# Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

Authors: Lingshu Wang; Guanghui Feng
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, whenτ=τ0. By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.


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## Body

## 1. Introduction

Since the pioneering work of Kermack-Mckendrick on SIRS [1], epidemiological models have received much attention from scientists. Mathematical models have become important tools in analyzing the spread and control of infectious disease. It is of more biological significance to consider the effect of interacting species when we study the dynamical behaviors of epidemiological models. Ecoepidemiology which is a relatively new branch of study in theoretical biology, tackles such situations by dealing with both ecological and epidemiological issues. It can be viewed as the coupling of an ecological predator-prey model and an epidemiological SI, SIS, or SIRS model. Following Anderso and May [2] who were the first to propose an ecoepidemiological model by merging the ecological predator-prey model introduced by Lotka and Volterra, the effect of disease in ecological system is an important issue from mathematical and ecological point of view. Many works have been devoted to the study of the effects of a disease on a predator-prey system [1–5]. In [5], Xiao and Chen have considered a ratio-dependent predator-prey system with disease in the prey. Consider
(1)dSdt=rS(1-S+IK)-βSI,dIdt=βSI-dI-bIYaY+I,dYdt=pbIYaY+I-cY,
where S(t) and I(t) represent the densities of susceptible and infected prey population at time t, respectively, and Y(t) represents the density of the predator population at time t. The parameters r, K, β, d, b, a, p, and c are positive constants representing the prey intrinsic growth rate, carrying capacity, transmission rate, the infected prey death rate, capturing rate, half capturing saturation constant, conversion rate, and the predator death rate, respectively. A periodic solution can occur whether the system (1) is permanent or not; that is, there are solutions which tend to disease-free equilibrium while bifurcating periodic solution exists.Recently, the qualitative analysis of predator-prey models incorporating a prey refuge has been done by many authors, see [3, 4]. In [3], Pal and Samanta incorporated a prey refuge (1-m)I into system (1). Sufficient conditions were derived for the stability of the equilibria of the system.We note that it is assumed in system (1) that each individual predator admits the same ability to feed on prey. This assumption seems not to be realistic for many animals. In the natural world, there are many species whose individuals pass through an immature stage during which they are raised by their parents, and the rate at which they attack prey can be ignored. Moreover, it can be assumed that their reproductive rate during this stage is zero. Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. Stage-structured models have received great attention in recent years (see, e.g., [6–9]).Time delays of one type or another have been incorporated into biological models by many researchers (see, e.g., [8–11]). In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause the population to fluctuate. Time delay due to gestation is a common example, because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays.Based on the above discussions, in this paper, we incorporate a prey refuge, stage structure for the predator, and time delay due to the gestation of predator into the system (1). To this end, we study the following differential equations:
(2)dSdt=rS(t)(1-S(t)+I(t)K)-βS(t)I(t),dIdt=βS(t)I(t)-dI(t)-b(1-m)I(t)Y2(t)aY2(t)+(1-m)I(t),dY1dt=pb(1-m)I(t-τ)Y2(t-τ)aY2(t-τ)+(1-m)I(t-τ)-(r1+d1)Y1(t),dY2dt=r1Y1(t)-d2Y2(t),
where Y1(t) and Y2(t) represent the densities of the immature and the mature predator population at time t, respectively, the parameters d1, d2, and r1 are positive constants in which d1 and d2 are the death rates of the immature and the mature predator, respectively, r1 denotes the rate of immature predator becoming mature predator, the constant proportion infected prey refuge is (1-m)I, where m∈[0,1) is a constant, and τ≥0 is a constant delay due to the gestation of the predator.The initial conditions for system (2) take the form
(3)S(θ)=ϕ1(θ)≥0,I(θ)=ϕ2(θ)≥0,Y1(θ)=φ1(θ)≥0,Y2(θ)=φ2(θ)≥0,θ∈[-τ,0),ϕ1(0)>0,ϕ2(0)>0,φ1(0)>0,φ2(0)>0,(ϕ1(θ),ϕ2(θ),φ1(θ),φ2(θ))∈C([-τ,0],R+04),
where R+04={(x1,x2,x3,x4):xi≥0,i=1,2,3,4}.It is well known by the fundamental theory of functional differential equations [12] that system (2) has a unique solution (S(t),I(t),Y1(t),Y2(t)) satisfying initial conditions (3).The organization of this paper is as follows. In the next section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3). In Section 3, we investigate the global stability of the predator-extinction equilibrium. In Section 4, we establish the local stability and the global attractivity of the coexistence equilibrium of system (2). Further, we study the existence of Hopf bifurcation for system (2) at the positive equilibrium. A brief discussion is given in Section 5 to conclude this work.


