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

Authors: Pei Yao; Zuocheng Wang; Lingshu Wang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, a ratio-dependent predator-prey model is investigated. The local stability and global stability of the nonnegative boundary equilibrium and positive equilibrium of the model are discussed, respectively. Sufficient condition is obtained for the existence of Hopf bifurcation at the positive equilibrium.


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## Body

## 1. Introduction

Recently, the predator-prey models have been studied by many authors [1–8]. In general, a predator-prey model has the following forms:(1)x˙=xfx−pxy,y˙=kpxy−ygy,where xt and yt are the densities of the prey and predator population at time t, respectively. The function fx represents the growth of the prey population rate, gy represents the growth rate of predator population, and px represents the functional response function of predator population to prey population. In [1], Xu et al. used the function px=x2/x2+my2 as the functional response function of predator population to prey population. The time delay due to the gestation of the predator is discussed in [1].It is noted that in model (1), each individual’s prey admits the same risk to be attacked by predators and each individual predator admits the same ability to feed on prey. This assumption seems not to be realistic for many animals. In natural world, there are many species whose individuals pass through an immature stage. Stage structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. In the last two decades, stage-structured models have received great attention [3–7, 9].Based on above discussion, we study the following predator-prey model:(2)x˙1t=rx2t−d1+r1x1t−a1x12ty2tx12t+my22t,x˙2t=r1x1t−d2x2t−ax22t,y1.t=a2x12t−τy2t−τx12t−τ+my22t−τ−r2+d3y1t,y2.t=r2y1t−d4y2t,where x1t and x2t are the densities of the immature and mature prey at time t and y1t and y2t are the densities of the immature and mature predators at time t. In model (2), all parameters are positive constants. τ≥0 is the time delay due to the gestation of the predator. x2/x2+my2 is the ratio-dependent functional response.Model (2) is of the following initial conditions:(3)x1θ=ϕ1θ≥0,x2θ=ϕ2θ≥0,y1θ=φ1θ≥0,y2θ=φ2θ≥0,θ∈−τ,0,ϕ10>0,ϕ20>0,φ10>0,φ20>0,ϕ1θ,ϕ2θ,φ1θ,φ2θ∈C−τ,0,R+04.The organization of this study is as follows. In Section2, we discuss the local stability of the nonnegative boundary equilibrium and the positive equilibrium of models (2) and (3). The existence of a Hopf bifurcation for models (2) and (3) at the positive equilibrium is also established. Sufficient conditions are derived for the global stability of the nonnegative boundary equilibrium and positive equilibrium of models (2) and (3) in Section 3, respectively.


