PERSISTENCE AND GLOBAL STABILITY OF A LESLIE-GOWER PREDATOR-PREY REFUGE SYSTEM WITH A COMPETITOR FOR THE PREY

Abstract. This paper analyzes the effect of refuge on the dynamics of a Leslie-Gower predator-prey model in which one predator feeds on one of two competing species. Existence conditions for equilibrium points are discussed. By using differential inequality argument, we developed persistence criterion. Sufficient condition for global stability of the unique positive equilibrium point is derived. Different type of local bifurcation near the equilibrium points has been investigated. The role of refuges have been shown on equilibrium densities of prey, competitor for prey and predator respectively. The results establish the fact that the effects of refuges used by prey increase the equilibrium density of prey population under certain restrictions, whereas opposite hold for competitor of prey population. However equilibrium density of predator may decrease or increase by increasing the amount of prey refuge. Some numerical simulations are performed to validate the results obtained.


Introduction
There has been a growing interest in the study of refuges in predator-prey system.González-Oilvares and Ramos-Jiliberto [6] studied a predator-prey system with Holling type-II functional response and a prey refuge.They showed that there is a trend from limit cycles through non-zero stable points up to predator extinction and prey stabilizing at high densities.Kar [9] investigated a Lotka-Volterra type predator-prey system incorporating a constant proportion of prey refuges with Holling type-II response function.He remarked that it is possible to break the cyclic behaviour of the system if harvesting effects as controls.Chen et al. [4] analysed the uniqueness of limit cycles and global stability of the unique positive equilibrium of predator-prey system with Holling type-II functional response and a constant number of refuges.Chen et al. [3] , and Yue [22] studied Leslie-Gower predator-prey system incorporating a constant proportion prey refuge and showed the global stability at the interior equilibrium point.More results on the effects of a prey refuge can be found in [2,5,7,10,12,14,15,16,17,19].Previous studies on Leslie-Gower predator-prey system are mainly confined into constant proportion of refuge which acts on the system as an external decreasing of the uptake rate and half saturation constant, does not change the dynamical behaviour of the prey-predator model.Thus our main object in this work to modify the refuge term.Recently, Mukherjee [14] studied the effect of immigration and refuge on the dynamics of three species system.He discussed about the persistence of the system and global stability.Model considered by him is of Lotka-Volterra type.In another paper [16] Mukherjee investigated same type of situation without immigration and predation process follows Holling-type II response function.Both of the papers, he did not addressed what will be dynamical consequence if Leslie-Gower form is taken.Further we are interested to know the dynamics consequence of the predator-prey system in presence of a competitor for the prey in a Leslie-Gower model.This paper is structured as follows.In Section 2, we propose our model.Some preliminaries (positivity and boundedness) is discussed in Section 3. In Section 4, we analyze our model with regard to equilibria and stabilities.Section 5, deals with the local bifurcation analysis around the equilibria.Persistence criterion is established in Section 6. Influence of prey refuge is given in Section 7. Numerical simulations is presented in Section 8.A brief discussion is presented in Section 9.

Mathematical Model
In Leslie-Gower prey-predator model, predator equation is taken logistic growth with carrying capacity proportional to the prey density.This type of situation are applicable in ecology [11,13,20] because the direct conversion of prey density into offspring is inappropriate for a small mammalian predator that uses most of its energy intake on generating heat and because model of Leslie's type assume interferences of predators which is justifiable for territorial predators [20].In this paper we introduce a predator-prey model with Leslie-Gower functional response incorporating a positive constant prey refuge with the presence of a competitor for the prey : with initial conditions x(0) > m, y(0) ≥ 0, z(0) ≥ 0.
Here x, y, z denotes the density of the prey, competitor for the prey and predator respectively.r 1 is the intrinsic growth rate of the prey species and r 2 is the intrinsic growth rate of the competitor for the prey species.b 1 is the infraspecific competition coefficient of the prey.α denotes the interspecific competition coefficient of the competitor for the prey.b 2 represents the infraspecific competition coefficient of the competitor for the prey.β corresponds to the intraspecific competition coefficient of the competitor for the prey.r 3 describes the growth rate of predator.a 1 is the per capita predator consumption rate.a 2 is the efficiency with which predators convert consumed prey.m is the constant number of prey using refuge.k is the half saturation constant.Specific example illustrates the above model: Consider two species of aphid (Acyrthosiphon pisum and Megoura viciae) competing for the same food plant and a species of a specialist parasitoid (Aphidius ervi) that attacks only one of the aphids (A.pisum).From experimental studies van Veen et al. [21]showed that (i) when the two aphid species compete for resources in the absence of parasitoid. A. pisum excludes M. viciae.(ii) When the aphid species and the parasitoid are all present, all three species can coexist.
Proof.Since the right hand side of system (1.1) is continuous and locally Lipschitzian on C, the solution (x(t), y(t), z(t)) of system (1.1) with initial conditions exists and is unique on [0, φ ), From system (1.1), we have Thus any trajectory starting in R 3 + cannot cross the co-ordinate axes.This completes the proof.
Proof.Let us define W (t) = x + y + z and λ > 0 be a constant.Then By using differential inequality [1] we obtain, Taking limit t → ∞, we have 0 < W (t) ≤ M ζ .This proves the Lemma.