## 2. Preliminaries

In this section, we show the positivity and the boundedness of solutions of system (2) with initial conditions (3).Theorem 1.
Solutions of system (2) with initial conditions (3) are positive, for all t≥0.Proof.
Let(S(t),I(t),Y1(t),Y2(t)) be a solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that
(4)S(t)=S(0)exp⁡{∫0t‍[r-rKS(t)-(rK+β)I(t)]dt}>0,(5)I(t)=I(0)exp⁡{∫0t‍[βS(t)-d-b(1-m)Y2(t)aY2(t)+(1-m)I(t)]dt}>0.
Let us considerY1(t) and Y2(t), for t∈[0,τ]. Since ψ2(θ)≥0 for θ∈[-τ,0], we derive from the third equation of system (2) that
(6)dY1dt≥-(r1+d1)Y1(t).
Since ψ1(0)>0, a standard comparison argument shows that
(7)Y1(t)≥Y1(0)e-(r1+d1)t>0;
that is, Y1(t)>0 for t∈[0,τ]. For t∈[0,τ], it follows from the fourth equation of (2) that
(8)dY2dt≥-d2Y2(t).
Since ψ2(0)>0, a standard comparison argument shows that
(9)Y2(t)≥Y2(0)e-d2t>0;
that is, Y2(t)>0 for t∈[0,τ]. In a similar way, we treat the intervals [τ,2τ],…,[nτ,(n+1)τ],n∈N. Thus, S(t)>0, I(t)>0, Y1(t)>0, and Y2(t)>0, for all t≥0. This completes the proof.Theorem 2.
Positive solutions of system (2) with initial conditions (3) are ultimately bounded.Proof.
Let(S(t),I(t),Y1(t),Y2(t)) be any positive solution of system (2) with initial conditions (3). Denote d^=min⁡{d,d1,d2}. Define
(10)V(t)=pS(t-τ)+pI(t-τ)+Y1(t)+Y2(t).
Calculating the derivative of V(t) along positive solutions of system (2), it follows that
(11)dVdt=prKS(t-τ)[K-S(t-τ)-I(t-τ)]-pdI(t-τ)-d1Y1(t)-d2Y2(t)≤-d^V(t)-prK[S(t-τ)-K(r+d^)2r]2+pK(r+d^)24r,
which yields
(12)limsup⁡t→∞V(t)≤pK(r+d^)24rd^.
If we choose M1=(K(r+d^)2)/4rd^ and M2=(pK(r+d^)2)/4rd^, then
(13)limsup⁡t→∞S(t)≤M1,limsup⁡t→∞I(t)≤M1,limsup⁡t→∞Yi(t)≤M2,(i=1,2).
This completes the proof.


## 3. Predator-Extinction Equilibrium and Its Stability

In this section, we discuss the stability of the predator-extinction equilibrium.It is easy to show that ifKβ>d, system (2) admits a predator-extinction equilibrium E1(S1,I1,0,0), where
(14)S1=dβ,I1=r(Kβ-d)β(Kβ+r).The characteristic equation of system (2) at the equilibrium E1 is of the form
(15)λ2+g1λ+g0+f0e-λτ=0,
where g1=(r1+d1+d2),g0=d2(r1+d1),f0=-pbr1. When τ=0, if d2(r1+d1)>pbr1, then E1 is locally asymptotically stable and if d2(r1+d1)<pbr1, then E1 is unstable. It is easily seen that
(16)g12-2g0=(r1+d1)2+d22>0,g0-f0=d2(r1+d1)+pbr1>0.
Hence, if d2(r1+d1)>pbr1, by Lemma B in [11], it follows that the equilibrium E1 is locally asymptotically stable for all τ≥0. If d2(r1+d1)<pbr1, then E1 is unstable for all τ≥0.Theorem 3.
LetKβ>d hold; the predator-extinction equilibrium E1 is globally stable provided that
(17)d2(r1+d1)>pbr1,0<(1-m)<ab(Kβ-d).Proof.