## 2. Local Stability and Hopf Bifurcation

In this section, by analyzing the corresponding characteristic equations, we study the local stability of each of nonnegative equilibria and the existence of a Hopf bifurcation at the positive equilibrium of models (2) and (3).Ifrr1>d2r1+d1, model (2) has a nonnegative boundary equilibrium E1x1′,x2′,0,0, where(4)x1′=rrr1−d2r1+d1ar1+d12,x2′=rr1−d2r1+d1ar1+d1.IfH1a2r2>d4r2+d3,rr1−d2r1+d1/a1d2>d4r2+d3/a2r2h, model (2) has a positive equilibrium E+x1+,x2+,y1+,y2+, where(5)x1+=rr1+d1+a1h/1+mh2x2+,x2+=1arr1r1+d1+a1h/1+mh2−d2,y1+=d4r2hx1∗+y2+=hx1+,h=a2r2−d4r2+d3md4r2+d3.The characteristic equation of model (2) at E1x1′,x2′,0,0 takes the following form:(6)λ2+r1+d1+d2+2ax2′λ+rr1−d2r1+d1λ2+g1λ+g0+h0e−λτ=0,where g1=r2+d3+d4,g0=d4r2+d3,h0=−a2r2. When rr1>d2r1+d1, all roots of equation,(7)λ2+r1+d1+d2+2ax2+λ+rr1−d2r1+d1=0,are negative. Now, we consider the roots of the following equation. λ2+g1λ+g0+h0e−λτ=0. By calculating, we obtain(8)g12−2g0=d42+r2+d32>0,g02−h02=d42r2+d32−a2r22.Whend4r2+d3>a2r2, we get g02−h02>0. Therefore, E1 is locally stable for all τ>0. When d4r2+d3<a2r2, we get g02−h02<0. Thus, E1 is unstable.The characteristic equation of model (2) at E+ is of the form(9)λ4+P3λ3+P2λ2+P1λ+P0+Q2λ2+Q1λ+Q0e−λτ=0,where P3=r1+d1+a1α+d2+2ax2++r2+d3+d4,P2=d4r2+d3+r2+d3+d4r1+d1+a1α+d2+2ax2++r1+d1+a1αd2+2ax2+−rr1,P1=d4r2+d3r1+d1+a1α+d2+2ax2++r2+d3+d4r1+d1+a1αd2+2ax2+−rr1,P0=d4r2+d3r1+d1+a1αd2+2ax2+−rr1,Q2=−a2r2β,Q1=−a2r2βr1+d1+d2+2ax2+,Q0=−a2r2βr1+d1d2+2ax2+−rr1,α=2mx1+y2+3/x1+2my2+22,β=x1+4/x1+2+my2+22.Letτ=0; then, (9) has the following form:(10)λ4+P3λ3+P2+Q2λ2+P1+Q1λ+P0+Q0=0.Note thatP3>0. When(11)H2P3P2+Q2>P1+Q1,P1+Q1P3P2+Q2−P1+Q1>P32P0+Q0>0,then positive equilibrium E+ is locally asymptotically stable.LetH1 and H2 hold. If iωω>0 is a solution of (9), by calculation, we can obtain(12)ω8+f3ω6+f2ω4+f1ω2+f0=0,where(13)f3=P32−2P2=d42+r2+d32+r1+d1+a1α2+d2+2ax2+2+2rr1>0,f2=P22+2P0−2P1P3−Q22=d42r2+d32−a2r2β2+r1+d1+a1αd2+2ax2+−rr12+d42+r2+d32r1+d1+a1α2+d2+2ax2+2+2rr1>0,f1=P12−2P0P2+2Q0Q2−Q12=d42r2+d32−a2r2β2r1+d12+d2+2ax2+2+2rr1+d42+r2+d32r1+d1+a1αd2+2ax2+−rr12+d42r2+d322a1αr1+d1+a1α2>0,f0=P02−Q02=P0+Q0P0−Q0,when P0>Q0, E+ is locally asymptotically stable for all τ>0. When P0<Q0, ω0 is the positive root of (12); in this case, (9) has a pair of roots ±iω0. By (12), we obtain(14)τk=2kπω0+1ω0arccosQ2ω02−Q0ω04−P2ω02+P0+Q1ω0P3ω03−P1ω0Q1ω02+Q2ω02−Q02,k=0,1,2,…,.Therefore,E+ remains stable for τ<τ0.Differentiating (9) with respect to τ, we obtain that(15)dλdτ−1=4λ3+3P3λ2+2P2λ+P1−λλ4+P3λ3+P2λ2+P1λ+P0+2Q2λ+Q1λQ2λ2+Q1λ+Q0−τλ.Hence, we get(16)sgndReλdτλ=iω0=sgnRedλdτ−1λ=iω0=sgn3P3ω02−P1P3ω02−P1+22ω02−P2ω04−P2ω02+P0ω02P1−P3ω022+ω04−P2ω02+P02+2Q2Q0−Q2ω02−Q12Q3ω03−Q1ω02+Q2ω02−q02=sgn4ω06+3f3ω04+2f2ω02+f1Q1ω02+Q2ω02−Q02>0.Therefore, asτ=τ0,ω=ω0, there is Hopf bifurcation.From above discussion, we have the following results.Theorem 1.
For model (2) with (3), we have the following:(i)
Letrr1>d2r1+d1; if a2r2<d4r2+d3, then E1 is locally asymptotically stable; if a2r2>d4r2+d3, then E1 is unstable.(ii)
AssumeH1 and H2 hold; if P0>Q0, then E+ is locally asymptotically stable for all τ≥0; if P0<Q0, then there exists a τ0>0, s.t., E+ is locally asymptotically stable if 0<τ<τ0 and unstable if τ>τ0. When τ=τ0, models (2) and (3) undergo Hopf bifurcation at E+.