Equilibria
Evidently, system (1.1) has at most five equilibrium points: the trivial equilibrium point E 0 = (0, 0, 0) which does not belongs to B. The axial equilibrium point equilibrium point E 13 = ( x, 0, z) where x is the positive root of the equation (4.1) . The interior equilibrium point is given by and x * is the positive root of the equation 2) Proof.Proof follows immediately by linearising around the equilibria.
Theorem 1.2.The interior equilibrium point E * of system ( 1) is locally asymptotically stable if Proof.The Jacobian matrix of system (1) at the equilibrium point E * is given by The characteristic equation about E * is given by where Now A 1 > 0, A 3 > 0 follows from the assumption of the Theorem 1.2.Also A 1 A 2 > A 3 .Therefore by Routh-Hurwitz criterion the result follows.

Local bifurcation analysis
In this section, we use the application of Sotomayor's theorem [18] to investigate the local bifurcation around the equilibrium points of system (1.1).As the existence of non-hyperbolic equilibrium point is a necessary but not sufficient condition for bifurcation to occur therefore we choose a parameter which gives zero eigenvalues to the Jacobian at the equilibria.Now rewrite system (1.1) in the form : dX dt = F(X) where X = (x, y, z) t and F = (F 1 , F 2 , F 3 ) where The local bifurcation near the equilibrium points is investigated in the following theorems: Theorem 1.3.System (1.1) undergoes a transcritical bifurcation at the axial equilibrium point E 1 but no saddle node bifurcation can occur when the parameter β crosses the critical value Proof.One of the eigenvalues of the Jacobian matrix J(E 1 ) will be zero if β = β * = b 1 r 2 r 1 .Now the Jacobian matrix of system (1.1) at E 1 with zero eigenvalue is given by ) t be the eigenvector corresponding to eigenvalue λ = 0. Thus V = (v 1 , − v 1 b 1 α , 0) t where v 1 be any non zero real number.Also, let W = (w 1 , w 2 , w 3 ) t represents the corresponding eigenvector of J(E 1 ) t to the eigenvalues of λ = 0. Hence J(E 1 ) t W = 0 gives that W = (0, w 2 , 0) t where w 2 be any non zero real number.Now Thus according to Sotomayor's theorem system (1.1) has no saddle-node bifurcation at β = β * .Again Thus according to Sotomayor's theorem system (1.1) has a transcritical bifurcation at E 1 when the parameter β crosses the critical value β * .Furthermore, as the Jacobian matrix of E 1 has three linear factors, so no Hopf bifurcation can occurs.

Numerical example for trnascritical bifurcation
Choose  Proof.Proof is similar to the proof of Theorem 1.3.
Remark 2: System (1) does not undergoes any Hopf-bifurcation around the interior equilibrium point E * as in equation ( 4), A 1 > 0 and A 1 A 2 − A 3 cannot be equal to zero.
Then the interior equilibrium point E * is globally asymptotically stable.
Proof.First note that, E 13 is unstable as r 2 > β x * and other boundary equilibrium points are always unstable whenever they exist.
Consider the following positive definite function about Differentiating V with respect to t along the solution of system (1.1), we get Thus the condition of Theorem 1.5 implies that V is a Lyapunov function and hence the theorem follows.

Persistence
Biologically persistence means the long time survival of all population in a future time whatever may be the initial populations.By differential inequality argument we state some result guaranteeing the persistence of all the populations of system (1.1).
Proof. when x(t) > m, lim t→∞ sup x(t) ≤ r 1 b 1 then from the second equation of (1) we have As, r 2 > β m, this implies that lim t→∞ inf y(t From third equation of (1.1), we have

Influence of prey refuge
In the following we shall discuss the influence of prey refuge on each population when the coexistence equilibrium point E * is exists and is stable.It is easy to see that x * , y * , z * are all continuous differential functions of parameter m.   1, r 3 = 2, k = 0.25, the system oscillates near this equilibrium point E * ( 5 7 , 17 7 , 13  21 ).Also some chaotic type oscillation is observed (see Fig. 5.).

Discussion
In this paper we have considered a prey-predator system where prey has a competitor.Due to predation pressure prey population uses refuge mechanism.The model is formulated according to Leslie-Gower.This type of model usually exhibits stable behaviour with or without refuge.
But the dynamics of the model may be changed if the interspecific competition is allowed.We have throughly investigated bifurcation analysis around the equilibria.We note that certain parameters of the system are very sensitive to give transcritical bifurcation.We further observe that Hopf-bifurcation cannot occur around the equilibria which is either interior or boundary.

. 1 )
Again dy* dm = − β b 2 dx * dm < 0 and dz * dm = r 3 a 2 ( dx * dm − 1).Clearly x * is strictly increasing function of parameter m whenever (7.1) holds and increasing the amount of prey refuge leads to the increasing density of the prey species.y * is strictly decreasing function of parameter m and increasing the amount of prey refuge leads to the decreasing density of the competitor prey species.The presence of negative term in the third equation indicates that increasing the amount of prey refuge may decrease the predator density as long asdx * dm < 1.

Fig. 5 .
Fig. 5.The system (1.1) is unstable and chaotic type oscillation observed Theorem 1.1 (i) Equilibrium points E 1 and E 12 are always unstable.(ii) E 13 is locally asymptotically stable if r 2 < β x.