Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

Hafizul Molla; Md. Sabiar Rahman; Sahabuddin Sarwardi

doi:10.1515/ijnsns-2017-0224

Article

Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

Hafizul Molla , Md. Sabiar Rahman and Sahabuddin Sarwardi

Published/Copyright: November 27, 2018

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 20 Issue 1

Abstract

We propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

Keywords: ecological model; refuge; permanent; stability; direction of Hopf bifurcation; numerical simulations

MSC 2010: 92D25; 92D30; 92D40; 70K50; 37L10

Appendix

’ Appendix A.1

Determination of equilibria: The system (3)-(4) has the trivial solution ψ0=(0,0), the predator free equilibrium ψ1=(k,0) and the coexistence equilibrium ψ2=(S∗,P∗), where P∗=(em−d)S∗−adδ(em−d)S∗ and S∗ is the root of the following cubic equation:

(25)X3−AX2+BX−C=0,

where A=k,B=kdδre,C=akd2δre(em−d). If em > d, by the Descarte’s rule of sign eq. (25) has at least one positive root or three positive roots. Let X=E be a root of the equation and then dividing (25) by the factor (X−E), we have

X3−AX2+BX−CX−E=X2+(E−A)X+(E2−AE+B)+E3−AE2+BE−CX−E.

Thus, we must have E3−AE2+BE−C=0 and then k=δreem−dE3E2rδe(em−d)−d(em−d)E+ad2. The other two roots of the equation, if they exist, must satisfy the following equation X2+(E−A)X+(E2−AE+B)=0 and putting the value of k, we get the other two solutions as

X±=(em−d)E−ad±4areE2(em−d)(δ1−δ)2re(em−d)E2(δ−δ2),

where δ1=(em−d)E+ad2−4a2d24are(em−d)E2 and δ2=d(em−d)E−adre(em−d)E2.

Proof of Lemma 1

When (E,δ)∈Ω1≡(E,δ)∈R+2:δ>maxδ1,δ2 and d<minem,emEa+E, then ∃ unique value for abscissa.

The unique interior equilibrium is ψ2I=(S∗I,P∗I), where S∗I=E,P∗I=(em−d)E−adδ(em−d)E. □

Proof of Lemma 2

When (E,δ) is on the curve δ=δ1, i.e., if (E,δ)∈Ω2≡(E,δ)∈R+2:δ=δ1>δ2) and d<minem,emEa+E, then ∃ two positive values on the abscissa, one of them of multiplicity 2. Therefore, the two equilibria are ψ2II+ and ψ2II−, where ψ2II+=ψ2I and the multiple equilibria are ψ2II−=(S∗II−,P∗II−), where S∗II−=(em−d)E−ad2re(em−d)E2(δ−δ2),P∗II−=(em−d)S∗II−−adδ(em−d)S∗II−. □

Proof of Lemma 3

If the condition (em−d)E−ad−4areE2(em−d)(δ1−δ)>0 holds, and (E,δ)∈Ω3≡(E,δ)∈R+2:δ2<δ<δ1 with d<minem,emEa+E, then there exist three distinct positive values on the abscissa. The three distinct interior equilibria are ψ2III and ψ2III±. Where ψ2III=ψ2I and ψ2III±=(S∗III±,P∗III±), where S∗III±=(em−d)E−ad±4are(em−d)E2(δ1−δ)2re(em−d)E2(δ−δ2),P∗III±=(em−d)S∗III±−adδ(em−d)S∗III±. □

’ Appendix A.2

The Jacobian matrix for the model system (3)-(4) at any arbitrary point (x,y) is given by

J=r1−2Sk−am1−δPPa+S1−δP2mS(2δP−1)a+S1−δP−δmS21−δPP(a+S1−δP)2\noalign\medskipem1−δPa+S1−δP−emS1−δP2a+S1−δP2PemδS2(1−δP)Pa+S(1−δP)2+emS1−δ−δPa+S1−δP−d.

’ Appendix A.3

The values of trace and determinant of the Jacobian matrix, used in Lemma 4:

tr(J2I)=δeE2(em−d){dekm(a+2E)−emE(emk+ra)−kd2(a+E)}−ad2k(ad+dE−Eem)ame2kE2δ(em−d)<0,ifδ>δ3=ad2kE(em−d)−adeE2(em−d){kd2(a+k)−dekm(a+2E)+emE(emk+ar)},anddet(J2I)={E(em−d)−ad}{δre(em−d)E3−dk(em−d)E+2kad2}δamke2E2=rE(em−d)E(em−d)−ad(δ−δ4)aekmδ>0,ifδ>δ4=kdE(em−d)−2adre(em−d)E3.