## 3. Global Stability

In this section, by using an iteration technique, we discuss the global stability of the nonnegative equilibriaE1 and E+ of models (2) and (3), respectively.Theorem 2.
Let(17)H3rr1>d2r1+d1+a1d22m,a2r2<d4r2+d3,hold; then, the nonnegative boundary equilibrium E1 of model (2) is globally stable.Proof.
It follows from the positive solution of model (2), and we can obtain(18)x˙1t⩽rx2t−d1+r1x1t,x˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.2 of [5] and comparison, we have(19)limsupt⟶+∞x1t≤rrr1−d2r1+d1ar1+d12,limsupt⟶+∞x2t≤rr1−d2r1+d1ar1+d1.
Therefore, there is a positive numbert1, for sufficiently small positive number ε, such that as t>t1, x1t≤x1′+ε. Hence, for t>t1+τ, we derive that(20)y˙1t≤a2x1′+ε2y2t−τx1′+ε2+my22t−τ−r2+d3y1t,uy.2t=r2y1t−d4y2t.
By Lemma 2.2 of [5] and comparison, we can obtain(21)limt⟶+∞y1t=0,limt⟶+∞y2t=0.
Therefore, there is a positive numbert2.t1, such that if t>t2, y2t<ε.
Fort>t2, we derive from model (2) that(22)x˙1t≥rx2t−r1+d1x1t−a12mx1tx˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.2 of [5] and comparison, we have(23)liminft⟶+∞x1t≥rar1+d1+a1/2mrr1r1+d1+a1/2m−d2≔x¯1,liminft⟶+∞x1t≥1arr1r1+d1+a1/2m−d2.
By model (2), it follows that(24)x˙1t≥rx2t−r1+d1x1t−a1εx¯1x1t,x˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.4 of [3] and comparison, we obtain that(25)liminft⟶+∞x1t≥rrr1−d2r1+d1ar1+d12,liminft⟶+∞x2t≥rr1−d2r1+d1ar1+d1,which together with (19) and (21) yields(26)limt⟶+∞x1t,x2t,y1t,y2t=x1′,x2′,0,0.
Hence, the equilibriumE1x1′,x2′,0,0 of model (2) is globally stable.Theorem 3.
AssumeH1, H2, and P0>Q0 hold; if(27)H4rr1−d2r1+d1a1d2>12m,a2r2r1+d1<a1d4r2+d3h,then the positive equilibrium E+x1+,x2+,y1+,y2+ of model (2) is global stability.Proof.
Let(28)Uxi=limsupt⟶+∞xit,Lxi=liminft⟶+∞xit,Uyi=limsupt⟶+∞yit,Lyi=liminft⟶+∞yit,i=1,2.
By the first two equations of model (2), we can obtain that(29)x˙1t≤rx2t−d1+r1x1t,x˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.2 of [5] and comparison, we have(30)Ux1=limsupt⟶+∞x1t≤rrr1−d2r1+d1ar1+d12≔M1x1,Ux2=limsupt⟶+∞x2t≤rr1−d2r1+d1ar1+d1≔M1x2.
So, for sufficiently small positive numberε, there exists a positive number t1, such that if t>t1, then x1t≤M1x1+ε.
Fort>t1+τ, by the last two equations of model (2), we get(31)y1.t≤a2M1x1+ε2y2t−τM1x1+ε2+my22t−τ−r2+d3y1t.x˙2t=r2x1t−d4x2t.
By Lemma 2.2 of [5] and comparison, we obtain(32)Uy1=limsupt⟶+∞y1t≤d4r2hM1x1≔M1y1,Uy2=limsupt⟶+∞y2t=hM1x1≔M1y2.
Hence,Uy1≤M1y1,Uy2≤M1y2, in which(33)M1y1=a2r2−d4r2+d3mr2r2+d3M1x1,M1y2=a2r2−d4r2+d3md4r2+d3M1x1.
