On the Comparison of Stability Analysis with Phase Portrait for a Discrete Prey-Predator System

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1 Intenational Jounal of Applied Engineeing Reseach ISSN Volume 12, Nume 24 (2017) pp Reseach India Pulications. On the Compaison of Staility Analysis with Phase Potait fo a Discete Pey-Pedato System a Ummu Atiqah Mohd Roslan and Syamimi Samsudin a, School of Infomatics and Applied Mathematics, Univesiti Malaysia Teengganu, Kuala Teengganu, Teengganu, Malaysia. Coesponding autho a Ocid: Astact Pey-pedato inteaction can e defined as consumption of pedato against pey. It is well known that when the pedato asent, the gowth of pey population will incease. In the asence of pey, the pedato population goes extinct. In this pape, we conside the inteaction etween pey and pedato in a discete pey-pedato system. Fo this system, we would like to study the dynamic ehavio of solutions (o in paticula fixed points) fo this system. The ojective of this poject is to investigate the staility of fixed points using staility analysis and phase potait. Afte that, we compae the esults of staility otained fom oth appoaches. Ou osevations indicate that esults of staility using oth appoaches ae same. This means that oth appoaches ae suitale to explain the ehavio of the solutions, especially when deal with eal wold polems. This eseach is impotant fo the sustaining of inteacting species of pey and pedato populations in an ecosystem. INTRODUCTION Dynamical system can e defined as a system that evolve with time and it consists of possile states, togethe with a ule that detemine the pesent state in tems of past state. Dynamical systems ae mathematical models of eal systems such as the climate, ain, electonic cicuit, lase, population, etc. In mathematics, dynamical system is a function which descies the time dependence of a point in geometical. Thee ae two types of dynamical system which ae flows and maps. Flows ae descied y continuous time while maps ae decied y discete time. In this pape, we focus on the discete-time model whee the geneal fom of equation is given as follows: x n+1 = f(x n ), (1) whee x is the state and f is the evolution o map. Ecology is the elationship etween oganisms and thei envionments. Pey-pedato is one of the inteaction in ecosystem. Pey is an oganism which the pedato eats and pedato is defined as an oganism that eats anothe oganism. Some examples of pey and pedato ae fish and shaks, at and snake and ait and fox. The fist mathematical model desciing inteacting population was poposed y Alfed Lotka in 1925 and Vito Voltea in The model of peypedato system is called as a Lotka-Voltea. The Lotka- Voltea model is expessed y the following equations (Reni Sagaya Raj et al., 2013): x = ax xy, y = cy + dxy, whee x is the population density of pey, y is the population density of pedato and a,, c, d ae all positive paametes. Inteaction etween pey and pedato is a cucial study in explaining the population dynamics, whee this efes to the changes in the size of populations of oganisms though time. Lynch (2014) states that thee ae two types of outcome fom the pey-pedato inteaction. Fist, thee is coexistence whee the two species live in hamony. Secondly, thee is mutual exclusion in which one of the species ecomes extinct. In this pape, we use the discete-time model. Thee ae easons why discete-time models ae ette than the continuous-time models. Accoding to Zhao et al. (2016), the discete-time models ae moe easonale to e used fo the case whee thee ae non-ovelapping geneations. Besides that, the discete time models ae moe efficient fo computation and numeical simulations. The aims of this pape ae to find the staility of fixed points of a discete pey-pedato system y means of staility analysis and phase potait. We will then compae the esults fom oth appoaches. MODEL In this poject, we conside the following system of diffeence equations which descies the inteactions etween two species fom Reni Sagaya Raj et al. (2013): x n+1 = x n (1 x n ) ax n y n, y n+1 = cy n + x n y n, (2) whee, a,, c > 0 and whee x n is the population of the pey at time n and y n is the population of the pedato at time n. The following ae the paametes used in the model: 15273