The values of trace and determinant of the Jacobian matrices, used in Lemma 4:

tr(J2II−)=(a+S∗II−)d2kP∗II−−rme2(S∗II−)2−δP∗II−S∗II−d2kP∗II−+S∗II−(km2e3−re2mS∗II−−kmde2)ke2ma+S∗II−(1−δP∗II−)<0,ifδ>(a+S∗II−)d2kP∗II−−rme2(S∗II−)2P∗II−S∗II−d2kP∗II−+S∗II−(km2e3−re2mS∗II−−kmde2)=δ5(say) anddet(J2II−)=(em−d)P∗II−δre2m(S∗II−)2−dk(em+d)P∗II−S∗II−+d2k(a+S∗II−)ke2mS∗II−a+S∗II−(a−δP∗II−)>0,ifδ<d2k(a+S∗II−)dk(em+d)P∗II−−re2m(S∗II−)2S∗II−=δ6(say).

The values of trace and determinant of the Jacobian matrices, used in Lemma 5:

tr(J2III+)=(a+S∗III+)d2kP∗III+−rme2(S∗III+)2−δP∗III+S∗III+d2kP∗III++S∗III+(km2e3−re2mS∗III+−kmde2)ke2ma+S∗III+(1−δP∗III+).tr(J2III+)<0,ifδ>(a+S∗III+)d2kP∗III+−rme2(S∗III+)2P∗III+S∗III+d2kP∗III++S∗III+(km2e3−re2mS∗III+−kmde2)=δ7(say) anddet(J2III+)=(em−d)P∗III+δre2m(S∗III+)2−dk(em+d)P∗III+S∗III++d2k(a+S∗III+)ke2mS∗III+a+S∗III+(a−δP∗III+)>0,ifδ<d2k(a+S∗III+)dk(em+d)P∗III+−re2m(S∗III+)2S∗III+=δ8(say).tr(J2III−)=(a+S∗III−)d2kP∗III−−rme2(S∗III−)2−δP∗III−S∗III−d2kP∗III−+S∗III−(km2e3−re2mS∗III−−kmde2)ke2ma+S∗III−(1−δP∗III−).

and tr(J2III−)<0,ifδ>(a+S∗III−)d2kP∗III−−rme2(S∗III−)2P∗III−S∗III−d2kP∗III−+S∗III−(km2e3−re2mS∗III−−kmde2)=δ9(say) anddet(J2III−)=(em−d)P∗III−δre2m(S∗III−)2−dk(em+d)P∗III−S∗III−+d2k(a+S∗III−)ke2mS∗III−a+S∗III−(a−δP∗III−)>0,ifδ<d2k(a+S∗III−)dk(em+d)P∗III−−re2m(S∗III−)2S∗III−=δ10(say).

’ Appendix A.4

The expressions for g11,g02,g21,andg21 appearing in (19) are calculated as follows:

Writing the dynamical system as dXdt=F(X), where X = (S,P) and F=F1,F2, which are the same expression as in the system (3)-(4). If we take δ=δc+σ, so that σ = 0 is the Hopf-bifurcation value for the system (3)-(4). Letting x1=(S−S∗), x2=(P−P∗) and translating the equilibrium point to the origin. Applying Taylor series expansion to the system (3)-(4), we have extended system as follows:

(27)dx1dt=a11x1+a12x2+∑m+n≥2∞αmnm!n!x1mx2n,dx2dt=a21x1+a22x2+∑m+n≥2∞βmnm!n!x1mx2n,

where m,n ≥ 0 and αmn=∂m+nF1∂mx∂ny|(S∗I,P∗I),βmn=∂m+nF2∂mx∂ny|(S∗I,P∗I), and aij are given in eq. (12).

Now we consider the new system (27) for further analysis in order to determine the nature of the Hopf-bifurcating periodic solution.

The eigenvector λ, for the eigenvalue η=ρ+iω is

λ=1\noalign\medskipa11−ρ−iω−a12,

where ρ=12(a11+a22) and ω=124(a12a21+a11a22)−(a11+a22)2.

