Mathematical Model of Iteroparous and Semelparous Species Interaction

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.


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

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.

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 ( ) and one semelparous species ( ) 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 densitydependent 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:

Mathematical Model of Iteroparous and Semelparous Species Interaction
Arjun Hasibuan 447 1 ( + 1) = 2 2 ( ) 2 ( + 1) = 1 (1 − ℎ 2 ) 1 + 1 ( ) + 1 ( ) 1 ( ) There are several parameters in the Model 1. First, 1 > 0 and 2 > 0 are the birth rates of the 1st and 2nd age classes of species , respectively. Second, 2 > 0 is the birth rate of the 2nd age classes of species . Third, 0 < 1 , 1 < 1 are the survival rates of the 1st age classes of species and , respectively. Fourth, 0 < ℎ 2 , ℎ 2 < 1 are the harvesting rates of the 2nd age classes of species and , respectively. In addition, the variables ( ), and ( ) represent the total population of each species and for the age class = {1,2}. Simply put, equation 1 in (1) means that the population of the first age class of species at time + 1 is obtained by adding the number of newborn from the first age class and the second class at time . 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 at time + 1 is obtained from the number of surviving populations which is influenced by density-dependent of the first age class at time . Equations 3 and 4 in (1) have the same meaning as equations 1 and 2 in (1) but in species 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.
1 ( + 1) = 1 1 + 1 ( ) + 1 ( ) 1 ( ) + 2 2 ( ) 2 ( + 1) = 1 (1 − ℎ 2 ) 1 + 1 ( ) + 1 ( ) 1 ( ) 1 ( + 1) = 2 2 ( ) 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 > 0 and > 0. In Model 2, to simplify the problem, we assume that the level of competition between the first age class in species and species has the same value, namely . Then, the level of competition between the first age class in species against species and vice versa has the same value, namely . The level of competition within the same species is referred to as intraspecific competition ( ) and the level of competition between different species is referred to as interspecific competition ( ). 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 harvesting rate in species which are influenced by density-dependent and competition. Then, species 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. where , , and are identity matrices with sizes + , , and , respectively, then matrix 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 and to the left of the Model 1 depending on time . The equilibrium Model 1 is obtained as follows:

Mathematical Model of Iteroparous and Semelparous Species Interaction
Arjun Hasibuan 449 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 .

The condition exists if it is fulfilled
is referred to as the expected number of offspring per individual per lifetime when densitydependent effects are neglected on harvest-influenced growth of species .
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 0 is locally stable asymptotically if < 1 and < 1. 2. The equilibrium point is locally stable asymptotically if > and > 1.

The equilibrium point
is locally stable asymptotically if > and > 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 The reason for using the M-Matrix theory in this study is due to the complexity of determining the spectral radius matrix ( ) at the corresponding equilibrium point . The use of Definition 1 and Theorem 1 guarantee that the spectral radius value of the ( ) matrix at the corresponding equilibrium is less than one.
Based on the ( ) matrix in (4)  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 0 is Next, determine the matrix = − ( ( 0 )) −1 and examine the element for ≠ that is non-positive and that all minor principals of are positive. If these conditions are met, then the matrix is an M-Matrix so that the spectral radius ( 0 ) is less than one. As a result, the equilibrium point 0 is locally stable asymptotically. Here we present the obtained matrix In this paper, we define ( = 1,2,3,4) as the -th minor principal of the matrix for each equilibrium point under consideration. It is clear that the element < 0 for ≠ in the matrix is nonpositive by recalling the previously defined parameters. Then, 2 and 4 will be positive if met < 1. In addition, < 1 implicitly results in 1 < 1 so that 1 > 0. Furthermore, 4 is positive if < 1 because it must be fulfilled that < 1. Therefore, is an M-Matrix, that is if it is filled with < 1 and < 1. Hence, according to Theorem 1, the equilibrium point 0 is locally stable asymptotically if < 1 and < 1. Based on the previously defined parameters, the element for ≠ will be nonpositive if > 1 is satisfied. Because of > 1, consequently 2 and 3 are positive. Besides that, 4 is also positive but with the additional condition that is > . Then, it is clear that 1 < 2 = ( 2 1 (1 − ℎ 2 ) + 1 ) 2 so that 1 > 0. Therefore, is an M-Matrix, if it is fulfilled > and > 1. Hence, according to Theorem 1, the equilibrium point is locally stable asymptotically if > and > 1. Note that all elements for ≠ are nonpositive except for 41 . Then, 41 will be negative if > 1. Next, focus on the minor principal terms of the matrix.

Mathematical Model of Iteroparous and Semelparous Species
. 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.

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 0 is locally stable asymptotically if < 1, and < 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 0 is By considering the matrix , it is clear that the values of all for ≠ are nonpositive. Next is focus on determining the conditions for > 0 ( = 1,2,3,4).
2 and 3 are positive if < 1 is satisfied. Because of < 1 so that 1 is positive and an additional condition for 4 to be positive is < 1. Therefore, is an M-matrix, and the equilibrium point 0 is locally stable asymptotically if < 1 and < 1. In the matrix , it can be seen that all for ≠ are non-positive because > 1 which is a condition for to exist. Therefore, the next step is to focus on determining the positive terms of the minor principal of the matrix. It is clear that 2 and 3 are positive because > 1. Then, it is clear that 1 is positive because in fact 1 < 2 = ( 2 1 (1 − ℎ 2 ) + 1 ) 2 . Since 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.

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 > which means the intensity of the intraspecific competition level is greater than the intensity of the interspecific level competition. The inequality of > 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