Based on above discussions, we only prove the global attractivity of the equilibriumE1. Let (S(t),I(t),Y1(t),Y2(t)) be any positive solution of system (2) with initial conditions (3). It follows from the first and the second equations of system (2) that
(18)dSdt=rS(1-S+IK)-βSI,dIdt≤βSI-dI.
Consider the following auxiliary equations:
(19)dx1dt=rx1(1-x1+x2K)-βx1x2,dx2dt=βx1x2-dx2.
If Kβ>d, then by Theorem 3.1 in [4], it follows from (19) that
(20)lim⁡t→+∞x1(t)=dβ,lim⁡t→+∞x2(t)=r(Kβ-d)β(Kβ+r).
By comparison, we obtain that
(21)limsup⁡t→+∞S(t)≤dβ,limsup⁡t→+∞I(t)≤r(Kβ-d)β(Kβ+r)=I1.
Hence, for ε>0 sufficiently small, there is a T1>0 such that if t>T1, then I(t)≤I1+ε.
It follows from the third and the fourth equations of system (2) that, for t>T1+τ,
(22)dY1dt≤pb(1-m)(I1+ε)Y2(t-τ)aY2(t-τ)+(1-m)(I1+ε)-(r1+d1)Y1(t),dY2dt=r1Y1(t)-d2Y2(t).
Consider the following auxiliary equations:
(23)du1dt=pb(1-m)(I1+ε)u2(t-τ)au2(t-τ)+(1-m)(I1+ε)-(r1+d1)u1(t),du2dt=r1u1(t)-d2u2(t).
If d2(r1+d1)>r1pb, then by Lemma 2.4 in [9], it follows from (23) that
(24)lim⁡t→+∞u1(t)=0,lim⁡t→+∞u2(t)=0.
By comparison, we obtain that
(25)lim⁡t→+∞Y1(t)=0,lim⁡t→+∞Y2(t)=0.
Hence, for ε>0 sufficiently small, there is a T2>0 such that if t>T2, then Y2(t)≤ε.
It follows from the first and the second equations of system (2) that for t>T2:
(26)dSdt=rS(1-S+IK)-βSI,dIdt≥βSI-dI-b(1-m)εIaε+(1-m)I.
Consider the following auxiliary equations:
(27)dx1dt=rx1(1-x1+x2K)-βx1x2,dx2dt=βx1x2-dx2-b(1-m)εx2aε+(1-m)x2.
If Kβ>d, and (1-m)<(a/b)(Kβ-d), then by Theorem 3.1 in [4], it follows from (27) that
(28)lim⁡t→+∞x2(t)=(1-m)r(Kβ-d)-εaβ(Kβ+r)2(1-m)β(Kβ+r)+(×(2(1-m)β(Kβ+r))-1([(1-m)r(Kβ-d)-εaβ(Kβ+r)]2+4ε(1-m)rβ(Kβ+r)×[a(Kβ-d)-b(1-m)][(1-m)r(Kβ-d)-εaβ(Kβ+r)]2)1/2×(2(1-m)β(Kβ+r))-1)≔x_2lim⁡t→+∞x1(t)=K-Kβ+rrx_2.
By comparison, for ε sufficiently small, we obtain that
(29)liminf⁡t→+∞S(t)≥dβ,liminf⁡t→+∞I(t)≥r(Kβ-d)β(Kβ+r),
which, together with (21), yields
(30)lim⁡t→+∞S(t)=dβ,lim⁡t→+∞I(t)=r(Kβ-d)β(Kβ+r).
Hence, ifKβ>d,d2(r1+d1)>pbr1,(1-m)<(a/b)(Kβ-d) hold, then the equilibrium E1(S1,I1,0,0) is globally stable.