Let M=ℜ(η),−ℑ(η)=10a11−ρ−a12−ωa12 and y1y2=M−1x1x2, then eq. (27) can be written in terms of y1, y2 as follows

(30)dy1dt=ρy1−ωy2+Φ(y1,y2;δ),dy2dt=ωy1+ρy2+Ψ(y1,y2;δ),

where

Φ(y1,y2;δ)=12α20y12+α11y1ϕ(y1,y2)+12α02ϕ2(y1,y2)+12α21y12ϕ(y1,y2)+12α12y1ϕ2(y1,y2)+16α30y13+16α03ϕ3(y1,y2)+h.o.t.,Ψ(y1,y2;δ)=−a12ω[12β20y12+β11y1ϕ(y1,y2)+12β02ϕ2(y1,y2)+12β12y1ϕ2(y1,y2)+12β21y12ϕ(y1,y2)+16β30y13+16β03ϕ3(y1,y2)]+−a11+ρω[12α20y12+α11y1ϕ(y1,y2)+12α02ϕ2(y1,y2)+12α21y12ϕ(y1,y2)+12α12y1ϕ2(y1,y2)+16α30y13+16α03ϕ3(y1,y2)]+h.o.t.,ϕ(y1,y2)=−a11+ρa12y1−ωa12y2,where h.o.t. stands for higher order terms.

As the system satisfy all the conditions of Hopf-bifurcation at ψ2I=(S∗I,P∗I), then we have ρ = 0 and ω=ω0=det(J∗I) and the system (27) is reduced to

(32)dy1dt=−ω0y2+Φ0(y1,y2;δ),dy2dt=ω0y1+Ψ0(y1,y2;δ),

where

Φ0(y1,y2;δ)=12α20y12+α11y1ϕ0(y1,y2)+12α02ϕ02(y1,y2)+12α21y12ϕ0(y1,y2)+12α12y1ϕ02(y1,y2)+16α30y13+16α03ϕ03(y1,y2)+h.o.t.,Ψ0(y1,y2;δ)=−a12ω0[12β20y12+β11y1ϕ0(y1,y2)+12β02ϕ02(y1,y2)+12β12y1ϕ02(y1,y2)+12β21y12ϕ0(y1,y2)+16β30y13+16β03ϕ03(y1,y2)]+−a11ω0[12α20y12+α11y1ϕ0(y1,y2)+12α02ϕ02(y1,y2)+12α21y12ϕ0(y1,y2)+12α12y1ϕ02(y1,y2)+16α30y13+16α03ϕ03(y1,y2)]+h.o.t.,

where ϕ0(y1,y2)=ξy1+ζy2, ξ=−a11a12, ζ=−ω0a12.

Now we calculate the following values of g11,g02,g21,andg21 at δ=δc and (y1,y2)=(0,0) as follows:

g11=14[∂2Φ0∂y12+∂2Φ0∂y22+i(∂2Ψ0∂y12+∂2Ψ0∂y22)]=14[α20+2ξα11+(ξ2+ζ2)α02+i(−a12ω0(β20+2ξβ11+(ξ2+ζ2)β02)−a11ω0(α20+2ξα11+(ξ2+ζ2)α02))],

g02=14[∂2Φ0∂y12−∂2Φ0∂y22−2∂2Ψ0∂y1∂y2+i(∂2Ψ0∂y12−∂2Ψ0∂y22+2∂2Φ0∂y1∂y2)]=14[α20+2ξα11+(ξ2−ζ2)α02+2ζa12ω0(β11+ξβ02)+2ζa11ω0(α11+ξα02)+i(−a12ω0(β20+2ξβ11+(ξ2−ζ2)β02)−a11ω0(α20+2ξα11+(ξ2−ζ2)α02)+2ζ(α11+ξα02))],

g20=14[∂2Φ0∂y12−∂2Φ0∂y22+2∂2Ψ0∂y1∂y2+i(∂2Ψ0∂y12−∂2Ψ0∂y22−2∂2Φ0∂y1∂y2)]=14[α20+2ξα11+(ξ2−ζ2)α02−2ζa12ω0(β11+ξβ02)−2ζa11ω0(α11+ξα02)+i(−a12ω0(β20+2ξβ11+(ξ2−ζ2)β02)−a11ω0(α20+2ξα11+(ξ2−ζ2)α02)−2ζ(α11+ξα02))],