2 Intenational Jounal of Applied Engineeing Reseach ISSN Volume 12, Nume 24 (2017) pp Reseach India Pulications. Paamete a c Desciption The natual gowth ate of the pey in the asence of pedatos The ate of pedation on the pey which depends on the likelihood that a pedato encountes a pey The efficiency and popagation ate of the of pedation in the pesence of pey The natual death ate of the pedato in the asence of pey Peviously, Elsadany et al. (2012) investigate system (2) whee they find local staility of fixed points, ifucation, chaotic ehavio, Lyapunov exponents and factal dimension of the solution of a discete pey-pedato system. In 2013, Reni Sagaya Raj et al. (2013a) who also consideed a discete model of pey-pedato, have plot the phase potait and ifucation diagam fo a cetain ange of paamete. Thei esults show that the pey population has chaotic ehaviou ove time. Recently, Zhao et al. (2016) consideed the same model whee necessay and sufficient conditions of the existence and staility of fixed points ae poved. Thei esults show that this model undegoes a flip ifucation. Moeove, they extend the analysis y stailizing the chaotic oits using feedack contol method. STABILITY ANALYSIS The solutions of discete-time model is called the fixed point. In this section, we find the fixed points fo system (2) and investigate thei staility. Definition 1 (Lynch, 2014). A fixed point of system (1) is a point which satisfies x n+1 = f(x n ) = x n, fo all n. Fom the aove definition, we ae ale to find the fixed points o in iological case, the nume of populations of pey and pedato in futue time. Fo each fixed point, we may find its staility using the following theoem: Theoem 1 (Lynch, 2014). Suppose that the map f μ (x) has a fixed point at x. Then the fixed point is stale if d dx f μ(x ) < 1 and it is unstale if d dx f μ(x ) < 1. The aove theoem is fo the one-dimensional system. Fo twodimensional system, we efe to the following poposition (Reni Sagaya Raj et al., 2013): Poposition 1. The chaacteistic oots λ 1 and λ 2 ae called eigenvalues of the fixed points (x, y ). Then, 1. (x, y ) is sink (stale) if λ 1,2 < 1; 2. (x, y ) is souce (unstale) if λ 1,2 > 1; 3. (x, y ) is saddle if λ 1 > 1 and λ 2 < 1 (o λ 1 < 1 and λ 2 > 1). Note that saddle implies unstale. Steps towads finding the staility of fixed points In this section, we study the ehaviou of the system (2) aout each fixed points. We compute the staility of the system (2) y using Jacoian matix coesponding to each fixed points. We will detemine whethe the fixed points ae stale, unstale o saddle fom the values of eigenvalues otained fom Jacoian matix. The pocess of finding the staility is discussed in the following steps: Step 1: Finding fixed points We solve system (2) as follows: x n (1 x n ) ax n y n = x n, cy n + x n y n = y n. Poposition 2. Then we otain that the system (2) has i) One extinction fixed point E 0 = (0,0), ii) One exclusion fixed point E 1 = ( 1, 0) and iii) One coexistence fixed point E 2 = ( c+1, ( c 1) 1 ). a a Fom the aove poposition, E 0 means that thee will e no pey and pedato in the futue in an ecosystem and E 2 means that only one species is suvive whee in this case the suvivo is the pey. Finally E 2 has oth solutions fo pey and pedato which indicates that oth species sustain foeve. All these outcomes have een expected y Lynch (2014) as we discussed in the intoduction. Accoding to Reni Sagaya Raj et al. (2013), we use the paamete = 2.41, a = 1.19, = 3.91 and c = Theefoe the fixed points ae E 0 = (0,0), E 1 = (0.59,0), E 2 = (0.37,0.43). Step 2: Finding the patial deivative The geneal fomula fo Jacoian matix fo system (2) is J(x n+1, y n+1 ) = ( x n+1 x n y n+1 x n x n+1 y n. y n+1 y n ) The geneal fom of Jacoian matix fo system (2) is 2x ay J(x, y) = ( ax y x c )

3 Intenational Jounal of Applied Engineeing Reseach ISSN Volume 12, Nume 24 (2017) pp Reseach India Pulications. Step 3: Classify eigenvalues fom step 2. We estimate the eigenvalues of Jacoian matix J at E 0. The Jacoian matix at E 0 is of the fom J(E 0 ) = ( 0 0 c ). Hence, the eigenvalues of J(E 0 ) ae λ 1 = and λ 2 = c. E 0 is stale if λ 1,2 < 1 which implies < 1 and c < 1. E 0 is unstale if λ 1,2 > 1which implies > 1 and c > 1. The Jacoian matix J at E 1 is given y J(E 1 ) = ( 2 a (1 ) 0 ( 1 ) c). Hence, the eigenvalues of the matix J(E 1 ) ae λ 1 = 2 and λ 2 = ( 1 ) c. Thus, E 1 is stale if λ 1,2 < 1 and unstale if λ 1,2 > 1 The fixed point E 2 has the Jacoian J(E 2 ) = ( (1 + c) 1 ( 1 c) a a ( 1 + c ) ). 1 By using the same values of paametes, we otain the following eigenvalues and thei coesponding staility fo E 0, E 1 and E 2 fo system (2). Tale 1: The esults on the staility of fixed points fo system (2) Fixed points Eigenvalues Staility E 0 λ 1 = 2.41, λ 2 = 0.45 Saddle E 1 λ 1 = 1.84, λ 2 = 0.41 Saddle E 2 λ 1,2 = ± i Stale We discuss the esults in Tale 1 ased on Poposition 2. E 0 is saddle since λ 1 > 1 and λ 2 < 1. E 1 is also saddle since λ 1 > 1 and λ 2 < 1. Meanwhile E 2 has a pai of complex conjugate eigenvalues with positive eal pats. Since oth eal pats of eigenvalues ae less than 1, theefoe E 2 is stale. PHASE PORTRAIT The ehavio of fixed points in a system can also e shown y plotting the phase potait. In this section, we plot the phase potait fo system (2) y using Maple. Definition 2 (Kenneth, 2008). Let (x n, y n ) e a solution to the discete system. As n vaies, the solution (x n, y n ) descies a cuve in the xy-plane called a tajectoy. The xy-plane is called phase plane. The phase potait is a epesentative sampling of tajectoies of the system. Definition 3. The phase potait is a two-dimensional figue showing how qualitative ehaviou of system (2) is detemined as x and y vay with n. Hence, the eigenvalues of the matix J(E 2 )ae and λ 1 = 1 2 λ 2 = (c + 1) c c c c c c 2 + (c + 1) c c c c c c Thus, the fixed point of E 2 is stale if λ 1,2 < 1 and unstale fixed point if λ 1,2 > 1. Some examples of sketch of staility using phase potait Hee we show some examples of phase potaits along with seveal gaphs of x 1 vesus x 2 which ae given elow. We show that the thee fixed points which ae stale, unstale and saddle points in Figue

