Dynamical Analysis of Predator-Prey Model Leslie-Gower with Omnivore

This article discussed a dynamical analysis on a model of predator-prey Leslie-Gower with omnivores which is modified by Lotka-Volterra model with omnivore. The dynamical analysis was done by determining the equilibrium point with its existing condition and analyzing the local stability of the equilibrium point. Based on the analysis, there are seven points of equilibrium. Three of them always exist while the four others exist under certain conditions. Four points of equilibrium, which are unstable, while the others three equilibrium point are local asymptotically stable under certain conditions. Moreover, numerical simulations were also conducted to illustrate the analytics. The results of numerical simulations agree with the results of the dynamical analysis.

(1) with state the population density of prey and state the population density of predator.
In 2015, Andayani and Kusumawinahyu [4] a three species predator -prey model, the third species are omnivores. This model is constructed by assumming there are just three species in such an ecosystem. The first species, called as prey (rice plant), the prey for the second and the third species. The second species, called as predator (carrion), only feeds on the first species and can extinct with prey. The third species, namely omnivores (mouse), eat the prey and carcasses of predator. Consequently, omnivores of predator only reduces the prey population but does not affect the predator growth. Assumed that the prey population grow logistically and any competition between omnivores [5]. Based on these assumption, the mathematical model representing those growth density of population rates by nonlinear ordinary differential equation system, namely In this model, and the density of prey, predator, and omnivore populations, respectively. All parameter of model (2) are positive. The death rates of the predator and omnivore are denoted by and , respectively. The parameter rivalry toward prey that effect increases of omnivore population, while the parameter rivalry toward prey that effect increases of omnivore population. Parameter and carrying the capacity of the prey and omnivore, respectively [5]. The aim of this study is a dynamical analysis on a model of predator-prey Leslie-Gower with omnivores which is modified by Lotka-Volterra model with omnivore.

MATERIAL AND METHOD
In this study, predator-prey model by Leslie-Gower with omnivore. This model is constructed by assumming the third species are omnivores. This model is constructed by assumming there are just three species in such an ecosystem. The first species, called as prey (rice plant), the prey for the second and the third species. The second species, called as predator (carrion), only feeds on the first species and can extinct with prey. The third species, namely omnivores (mouse), eat the prey and carcasses of predator. Consequently, omnivores of predator only reduces the prey population but does not affect the predator growth. Assumed that the prey population grow logistically and any competition between omnivores.

Literature Study
Literature study related to the research process, such as the literature discussing the Leslie-Gower model, Lotka-Volterra model, omnivore, and forward-backward sweep method [7][8][9][10][11][12]. We also used other supporting references in problem solving in this study. In the Lotka-Volterra predator-prey model with omnivore, namely: on this system has only five equilibrium point's, namely: To accommodate biological meaning, the existence conditions for the equilibria require that they are nonnegative. It is obvious that dan always exist, exist if , exist if and the densities of omnivores and predators has to be positive. Then, exist if and . While, predator-prey model by Leslie-Gower with omnivore has seven equilibrium points (Table 1). So the predator-prey model by Leslie-Gower model with omnivores is more concrete in this case.

MATHEMATICAL MODEL
This study constructs Lotka-Volterra's predator-prey model with omnivore (2). This model is developed by modify the predator that previously used Lotka-Volterra's form to Leslie-Gower's, which was examined by Leslie-Gower. This is based on the fact that predator depends on the available number of prey to establish. Therefore, the model is stated to be in the following equation system (3): with and state the population density of prey, predator, and omnivore. All of the parameters are positive in value. Parameters and respectively show intrinsic growth of prey and predator.
is the coeffcient of competition between prey, is the predator interaction coefficient between predator to prey and is a predatory interaction between omnivores against prey. Whereas is an interaction parameter between predators and parameter is a parameter of protection against can be interpreted as scarcity of prey may stimulate predators to replace foot sources with other alternatives. Therefore, it is assumed that predators depend not only on prey, but predators can eat other than prey in the prey environment. So in this article it is modeled by adding a positive constant to the division.

RESULT AND DISCUSSION
All parameters of Equation (3.1) in this study are assumed positive in value. Parameters and consecutively show intrinsic growth of prey and predator.
is the competition coefficient among preys, is the predation interaction coefficient between predator and prey, and is the predation interaction between omnivore and prey. Meanwhile, is the interaction parameter among predators, and parameter is the protection parameter against predator. Parameter is the natural death of omnivore, is the predation coefficient of omnivore on prey, is the predation coefficient on the carcass of predator, and is the competition among omnivore populations.

Equilibrium Point and Existence
The point of Equilibrium (3) is solution for sytem:

Numerical Simulation
Numerical simulations are performed to see the validity of numerical analysis using the fourth-orde Runge-Kutta method to ilustrate the results of the analysis. There are several cases that are simulated in the discussion of this study, as follows.

Simulation I
Simulation I (Fig. 1) show exists, and conditions of stable are and , Parameter being used Thus, the equilibrium points of exists, exist, exists, and exists. The numerical simulation to equilibrium point . This is relevant to the analysis result which states that equilibrium is stable.

Simulation II
Simulation II, the stability conditions of are changed into and parameter . Thus, it produces exists, exists, exists. Then, exists and is stable toward equilibrium point, so the initial value shows that is stable.

Simulation III
The stability conditions in simulation III of are changed into dan parameter exists, exists, exists, exists, and exists, but, in this case, it does not go to any point, so it exists but is unstable.

CONCLUSION
The conclusions that are drawn based on the discussion of the thesis are as follows. Predatorprey model by Leslie-Gower with omnivore is obtained in a form of common differential equation system. There are seven equilibrium points in the model, there are three of them, i.e. and , unconditionally exist and the other four, i.e. and , conditionally exist. Of the seven equilibrium points, three of them, dan , have stability condition.