g21=18[∂3Φ0∂3y1+∂3Φ0∂y1∂2y2+∂3Ψ0∂2y1∂y2+∂3Ψ0∂3y2+i(∂3Ψ0∂3y1+∂3Ψ0∂y1∂2y2−∂3Φ0∂2y1∂y2−∂3Φ0∂3y2)]=18[α30+3ξα21+(3ξ2+ζ2)α12+ξ2(ξ+ζ)α03−ζa12ω0(β21+2ξβ12+(ξ2+ζ2)β03)−ζa11ω0(α21+2ξα12+(ξ2+ζ2)α03)+i(−a12ω0(β30+3ξβ21+(3ξ2+ζ2)β12+ξ2(ξ+ζ)β03)−ζa11ω0(α30+3ξα21+(3ξ2+ζ2)α12+ξ2(ξ+ζ)α03)−ζ(α21+2ξα12+(ξ2+ζ2)α03))].

Acknowledgements

The present form of the paper owes much to the helpful suggestions of the referees, whose careful scrutiny we are pleased to acknowledge. The corresponding author Dr. Sarwardi is thankful to the Department of Mathematics, Aliah University for providing opportunities to perform the present work. He is also thankful to his Ph.D. supervisor Prof. Prashanta Kumar Mandal, Department of Mathematics, Visva-Bharati (A Central University) for his continuous encouragement and inspiration. First author Mr. H. Molla is thankful to Mr. M. Haque, Research Scholar, Department of Mathematics, Aliah University for his generous help in performing numerical simulations.

References

[1] N. Rashevsky, Mathematical biology of social behavior, University of Chicago Press, 1951.Search in Google Scholar

[2] R. M. May, Stability and complexity in model ecosystems, Princeton University Press, 1973.10.2307/1935352Search in Google Scholar

[3] S. I. Rubinow, Introduction to mathematical biology, Wiley, 1975.Search in Google Scholar

[4] R. M. May and A. R. McLean, Theoretical ecology: principles and applications, Blackwell Scientific Publications, Oxford University Press, 1976.Search in Google Scholar

[5] H. I. Freedman, Deterministic mathematical models in population ecology, Marcel Debber, New York, 1980.Search in Google Scholar

[6] J. M. Epstein, Nonlinear dynamics, mathematical biology, and social science, Westview Press, 1997.Search in Google Scholar

[7] M. Kot, Elements of mathematical ecology, Cambridge University Press, New York, 2001.10.1017/CBO9780511608520Search in Google Scholar

[8] N. Britton, Essential mathematical biology, Springer Science & Business Media, 2012.Search in Google Scholar

[9] P. Cartigny, W. Gómez and H. Salgado, The spatial distribution of small-and large-scale fisheries in a marine protected area, Ecol. Model. 212(3) (2008), 513–521.10.1016/j.ecolmodel.2007.11.001Search in Google Scholar

[10] B. Dubey, P. Chandra and P. Sinha, A model for fishery resource with reserve area, Nonl. Anal.: Real World Appl. 4(4) (2003), 625–637.10.1016/S1468-1218(02)00082-2Search in Google Scholar

[11] Holling, C. S., The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, The Can. Entomologist 91 (1959), 293–320. Cambridge University Press.10.4039/Ent91293-5Search in Google Scholar

[12] C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulations, Memoirs Entomol. Soc. Canada. 97(545) (1965), 5–60.10.4039/entm9745fvSearch in Google Scholar

[13] R. J. Taylor, Predation, Chapman and Hall Ltd., New York, 1984.10.1007/978-94-009-5554-7Search in Google Scholar

[14] M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator–prey interactions, The Am. Nat., Sci. Press. 97(895) (1963), 209–223.10.1086/282272Search in Google Scholar

[15] S. L. Lima, Putting predators back into behavioral predator–prey interactions, Trends Ecol. & Evol. 17(2) (2002), 70–75.10.1016/S0169-5347(01)02393-XSearch in Google Scholar

[16] E. González-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model. 166(1–2) (2003), 135–146.10.1016/S0304-3800(03)00131-5Search in Google Scholar

[17] A. Sih, Prey refuges and predator–prey stability, Theor. Popul. Biol. 31 (1987), 1-12.10.1016/0040-5809(87)90019-0Search in Google Scholar

[18] R. Cressman and J. Garay, A predatorprey refuge system: evolutionary stability in ecological systems, Theor. Popul, Biol. 76 (2009), 248–257.10.1016/j.tpb.2009.08.005Search in Google Scholar