## 4. Coexistence Equilibrium and Its Stability

In this section, we discuss the stability of the coexistence equilibrium and the existence of a Hopf bifurcation. It is easy to show that if the following holds:(H1)
d2(r1+d1)<r1pb, 0<(1-m)<(ar1p(Kβ-d))/(r1pb-d2(r1+d1)),then system (2) has a unique coexistence equilibrium E*(S*,I*,Y1*,Y2*), where
(31)S*=dar1p+(1-m)[r1pb-d2(r1+d1)]βar1p,I*=rr+Kβ(K-S*),Y1*=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)I*,Y2*=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)I*.The characteristic equation of system (2) at the equilibrium E* takes the form
(32)λ4+p3λ3+p2λ2+p1λ+p0+(q2λ2+q1λ+q0)e-λτ=0,
where
(33)p3=r1+d1+d2+rKS*-α1,p2=d2(r1+d1)+(r1+d1+d2)(rKS*-α1)+S*[β(rK+β)I*-rKα1],p1=S*(r1+d1+d2)[β(rK+β)I*-rKα1]+d2(r1+d1)(rKS*-α1),p0=d2(r1+d1)S*[β(rK+β)I*-rKα1],q2=-pr1α2,q1=-pr1α2(rKS*-α1)+pr1α2α3,q0=-pr1α2S*[β(rK+β)I*-rKα1]+pr1α2α3rKS*,α1=b(1-m)2I*Y2*[aY2*+(1-m)I*]2,α2=b(1-m)2(I*)2[aY2*+(1-m)I*]2,α3=ab(1-m)(Y2*)2[aY2*+(1-m)I*]2.Whenτ=0, (32) becomes
(34)λ4+p3λ3+(p2+q2)λ+p0+q0=0.
If the following holds:(H2)
(r/K)S*-α1>0, β((r/K)+β)I*-(r/K)α1>0,then it is easy to show that
(35)p3>0,p0+q0>0,p1+q1>0,p2+q2>0.
If (p1+q1)[p3(p2+q2)-(p1+q1)]>p32(p0+q0), then, by the Routh-Hurwitz theorem, when τ=0, the coexistence equilibrium E* of system (2) is locally asymptotically stable and E* is unstable if (p1+q1)[p3(p2+q2)-(p1+q1)]<p32(p0+q0).Ifiω(ω>0) is a solution of (34), separating real and imaginary parts, we have
(36)(q2ω2-q0)sinωτ+q1ωcos⁡ωτ=p3ω3-p1ω,(q2ω2-q0)cos⁡ωτ-q1ωsinωτ=ω4-p2ω2+p0.
By squaring and adding the two equations of (36), it follows that
(37)ω8+h3ω6+h2ω4+h1ω2+p02-q02=0,
where
(38)h3=p32-2p2,h2=p22+2p0-q22-2p1p3,h1=p12+2q0q2-q12-2p0p2.
If h3>0,h2>0,h1>0 and p0-q0>0, by the general theory on characteristic equation of delay differential equation from [13] (Theorem 4.1), E* remains stable for all τ>0.Ifhi>0,(i=1,2,3) and p0-q0<0, then (37) has a unique positive root ω0; that is, (34) admits a pair of purely imaginary roots of the form ±ω0. From (36), we see that
(39)τn=2nπω0+1ω0arccos×((q2ω02-q0)(ω04-p2ω02+p0)+q1ω0(p3ω03-p1ω0))×((q1ω0)2+(q2ω02-q0)2)-1,n=0,1,2,….
By Theorem 3.4.1 in [13], we see that E* remains stable for τ<τ0.In the following, we claim that(40)d(Re(λ))dτ|τ=τ0>0.
This will show that there exists at least one eigenvalue with a positive real part for τ>τ0. Moreover, the conditions for the existence of a Hopf bifurcation (Theorem 2.9.1 in [13]) are then satisfied yielding a periodic solution. To this end, by differentiating equation (34) with respect to τ, it follows that
(41)(dλdτ)-1=4λ3+3p3λ2+2p2λ+p1-λ(λ4+p3λ3+p2λ2+p1λ+p0)+2q2λ+q1λ(q2λ2+q1λ+q0)-τλ.
Hence, a direct calculation shows that
(42)sgn⁡{d(Reλ)dτ}λ=iω0=sgn⁡{Re(dλdτ)-1}λ=iω0=sgn⁡{-q12+2q2(q0-q2ω02)(q1ω0)2+(q2ω02-q0)2((3p3ω02-p1)(p3ω02-p1)+2(2ω02-p2)(ω04-p2ω02+p0))×(ω02(p1-p3ω02)2+(ω04-p2ω02+p0)2)-1+-q12+2q2(q0-q2ω02)(q1ω0)2+(q2ω02-q0)2}.
We derive from (36) that
(43)ω02(p1-p3ω02)2+(ω04-p2ω02+p0)2=(q1ω0)2+(q2ω02-q0)2.