Therefore, for sufficiently small positive numberε, there is t2≥t1+τ, such that if t>t2, y2t≤M1y2+ε.
Fort>t2, by the first two equations of model (2), we have(34)x˙1t≥rx2t−r1+d1x1t−a12mx1t,x˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.4 of [3] and comparison, we derive that(35)Lx1=liminft⟶∞x1t≥rrr1−d2r1+d1+a1/2mar1+d1+a1/2m2≔N1x1,Lx2=liminft⟶∞x2t≥rr1−d2r1+d1+a1/2mar1+d1+a1/2m≔N1x2.
Hence, for sufficiently small positive numberε, there is t3≥t2, such that if t>t3, x1t≥N1x1−ε.
Fort>t3+τ, it follows from the last two equations of model (2) that(36)y1.t≥a2N1x1−ε2y2t−τN1x1−ε2+my22t−τ−d3+r2y1t.x˙2t=r2x1t−d4x2t.
By Lemma 2.4 of [3] and comparison, we can obtain(37)Ly1=liminft⟶+∞y1t≤d4r2hN1x1≔N1y1,Ly2=limsupt⟶+∞y2t=hN1x1≔N1y2.
Therefore, for sufficiently small positive numberε, there is a positive number t4≥t3+τ, such that if t>t4, y2t≥N1y2−ε. In this case, by the first two equations of model (2), we have(38)x˙1t≤rx2t−d1+r1x1t−a1N1x1−εN1y2−εM1x1+ε2+mM1y2+ε2x1t,x˙2t=r1x1t−d2x2t−ax22t.
For sufficiently small positive numberε, if H4 holds, by Lemma 2.2 of [5] and a comparison argument, we can obtain(39)Ux1=limsupt⟶+∞x1t≤rrr1−d2r1+d1+a1N1x1N1y2/M1x12+mM1y22ar1+d1+a1N1x1N1y2/M1x12+mM1y222≔M2x1,Ux2=limsupt⟶+∞x2t≤rr1−d2r1+d1+a1N1x1N1y2/M1x12+mM1y22ar1+d1+a1N1x1N1y2/M1x12+mM1y22≔M2x2.
Therefore, for sufficiently small positive numberε, there is t5≥t4, such that if t>t5, x1t≤M2x1+ε.
From the last two equations of model (2), we obtain that for t>t5+τ,(40)y1.t≤a2M2x1+ε2y2t−τM2x1+ε2+my22t−τ−d3+r2y1t,x˙2t=r2x1t−d4x2t.
By Lemma 2.2 of [5] and comparison, if a2r2>d4r2+d3 holds, we have(41)Uy1=limsupt⟶+∞y1t≤d4r2M2x1≔M2y1,Uy2=limsupt⟶+∞y2t≤hM2x1≔M2y2.
Hence, forε>0 sufficiently small, there is a T6≥T5+τ, such that if t>T6, y2t≤M2y2+ε.
Again, for sufficiently small positive numberε and t>t6, by the first two equations of model (2), we have(42)x˙1t≥rx2t−d1+r1x1t−a1M2x1+εM2y2+εN1x1−ε2+mN1y2−ε2x1t,x˙2t=r1x1t−d2x2t−ax22t.
By Lemma 2.4 of [3] and comparison, if H4 holds, we can obtain(43)Lx1=liminft⟶+∞x1t≥rrr1−d2r1+d1+a1M2x1M2y2/N1x12+mN1y22ar1+d1+a1M2x1M2y2/N1x12+mN1y222≔N2x1,Lx2=liminft⟶+∞x2t≥rr1−d2r1+d1+a1M2x1M2y2/N1x12+mN1y22ar1+d1+a1M2x1M2y2/N1x12+mN1y22≔N2x2.
So, there is a positive numbert7≥t6, for t>t7, x1t≥N2x1−ε.
For sufficiently small positive numberε and t>7+τ, from the last two equations of model (2), we can derive(44)y1.t≥a2N2x1−ε2y2t−τN2x1−ε2+my22t−τ−d3+r2y1t,x˙2t=r2x1t−d4x2t.
By Lemma 2.4 of [3] and comparison, if a2r2>d4d3+r2, we have(45)Uy1=limsupt⟶+∞y1t≥d4r2N2x1≔N2y1,Uy2=limsupt⟶+∞y2t≥hN2x1≔N2y2.
