March  2016, 21(2): 537-555. doi: 10.3934/dcdsb.2016.21.537

Oscillations in age-structured models of consumer-resource mutualisms

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875

2. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence

3. 

Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250

Received  February 2015 Revised  September 2015 Published  November 2015

In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource interactions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation parameter passes through some critical values.
Citation: Zhihua Liu, Pierre Magal, Shigui Ruan. Oscillations in age-structured models of consumer-resource mutualisms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 537-555. doi: 10.3934/dcdsb.2016.21.537
References:
[1]

A. Barkai and C. McQuaid, Predator-prey role reversal in a marine benthic ecosystem,, Science, 242 (1988), 62. doi: 10.1126/science.242.4875.62. Google Scholar

[2]

J. M. Cushing, Equilibria in systems of interacting strustured populations,, J. Math. Biol., 24 (1987), 627. doi: 10.1007/BF00275507. Google Scholar

[3]

J. M. Cushing, An Introduction to Structured Population Dynamics,, SIAM, (1998). doi: 10.1137/1.9781611970005. Google Scholar

[4]

J. M. Cushing and M. Saleem, A predator prey model with age structure,, J. Math. Biol., 14 (1982), 231. doi: 10.1007/BF01832847. Google Scholar

[5]

A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems,, J. Math. Anal. Appl., 341 (2008), 501. doi: 10.1016/j.jmaa.2007.09.074. Google Scholar

[6]

M. E. Gurtin and D. S. Levine, On predator-prey interactions with predation dependent on age of prey,, Math. Biosci., 47 (1979), 207. doi: 10.1016/0025-5564(79)90038-5. Google Scholar

[7]

J. N. Holland and D. L. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions,, Ecol. Lett., 12 (2009), 1357. Google Scholar

[8]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. Google Scholar

[9]

D. S. Levine, Bifurcating periodic solutions for a class of age-structured predator-prey systems,, Bull. Math. Biol., 45 (1983), 901. doi: 10.1007/BF02458821. Google Scholar

[10]

J. Li, Dynamics of age-structured predator-prey population model,, J. Math. Anal. Appl., 152 (1990), 399. doi: 10.1016/0022-247X(90)90073-O. Google Scholar

[11]

Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Z. Angew. Math. Phys., 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. Google Scholar

[12]

Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models,, J. Differential Equations, 257 (2014), 921. doi: 10.1016/j.jde.2014.04.018. Google Scholar

[13]

R. H. MacArthur, Geographical Ecology,, Harper and Row, (1972). Google Scholar

[14]

P. Magal, Compact attractors for time-periodic age structured population models,, Electron. J. Differential Equations, 65 (2001), 1. Google Scholar

[15]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Adv. Differential Equations, 14 (2009), 1041. Google Scholar

[16]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[17]

S. Magalhães, A. Janssen, M. Montserrat and M. W. Sabelis, Prey attack and predators defend: Counterattacking prey triggers parental care in predators,, Proc. R. Soc. B, 272 (2005), 1929. Google Scholar

[18]

R. M. May, Limit cycles in predator-prey communities,, Science, 177 (1972), 900. doi: 10.1126/science.177.4052.900. Google Scholar

[19]

R. J. Mitchell, R. E. Irwin, R. J. Flanagan and J. D. Karron, Ecology and evolution of plant pollinator interactions,, Ann. Bot., 103 (2009), 1355. doi: 10.1093/aob/mcp122. Google Scholar

[20]

W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics,, Princeton University Press, (2003). Google Scholar

[21]

G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that eat each other,, Annu. Rev. Ecol. Syst., 20 (1989), 297. doi: 10.1146/annurev.es.20.110189.001501. Google Scholar

[22]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, Am. Nat., 97 (1963), 209. doi: 10.1086/282272. Google Scholar

[23]

M. Saleem, Predator-prey relationships: Indiscriminate predation,, J. Math. Biol., 21 (1984), 25. doi: 10.1007/BF00275220. Google Scholar

[24]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in , 6 (1997), 691. Google Scholar

[25]

E. Venturino, Age-structured predator-prey models,, Math. Modelling, 5 (1984), 117. doi: 10.1016/0270-0255(84)90020-4. Google Scholar

[26]

Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a uni-directional consumer-resource system,, J. Theoret. Biol., 280 (2011), 43. doi: 10.1016/j.jtbi.2011.03.038. Google Scholar

[27]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional consumer-resource theory characterizing transitions of interaction outcomes,, Ecol. Complexity, 8 (2011), 249. doi: 10.1016/j.ecocom.2011.04.002. Google Scholar

[28]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional Interaction and Plant pollinator robber Coexistence,, Bull. Math. Biol., 74 (2012), 2142. doi: 10.1007/s11538-012-9750-0. Google Scholar

show all references

References:
[1]