Hence, it follows that
(44)sgn⁡{d(Reλ)dτ}λ=iω0=sgn⁡{4ω06+3h3ω04+2h2ω02+h1(q1ω0)2+(q2ω02-q0)2}>0.
Therefore, the transversal condition holds and a Hopf bifurcation occurs at ω=ω0,τ=τ0.In conclusion, we have the following results.Theorem 4.
For system (2), let (H1) and (H2) hold; we have the following:(i)
if(p1+q1)[p3(p2+q2)-(p1+q1)]>p32(p0+q0), hi>0, and p0-q0>0, then the coexistence equilibrium E* is locally asymptotically stable, for all τ≥0;(ii)
if(p1+q1)[p3(p2+q2)-(p1+q1)]>p32(p0+q0), hi>0, and p0-q0<0, then there exists a positive number τ0, such that the coexistence equilibrium E* is locally asymptotically stable if 0≤τ<τ0 and is unstable for τ>τ0; further, system (2) undergoes a Hopf bifurcation at E* when τ=τn,n=0,1,2,…;(iii)
if(p1+q1)[p3(p2+q2)-(p1+q1)]<p32(p0+q0), then the coexistence equilibrium E* is unstable, for all τ≥0.We now give some examples to illustrate the main results above.Example 5.
In (2), we let a=2,p=0.9, r=10, r1=0.5,k=1,β=1, d=d1=d2=0.1, b=1, and m=0.5. System (2), with the above coefficients, has a unique coexistence equilibrium E*(0.3167,0.6212,0.2019,1.0095). It is easy to show that (r/K)S*-α1≈3.1468>0, β((r/K)+β)I*-(r/K)S*α1≈6.6343>0, that is the condition (H2) holds. We can get (p1+q1)[p3(p2+q2)-(p1+q1)]-p32(p0+q0)≈23.1510>0,h3≈6.0704>0,   h2≈6.5263>0,   h1≈1.6535>0, and p0-q0≈0.1310>0. By Theorem 4(i), the coexistence equilibrium E* is locally asymptotically stable, for all τ≥0. Numerical simulation illustrates our result (see Figure 1).Figure 1
The temporal solution found by numerical integration of system (2) with τ=1.Example 6.
In (2), we let a=0.55,p=0.95,r=20,r1=0.5,k=1,β=1,d=d1=d2=0.1,b=1, and m=0.5. System (2), with the above coefficients, has a unique coexistence equilibrium E*(0.8946,0.1007,0.1266,0.6332). It is easy to show that (r/K)S*-α1≈17.7848>0, and β((r/K)+β)I*-(r/K)S*α1≈0.1083>0; that is, the condition (H2) holds. We can get (p1+q1)[p3(p2+q2)-(p1+q1)]-p32(p0+q0)≈199.2429>0,h3≈316.4769>0,h2≈116.9725>0,h1≈1.1233>0, and p0-q0≈-0.0875<0. By Theorem 4(ii), there exists a positive number τ0≈11.4092, such that the coexistence equilibrium E* is locally asymptotically stable if 0≤τ<τ0 and is unstable for τ>τ0. Numerical simulations illustrate our results (see Figure 2).The temporal solution found by numerical integration of system (2) with (a) τ=1 and (b) τ=15.
(a)
(b)Now, we are concerned with the global attractiveness of the coexistence equilibriumE*.Theorem 7.
The coexistence equilibriumE*(S*,I*,Y1*,Y2*) of system (2) is globally attractive provided that the following conditions hold:(i)
0<(1-m)<(a/b)(Kβ-d);(ii)
d2(r1+d1)<r1pb<2d2(r1+d1).That is, the system (2) is persistent, if conditions (i) and (ii) hold.Proof.
Let(S(t),I(t),Y1(t),Y2(t)) be any positive solution of system (2) with initial conditions (3). Let
(45)US=limsup⁡t→+∞S(t),LS=liminf⁡t→+∞S(t),UI=limsup⁡t→+∞I(t),LI=liminf⁡t→+∞I(t),UYi=limsup⁡t→+∞Yi(t),LYi=liminf⁡t→+∞Yi(t).
We now claim that US=LS=S*,UI=LI=I*, UYi=LYi=Yi*  (i=1,2). The strategy of the proof is to use an iteration technique.
We derive from the first and the second equations of the system (2) that
(46)dSdt=rS(1-S+IK)-βSI,dIdt≤βSI-dI.