Repeat the above process; forn≥2, we can obtain eight sequences:(46)Mnx1,Mnx2,Mny1,Mny2,Nnx1,Nnx2,Nny1,Nny2n=1,2,,in which(47)Mnx1=rr1+d1+a1Nn−1x1Nn−1y2/Mn−1x12+mMn−1y22Mnx2,Mnx2=rr1−d2r1+d1+a1Nn−1x1Nn−1y2/Mn−1x12+mMn−1y22ar1+d1+a1Nn−1x1Nn−1y2/Mn−1x12+mMn−1y22,Mny1=d4r2hMnx1,Mny2=hMnx1,Nnx1=rr1+d1+a1Nn−1x1Nn−1y2/Mn−1x12+mMn−1y22Nnx2,Nnx2=rr1−d2r1+d1+a1Mnx1Mny2/Nn−1x12+mNn−1y22ar1+d1+a1Mnx1Mny2/Nn−1x12+mNn−1y22,Nny1=d4r2hNnx1,Nny2=hNnx1.
It is noted that(48)Nnxi≤Lxi≤Uxi≤Mnxi,Nnyi≤Lyi≤Uyi≤Mnyi,i=1,2.
Direct calculation, we haveMnxi and Mnyi as nonincreasing, and Nnxi and Nnyi as nondecreasing. Therefore, the limits of sequences in Mnxi, Mnyi, Nnxi, and Nnyi exist. Let(49)limt⟶+∞Mnxi=x¯i,limt⟶+∞Nnxi=x¯i,limt⟶+∞Mnyi=y¯i,limt⟶+∞Nnyi=y¯i,i=1,2.
We have(50)x¯1=rr1+d1+a1x¯1y¯2/x¯12+my¯22x¯2,x¯2=rr1−d2r1+d1+a1x¯1y¯2/x¯12+my¯22ar1+d1+a1x¯1y¯2/x¯12+my¯22,y¯1=d4r2hx¯1,y¯2=hx¯1,x¯1=rr1+d1+a1x¯1y¯2/x¯12+my¯22x¯2,x¯2=rr1−d2r1+d1+a1x¯1y¯2/x¯12+my¯22ar1+d1+a1x¯1y¯2/x¯12+my¯22,y¯1=d4r2hx¯1,y¯2=hx¯1.
Now, we prove thatx¯i=x¯i,y¯i=y¯i,i=1,2. By (50), we can obtain(51)ar1+d11+mh2x¯12+a1hx¯12=r1+mh2rr1−d2r1+d1x¯13−rd2a1h1+mh2x¯1x¯12,ar1+d11+mh2x¯12+a1hx¯12=r1+mh2rr1−d2r1+d1x¯13−rd2a1h1+mh2x¯1x¯12.
From above two equations, we have(52)ar1+d121+mh22−a1h2x¯12+x¯12x¯1+x¯1x¯1−x¯1=1+mh2rr1−d2r1+d1x¯12+x¯1x¯1+x¯12+rd2a1h1+mh2x¯1x¯1x¯1−x¯1.
Ifx¯1≠x¯1, then we obtain(53)ar1+d121+mh22−a1h2x¯12+x¯12x¯1+x¯1=1+mh2rr1−d2r1+d1x¯12+x¯1x¯1+x¯12+rd2a1h1+mh2x¯1x¯1.
Sincerr1>d2r1+d1,x¯1>0,x¯1>0, therefore, r1+d11+mh2>a1h. This is a contradiction. So, x¯1=x¯1. By (50), we have x¯2=x¯2y¯1=y¯1 and y¯2=y¯2. Therefore, the positive equilibrium E+ is globally stable.


## 4. Discussion

In this study, we have studied a ratio-dependent predator-prey model with stage structure for the prey and predator. A time delay due to the gestation of the predator is considered. By using the eigenvalue theory, we have obtained the sufficient conditions for the local stability of the nonnegative equilibria of model (2). The existence of Hopf bifurcation is given. By the iteration technique and comparison arguments, sufficient conditions have been established for the global stability of the nonnegative equilibria. From Theorem 2, we know that if H3 holds, the predator population will go to extinction. By Theorem 3, we learn that if H1 and H4 hold, then both the predator and prey species of model (2) are permanent [10, 11].



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