Stability analysis of two predator-one stage-structured prey model incorporating a prey refuge

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1 IOSR Jornal of Mathematics (IOSR-JM) e-issn: p-issn: X. Volme Isse 3 Ver. II (Ma - Jn. 5) PP -5 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge Zahraa Jawad Kadhim Azhar Abbas Majeed and Raid Kamel Naji Department of Mathematics College of Science Universit of Baghdad Baghdad Iraq. Abstract: In this paper a food web model consisting of two predator-one stage strctred pre involving Lotka-Voltera tpe of fnctional response and a pre refge is proposed and analzed. It is assmed that the pre growth logisticall in the absence of predator. The role of pre refges in predator-pre model is investigated. The existence niqeness and bondedness of the soltion are stdied. The existence and the stabilit analsis of all possible eqilibrim points are stdied. Sitable Lapnov fnctions are sed to std the global dnamics of the proposed model. Nmerical simlation for different sets of parameter vale and for different sets of initial conditions are carried ot to investigate the inflence of parameters on the dnamical behavior of the model and to spport the obtained analtical reslts of the model. Kewords: food web Lapnov fnction refge stabilit analsis stage-strctre. I. Introdction The dnamic relationship between predators and their pre has long been and will contine to be one of the dominant themes in both ecolog and mathematical ecolog de to its niversal existence and importance. Over the past decades Mathematics has made a considerable impact as a tool to model and nderstand biological phenomena. In retrn biologists have confronted the mathematics with variet of challenging problems which have simlated developments in the theor of nonlinear differential eqations. Sch differential eqations have long plaed important role in the field of theoretical poplation dnamics and the will no dobt contine to serve as indispensable tools in ftre investigations. Differential eqation models for interactions between species are one of the classical applications of mathematics to biolog. The development and se of analtical techniqes and the growth of compter power have progressivel improved or nderstanding of these tpes of models. Althogh the predator-pre theor has been mch progress man long standing mathematical and ecological problems remain open. Food chains and food webs depict the network of feeding relationship within ecological commnities. Dring the last few decades a large nmber of food-chain and food-web sstems have been proposed to describe the food transition patterns and processes.living organisms enter into a variet of relationships sch as Pre-Predator Competition Mtalism Commensalism and so on among themselves according to the needs of individals as well as those of species grops. Food webs are one example of interactions that go beond feeding relationships. Recentl nmber of researchers have been proposed and stdied the dnamics of food webs involving some tpes of these relationships for example see 5 8 and the references their in.the std of the conseqences of hiding behavior of pre on the dnamics of predator pre interactions can be recognized as a major isse in applied mathematics and theoretical ecolog 9. Some of the empirical and theoretical work have investigated the effect of pre refges and drawn a conclsion that the refges sed b pre have a stabilizing effect on the considered interactions and pre extinction can be prevented b the addition of refges.in fact the effect of pre refges on the poplation dnamics are ver complex in natre bt for modeling prposes it can be considered as constitted b two components the first effects which affect positivel the growth of pre and negativel that of predators. Comprise the redction of pre mortalit de to decrease in predation sccess. The second one ma be the trad-offs and b-prodcts of the hiding behavior of pre which cold be advantageos or detrimental for all the interacting poplations. A classic secondar effect is the redction in the birth rate of pre poplation becase refges are safe bt rarel offer feeding or mating opportnities 3.Z. Ma and et.al 5 derived a predator-pre model with Lotka-Voltera fnctional response incorporating pre refges the refges are considered as two tpes: a constant proportion of pre and a fixed nmber of pre sing refges. The evalate the effect with regard to the stabilit of the interior