Consider the following auxiliary equations:
(47)dx1dt=rx1(1-x1+x2K)-βx1x2,dx2dt=βx1x2-dx2.
If Kβ>d, then by Theorem 3.1 in [4], it follows from (47) that
(48)lim⁡t→+∞x1(t)=dβ,lim⁡t→+∞x2(t)=r(Kβ-d)β(Kβ+r).
By comparison, we obtain that
(49)US=limsup⁡t→+∞S(t)≤dβ≔M1S,UI=limsup⁡t→+∞I(t)≤r(Kβ-d)β(Kβ+r)≔M1I.
Hence, for ε>0 sufficiently small, there is a T1>0 such that if t>T1, then I(t)≤M1I+ε.
It follows from the third and the fourth equations of system (2) that, for t>T1+τ,
(50)dY1dt≤pb(1-m)(M1I+ε)Y2(t-τ)aY2(t-τ)+(1-m)(M1I+ε)-(r1+d1)Y1(t),dY2dt=r1Y1(t)-d2Y2(t).
Consider the following auxiliary equations:
(51)du1dt=pb(1-m)(M1I+ε)u2(t-τ)au2(t-τ)+(1-m)(M1I+ε)-(r1+d1)u1(t),du2dt=r1u1(t)-d2u2(t).
If r1pb>d2(r1+d1), then by Lemma 2.4 in [9], it follows from (51) that
(52)lim⁡t→+∞u1(t)=(1-m)(M1I+ε)[r1pb-d2(r1+d1)]ar1(r1+d1),lim⁡t→+∞u2(t)=(1-m)(M1I+ε)[r1pb-d2(r1+d1)]ad2(r1+d1).
By comparison, we obtain that
(53)UY1=limsup⁡t→+∞Y1(t)≤(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)M1I≔M1Y1,UY2=limsup⁡t→+∞Y2(t)=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)M1I≔M1Y2.
Hence, for ε>0 sufficiently small, there is a T2>T1+τ such that if t>T2, then Y2(t)≤M1Y2+ε.
We derive from the first and the second equations of system (2) that
(54)dSdt=rS(1-S+IK)-βSI,dIdt≥βSI-dI-ba(1-m)I.
Since (1-m)<(a/b)(Kβ-d) holds, by Theorem 3.1 in [4], it follows from (54) and comparison argument that
(55)LS=liminf⁡t→+∞≥dβ+b(1-m)aβ≔N1S,LI=liminf⁡t→+∞≥rr+βK(K-ad+b(1-m)aβ)≔N1I.
Hence, for ε>0 sufficiently small, there is a T3>T2 such that if t>T3, then I(t)≥N1I-ε. We derive from the third and the fourth equations of system (2) that, for t>T3+τ,
(56)dY1dt≥pb(1-m)(N1I-ε)Y2(t-τ)(1-m)(N1I-ε)+aY2(t-τ)-(d1+r1)Y1(t)dY2dt=r1Y1(t)-d2Y2(t).
Since r1pb>d2(r1+d1) holds, by Lemma 2.4 of [9], it follows from (56) and comparison argument that
(57)LY1=liminf⁡t→+∞Y1(t)≥(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)(N1I-ε)LY2=liminf⁡t→+∞Y2(t)≥(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)(N1I-ε).
Since these two inequalities hold, for arbitrary ε>0 sufficiently small, we conclude that LY1≥N1Y1,LY2≥N1Y2, where
(58)N1Y1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)N1I,N1Y2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)N1I.
Hence, for ε>0 sufficiently small, there is a T4≥T3+τ, such that if t>T4, Y2(t)≥N1Y2-ε.
Forε>0 sufficiently small, we derive from the first and the second equations of system (2) that, for t>T4,
(59)dSdt=rS(1-S+IK)-βSI,dIdt≤βSI-dI-b(1-m)(N1Y2-ε)a(N1Y2-ε)+(1-m)(M1I+ε)I.
By comparison and Theorem 3.1 in [4], it follows that
(60)US=limsup⁡t→+∞S(t)≤dβ+b(1-m)(N1Y2-ε)β[a(N1Y2-ε)+(1-m)(M1I+ε)]≔M¯2S,UI=limsup⁡t→+∞I(t)≤rr+Kβ(K-M¯2S).
