# American Institute of Mathematical Sciences

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 and 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. [2] J. M. Cushing, Equilibria in systems of interacting strustured populations, J. Math. Biol., 24 (1987), 627-649. doi: 10.1007/BF00275507. [3] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] R. H. MacArthur, Geographical Ecology, Harper and Row, New York, 1972. [14] P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. [15] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084. [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. [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. [18] R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. doi: 10.1126/science.177.4052.900. [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. [20] W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics, Princeton University Press, Princeton, 2003. [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. [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. [23] M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220. [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. [25] E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. doi: 10.1016/0270-0255(84)90020-4. [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. [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. [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.

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##### 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. [2] J. M. Cushing, Equilibria in systems of interacting strustured populations, J. Math. Biol., 24 (1987), 627-649. doi: 10.1007/BF00275507. [3] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [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. [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. [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. [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. [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. [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. [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. [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. [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. [13] R. H. MacArthur, Geographical Ecology, Harper and Row, New York, 1972. [14] P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. [15] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084. [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. [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. [18] R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. doi: 10.1126/science.177.4052.900. [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. [20] W. M. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics, Princeton University Press, Princeton, 2003. [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. [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. [23] M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220. [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. [25] E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. doi: 10.1016/0270-0255(84)90020-4. [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. [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. [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.
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