eqilibrim. The reslts show that the refges sed b pre can increase the eqilibrim densit of the pre poplation while decrease that of predator.on the other hand it is well known that the age factor is importance for the dnamics and evoltion of man mammals. The rate of srvival growth and reprodction almost depend on age or development stage and it has been noticed that the life histor of man species is composed of at least two stages immatre and matre and the species in the first stage ma often neither interact with other species nor reprodce being raised b their matre parents. Most of classical pre-predator models of two species in the literatre assmed that all predators are able to attack their pre and reprodce ignoring the fact that the life ccle of most animals consists of at least DOI:.979/ Page

2 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge two stages ( immatre and matre ). Recentl several of the pre-predator models with stage-strctre of species with or withot time delas are proposed and analzed 6 9. In this paper the food web prepredator model involving pre s refges is proposed and analzed so that the pre growing logisticall in the absence of predators. The effect of pre s refges and pre stage-strctre on the dnamical behavior of the food web model is investigated theoreticall as well as nmericall. II. The mathematical model Consider the food web model consisting of two predators-stage strctre pre in which the pre species growth logisticall in the absence of predation while the predators deca exponentiall in the absence of pre species. It is assmed that the pre poplation divides into two compartments: immatre pre poplation N (t) that represents the poplation size at time t and matre pre poplation N (t) which denotes to poplation size at time t. Frthermore the poplation size of the first predator at time t is denoted b N 3 (t) while N (t) represents the poplation size of second predator at time t. Now in order to formlate the dnamics of sch sstem the following assmptions are considered :. The immatre pre depends completel in its feeding on the matre pre that growth logisticall with intrinsic growth rate α > and carring capacit k >. The immatre pre individals grown p and becomes matre pre individals with grown p rate β >. However the matre pre facing death with natral death rate d >.. There is tpe of protection of the pre species from facing predation b first and second predators with refge rate constant m ). 3. The first and second predators consmed the matre pre individals onl according to the Lotka-Voltera tpe of fnctional response with predation rates c > and c > respectivel and contribte a portion of sch food with conversion rates < e < and < e < respectivel. Moreover there is an enter specific competition between these two predators with competition force rate c 3 > and c > respectivel. Finall in the absence of food the first and second predators facing death with natral death rate d > and d 3 >. Therefore the dnamics of this model can be represented b the set of first order nonlinear differential eqations: dn = α N N K β N dn = β N d N c m N N 3 c m N N dn 3 = d N 3 + e c m N N 3 c 3 N 3 N dn = d 3 N + e c m N N c N 3 N with initial conditions N i (). Note that the above proposed model has thirteen parameters in all which make the analsis difficlt. So in order to simplif the sstem the nmber of parameters is redced b sing the following dimensionless variables and parameters : t = α T = β α = d α 3 = d α = e c K α 5 = c 3 c 6 = d 3 α 7 = e c K α 8 = c c x = N = N z = c N 3 w = c N. K K α α Then the non-dimensional form of sstem can be written as : dx = x x = x f x z w d = x m z m w = f x z w dz = z 3 + m 5 w = z f 3 x z w dw = w m 8 z = w f x z w with x z and w. It is observed that the nmber of parameters have been redced from thirteen in the sstem to eight in the sstem.obviosl the interaction fnctions of the sstem are continos and have continos partial derivatives on the following positive for dimensional space. R + = x z w R x z w. Therefore these fnctions are DOI:.979/ Page