Since these two inequalities hold, for arbitrary ε>0 sufficiently small, we conclude that US≤M2S,UI≤M2I, where
(61)M2S=dβ+b(1-m)N1Y2β[aN1Y2+(1-m)M1I],M2I=rr+Kβ(K-M2S).
Therefore, for ε>0 sufficiently small, there is a T5≥T4 such that if t>T5, I(t)≤M2I+ε.
Forε>0 sufficiently small, we derive from the third and the fourth equations of system (2) that, for t>T5+τ,
(62)dY1dt≤pb(1-m)(M2I+ε)Y2(t-τ)(1-m)(M2I+ε)+aY2(t-τ)-(d1+r1)Y1(t),dY2dt=r1Y1(t)-d2Y2(t).
Since pbr1>d2(r1+d1) holds, by Lemma 2.4 of [9], it follows from (62) that
(63)UY1=limsup⁡t→+∞Y1(t)≤(1-m)[r1pb-d2(d1+r1)]ar1(d1+r1)(M2I+ε),UY2=limsup⁡t→+∞Y2(t)≤(1-m)[r1pb-d2(d1+r1)]ad2(d1+r1)(M2I+ε).
Since these two inequalities hold, for arbitrary ε>0 sufficiently small, we conclude that UY1≤M2Y1,UY2≤M2Y2, where
(64)M2Y1=(1-m)[r1pb-d2(d1+r1)]ar1(d1+r1)M2I,M2Y2=(1-m)[r1pb-d2(d1+r1)]ad2(d1+r1)M2I.
Therefore, for ε>0 sufficiently small, there is a T6≥T5+τ such that if t>T6, y2(t)≤M2Y2+ε.
Forε>0 sufficiently small, it follows from the first and the second equations of system (2) that, for t>T6,
(65)dSdt=rS(1-S+IK)-βSI,dIdt≥βSI-dI-b(1-m)(M2Y2+ε)a(M2Y2+ε)+(1-m)(N1I-ε)I(t).
By Theorem 3.1 in [4] and comparison argument, we can obtain
(66)LS=liminf⁡t→+∞S(t)≥dβ+b(1-m)(M2Y2+ε)β[a(M2Y2+ε)+(1-m)(N1I-ε)],LI=liminf⁡t→+∞I(t)≥rr+βK[K-dβ-b(1-m)(M2Y2+ε)β[a(M2Y2+ε)+(1-m)(N1I-ε)]].
Since these two inequalities hold, for arbitrary ε>0 sufficiently small, we conclude that LS≥N2S,LI≥N2I, where
(67)N2S=dβ+b(1-m)M2Y2β[aM2Y2+(1-m)N1I],N2I=rr+βK(K-N2S).
Hence, for ε>0 sufficiently small, there is a T7≥T6 such that if t>T7, I(t)≥N2I-ε. We therefore obtain from the third and the fourth equations of system (2) that, for t>T7+τ,
(68)dY1dt≥pb(1-m)(N2I-ε)Y2(t-τ)(1-m)(N2I-ε)+aY2(t-τ)-(d1+r1)Y1(t),dY2dt=r1Y1(t)-d2Y2(t).
Since pbr1>d2(r1+d1) holds, by Lemma 2.4 in [9] and comparison argument, we derive that
(69)LY1=liminf⁡t→+∞Y1(t)≥(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)(N2I-ε),LY2=liminf⁡t→+∞Y2(t)≥(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)(N2I-ε).
Since these inequalities hold for arbitrary ε>0 sufficiently small, we conclude that LY1≥N2Y1,LY2≥N2Y2, where
(70)N2Y1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)N2I,N2Y2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)N2I.
Continuing this process, we derive eight sequences MkS,MkI,MkY1,MkY2,NkS,NkI,NkY1, and NkY2(k=1,2,…) such that, for k≥2,
(71)MkS=1β[d+b(1-m)Nk-1Y2aNk-1Y2+(1-m)Mk-1I],MkI=rr+Kβ(K-MkS),MkY1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)MkI,MkY2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)MkI,NkS=1β[d+b(1-m)MkY2aMkY2+(1-m)Nk-1I],NkI=rr+Kβ(K-NkS),NkY1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)NkI,NkY2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)NkI.
It is readily seen that
(72)NkS≤LS≤US≤MkS,NkI≤LI≤UI≤MkI,NkYi≤LYi≤UYi≤MkYi,(i=1,2).
