in both iteroparous species, density dependence occurs only in the first age class. In addition, we also assumed that intraspecific and interspecific competition affected the growth of populations of both iteroparous species. Competition in multispecies can be divided into two competitions: interspecific competition and intraspecific competition. Interspecific competition occurs between species, while intraspecific competition occurs within the same species [40]. Furthermore, we divide this problem into two cases: iteroparous species with the same and different levels of intraspecific ( > 0) and interspecific ( > 0) competition. Hence, there are two models formed from the two problems. In both models, we derived the inherent net reproductive number which is often applied in research related to the Leslie matrix model [35]- [39], [41]. Next, we determine the equilibrium points of both models and their existence conditions. Finally, we analyze the asymptotic local stability of each equilibrium point of the two models using the M-matrix. Our aim is to investigate the impact of the level of intraspecific and interspecific competition along with inherent net reproductive number on the existence conditions and asymptotic local stability of each equilibrium point in both models.

A Leslie Matrix Model of Two Iteroparous Species in a Community with Same Level of
Intraspecific and Interspecific Competition.
In this section, we present a multispecies Leslie matrix model for two iteroparous species with the same level of intraspecific and interspecific competition in a community. It means that = . Because the species studied are iteroparous, both age classes of each species are assumed to be able to give birth. This problem is modelled with equations (1), and we refer to it as Model A.
In Table 1, a description of the Model A parameters is given. The total population of age class of the species and , respectively, is represented by ( ) and ( ) for = 1, 2.

> 0
The birth rate at age for = 1,2 of species .

> 0
The birth rate at age for = 1,2 of species .
Parameters and the details regarding the explanation of Model B are almost the same as Model A. The difference between the two models is the competition that affects the birth rate and survival of the first age class in both species and .

-matrix and Asymptotically Local Stability Criterion Using the -matrix
The asymptotic determination of the local stability of a discrete system or model can be seen through the absolute values of all the eigenvalues of the Jacobian matrix. Nevertheless, working with a system's eigenvalues from its Jacobian matrix to determine the asymptotic local stability is not easy. Therefore, we employ another method using asymptotic local stability introduced by Travis et al. [34]. The definition and theorem regarding the -matrix can be seen below in Definition 1 and Theorem 1.

Definition 1 [34]:
A square matrix of size is said to be an -Matrix if it satisfies two conditions. First, element ≤ 0 for ≠ . Second, one of the following five conditions is met i) All minor principals of matrix are positive.
ii) All real parts of the eigenvalues of matrix are positive.
iii) Matrix is a non-singular matrix and −1 is a positive matrix.
iv) There is a vector > 0 so that > 0.
v) There is a vector > 0 so that > 0.

Inherent Net Reproductive Number from Model A and Model B
The inherent net reproductive number, which has been studied in research [35]- [39], [41], is one of the frequently applied essential aspects, particularly in the study of the Leslie matrix model. This quantity refers to the number of offspring expected per individual over a lifetime.
There are two inherent net reproductive numbers, denoted by the letters and because our attention is on the situation of two species, species and . The detailed step-by-step explanation of the inherent net reproductive number for species and can be found in [42].
The fertility matrix and transition matrix of Model A for species , i.e.
Second, the fertility matrix and the transition matrix of Model A for species , i.e.
By using the same method as for spesies , we obtained Then, the dominant eigenvalue of ( 2 − ) −1 ( ) is 1 + 2 1 (1 − ℎ 2 ). It is therefore known as the , or the inherent net reproductive number of species .
Next, the inherent net reproductive number of Model B for both species is determined. The and matrices for each species, i.e.
Our results show that the dominant eigenvalues of ( 2 − ) −1 (0) and ( The next step is to find solutions from (3) and the solutions are: i) The extinction equilibrium point for the species and is ii) The equilibrium point with species extinct, i.e.
iii) The equilibrium point with species extinct, i.e.
The thing that is often studied at the equilibrium point is to determine the existing condition at the equilibrium point. Only the and equilibrium points, according to model (3) solutions, do not have all of the element values equal to zero. The conditions for the existence of equilibrium points and of Model A are provided in Theorem 2 below.

Theorem 2
For Model A that i) The equilibrium point exist if > 1.
ii) The equilibrium point exist if > 1.

Proof.
In and , it can be seen that the nonpositive generators are − 1 and − 1.

Theorem 3
For Model A that i) If < 1 and < 1, the equilibrium point 0 is asymptotically stable locally.
ii) If > 1 and > , the equilibrium point is asymptotically stable locally.
iii) If > 1 and > , the equilibrium point is asymptotically stable locally.
Proof. The first thing to do in the local stability problem is to determine the Jacobian matrix of Referring to the existence of equilibrium points, it follows that ( * ) 1 ≤ 0, ( * ) 2 ≤ 0, .
After that, the matrix .
It should be noted that all ≤ 0 for ≠ meet Theorem 1's first condition. The next step is to establish the requirement that all of matrix 's minor principles are positive. Take note that if < 1, 2 > 0. Because 0 < < 1 consequently 1 < 1 so that 1 > 0. Then, since < 1, it follows that 4 > 0 in the case of < 1.
After that, the matrix .
Let's note that, since     Finally, Theorem 4 presents the conditions needed for the , , and equilibrium points of Model B to exist.