3 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge lipschitzian on R + and hence the soltion of the sstem exists and is niqe. Frther all the soltions of sstem with non-negative initial conditions are niforml bonded as shown in the following theorem. Theorem : All the soltions of sstem which initiate in R + are niforml bonded. Proof: Let x t t z t w t be an soltion of the sstem with non-negative initial condition x z w R +. Now according to the first eqation of sstem we have : dx = x. So b sing the comparison theorem on the above differential ineqalit with the initial point x = x we get : x t + x e t. Ths lim t x t and hence sp x t t > >. Now define the fnction : M t = x t + t + z t + w t and 7 then taken the time derivative of M t along the soltion of the sstem. we get : dm + x s M where s = min dm 3 6. Then + s M H where H = +. Again b solving this differential ineqalit for the initial vale M = M we get : M t H + M s H e st. Then lim s t M t H.So M t H s s of sstem are niforml bonded and the proof is complete. t >.Hence all the soltions III. The existence of eqilibrim points In this section the existence of all possible eqilibrim points of sstem is discssed. It is observed that sstem has at most five eqilibrim points which are mentioned in the following : The eqilibrim point E = which known as the vanishing point is alwas exists. The first eqilibrim point E = x where = and x = exists niqel in Int. R + (Interior of R + ) of x plane nder the following necessar and sfficient condition : < 3 The first three species eqilibrim point E = x z where x = 3 m m 3 m = 3 and z = m 3 3 exists niqel in Int. R m m + of xz spacender the following necessar and sfficient condition : 3 < min m m The second three species eqilibrim point E 3 = x w where x = = 6 7 m 6 7 m 7 m 6 7 m and w = 7 m 6 7 m exists niqel in Int.R + 3 of xw space nder the following necessar and sfficient condition : 6 < min 7 m 7 m 5 Finall the positive eqilibrim point E = x z w where x = DOI:.979/ Page = m + 3 m m + m z = 7 m 6 and w = m 3 exists in Int.R nder the 5 5 following necessar and sfficient conditions : + 7 m + m > m + 3 m 6 a m + 6 m + 3 m > m + m 6 b m + 6 m + 3 m > m + m 6 c IV. Local stabilit analsis In this section the local stabilit analsis of sstem arond each of the above eqilibrim points are discssed throgh compting the Jacobian matrix J x z w of sstem at each of them which given b : J = a ij where a = a = a 3 = a = a = a = m z m w a 3 = m a = m a 3 = a 3 = m z a 33 = 3 + m 5 w a 3 = 5 z a = a = 7 m w a 3 = 8 w a = m 8 z. The Local stabilit analsis at E The Jacobian matrix of sstem at E can be written as : J E = 3 6 Then the characteristic eqation of J E is given b : λ + A λ + A 3 λ 6 λ =

4 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge where A = + A =. So either 3 λ 6 λ = 7 a which gives two of the eigenvales of J E b : λ z = 3 < and λ w = 6 < Or λ + A λ + A = 7 b which gives the other two eigenvales of J E b : λ x = A + A A and λ = A A A Therefore if the following condition holds : > 7 c E is locall asmptoticall stable in the R +. However it is a saddle point otherwise. The Local stabilit analsis at E The Jacobian matrix of sstem at E can be written as : J E = m m 3 + m m Then the characteristic eqation of J E is given b : λ + A λ + A 3 + m λ m λ = where A = + and A = + So either 3 + m λ m λ = 8 a which gives two of the eigenvales of J E b : λ z = 3 + m and λ w = m Or λ + A λ + A = 8 b which gives the other two eigenvales of J E b : λ x = A holds : + A A and λ = A < < min 3 m 6 7 m A A.Therefore if the following condition Under condition 3. E is locall asmptoticall stable in the R +. However it is a saddle point otherwise. The Local stabilit analsis at E The Jacobian matrix of sstem at E can be written as : J E = b ij where : b = < b = b 3 = b = b = > b = m z < b 3 = m < b = m < b 3 = b 3 = m > b 33 = 3 + m b 3 = 5 z < b = b = b 3 = b = m 8 z. Then the characteristic eqation of J E is given b : λ 3 + A λ + A λ + A m 8 z λ = where 3 A = + m A = m + 3 m m 3 m A 3 = 3 m 3. m So either m 8 z λ = 9 a Or λ 3 + A λ + A λ + A 3 = 9 b Hence from eqation.9 a we obtain that : λ w = m 8 z which is negative provided that : 7 m < z 9 c On the other hand b sing Roth-Hawirtiz criterion eqation 9 b has roots ( eigenvales ) with negative real parts if and onl if : A > A 3 > and = A A A 3 >. Now direct comptation gives that : = 3 m m + 3 m 3 + m m So > nder the following condition : m m + 3 > 3 + m Now it is eas to verif that A > and A 3 > nder condition.then all the eigenvales λ x λ and λ z of eqation 9 b have negative real parts. So E is locall asmptoticall stable if and onl if conditions 9 c and are hold. However it is a saddle point otherwise. The Local stabilit analsis at E 3 The Jacobian matrix of sstem at E 3 can be written as : J E 3 = c ij where c = < c = c 3 = c = c = > c = m w > c 3 = m > c = m < c 3 = c 3 = c 33 = 3 + m 5 w DOI:.979/ Page 8 c