Noting that the sequences MkS,MkI,MkY1,MkY2 are nonincreasing, and the sequences NkS,NkI,NkY1,NkY2 are nondecreasing. Hence, the limit of each sequence in MkS,MkI,MkY1,MkY2,NkS,NkI,NkY1, and NkY2 exists. Denote
(73)lim⁡k→+∞MkS=S¯,lim⁡k→+∞MkI=I¯,lim⁡k→+∞MkYi=Y¯i,(i=1,2),lim⁡k→+∞NkS=S_,lim⁡k→+∞NkI=I_,lim⁡k→+∞NkYi=Y_i,(i=1,2).
From (71), we can obtain
(74)S¯=1β[d+b(1-m)Y_2aY_2+(1-m)I¯],I¯=rr+Kβ(K-S¯),Y¯1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)I¯,Y¯2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)I¯,S_=1β[d+b(1-m)Y¯2aY¯2+(1-m)I_],I_=rr+Kβ(K-S_),Y_1=(1-m)[r1pb-d2(r1+d1)]ar1(r1+d1)I_,Y_2=(1-m)[r1pb-d2(r1+d1)]ad2(r1+d1)I_.
It follows from (74) that
(75)αβ(r+Kβ)d2(r1+d1)(I¯)2+αβ(r+Kβ)[r1pb-d2(r1+d1)]I¯I_=r(Kβ-d)ad2(r1+d1)I¯+r[a(Kβ-d)-b(1-m)][r1pb-d2(r1+d1)]I_,(76)αβ(r+Kβ)d2(r1+d1)(I_)2+αβ(r+Kβ)[r1pb-d2(r1+d1)]I¯I_=r(Kβ-d)ad2(r1+d1)I_+r[a(Kβ-d)-b(1-m)][r1pb-d2(r1+d1)]I¯.
(75) minus (76),
(77)αβ(Kβ+r)d2(r1+d1)[(I¯)2-(I_)2]=r(Kβ-d)ad2(r1+d1)(I¯-I_)-r[a(Kβ-d)-b(1-m)]×[r1pb-d2(r1+d1)](I¯-I_).
Assume that I¯≠I_. Then we derive from (77) that
(78)αβ(Kβ+r)d2(r1+d1)(I¯+I_)=r(Kβ-d)ad2(r1+d1)-r[a(Kβ-d)-b(1-m)][r1pb-d2(r1+d1)].
(75) plus (76),
(79)αβ(Kβ+r)d2(r1+d1)(I¯+I_)2+2αβ(Kβ+r)[r1pb-2d2(r1+d1)]I¯I_=[r(Kβ-d)ad2(r1+d1)+r(a(Kβ-d)-b(1-m))×(r1pb-d2(r1+d1))](I¯+I_).
On substituting (78) into (79), it follows that
(80)αβ(Kβ+r)[r1pb-2d2(r1+d1)]I¯I_=r[a(Kβ-d)-b(1-m)]×[r1pb-d2(r1+d1)](I¯+I_).
Note that I¯>0 and I_>0. If d2(r1+d1)<r1pb<2d2(r1+d1), we derive that 1-m>a(Kβ-d)/b. This is a contradiction. Hence, we have S¯=S_. It therefore follows from (74) that I¯=I_,Y¯1=Y_1,and Y¯2=Y_2. We therefore conclude that E* is globally attractive. The proof is complete.


## 5. Conclusion

In this paper, we have incorporated a prey refuge, stage structure for the predator and time delay due to the gestation of the predator into a predator-prey system. Incorporating a refuge into system (1) provides a more realistic model. A refuge can be important for the biological control of a pest; however, increasing the amount of refuge can increase prey densities and lead to population outbreaks. By using the iteration technique and comparison arguments, respectively, we have established sufficient conditions for the global stability of the predator-extinction equilibrium and the globally attractivity for the coexistence equilibrium. As a result, we have shown the threshold for the permanence and extinction of the system. By Theorem 3, we see that the predator population go to extinction if 0<(1-m)<(a/b)(Kβ-d) and kβ>d, d2(r1+d1)>pbr1. By Theorem 7, we see that if 0<(1-m)<(a/b)(Kβ-d) and d2(r1+d1)<r1pb<2d2(r1+d1), then both the prey and predator species of system (2) are permanent.



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*Source: 978758-2014-03-23.xml*