Journl of Mthemtics nd Computer Applictions Reserch (JMCAR) ISSN(P): 5-48; ISSN(E): Applied Vol 3, Issue 1, Dec 16, 7-34 TJPRC Pvt Ltd ON PREY-PREDATOR MODEL WITH HOLLING-TYPE II AND LESLIE-GOWER SCHEMES ABSTRACT AHMED BUSERI ASHINE Deprtment of Mthemtics Mdd Wlbu University Ble Robe, Ethiopi A predtor-prey system with Holling type II functionl response nd modified Leslie Gower type dynmics is considered By non-dimensionlize the system, condition for locl symptotic stbility of positive equilibrium point of the system is discussed nd globl symptotic stbility is proved by defining pproprite Dulc function Numericl simultions re lso crried out to verify the nlyticl results KEYWORDS: Locl Stbility; Globl Stbility; Dulc Function; Limit Cycle; Modified Leslie-Gower Received: Oct 7, 16; Accepted: Nov 6, 16; Published: Dec 8, 16; Pper Id: JMCARDEC163 1 INTRODUCTION Bckground The dynmic reltionships between species nd their complex properties re t the hert of mny ecologicl nd biologicl processes [1] The historicl origin nd pplicbility of this model is discussed in detilin [,3,4,5] THE MATHEMATICAL MODEL Originl Article Let x(t)nd y(t) denote densities of prey nd predtors t t respectively Consider the model incorportes the Holling-type-II nd modified Leslie-Gower functionl responses Y( ) dx X c = 1XY r 1 X dt K k1 + X dy cy = Y s dt k + X Where ll the prmeters in the model ssumes positive vlues nd with initil vlue X ( ) (1) nd This two species food chin model describes prey popultion x which serves s food for predtor y The model prmeters r, s, K, k 1, k, c 1 nd c re ssuming only positive vlues These prmeters re defined s follows: r is per cpit intrinsic growth rtes for prey, s is gives the mximl per-cpit growth rte of predtor, K is the crrying cpcity of the environment, k1 (respectively, k ) mesures the extent to which environment wwwtjprcorg editor@tjprcorg
8 Ahmed Buseri Ashine c provides protection to prey x (respectively, to the predtor y), 1 is the mximum vlue which per cpit reduction rte of c prey nd is the crowding effect for the predtor The following non-dimensionl stte vribles nd prmeters re chosen x = X k Y y = k t = rt c α = 1 r k β = 1 K s γ = r c σ = r k ω = K The system (1) tkes the following non-dimensionl form dx αy = 1 x x dt β + x dy σy = y γ dt ω + x F( x, y) G( x, y) () x ( ) = x ; y ( ) = y Theorem 1: All the solution ( x( t), y( t)) Proof: The first eqution of () gives us dx dt αy = x 1 x x β + < x(1 x) of the system () re bounded lim sup x( t) < 1 Therefore, t Hence, x(t) is lwys bounded Similrly, dy dt σy = y γ ω + x σy y γ ω + 1 y σ = γy 1 ω + 1 γ y = γy 1 λ ω + 1 y = y 1 ( ω + 1) / λ wwwtjprcorg editor@tjprcorg
On -Predtor Model With Holling-Type Ii nd Leslie-Gower Schemes 9 Therefore, we hve ω + 1 y( t) mx, y() L λ, σ λ = γ Hence, the solutions EQUILIBRIUM POINTS ( x( t), y( t)) of the system () with the given initil conditions re bounded We now study the existence of equilibri of system () All possible equilibri re The trivil equilibrium E (, ) Equilibrium in the bsence of predtor (y = ) E 1 (1, ) Equilibrium in the bsence of prey (x= ) E E ( x, y ) The interior (positive) equilibrium ωγ, σ 3 where x is the unique positive root of the qudrtic eqution σx + ( αγ + σβ σ ) x + αγω σβ = ; x = B + B 4σC σ γ ω ) ( + x y =, σ Where B = αγ + σβ σ C = αγω σβ, STABILITY Locl Stbility of the Equilibrium Points The locl symptotic stbility of ech equilibrium point is studied by computing the Jcoben mtrix nd finding the eigenvlues evluted t ech equilibrium point For stbility of the equilibrium points, the rel prts of the eigenvlues of the Jcoben mtrix must be negtive From equtions (), the Jcoben mtrix of the system is given by F x J ( E ) i = G x F y G y Which gives αβy αx 1 x ( β + x) β + x J ( E = i ) σy σy γ ( ω + x) ω + x The locl symptotic stbility for ech equilibrium point is nlyzed s below: wwwtjprcorg editor@tjprcorg