5 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge c 3 = c = c = 7 m w > c 3 = 8 w < c = m. Then the characteristic eqation of J E 3 is given b : λ 3 + B λ + B λ + B m 5 w λ = where 6 B = + 7 m B = 7 m + 7 m m 6 7 m B 3 = 6 7 m m 6. 7 m 7 m while Δ = B B B 3 = 6 7 m m 7 m m 7 m So either 3 + m 5 w λ = a Or λ 3 + B λ + B λ + B 3 = b From eqation a we obtain that : λ 3z = 3 + m 5 w which is negative provided that: m < w On the other hand it is eas to verif that B > and B 3 > nder condition 5 while > nder the following condition : 7 m m > m 3 Then all the eigenvales λ 3x λ 3 and λ 3w of eqation b have negative real parts. So E 3 is locall asmptoticall stable if and onl if conditions and 3 are hold. However it is a saddle point otherwise. Theorem : Assme that the positive eqilibrim point E = x z w of sstem exists in IntR +. Then it is locall asmptoticall stable provided that the following conditions hold : P < 3 P P a P 3 < 3 P P 33 b P < 3 P P P 33 P < P 3 Proof : It is eas to verif that the linearized sstem of sstem can be written as : dx = dr = J E R here X = x z w and R = r r r 3 r t where r = x x r = x x r 3 = x 3 x 3 and r = x x. Moreover J E = d ij where d = d = d 3 = d = d = d = m m w d 3 = m d = m d 3 = d 3 = m d 33 = 3 + m 5 w d 3 = 5 z d = d = 7 m w d 3 = 8 w d = m 8 z. Now consider the following positive definite fnction : V = r + r + r 3 + r. It is clearl that V R + R and is a C positive definite fnction. Now b differentiating V with respect to time t and doing some algebraic maniplation gives that : dv = P r + P r r P r + P 3 r r 3 + P r r P 33 r 3 + P 3 r 3 r P r where P = P = + P = + m z + m w P 3 = m z P = m 7 w P 33 = 3 m + 5 w P 3 = 5 z 8 w P = 6 7 m + 8 z. Now it is eas to verif that the above set of conditions garantee the qadratic terms give below : dv P r P 3 r P 3 r P 33 r 3 P 3 r P r P 33 r 3 + P r So dv is negative definite and hence V is a Lapnov fnction.ths E is locall asmptoticall stable and the proof is complete. V. Global stabilit analsis In this section the global stabilit analsis for the eqilibrim points which are locall asmptoticall stable of sstem is stdied analticall with the help of Lapnov method as shown in the following theorems. c d DOI:.979/ Page

6 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge Theorem 3 : Assme that the eqilibrim point E = of sstem is locall asmptoticall stable in the R +. Then the eqilibrim point E of sstem is globall asmptoticall stable. Proof: Consider the following fnction : V x z w = c x + c + c 3 z + c w where c c c 3 and c are positive constants to be determine.clearl V : R + R is a C positive definite fnction. Now b differentiating V with respect to time t and doing some algebraic maniplation gives that : dv = c + c c x c + c 3 c m z c + c 7 c m w c 3 3 z c 6 w c c 8 w z. B choosing c = c = c 3 = c = dv we get : dv 7. Then we obtain that is negative definite and hence V is a Lapnov fnction. Ths E is globall asmptoticall stable and the proof is complete. Theorem Assme that the eqilibrim point E = x of sstem is locall asmptoticall stable in the Int R + Then E is globall asmptoticall stable on an region Ω Int R + that satisfies the following conditions : + + x + m m x < x + m x x < x x Proof: Consider the following fnction : x m V x z w = c x x x ln x + c x ln + c 3 z + c w where c c c 3 and c are positive constants to be determine.clearl V : R + R is a C positive definite fnction. Now b differentiating V with respect to time t and doing some algebraic maniplation gives that : dv = c x x + c x c + + c x x c x x x c x +c x x c m z c m w c 3 3 z +c 3 m c 6 w + c 7 m w c c 8 w z. B choosing c = c = c m 3 = c = we get : 3 6 dv x x 6 7 m 6 x m w So according to condition 8 c we obtain that : dv DOI:.979/ Page + x x xx x x 3 m 3 z 5 a 5 b 5 c x + x x m However conditions 5 a and 5 b garantee the completeness of the qadratic term between x and. So if condition 5 c holds then we obtain that is negative definite on the region Ω and hence V is a Lapnov fnction defined on the region Ω. Ths E is globall asmptoticall stable on the region Ω and the proof is complete. Theorem 5 : Assme that the eqilibrim point E = x z of sstem is locall asmptoticall stable in IntR + 3. Then E is globall asmptoticall stable on an region Ω Int R + 3 that satisfied the following conditions : x + + x + x < x + x x < x x x x dv 6 a 6 b 6 c.

