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

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Dynamical Analysis of Predator-Prey Model Leslie-Gower with Omnivore Predator-Prey Model Leslie-Gower with Omnivore 126 (Exviani et al) Dynamical Analysis of Predator-Prey Model Leslie-Gower with Omnivore Rina Exviani1*, Wuryansari Muharini Kusumawinahyu2, Noor Hidayat3 1Master Program of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia 2Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia Abstract 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. Keywords: local stability, omnivore, predator-prey models, the equilibrium point. INTRODUCTION species are omnivores. This model is constructed Lotka-Volterra model was firstly introduced by assumming there are just three species in such Lotka in 1925 and Volterra in 1926 [1]. Lotka- an ecosystem. The first species, called as prey Volterra’s study has produced a simple model of (rice plant), the prey for the second and the third predation or interaction between two species in species. The second species, called as predator an ecosystem. They also have introduced classi- (carrion), only feeds on the first species and can cal Lotka-Volterra model, which is currently de- extinct with prey. The third species, namely om- veloped by researchers [2]. nivores (mouse), eat the prey and carcasses of In 1948, Leslie discussed Lotka-Volterra mod- predator. Consequently, omnivores of predator el and found impossibility in a model, which is only reduces the prey population but does not infinity in predator growth [3]. Therefore, Leslie affect the predator growth. Assumed that the and Gower introduced a new name’s predator- prey population grow logistically and any compe- prey model, which is modification of Lotka- tition between omnivores [5]. Based on these Volterra’s model. The model is known as Leslie- assumption, the mathematical model represent- Gower Predator-Prey Model. Leslie-Gower have ing those growth density of population rates by modified Lotka-Volterra’s predator-prey model nonlinear ordinary differential equation system, by assuming that the predation of predator is namely limited, which means that the predation of pred- ator will not more carrying capacity of prey. Leslie-Gower two dimension models as: (2) (1) In this model, and the density with state the population density of prey of prey, predator, and omnivore populations, and state the population density of preda- respectively. All parameter of model (2) are posi- tor. tive. The death rates of the predator and omni- In 2015, Andayani and Kusumawinahyu [4] a vore are denoted by and , respectively. The three species predator – prey model, the third parameter rivalry toward prey that effect in- creases of omnivore population, while the pa- rameter rivalry toward prey that effect in- Correspondence address: creases of omnivore population. Parameter and Rina Exviani Email : [email protected] carrying the capacity of the prey and omni- Address : Dept. Mathematics, Faculty of Mathematics and vore, respectively [5]. The aim of this study is a Natural Science, University of Brawijaya, Veteran dynamical analysis on a model of predator-prey Malang, Malang 65145. J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655 Predator-Prey Model Leslie-Gower with Omnivore 127 (Exviani et al) Leslie-Gower with omnivores which is modified predators has to be positive. Then, by Lotka-Volterra model with omnivore. exist if and . MATERIAL AND METHOD While, predator-prey model by Leslie-Gower In this study, predator-prey model by Leslie- with omnivore has seven equilibrium points (Ta- Gower with omnivore. This model is constructed ble 1). So the predator-prey model by Leslie- by assumming the third species are omnivores. Gower model with omnivores is more concrete in This model is constructed by assumming there this case. are just three species in such an ecosystem. The Table 1. Equilibrium Points of predator-prey model by first species, called as prey (rice plant), the prey Leslie-Gower with omnivore for the second and the third species. The second Equilibrium Existence Requirement species, called as predator (carrion), only feeds Points 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 pro- cess, such as the literature discussing the Leslie- Gower model, Lotka-Volterra model, omnivore, and forward-backward sweep method [7-12]. We MATHEMATICAL MODEL also used other supporting references in problem This study constructs Lotka-Volterra’s preda- solving in this study. In the Lotka-Volterra tor-prey model with omnivore (2). This model is predator-prey model with omnivore, namely: developed by modify the predator that previous- ly 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 equa- tion system (3): on this system has only five equilibrium point’s, namely: (3) with and state the population density of prey, predator, and omni- vore. 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 To accommodate biological meaning, the interaction coefficient between predator to prey existence conditions for the equilibria require and is a predatory interaction between that they are nonnegative. It is obvious that omnivores against prey. Whereas is an dan always exist, exist if , exist if interaction parameter between predators and and the densities of omnivores and parameter is a parameter of protection against J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655 Predator-Prey Model Leslie-Gower with Omnivore 128 (Exviani et al) predators. Parameter is omnivorous natural death, partially omnivorous predictor coefficient to prey, as predator coefficient of predator carcass, is competition between omnivorous population. While, tribal form exists if . can be interpreted as scarcity of prey may stimulate predators to replace foot sources with Stability Analysis other alternatives. Therefore, it is assumed that The local stability of system (3) for each predators depend not only on prey, but equilibrium point is as follows. predators can eat other than prey in the prey a) is unstable environment. So in this article it is modeled by adding a positive constant to the division. b) is stable if RESULT AND DISCUSSION and All parameters of Equation (3.1) in this study c) is unstable are assumed positive in value. Parameters and consecutively show intrinsic growth of prey d) is stable if and predator. is the competition coefficient among preys, is the predation interaction co- efficient between predator and prey, and is e) is unstable the predation interaction between omnivore and prey. Meanwhile, is the interaction parameter f) is unstable among predators, and parameter is the protec- tion parameter against predator. Parameter is g) . the natural death of omnivore, is the preda- tion coefficient of omnivore on prey, is the This stability analysis uses the criteria of predation coefficient on the carcass of predator, Routh-Hurwitz . The and is the competition among omnivore popu- characteristic equation will have negative lations. roots if and only if and . Equilibrium Point and Existence Therefore, it can be concluded that The point of Equilibrium (3) is solution for . If the condition is met, the equi- sytem: librium point of will be stable. Proof i. Jacobi matrix system (3) for is . The three Eigen The system has seven points of equilibrium, values of the matrix are positive, namely and so is unstable. in which the three points exist ii. Jacobi matrix is unconditionally, equilibrium point exists with the condi- tion of , equilibrium point exists if , equilibrium point exists if which has eigen values Equilibrium point with J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655 Predator-Prey Model Leslie-Gower with Omnivore 129 (Exviani et al) . The equilibrium point stable, if The equilibrium point result and . so unstable l. iii. Jacobi matrix on equilibrium point is vi. Jacobi matrix on the equilibrium point with, has eigen values so unstable. iv. Jacobi matrix on equilibrium point so is because and so sta- ble if result with used characeristic as follow proven that unstable. v. Jacobi matrix on the equilibrium point vii. Jacobi matrix on equilibrium point with, so Same of characteristic with use cofactor expansion second line, J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655 Predator-Prey Model Leslie-Gower with Omnivore 130 (Exviani et al) With declaration as: with, Based on criteria Routh-Hurwitz, the equilibrium point local asymptotically stable if and . Numerical Simulation , Parameter being used 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, Thus, the equilibrium points of exists, exist, as follows.
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