Analysis of a negative binomial host-parasitoid model with two maturation delays and impulsive resource input. (English) Zbl 1448.92197
Summary: To study the interaction of parasitoids and their insect hosts in laboratory environment, we propose a mathematical model incorporating impulsive resource inputs, stage-structure, maturation times and negative binomial distribution of parasitoid attacks. According to the adaptability of the insect host to the environment, we obtain conditions under which the system is uniformly permanent in two cases, which guarantee that the host and its parasitoid can coexist. By applying fixed point theory, we show existence of the positive periodic solution where the host and its parasitoid can coexist, and also obtain the conditions that ensure the existence of the parasitoid-extinction periodic solution. Our numerical analysis confirms and extends our theoretical results. The simulations show that when the total amount of resource is fixed, a smaller amount of recourse inputs with a shorter period of impulsive delivery results in smaller oscillation amplitude in the insect host population. However, the development of parasitoid population is not affected by the resource management strategy. It is also demonstrated that larger maturation times, either the host’s or the parasitoid’s, lead to the decline of the parasitoid population. But larger parasitoid’s maturation time does accelerate the host’s population growth. These are helpful for us to acquire a deeper knowledge of the host-parasitoid interaction in laboratory environment.
MSC:
| 92D25 | Population dynamics (general) |
| 34K13 | Periodic solutions to functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
References:
| [1] | R.R. Askew, Parasitic Insects, Heinemann, London, 1971. [Google Scholar] |
| [2] | R.R. Askew and M.R. Shaw, Parasitoid communities: Their size, structure and development, in Insect Parasitoids, 13th Symposium of Royal Entomological Society of London, Academic Press, London, 1986. [Google Scholar] |
| [3] | V.A. Bailey, A.J. Nicholson and E.J. Williams, Interactions between hosts and parasites when some host individuals are more difficult to find than others, J. Theoret. Biol. 3 (1962), pp. 1-18. doi: 10.1016/S0022-5193(62)80002-2[Crossref], [Web of Science ®], [Google Scholar] |
| [4] | N.E. Beckage and D.B. Gelman, Wasp parasitoid disruption of host development: Implications for new biologically based strategies for insect control, Annu. Rev. Entomol. 49 (2004), pp. 299-330. doi: 10.1146/annurev.ento.49.061802.123324[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [5] | O.N. Bjørnstad and J. Bascompte, Synchrony and second-order spatial correlation in host-parasitoid systems, J. Anim. Ecol. 70 (2001), pp. 924-933. doi: 10.1046/j.0021-8790.2001.00560.x[Crossref], [Web of Science ®], [Google Scholar] |
| [6] | C.J. Briggs, R.M. Nisbet, W.W. Murdoch, T.R. Collier and J.A.J. Metz, Dynamical effects of host-feeding in parasitoids, J. Anim. Ecol. 64 (1995), pp. 403-416. doi: 10.2307/5900[Crossref], [Web of Science ®], [Google Scholar] |
| [7] | C.J. Briggs, S.M. Sait, M. Begon, D.J. Thompson and H.C.J. Godfray, What causes generation cycles in populations of stored-product moths?, J. Anim. Ecol. 69 (2000), pp. 352-366. doi: 10.1046/j.1365-2656.2000.00398.x[Crossref], [Web of Science ®], [Google Scholar] |
| [8] | P.L. Chesson and W.W. Murdoch, Aggregation of risk: relationships among host-parasitoid models, Am. Nat. 127 (1986), pp. 696-715. doi: 10.1086/284514[Crossref], [Web of Science ®], [Google Scholar] |
| [9] | J.C. Corley and A.F. Capurro, The persistence of simple host-parasitoid systems with prolonged diapause, Ecologia Austral. 10 (2000), pp. 37-45. [Google Scholar] |
| [10] | J.T. Cronin, J.D. Reeve, D.S. Xu, M.Q. Xiao and H.N. Stevens, Variable prey development time suppresses predator-prey cycles and enhances stability, Ecol. Lett. 19 (2016), pp. 318-327. doi: 10.1111/ele.12571[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [11] | B. Emerick and A. Singh, Host-feeding enhances stability of discrete-time host-parasitoid population dynamic models, Math. Biosci. 272 (2015), pp. 54-63. doi: 10.1016/j.mbs.2015.11.011[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1369.92094 |
| [12] | B. Emerick and A. Singh, The effects of host-feeding on stability of discrete-time host-parasitoid population dynamic models, Math. Biosci. 272 (2016), pp. 54-63. doi: 10.1016/j.mbs.2015.11.011[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1369.92094 |