[19] S. Sarwardi, P. K. Mandal and S. Ray, Analysis of a competitive prey–predator system with a prey refuge, Biosystems, 110(3) (2012), 133–148.10.1016/j.biosystems.2012.08.002Search in Google Scholar

[20] S. Sarwardi, P. K. Mandal and S. Ray, Dynamical behaviour of a two-predator model with prey refuge, J. Biol. Phys. 39 (2013), 701–722.10.1007/s10867-013-9327-7Search in Google Scholar

[21] A. Gkana and L. Jachilas, Non-overlapping generation species: complex prey-predator interactions, Int. J. Nonl. Sci. Num. Simul. 16(5) (2015), 207–219.10.1515/ijnsns-2014-0121Search in Google Scholar

[22] K. P. Hadeler and I. Gerstmann, The discrete Rosenzweig model, Math. Biosci. 98(1) (1990), 49–72.10.1016/0025-5564(90)90011-MSearch in Google Scholar

[23] M. Danca, S. Codreanu and B. Bako, Detailed analysis of a nonlinear prey–predator model, J. Biol. Phys. 23(1) (1997), 11–20.10.1023/A:1004918920121Search in Google Scholar

[24] V. Volterra, La concorrenza vitale tra le specie nell’ambiente marino, Soc. nouv. de l’impr. du Loiret., 1931.Search in Google Scholar

[25] A. J. Lotka, Elements of mathematical biology, Dover Publication, New York, 1956.Search in Google Scholar

[26] J. N. McNair, The effects of refuges on predator–prey interactions: a reconsideration, Theor. Popul. Biol. 29(1) (1986), 38–63.10.1016/0040-5809(86)90004-3Search in Google Scholar

[27] J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator–prey interaction model incorporating a prey refuge, Bull. Math. Biol. 57(1) (1995), 63–76.10.1007/BF02458316Search in Google Scholar PubMed

[28] Y. Huang, F. Chen and L. Zhong, Stability analysis of preypredator model with Holling type-III response function incorporating a prey refuge, Appl. Math. Comput. 182 (2006), 672–683.10.1016/j.amc.2006.04.030Search in Google Scholar

[29] J. Kolasa and C. D. Rollo, Introduction: the heterogeneity of heterogeneity: a glossary, Ecol. Heterogeneity. 86 (1991), 1–23. Springer-Verlag, New York.10.1007/978-1-4612-3062-5_1Search in Google Scholar

[30] J. H. Micheal, A. Elizabeth and A. John, The Ecological Consequences of Environmental Heterogeneity, Cambridge University Press, 1999.Search in Google Scholar

[31] W. W. Murdoch and A. Oaten, Predation and population stability, Adv. Ecol. Res. 9 (1975), 1–131.10.1016/S0065-2504(08)60288-3Search in Google Scholar

[32] R. A. Stein, Selective predation, optimal foraging, and the predator–prey interaction between fish and crayfish, Ecology, Wiley Online Library. 58(6) (1977), 1237–1253.10.2307/1935078Search in Google Scholar

[33] G. W. Harrison, Global stability of predator–prey interactions, J. Math. Biol. 8(2) (1979), 159–171.10.1007/BF00279719Search in Google Scholar

[34] A. A. Berryman, The origins and evolution of predator–prey theory, Ecology. 75(5) (1992), 1530–1535. Wiley Online Library.10.2307/1940005Search in Google Scholar

[35] V. Krivan, Effect of optimal antipredator behaviour of prey on predator–prey dynamics: the role of refuges, Theor. Popul. Biol. 53(2) (1998), 131–142.10.1006/tpbi.1998.1351Search in Google Scholar

[36] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges, Math. Bios. 218(2) (2009), 73–79.10.1016/j.mbs.2008.12.008Search in Google Scholar

[37] Y. Wang, J. Wang, Influence of prey refuge on predator–prey dynamics, Nonl. Dyn. 67(1) (2012), 191–201.10.1007/s11071-011-9971-zSearch in Google Scholar

[38] T. K. Kar, Stability analysis of a prey–predator model incorporating a prey refuge, Commun. Nonl. Sci. Num. Simul. 10 (2005), 681–69110.1016/j.cnsns.2003.08.006Search in Google Scholar

[39] M. J. Smith, Models in ecology, Cambridge University Press, UK, 1974.Search in Google Scholar

[40] P. W. Price, Price, P.W. Insect ecology, Wiley, New York. QL 463. P74, 1975.Search in Google Scholar