Theorem 4
For Model B that i) The equilibrium point exist if > 1.
ii) The equilibrium point exist if > 1.
iii) The equilibrium point exists if > , > 0, and > 0 or < , < 0, and < 0. that the equilibrium point has the same denominator. Also note that the first element of is included in the second element, and the third element of is also included in the fourth element of , which must be guaranteed to be positive. Therefore, is positive if and have the same sign as . In other words, > 0 if > 0 is > , > 0, and > 0 or < , < 0, and < 0.
In this model, we obtain a co-existence equilibrium point, namely an equilibrium point with both species existing, which shows the influence of competition. This condition should be expected in a community so that the two species can coexist in one community.

Asymptotically Local Stability at Equilibrium Points of Model B
The asymptotic local stability analysis is also performed on the equilibrium points of Model B. Theorem 5 below provides the conditions that the asymptotic local stability of the equilibrium points of Model B must satisfy.

Theorem 5
For Model B that i) If < 1 and < 1, the equilibrium point 0 is asymptotically stable locally.
iii) If  Referring to the existence of equilibrium points of Model B, it follows that ( * ) 1  .
After that, the matrix .
It should be noted that all ≤ 0 for ≠ meet Theorem 1's first condition. The next step is to establish the requirement that all of matrix 's minor principles are positive. Take note that if < 1, 2 > 0. Because 0 < < 1 consequently 1 < 1 so that 1 > 0. Then, since < 1, it follows that 4 > 0 in the case of < 1.
Furthermore, due to < 1 and < 1 it follows that 3 > 0. Therefore, is an -matrix. Then, the equilibrium point 0 is locally asymptotically stable if < 1 and < 1.

iv) For
, the Jacobian matrix is Note that 2 > 0 and 4 > 0 if , , and > 0 . Because > 0 which results in > so that 3 > 0 if ( 2 − 1 ) > 1 ( − 1) or 2 > where it is clear that 2 = ( 2 1 (1 − ℎ 2 ) + 1 ) 2 > 1 . Then, Therefore, is an -Matriks and equilibrium point is asymptotically stable  Table 2.  Based on the parameters presented in Table 2, for the case I that = 0.94 < 1 and = 0.87 < 1. The simulation results from the case I in Table I are presented in Figure 1, where Figure 1 interprets that when < 1 and < 1, the system is asymptotically stable locally towards the equilibrium point 0 . In that sense, the populations of both species and are extinct. Furthermore, for case II in Table 2 Figure 3. In the sense that the population exists because it has an inherent net reproductive number greater than and exceeds the threshold. In case III from Table 2, the population exists where the system is asymptotically stable locally towards the equilibrium point = [0,0,577.1,0.94] , which is shown graphically in Figure 3.
This is because = 369.19 > = 578.10 and > 1. In a sense, species has an inherent net reproductive number greater than and exceeds the threshold.  Table 3.
Furthermore, the intraspecific and interspecific competition levels are assumed to be = 0.002 and = 0.001, respectively.     Table 3.
Because the parameter values in case I from Table 3 satisfy the first condition of Theorem 5, Figure 4 shows that the system is asymptotically stable locally towards the equilibrium point where all species become extinct or 0 . The parameter values in case II Table 3 fulfil the second condition in Theorem 5, where the value of exceeds the threshold and fulfils the condition ( − 1) < ( − 1). Consequently, the numerical simulation results in Figure 5 show that the system is asymptotically stable locally towards the equilibrium point = [498750, 498, 0,0] . In a sense, the population that survives is only the population in species for both age classes. Then, Figure 6 shows that the system is asymptotically stable locally towards the equilibrium point = [0,0,592030,487] . In a sense, the population that survives is only the population in species for both age classes. This is because the selected parameter values in case III Table 3 fulfil the third condition of Theorem 5 where exceeds the threshold and ( − 1) < ( − 1). Next, Figure 7 shows the asymptotically stable system towards the

CONCLUSION
The problem of growth dynamics of two iteroparous species with two age classes for each species is developed using two models in this work. The two models consist of models on the growth of species affected by same and different levels of intraspecific and interspecific competition. The two models are referred to as Model A and Model B, respectively. Density dependency and harvesting were taken into account in these models. These models were established using the Leslie Matrix model for multispecies. In this paper, the equilibrium points of both models were found, and the asymptotically local stability for each equilibrium point was also analyzed using M-matrix theory. There were three equilibrium points obtained from Model A where no co-existence equilibrium point was found. Unlike the case in Model B, there was an additional one type of equilibrium point, namely the co-existence equilibrium point. The existence and stability of each equilibrium point in models A and B were characterized by the inherent net reproductive number of each species, namely and ; for a species to exist and be locally asymptotically stable, its value must exceed a threshold of one. Conversely, if and 27 STABILITY ANALYSIS OF THE POPULATION MATRIX MODEL are smaller than one, both species would become extinction in the long term. However, this condition is not enough, so there are other conditions. The Model A showed that the equilibrium point with one species existing will be asymptotically locally stable if that species has a larger inherent net reproductive number. This demonstrates the existence of the competition exclusion principle. A species that is dominant no matter how small will also dominate in the long run over other species. The Model B showed that the equilibrium point with one species existing will be asymptotically stable if it satisfies the other conditions stated in Theorem 5. Then, the co-existence equilibrium point will be asymptotically stable if the degree of intraspecific competition is greater than interspecific competition and other conditions which are complex enough to be interpreted biologically.
The model presented and studied in this research is still open for development into a more realistic and in-depth model. The research conducted in this study is the basis to be used as a reference for us or other researchers to be able to develop a more general model as done by Kon [37], namely on an arbitrary number of iteroparous species with an arbitrary number of age classes. In the end, the more general model is more applicable to various species. Further developments that can be made from the generalization of the model include studying global stability, studying bifurcations in the model, and many more.