| [13] | L. Fors, I. Liblikas, P. Andersson, A.-K. Borg-Karlson, N. Cabezas, R. Mozuraitis and P.A. Hamb”ack, Chemical communication and host search in Galerucella leaf beetles, Chemoecology 25 (2015), pp. 33-45. doi: 10.1007/s00049-014-0174-1[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [14] | L. Fors, R. Markus, U. Theopold and P.A. Hamb”ack, Differences in cellular immune competence explain parasitoid resistance for two coleopteran species, PLoS One 9 (2014), e108795. doi:10.1371/journal.pone.0108795. [Crossref], [PubMed], [Web of Science ®], [Google Scholar] · doi:10.1371/journal.pone.0108795 |
| [15] | M.A. Ghani and H.L. Sweetman, Ecological studies on the granary weevil parasite, Aplastomorpha calandrae (Howard), Biologia 1 (1995), pp. 115-139. [Google Scholar] |
| [16] | H.C.J. Godfray, Parasitoids: Behavioral and Evolutionary Ecology, Princeton University Press, Princeton, 1994. [Crossref], [Google Scholar] |
| [17] | H.C.J. Godfray and M.P. Hassell, Discrete and continuous insect populations in tropical environment, J. Anim. Ecol. 58 (1989), pp. 153-174. doi: 10.2307/4992[Crossref], [Web of Science ®], [Google Scholar] |
| [18] | D.M. Gordon, R.M. Nisbet, A. DeRoos, W.S.C. Gurney and R.K. Stewart, Discrete generations in host-parasitoid models with contrasting life-cycles, J. Anim. Ecol. 60 (1991), pp. 295-308. doi: 10.2307/5461[Crossref], [Web of Science ®], [Google Scholar] |
| [19] | M.P. Hassell, Density-dependence in single-species populations, J. Anim. Ecol. 44 (1975), pp. 283-295. doi: 10.2307/3863[Crossref], [Web of Science ®], [Google Scholar] |
| [20] | M.P. Hassell, Host-parasitoid population dynamics, J. Anim. Ecol. 69(4) (2000), pp. 543-566. doi: 10.1046/j.1365-2656.2000.00445.x[Crossref], [Web of Science ®], [Google Scholar] |
| [21] | M.P. Hassell, R.M. May, S.W. Pacala and P.L. Chesson, The persistence of host-parasitoid associations in patchy environments. I. A general criterion, Am. Nat. 183 (1991), pp. 568-583. doi: 10.1086/285235[Crossref], [Google Scholar] |
| [22] | A.H. Hirzel and W.W. Murdoch, Host-parasitoid spatial dynamics in heterogeneous landscapes, Oikos, 116 (2007), pp. 2082-2096. doi: 10.1111/j.2007.0030-1299.15976.x[Crossref], [Web of Science ®], [Google Scholar] |
| [23] | M.Z. Huang, J.X. Li, X.Y. Song and H.J. Guo, Modeling impulsive injections of insulin analogues: towards artificial pancreas, SIAM J. Appl. Math. 72 (2012), pp. 1524-1548. doi: 10.1137/110860306[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1325.92045 |
| [24] | M.Z. Huang, S.Z. Liu, X.Y. Song and L.S. Chen, Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting, Nonlinear Dynam. 73 (2013), pp. 815-826. doi: 10.1007/s11071-013-0834-7[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1281.92069 |
| [25] | M.Z. Huang and X.Y. Song, Modeling and qualitative analysis of diabetes therapies with state feedback control, Int. J. Biomath 7 (2014), pp. 1450035. doi: 10.1142/S1793524514500351[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1329.92057 |
| [26] | J. Ji, W.I. Choi and M.I. Ryoo, Fitness and sex allocation of Anisopteromalus calandrae (Hymenoptera: Pteromalidae): relative fitness of large females and males in a multi-patch system, Ann. Entomol. Soc. Am. 97 (2004), pp. 825-830. doi: 10.1603/0013-8746(2004)097[0825:FASAOA]2.0.CO;2[Crossref], [Web of Science ®], [Google Scholar] |
| [27] | P.M. Kareiva, Renewing the dialogue between theory and experiments in population ecology, in Perspectives in Ecological Theory, J. Roughgarden, R. M. May, and S. A. Levin, eds., Princeton University Press, Princeton, 1989, pp. 68-88. [Google Scholar] |
| [28] | M. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. [Google Scholar] · Zbl 0121.10604 |
| [29] | Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [Google Scholar] · Zbl 0777.34002 |
| [30] | S. Lebreton, C. Chevrier and E. Darrouzet, Sex allocation strategies in response to conspecifics’ offspring sex ratio in solitary parasitoids, Behav. Ecol. 21 (2010), pp. 107-112. doi: 10.1093/beheco/arp156[Crossref], [Web of Science ®], [Google Scholar] |
| [31] | B. Lemaitre and J. Hoffmann, The host defense of Drosophila melanogaster, Annu. Rev. Entomol. 25 (2007), pp. 697-743. [Google Scholar] |
| [32] | J.Q. Li and Y.L. Yang, SIR-SVS epidemic models with continuous and impulsive vaccination strategies, J. Theor. Biol. 280 (2011), pp. 108-116. doi: 10.1016/j.jtbi.2011.03.013[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1397.92646 |
| [33] | H. Liu, Z.Z. Li, M. Gao, H.W. Dai and Z.G. Liu, Dynamics of a host-parasitoid model with Allee effect for the host and parasitoid aggregation, Ecol. Complex. 6 (2009), pp. 337-345. doi: 10.1016/j.ecocom.2009.01.003[Crossref], [Web of Science ®], [Google Scholar] |