7 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge In addition that the following condition holds : 5 z + m < 6 7 Proof: Consider the following fnction : V 3 x z w = c x x x ln + c ln + c 3 z z z ln z + c z w where c c c 3 and c are positive constants to be determine.clearl V 3 : R + R is a C positive definite fnction. Now b differentiating V 3 with respect to time t and doing some algebraic maniplation gives that : dv 3 = c x x + c c + + c x x + c x x x x c c m z z c m w + c 3 m z z c 3 5 z z w c 6 w + c 7 m w c 8 w z. B choosing c = c = c 3 = c = we get : 7 dv 3 x x x Now according to condition 6 d we obtain that : + x x 6 7 5z m w DOI:.979/ Page dv 3 x x x + 6 d x x. However conditions 6 a and 6 b garantee the completeness of the qadratic term between x and. So if condition 6 c holds. Therefore dv 3 is negative on the region Ω and hence V 3 is a Lapnov fnction defined on the region Ω. Ths E is globall asmptoticall stable and the proof is complete. Theorem 6 : Assme that the eqilibrim point E 3 = x w of sstem is locall asmptoticall 3 3 stable in Int R +.Then E 3 is globall asmptoticall stable on an region Ω 3 Int R + that satisfies the following conditions : + + x < x + x x < + x x x x In addition that the following condition holds : 8 w + m < 3 7 Proof: Consider the following fnction : V x z w = c x x x ln + c ln + c 3 z + c w w w ln w w where c c c 3 and c are positive constants to be determine.clearl V : R + R is a C positive definite fnction. Now b differentiating V with respect to time t and doing some algebraic maniplation gives that : dv = c x x + c x c + + c x x + c x x x x c c m z c m w w c 3 3 z + c 3 m z c 3 5 w z + c 7 m w w c 8 w w z. B choosing c = c = c 3 = c = we get : 7 dv x x x m Now according to condition 7 d we obtain that : dv 3 m 8 7 w z + 7 a 7 b 7 c 7 d x x x x x + m x x. However conditions 7 a and 7 b garantee the completeness of the qadratic term between x and.so if condition 7 c holds then we obtain that is negative definite on the region Ω 3 and hence V is a Lapnov fnction defined on the region Ω 3. Ths E 3 is globall asmptoticall stable on the region Ω 3 and the proof is complete. dv

8 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge Theorem 7 : Assme that the eqilibrim point E = x z w of sstem is locall asmptoticall stable in the R +. Then E is globall asmptoticall stable on an region Ω Int R + that satisfies the following conditions : + + x + x < x + 8 b x x w z + w z < β where 8 c β = x x Proof: Consider the following fnction : x 8 a V 5 x z w = c x x x ln + c ln + c 3 z z z ln z z + c w w w ln w w where c c c 3 and c are positive constants to be determine.clearl V 5 : R + R is a C positive definite fnction. Now b differentiating V 5 with respect to time t and doing some algebraic maniplation gives that : dv 5 = c x x + c x c + x + c x x c x + c x x c 8 + c 3 w w z z + c 7 c m w w + c 3 c m z z B choosing c = c = c 3 = c = we get : 7 dv 5 x x x + x x w z + w z However conditions 8 a and 8 b garantee the completeness of the qadratic term between x and. So if condition 8 c holds then we obtain that dv 5 is negative definite on the region Ω and hence V 5 is a Lapnov fnction defined on the region Ω. Ths E is globall asmptoticall stable on the region Ω and the proof is complete. VI. Nmerical simlation In this section the dnamical behavior of sstem is stdied nmericall for different sets of parameters and different sets of initial points. The objectives of this std are : first investigate the effect of varing the vale of each parameter on the dnamical behavior of sstem and second confirm or obtained analtical reslts. It is observed that for the following set of hpothetical parameters that satisfies stabilit conditions of the positive eqilibrim point sstem has a globall asmptoticall stable positive eqilibrim point as shown in Fig.. =. =. 3 =. =.8 5 =. 6 =. 7 =.8 8 =. m =.8 9 DOI:.979/ Page

