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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 |
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. |
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. |
[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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