American Institute of Mathematical Sciences

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

Oscillations in age-structured models of consumer-resource mutualisms

Zhihua Liu 1, , Pierre Magal 2, and  Shigui Ruan 3,

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.
Keywords: Consumer-resource interaction, age-structure, stability, Hopf bifurcation, periodic solutions..
Mathematics Subject Classification: Primary: 35B32, 35Q92; Secondary: 92D3.
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), 62-64. 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), 627-649. doi: 10.1007/BF00275507.  Google Scholar

[3]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 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), 231-250. 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), 501-518. 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), 207-219. 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), 1357-1366. 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), 1286-1295. Google Scholar

[9]

D. S. Levine, Bifurcating periodic solutions for a class of age-structured predator-prey systems, Bull. Math. Biol., 45 (1983), 901-915. doi: 10.1007/BF02458821.  Google Scholar

[10]

J. Li, Dynamics of age-structured predator-prey population model, J. Math. Anal. Appl., 152 (1990), 399-415. 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), 191-222. 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), 921-1011. doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[13]

R. H. MacArthur, Geographical Ecology, Harper and Row, New York, 1972. Google Scholar

[14]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.  Google Scholar

[15]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.  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), vi+71 pp. 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), 1929-1933. Google Scholar

[18]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. 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), 1355-1363. doi: 10.1093/aob/mcp122.  Google Scholar

[20]

W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics, Princeton University Press, Princeton, 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), 297-330. 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), 209-223. doi: 10.1086/282272.  Google Scholar

[23]

M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220.  Google Scholar

[24]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in "Advances in Mathematical Population Dynamics: Molecules, Cells and Man'', O. Arino, D. Axelrod and M. Kimmel (Eds), World Sci. Publ., River Edge, NJ, 6 (1997), 691-711.  Google Scholar

[25]

E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. 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), 43-49. 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), 249-257. 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), 2142-2164. 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), 62-64. 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), 627-649. doi: 10.1007/BF00275507.  Google Scholar

[3]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 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), 231-250. 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), 501-518. 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), 207-219. 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), 1357-1366. 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), 1286-1295. Google Scholar

[9]

D. S. Levine, Bifurcating periodic solutions for a class of age-structured predator-prey systems, Bull. Math. Biol., 45 (1983), 901-915. doi: 10.1007/BF02458821.  Google Scholar

[10]

J. Li, Dynamics of age-structured predator-prey population model, J. Math. Anal. Appl., 152 (1990), 399-415. 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), 191-222. 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), 921-1011. doi: 10.1016/j.jde.2014.04.018.  Google Scholar

[13]

R. H. MacArthur, Geographical Ecology, Harper and Row, New York, 1972. Google Scholar

[14]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.  Google Scholar

[15]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084.  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), vi+71 pp. 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), 1929-1933. Google Scholar

[18]

R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. 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), 1355-1363. doi: 10.1093/aob/mcp122.  Google Scholar

[20]

W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics, Princeton University Press, Princeton, 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), 297-330. 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), 209-223. doi: 10.1086/282272.  Google Scholar

[23]

M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220.  Google Scholar

[24]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in "Advances in Mathematical Population Dynamics: Molecules, Cells and Man'', O. Arino, D. Axelrod and M. Kimmel (Eds), World Sci. Publ., River Edge, NJ, 6 (1997), 691-711.  Google Scholar

[25]

E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. 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), 43-49. 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), 249-257. 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), 2142-2164. doi: 10.1007/s11538-012-9750-0.  Google Scholar

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