9 Poplation Poplation Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge Fig. : Time series of the soltion of sstem that started from for different initial points and.7... for the data given b Eq..9. a trajectories of x as a fnction of time b trajectories of as a fnction of time c trajectories of z as a fnction of time d trajectories of w as a fnction of time. Clearl Fig. shows that sstem has a globall asmptoticall stable as the soltion of sstem approaches asmptoticall to the positive eqilibrim point E = starting from for different initial points and this is confirming or obtained analtical reslts..now in order to discss the effect of the parameters vales of sstem on the dnamical behavior of the sstem the sstem is solved nmericall for the data given in Eq. 9 with varing one parameter at each time. It is observed that for the data as given in Eq. 9 with..9 the soltion of sstem approaches asmptoticall to the positive eqilibrim point as shown in Fig immatre pre matre pre first predator second predator Time Fig. : Time series of the soltion of sstem for the data given b Eq. 9 which approaches to in the interior of R +. B varing the parameter and keeping the rest of parameters vales as in Eq. 9 it is observed that for.. the soltion of sstem approaches asmptoticall to a positive eqilibrim point E while for.5.9 the soltion of sstem approaches asmptoticall to E = x in the interior of the positive qadrant of x plane as shown in Fig immatre pre matre pre first predator second predator Time Fig. 3 : Times series of the soltion of sstem for the data given b Eq. 9 which approaches to.97.8 in the interior of the positive qadrant of x plane. On the other hand varing the parameter 3 keeping the rest of parameters vales as in Eq. 9 it is observed that for. 3.9 sstem approaches asmptoticall to E = x in the interior of the positive qadrant of x plane.moreover varing the parameter and keeping the rest of parameters vales as in Eq. 9 it is observed that for.. the soltion of sstem approaches asmptoticall to E = x in the interior of the positive qadrant of x plane while for.5.9 the soltion of sstem approaches asmptoticall to a positive eqilibrim point E.For the parameter. 5.9 and keeping the rest of parameters vales as in Eq. 9 showed that the soltion of sstem approaches asmptoticall to a positive eqilibrim point E.Moreover for the parameters vales given in Eq. 9 with. 6.9 the soltion of sstem approaches asmptoticall to a positive eqilibrim point E. For the parameters vales given in Eq. 9 with. 7.9 the soltion of sstem approaches asmptoticall to a positive eqilibrim point E. For the parameter. 8.9 and keeping the rest of parameters vales as in Eq. 9 showed that the soltion of sstem approaches DOI:.979/ Page

10 Poplation Poplation Poplation Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge asmptoticall to a positive eqilibrim point E.For the parameters vale given in Eq. 9 and varing the parameter m in the rang. m.9 showed that the soltion of sstem approaches asmptoticall to a positive eqilibrim point E.For the parameters vales given in Eq. 9 with =.9 5 =. 7 =. and 8 =.8 the soltion of sstem approaches asmptoticall to E = x z in the interior of the positive qadrant of xz plane as shown in Fig...5 immatre pre matre pre first predator second predator Time Fig. : Time series of the soltion of sstem for the data given b Eq. 9 with =.9 5 =. 7 =. and 8 =.8 which approaches asmptoticall to in the interior of the positive qadrant of xz space. Moreover for the parameters vales given in Eq. 9 with =. 5 =.8 6 =. the soltion of sstem approaches asmptoticall to E 3 = x w in the interior of the positive qadrant of xw space as shown in Fig immatre pre matre pre first predator second predator Time Fig. 5 : Time series of the soltion of sstem for the data given b Eq. 9 with =. 5 =.8 6 =. which approaches asmptoticall to in the interior of the positive qadrant of xw space. Finall the dnamical behavior at the vanishing eqilibrim point E = is investigated b choosing = and keeping other parameters fixed as given in Eq. 9 and then the soltion of sstem is drawn in Fig immatre pre matre pre first predator second predator Time Fig. 6 : Time series of the soltion of sstem for the data given b Eq. 9 with = which approaches to. Obviosl Fig. 6 shows clearl the convergence of the soltion of sstem to the vanishing eqilibrim point E = when the parameters increase p to a specific vales. clearl the sed valed in Fig. 6 satisf the stabilit condition of the vanishing eqilibrim point. VII. Conclsion and discssion In this chapter we have considered a pre-predator sstem incorporating a stage strctre of pre with refge. It is assmed that the predator species pre pon the pre according to Lotka-Voltera tpe of fnctional response. The existence niqeness and bondedness of the soltion of the sstem are discssed. The existence of all possible eqilibrim points is stdied. The local and global dnamical behaviors of the sstem are stdied analticall as well as nmericall. Finall to nderstand the effect of varing each parameter on the global dnamics of sstem has been solved nmericall for a biological feasible set of hpothetical parameters vales and the following reslts are obtained :. For the set of hpothetical parameters vales given in Eq. 9 the sstem approaches asmptoticall to global stable positive eqilibrim point. DOI:.979/ Page