A. Barkai and C. McQuaid, Predator-prey role reversal in a marine benthic ecosystem,, Science, 242 (1988), 62. doi: 10.1126/science.242.4875.62. Google Scholar

[2]

J. M. Cushing, Equilibria in systems of interacting strustured populations,, J. Math. Biol., 24 (1987), 627. doi: 10.1007/BF00275507. Google Scholar

[3]

J. M. Cushing, An Introduction to Structured Population Dynamics,, SIAM, (1998). doi: 10.1137/1.9781611970005. Google Scholar

[4]

J. M. Cushing and M. Saleem, A predator prey model with age structure,, J. Math. Biol., 14 (1982), 231. doi: 10.1007/BF01832847. Google Scholar

[5]

A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems,, J. Math. Anal. Appl., 341 (2008), 501. doi: 10.1016/j.jmaa.2007.09.074. Google Scholar

[6]

M. E. Gurtin and D. S. Levine, On predator-prey interactions with predation dependent on age of prey,, Math. Biosci., 47 (1979), 207. doi: 10.1016/0025-5564(79)90038-5. Google Scholar

[7]

J. N. Holland and D. L. DeAngelis, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions,, Ecol. Lett., 12 (2009), 1357. Google Scholar

[8]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. Google Scholar

[9]

D. S. Levine, Bifurcating periodic solutions for a class of age-structured predator-prey systems,, Bull. Math. Biol., 45 (1983), 901. doi: 10.1007/BF02458821. Google Scholar

[10]

J. Li, Dynamics of age-structured predator-prey population model,, J. Math. Anal. Appl., 152 (1990), 399. doi: 10.1016/0022-247X(90)90073-O. Google Scholar

[11]

Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, Z. Angew. Math. Phys., 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. Google Scholar

[12]

Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models,, J. Differential Equations, 257 (2014), 921. doi: 10.1016/j.jde.2014.04.018. Google Scholar

[13]

R. H. MacArthur, Geographical Ecology,, Harper and Row, (1972). Google Scholar

[14]

P. Magal, Compact attractors for time-periodic age structured population models,, Electron. J. Differential Equations, 65 (2001), 1. Google Scholar

[15]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, Adv. Differential Equations, 14 (2009), 1041. Google Scholar

[16]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models,, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[17]

S. Magalhães, A. Janssen, M. Montserrat and M. W. Sabelis, Prey attack and predators defend: Counterattacking prey triggers parental care in predators,, Proc. R. Soc. B, 272 (2005), 1929. Google Scholar

[18]

R. M. May, Limit cycles in predator-prey communities,, Science, 177 (1972), 900. doi: 10.1126/science.177.4052.900. Google Scholar

[19]

R. J. Mitchell, R. E. Irwin, R. J. Flanagan and J. D. Karron, Ecology and evolution of plant pollinator interactions,, Ann. Bot., 103 (2009), 1355. doi: 10.1093/aob/mcp122. Google Scholar

[20]

W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics,, Princeton University Press, (2003). Google Scholar

[21]

G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: Potential competitors that eat each other,, Annu. Rev. Ecol. Syst., 20 (1989), 297. doi: 10.1146/annurev.es.20.110189.001501. Google Scholar

[22]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions,, Am. Nat., 97 (1963), 209. doi: 10.1086/282272. Google Scholar

[23]

M. Saleem, Predator-prey relationships: Indiscriminate predation,, J. Math. Biol., 21 (1984), 25. doi: 10.1007/BF00275220. Google Scholar

[24]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, in , 6 (1997), 691. Google Scholar

[25]

E. Venturino, Age-structured predator-prey models,, Math. Modelling, 5 (1984), 117. doi: 10.1016/0270-0255(84)90020-4. Google Scholar

[26]

Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a uni-directional consumer-resource system,, J. Theoret. Biol., 280 (2011), 43. doi: 10.1016/j.jtbi.2011.03.038. Google Scholar

[27]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional consumer-resource theory characterizing transitions of interaction outcomes,, Ecol. Complexity, 8 (2011), 249. doi: 10.1016/j.ecocom.2011.04.002. Google Scholar

[28]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional Interaction and Plant pollinator robber Coexistence,, Bull. Math. Biol., 74 (2012), 2142. doi: 10.1007/s11538-012-9750-0. Google Scholar

[1]

Robert Stephen Cantrell, Chris Cosner, Shigui Ruan. Intraspecific interference and consumer-resource dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 527-546. doi: 10.3934/dcdsb.2004.4.527

[2]

Wonlyul Ko, Inkyung Ahn, Shengqiang Liu. Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1715-1733. doi: 10.3934/dcdsb.2015.20.1715

[3]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819

[4]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[5]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

[6]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657

[7]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046

[8]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[9]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[10]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[11]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291

[12]

C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008

[13]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264

[14]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[15]

George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations & Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563

[16]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

[17]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[18]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045

[19]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[20]

Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1859-1867. doi: 10.3934/dcdsb.2016026

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]