[41] N. J. Gotelli, A primer of ecology, Sinauer Associates Incorporated, Massachusetts, 1995.Search in Google Scholar

[42] J. N. McNair, Stability effects of The effect of prey refuges with entry-exit dynamics, J. Theor. Biol. 125(4) (1987), 449–464.10.1016/S0022-5193(87)80213-8Search in Google Scholar

[43] M. Haque, M. S. Rahman, E. Venturino, B. Li, Effect of a functional response-dependent prey refuge in a predator–prey model, Ecol. Complex. 20 (2014), 248–256.10.1016/j.ecocom.2014.04.001Search in Google Scholar

[44] J. M. Smith, M. Slatkin, The stability of predator–prey systems, Ecology 125(4) (1987), 449–464. Wiley Online Library.Search in Google Scholar

[45] A. R. Ives, A. P. Dobson, Antipredator behavior and the population dynamics of simple predator–prey systems, The Am. Nat. 154(2) (1973), 384–391. University of Chicago Press.10.1086/284719Search in Google Scholar

[46] G. D. Ruxton, Short term refuge use and stability of predator-prey model, Theor. Popul. Biol. 47 (1995).10.1006/tpbi.1995.1001Search in Google Scholar

[47] S. Hsu, T. Huang, Global stability for a class of predator–prey systems, SIAM. 55(3) (1975), 763–783.10.1137/S0036139993253201Search in Google Scholar

[48] R. M. Anderson, R. M. May, Regulation and stability of host–parasite population interactions: I. Regulatory processes, J. Anim. Ecol. 47 (1978), 219–247.10.2307/3933Search in Google Scholar

[49] X. Song, Y. Li, Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonl. Anal.: Real World Appl. 9(1) (2008), 64–79.10.1016/j.nonrwa.2006.09.004Search in Google Scholar

[50] L. Chen, F. Chen, L. Chen, Regulation and stability of host-parasite population interactions: I. Regulatory processes, Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a constant prey refuge, Nonl. Anal.: Real World Appl. 11(1) (2010), 246–252.10.1016/j.nonrwa.2008.10.056Search in Google Scholar

[51] M. P. Hassell, The dynamics of arthropod predator-prey systems, Princeton University Press, Princeton, NJ, 1978.Search in Google Scholar

[52] R. Yafia, F. El Adnani, H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling-type II schemes, Nonl. Anal.: Real World Appl. 9(5) (2008), 2055–2067.10.1016/j.nonrwa.2006.12.017Search in Google Scholar

[53] G. Birkhoff, G. C. Rota, Ordinary differential equations, Wiley, New York, 1978.Search in Google Scholar

[54] T. C. Gard, T. G. Hallam, Persistence in food web -1, Lotka- Volterra food chains, Bull. Math., Biol. (1979).10.1016/S0092-8240(79)80024-5Search in Google Scholar

[55] H. Freedman, P. Waltman, Persistence in models of three interacting predator–prey populations, Math. Bios. 68(2) (1984), 213–231.10.1016/0025-5564(84)90032-4Search in Google Scholar

[56] B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge. 41, 1981.Search in Google Scholar

[57] P. J. Pal, S. Sarwardi, T. Saha, P. K. Mandal, Mean square stability in a modified Leslie-Gower and Holling-Type II predator–pray model, J. Appl. Math. & Inf. 29 (2011), 781–802.Search in Google Scholar

[58] A. M. Alaoui, D. M. Okiye, Boundedness and global stability for a predator–prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett. 16(7) (2003), 1069–1075.10.1016/S0893-9659(03)90096-6Search in Google Scholar

[59] J. Hale, Ordinary differential equation, Klieger Publishing Company, Malabar, 1989.Search in Google Scholar

[60] D. Xiao, S. Ruan, Multiple bifurcations in a delayed predator–prey system with nonmonotonic functional response, J. Diff. Eq. 176 (2001), 494–510.10.1006/jdeq.2000.3982Search in Google Scholar

[61] P. Y. H. Pang, M. X. Wang, Non-constant positive steady states of a predatorprey system with non-monotonic functional response and diffusion, Proc. Lond. Math. Soc. 88 (2004), 135–157.10.1112/S0024611503014321Search in Google Scholar

Received: 2017-10-17

Accepted: 2018-11-03

Published Online: 2018-11-27

Published in Print: 2019-02-23

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2017-0224

Keywords for this article

ecological model; refuge; permanent; stability; direction of Hopf bifurcation; numerical simulations