| [34] | J.L. Maron and S. Harrison, Spatial pattern formation in an insect host-parasitoid system, Science278 (1997), pp. 1619-1621. doi: 10.1126/science.278.5343.1619[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [35] | R.M. May, Host-parasitoid systems in patchy environments: a phenomenological model, J. Anim. Ecol. 47 (1978), pp. 833-844. doi: 10.2307/3674[Crossref], [Web of Science ®], [Google Scholar] |
| [36] | W.W. Murdoch, C.J. Briggs and S. Swarbrick, Host suppression and stability in a parasitoid-host system: experimental demonstration, Science 309 (2005), pp. 610-613. doi: 10.1126/science.1114426[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [37] | W.W. Murdoch, R.M. Nisbet, S.P. Blythe, W.S.C. Gurney and J.D. Reeve, An invulnerable age class and stability in delay-differential parasitoid-host models, Am. Nat. 129 (1987), pp. 263-282. doi: 10.1086/284634[Crossref], [Web of Science ®], [Google Scholar] |
| [38] | R.M. Penczykowski, A. Laine and B. Koskella, Understanding the ecology and evolution of host-parasite interactions across scales, Evolut. Appl. 9 (2016), pp. 37-52. doi: 10.1111/eva.12294[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [39] | J.D. Reeve, J.T. Cronin and D.R. Strong, Parasitism and generation cycles in a saltmarsh planthopper, J. Anim. Ecol. 63 (1994), pp. 912-920. doi: 10.2307/5268[Crossref], [Web of Science ®], [Google Scholar] |
| [40] | J.D. ReeveD.J. Rhodes, and P. Turchin, Scramble competition in the southern pine beetle, Dendroctonus frontalis, Ecol. Entomol. 23 (1998), pp. 433-443. doi: 10.1046/j.1365-2311.1998.00143.x[Crossref], [Web of Science ®], [Google Scholar] |
| [41] | X.Y. Song, M.Z. Huang and J.X. Li, Modeling impulsive insulin delivery in insulin pump with time delays, SIAM J. Appl. Math. 74 (2014), pp. 1763-1785. doi: 10.1137/130933137[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1330.92060 |
| [42] | S.Y. Tang and L.S. Chen, Chaos in functional response host-parasitoid ecosystem models, Chaos Solitons Fract. 13 (2002), pp. 875-884. doi: 10.1016/S0960-0779(01)00063-7[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1022.92042 |
| [43] | M. Tuda, Temporal/spatial structure and the dynamical property of laboratory host-parasitoid systems, Res. Popul. Ecol. 38 (1996), pp. 133-140. doi: 10.1007/BF02515721[Crossref], [Google Scholar] |
| [44] | M. Tuda and M. Shimada, Developmental schedules and persistence of experimental host-parasitoid systems at two different temperatures, Oecalogia 103 (1995), pp. 283-291. doi: 10.1007/BF00328616[Crossref], [PubMed], [Web of Science ®], [Google Scholar] |
| [45] | M. Tuda and M. Shimada, Complexity, evolution, and persistence in host-parasitoid experimental systems with Callosobruchus beetles as the host, Adv. Ecol. Res. 37 (2005), pp. 37-75. doi: 10.1016/S0065-2504(04)37002-9[Crossref], [Web of Science ®], [Google Scholar] |
| [46] | S. Utida, Studies on experimental population of the azuki bean weevil, Callosobruchus chinensis (L.). I. The effect of population density on the progeny population, Memiors of the College of Agriculture, Kyoto Imperial University 48 (1941), pp. 1-30. [Google Scholar] |
| [47] | S. Utida, Studies on experimental population of the azuki bean weevil, Callosobruchus chinensis (L.). VIII-IX., Memiors of the College of Agriculture, Kyoto Imperial University 54 (1943), pp. 1-40. [Google Scholar] |
| [48] | S. Utida, “Phase” dimorphism observed in the laboratory population of the cowpea weevil Callosobruchus quadrimaculatus, J. Appl. Ecol. 18 (1954), pp. 161-168. [Google Scholar] |
| [49] | S. Utida, Cyclic fluctuations of population density intrinsic to the host-parasite system, Ecology 38 (1957), pp. 442-449. doi: 10.2307/1929888[Crossref], [Web of Science ®], [Google Scholar] |
| [50] | L.M. Wang, L.S. Chen and J.J. Nieto, The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Anal. Real World Appl. 11 (2010), pp. 1374-1386. doi: 10.1016/j.nonrwa.2009.02.027[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1188.93038 |
| [51] | H.J. Wearing, S.M. Sait, T.C. Cameron and P. Rohani, Stage-structured competition and the cyclic dynamics of host-parasitoid populations, J. Anim. Ecol. 73 (2004), pp. 706-722. doi: 10.1111/j.0021-8790.2004.00846.x[Crossref], [Web of Science ®], [Google Scholar] |
| [52] | D.S. Xu, J.D. Reeve, X.Q. Wang and M.Q. Xiao, Developmental variability and stability in continuous-time host C parasitoid models, Theor. Popul. Biol. 78 (2010), pp. 1-11. doi: 10.1016/j.tpb.2010.03.007[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1403.92276 |
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