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

Predator-Prey Model Leslie-Gower with Omnivore 126 (Exviani et al)

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 predator 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 parameter 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 equation system (3):

on this system has only five equilibrium point’s, namely:

(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 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 coefficient 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 protection parameter against predator. Parameter is g) . the natural death of omnivore, is the predation 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 condition 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 stable 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. exists, and exists. The numerical simulation to equilibrium point . This is rel- Simulation I evant to the analysis result which states that Simulation I (Fig. 1) show exists, and con- equilibrium is stable. ditions of stable are and

z(t) 2

0 20 15 E3=(0,321,0,0) 14 12 10 10 8 6 5 E5=(0.145,0,0.452)4 E2=(0,1,0) 2 0 E1=(0,0,0)0 y(t) x(t) Figure 1. Portrait Phase System Equation 3.1 for simulation I

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

J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655

Predator-Prey Model Leslie-Gower with Omnivore 131 (Exviani et al)

E1=(0,0,0)E7=(4.05,8.71,2.53) E5=(15.24,0,13.48) 300 E3=(16,0,0)

200

z(t) 100

0 0 0 50 100 100 150 200 200 250 300 300 350 400 400 450 500 500 x(t) y(t) Figure 2. Portrait Phase System Equation 3.1 for Simulation II

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.

z(t) 20

0 E5=(7.143,0,4.429) 50 E2=(0,1.05,0) 40 50 45 30 E6=(0.263,1.967,0) E3=(16,0,0) 40 35 30 20 E1=(0,0,0) 25 20 10 15 10 5 0 0 y(t) x(t) Figure 3. Portrait Phase System Equation 3.1 for Simulation III

Simmulation IV The stability conditions in simulation IV, of are changed into dan Thus, it produces exists, parameter exists, exists, exists, and exists and unstable.

z(t) 20

0 50

40 E4=(0,0.234,0.571) 50 45 30 40 35 E5=(7.321,0,4.431) 30 20 E2=(0,1.05,0) E3=(10,0,0) 25 20 10 15 E1=(0,0,0) 10 5 0 0 y(t) x(t) Figure 4. portrait phase system equation 3.1 for simulation IV

J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655

Predator-Prey Model Leslie-Gower with Omnivore 132 (Exviani et al)

Simulation V: Stability condition

the fifth sim- Thus, it produces exists, ulation is conducted to show the stability charac- exists, exists, exists, and exists to- teristics of equilibrium point Based on the wards equilibrium point This is relevant to the existence condition and the stability of equilibri- analysis result which states that the equilibrium um point parameter point is stable.

1000

500

z(t)

0 500 450 400 350 E5=(1.538,0,4.321) 1000 300 900 250 800 700 200 E1=(0,0,0) E2=(0,1.05,0) E3=(10,0,0) 600 150 500 400 100 E4=(0,2.25,52.5 300 50 200 100 y(t) 0 0 x(t) Figure 5. Portrait Phase System Equation 3.1 for Simulation V

CONCLUSION Gower Predator- Prey Model. Appl. Math. The conclusions that are drawn based on the Article ID 149341. discussion of the thesis are as follows. Predator- [7] Chen, L., F. Chen. 2009. Global stability in prey model by Leslie-Gower with omnivore is Leslie-Gower Predator-prey model With obtained in a form of common differential feedback controls. 22(9). 1330-1334. equation system. There are seven equilibrium [8] Tanabe, K., T. Namba. 2005. Omnivory points in the model, there are three of them, i.e. creates chaos in simple food web models. and , unconditionally exist and the other Ecology. 86(12). 3411-3414. four, i.e. and , conditionally exist. Of [9] Nagle, R.K., E.B. Saff, A. D. Snider. 2004. the seven equilibrium points, three of them, Fundamentals of differential equations and dan , have stability condition. boundary value problems, 6th Ed. Pearson Education Inc. USA. REFERENCES [10] Mattheij, R., J. Molenaar. 2002. Ordinary [1] Jeffrey, R. C. 2012. Introduction to differential equation in theory and practice. numerical methods. Lecture Notes for Society for Industrial and Applied Matth 3311. The Hong Kong University of Mathematics. New York. Science and Technology. Kowloon, [11] Korobeinikov, A. 2001. A Lyapunov Function Hongkong. for Leslie-Gower Predator-Prey Models. [2] Murray, J.D. 2002. Mathematical biology I: Appl. Math. Lett. 14(6). 697-699. an introduction, 3rd Ed. Springer. New York. [12] Polis, G.A., C.A. Myers, R.D. Holt. 1989. The [3] Mohammadi, H., M. Mahzoon. 2013. Effect ecology and evolution of intraguild of weak prey in Leslie-Gower Predator-Prey predation: potential competitors that eat Model. Appl. Math. Comput. 224, 196-204. each other. Annu. Rev. Ecol. Evol. Syst. [4] Andayani, P., W.M. Kusumawinahyu. 2015. 20(1). 297-330. Global Stability Analysis on a Predator-Prey

Model with Omnivora. Appl. Math. Sci. 9(36). 1771-1782. [5] Previte, J.P., K.A. Hoffman. 2010. Chaos in Predator-Prey Model with an Omnivore. Math. Biol. 92(1). 1-7. [6] Huo, H.F., X. Wang, C. Carlos. 2011. Dynamics of a stage structured Leslie-

J.Exp. Life Sci. Vol. 8 No. 2, 2018 ISSN. 2087-2852 E-ISSN. 2338-1655