11 Stabilit analsis of two predator-one stage-strctred pre model incorporating a pre refge. It is observed that the sstem has no effect on the dnamical behavior for the data given in Eq. 9 with varing the parameter vale and the sstem still approaches to positive eqilibrim point. 3. As the natral death rate of matre pre increasing in the range. < <. and keeping other parameters fixed as in Eq. 9 then again the soltion of sstem approaches asmptoticall to the positive eqilibrim point.however increasing in the range.5 < <.9 will case extinction in the predators and the soltion of sstem approaches asmptoticall to E = x.. As the first predator natral death rate 3 increasing keeping the rest of parameters as in Eq. 9 then again the soltion of sstem approaches asmptoticall to E = x. It is observed that the natral death rate 6 of the second predator has the same effect as. 5. As the predation rate increasing in the range..7 cases extinction in the predators and the soltion of sstem approaches asmptoticall to E = x. However increasing in the range.5.9 then again the soltion of sstem approaches asmptoticall to the positive eqilibrim point. 6. As the competition rate 5 increasing keeping the rest of parameters as in Eq. 9 then again the soltion of sstem approaches asmptoticall to the positive eqilibrim point. It is observed that the competition rate 8 and the nmber of pres inside refge m have the same effect as. 7. As the predation rate 7 increasing in the range. 7.9 and keeping the rest of parameters as in Eq. 9 then again the soltion of sstem approaches asmptoticall to the positive eqilibrim point. References []. T. K. Kar and Ashim Batabal Persistence and stabilit of a two pre one predator International Jornal of Engineering Science and Technolog Vol. No. pp.7-9. []. Weiming Wang Hailing Wan and Zhenqing Li Chaotic behavior of a three-species beddington-tpe sstem with implsive pertrbations Chaos Soltion and Fractals [3]. Snita Gakkar Brahampal Singh and Raid Kamel Naji Dnamical behavior of two predators competing over a single pre BioSstems []. Jawdat Alebraheem and YahYa Ab-Hasan Presistence of predators in a two predators-one pre model with non-preiodic soltion App.Math.Scie.Vol.6 No [5]. Madhsdanan V. Vijaa S. and Gnasekaran M Impact of harvesting in three species food web model with two distinct fnctional response International Jornal of Innovative Research in Science Engineering and TechnologVol.3 Isse p [6]. R. K. Naji and I.H. Kasim The dnamics of food web model with defensive switching propert Nonlinear Analsis: Modelling and Control Vol.3 No [7]. B-Dbe and R.K. Upadha Persistence and extinction of one-pre and two-predators sstem Nonlinear Analsis: Modeling and Control Vol.9 No. ew [8]. Sze-Bi Hs Shigi Ran and Ting-Hi Yang Analsis of three species Lotka-Voltera food web models with omnivor J. Math. Anal. Applic [9].. Manard Smith Models in ecolog Cambridge Universit Cambridge 97. []. M. P. Hassell The dnamics of arthropod predator-pre sstems Princeton Universit Princeton NJ 978. []. T. K. Kar Stabilit analsis of a per-predator model incorporating a pre refge Commnications Nonlinear Science and Nmerical Simlation []. Vlastimil Krivan On the Gase predator-pre model with a refge: A fresh look at the histor Jornal of Theoretical Biolog [3]. R. J. Talor Predation Chapman and Hall NewYork 98. []. Xiao lei Y and Fqin Sn Global dnamics of a predator-pre model incorporating a constant pre refge Electronic Jornal of differential eqations Vol.3 No. 3 pp.-8. [5]. Zhihi Ma Wenlong Li Y. ZhaoWenting Wang Hi Zhang and Zizhen Li Effects of pre refges on a predator-pre model with a class of fnctional responses: The role of refges Mathematical BioSciences [6]. Kai Hong and Peixam Weng Stabilit and traveling waves of diffsive predator-pre model with age-strctre and nonlocal effect Jornal of Applied Analsis and Comptation Vol. No. p [7]. Ri X Global stabilit and Hopf bifrcation of a predator-pre model with stage strctre and delaed predator response Nonlinear Dnamics Vol.67 Isse pp [8]. O. P. Misra Poonam Sinha and Chhatrapal Singh Stabilit and bifrcation analsis of a pre-predator model with age based predation Applied Mathematical Modelling [9]. Ye Kaili and Song XinY Predator-pre sstem with stage strctre and dela Appl. Math. J. Chinese Univ. Ser. B 8() []. Pal-Georgesc and Ying-Hen Hsieh Global dnamics of a predator-pre model with stage strctre for the predator SIAMJ.App.Math.Vol.67 No.5 7 pp DOI:.979/ Page