3 Ahmed Buseri Ashine Proposition E (,) E ( 1, ) nd 1 re unstble E ωγ, σ is loclly symptoticlly stble if σβ γ > αω For positive equilibrium E 3 (x *, y * ), J(E 3 )cn be simplified to ( ) J E3 = 1 Where 1 while it is unstble if γ < σβ αω αγbx β + x αx β + x = 1 x = 1 γ 1 =, σ = γ, E Theorem 4: The unique positive equilibrium point 3 ( x, y ) is loclly symptoticlly stble provided > γ Proof: The chrcteristic eqution is ( + ) λ + ( ) = λ The equilibrium point E 3 ( x, y ) is stble when trce( J ( E ) = + Now trce( J ( E ) = = < + γ < nd det( J ( E ) = > γ > det( J ( E ) = = δx 3 1 1 + ( γ α - δ + 4 β δ ) x + β ( αγ - δ + βδ ) x + β ( αγω - βδ ) ( x + β ) δ > γ > Hence, the unique positive equilibrium point E 3 (x *, y * ) is loclly symptoticlly stble provided wwwtjprcorg editor@tjprcorg
On -Predtor Model With Holling-Type Ii nd Leslie-Gower Schemes 31 Globl Stbility Theorem 41: The system () does not dmit ny periodic solution for ( ) Proof: Let x( t), y( t) H Then ( x, y) Q = x + x = β xy ( HF ) ( HG) + y ( β 1) + x = + y >1 β be solutions of the system () Define Dulc function ( x + β ) σ ( ) x x + ω solutions It is observed tht Q < for β >1 Therefore, by Dulc criterion, the system () hs no non-trivil periodic Corollry: If β >1 then the locl symptoticl stbility of the system () ensures its globl symptoticl stbility round the unique positive interior equilibrium point NUMERICAL SIMULATION E ( x In this section we will solve the system eqution () by using the in-built ordinry differentil eqution solver MtLb function ode45 For the system eqution (), tht is the system, we hve used the following prmetric vlues s fixed nd the prmeter γ s control prmeter These vlues re vlues the coexistence equilibrium point exists whenever, y ) α = 1, ω =, σ = 1, β = γ < 1 For these set of prmetric otherwise The coexistence equilibrium point is loclly symptoticlly stble for γ < 65134 nd hence unstble Figures 1 to 3: Shows the Stbility of the Coexistence Equilibrium Point Tht is; the Solution, Trjectory, of the nd Predtor Species Approches to the Coexistence Equilibrium Point wwwtjprcorg editor@tjprcorg
3 Ahmed Buseri Ashine 1, Predtor density 8 6 4 Predtor 4 6 8 1 Figure 1: Series Plot of the nd Predtor t γ =, Predtor density 7 65 6 55 5 45 4 35 Predtor 4 6 8 1 Figure : Time Series Plot of nd Predtor t γ =4, Predtor density 65 6 55 5 45 4 35 3 Predtor 5 4 6 8 1 Figure 3: Time Series Plot of nd Predtor t γ =6 wwwtjprcorg editor@tjprcorg
On -Predtor Model With Holling-Type Ii nd Leslie-Gower Schemes 33 4 35 Predtor 3 5 1 3 4 5 6 7 8 Figure 4: Phse Portrit of nd Predtor t γ =7 prey, predtor density 8 7 6 5 4 3 1 4 6 8 1 Figure 5: Time Series Plot of nd Predtor t γ =7 7 6 Predtor prey, predtor density 5 4 3 1 4 6 8 1 Figure 6: Time Series Plot of nd Predtor t γ =9 A figure 4 shows the existence of limit cycle, periodic solution Figure 5 lso shows the oscilltory nture of the predtor prey system Figure 6 represents the instbility of the coexistence equilibrium point wwwtjprcorg editor@tjprcorg
34 Ahmed Buseri Ashine CONCLUSIONS In this pper, considering modified Leslie Gower predtor prey model with Holling-type II schemes The structure of equilibri nd their linerized stbility is investigted Moreover, by defining Dulc Criterion, sufficient conditions on the globl symptoticl stbility of positive equilibrium re obtined REFERENCES 1 Cmr, BI nd Aziz-Aloui MA(7), Complexity in prey predtor model, 7 interntionl Confrence in Honor of Clude Lobry MA Aziz-Aloui nd M Dher Okiye, Boundedness nd globl stbility for predtor - prey model with modified Leslie-Gower nd Holling type II schemes, Applied Mth Lets, 16, (3) 169-175 3 M Dher Okiye nd MA Aziz-Aloui, On the dynmics of predtor-prey modelwith the Holling-Tnner functionl response, MIRIAM Editions, Editor V Cpsso, Proc ESMTB conf () 7-78 4 AF Nindjin, MA Aziz-Aloui nd M Cdivel, Anlysis of Predtor- Modelwith Modified Leslie-Gower nd Holling-Type II Schemes with Time Dely, Non Liner Anlysis Rel World Applictions, 7(5), (6) 4-18 5 AF Nindjin nd MA Aziz-Aloui, Persistence nd globl stbility in delyed Leslie-Gower type three species food chin, Journl of Mthemticl Anlysis nd Applictions, 34(1), (8) 34-357 wwwtjprcorg editor@tjprcorg