4 Intenational Jounal of Applied Engineeing Reseach ISSN Volume 12, Nume 24 (2017) pp Reseach India Pulications. a) stale ) unstale c) saddle point Figue 1: Schematic diagams fo stale, unstale and saddle points a) spial in ) spial out c) cente Figue 2: Schematic diagams fo points with complex eigenvalues Fo the case of complex eigenvalues, thee ae some examples of gaph staility. Suppose the eigenvalues ae a ± i, whee a and ae eal with a 0 and > 0. We show fo this case in Figue 2. Result fo phase potait fo system (2) In this section, we discuss aout the staility of fixed points etween pey and pedato ased on phase potait diagam otained using Maple. We mak the thee fixed points E 0, E 1 and E 2 in Figue 3 with small empty cicle. The small aows filled thoughout the figue epesent the tajectoies of initial conditions chosen in the system. Fom this figue, the tajectoies appoach E 0 along y-axis ut epel fom E 0 along x-axis. Since thee ae mixed of diections of in and out, theefoe we say that E 0 is a saddle point. Fo E 1, we can see that the tajectoies appoach E 1 fom oth left and ight. Howeve, thee ae also tajectoies that move away fom E 1. Since E 1 has oth in and out diections of the tajectoies, E 1 is also a saddle point. Meanwhile E 2 is a stale point since the tajectoies move towads E 2 fom all diections. This indicates that the nume of populations of pey and pedato stailize to a constant value afte long peiod of time. Fom the aove discussion, we oseve that the esults of staility fo fixed points otained using phase potait ae the same as y using the staility analysis appoach

5 Intenational Jounal of Applied Engineeing Reseach ISSN Volume 12, Nume 24 (2017) pp Reseach India Pulications. Figue 3: The points of thee fixed points fo system (2). CONCLUSION In the pesent wok, we have consideed a 2-dimensional discete pey-pedato model and otained thee fixed points fo the system (2). The staility of the fixed points otained fom two appoaches has een discussed. The esults show that system (2) has two saddle and one stale points. Based on the esults of this model, we hope to apply this model in eal cases of maine ecosystem whee we will e ale to povide the values of paametes that affecting the pey and pedato populations. ACKNOWLEDGMENTS The authos would like to thank Univesiti Malaysia Teengganu fo the financial suppot fo this wok unde TPM gant. REFERENCES [1] Elsadany, A.E., EL-Metwally, H.A., Elaasy, E.M. and Agiza, H.N., Computational Ecology and Softwae 2(3), (2012). [2] Kenneth, H., Math 216 Diffeential Equations. Michigan: Depatment of Mathematic Univesity of Michigan (2008). [3] Lynch, S. Dynamical Systems with Applications using Matla. New Yok: Spinge (2014). [4] ReniSagayaRaj, M., Selvam, A.G.M. and Janagaaj, R., Intenational Jounal of Latest Reseach in Science and Tecnology, 2(1), (2013a). [5] ReniSagayaRaj, M., Selvam, A.G.M. and Meganathan, M., Intenational Jounal of Engineeing Reseach and Development, 6(5), 1-5 (2013). [6] Zhou, M., Xuan, Z. and Li, C., Advances in Diffeence Equations, 191,1-18 (2016)