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A mutualismparasitism system modeling host and parasite with mutualism at low density
1.  School of Mathematics and Computational Science, Sun Yatsen University, Guangzhou 510275, China 
2.  US Geological Survey/Biological Resources Division and Department of Biology, University of Miami, Coral Gables, FL 33124, United States 
References:
[1] 
A. J. Belsky, Does herbivory benefit plants? A review of the evidence, American Naturalist, 127 (1986), 870892. doi: 10.1086/284531. 
[2] 
R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumerresource dynamics, Discrete and Continuous Dynamical Systems Ser. B, 4 (2004), 527546. doi: 10.3934/dcdsb.2004.4.527. 
[3] 
M. I. Dyer, M. L. Turner and T. R. Seastedt, Herbivory and its consequences, Ecological Applications, 3 (1993), 1016. doi: 10.2307/1941781. 
[4] 
A. J. Lotka, "Elements of Physiological Biology," Dover Publications, New York, 1956. 
[5] 
J. Herrera, Acorn predation and seedling production in a lowdensity population of cork oak (Quercus suber L.), Forest Ecology and Management, 76 (1995), 197201. doi: 10.1016/03781127(95)03566S. 
[6] 
D. W. Hilbert, D. M. Smith, J. K. Detling and M. I. Dyer, Relative growth rates and the grazing optimization hypothesis, Oecologia, 51 (1981), 1418. doi: 10.1007/BF00344645. 
[7] 
J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998. 
[8] 
J. N. Holland and D. L. DeAngelis, Consumerresource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecological Letters, 12 (2009), 13571366. 
[9] 
J. N. Holland and D. L. DeAngelis, A consumerresource approach to the densitydependent population dynamics of mutualism, Ecology, 5 (2010), 12861295. doi: 10.1890/091163.1. 
[10] 
K. Kielland, J. P. Bryant and R. W. Ruess, Moose herbivory and carbon turnover of early successional stands in interior Alaska, Oikos, 80 (1997), 2530. doi: 10.2307/3546512. 
[11] 
Y. Li, Z. L. Feng, R. Swihart, J. P. Bryant and N. Huntly, Modeling the impact of plant toxicity on plantherbivore dynamics, Journal of Dynamics and Differential Equations, 18 (2006), 10211024. doi: 10.1007/s108840069029y. 
[12] 
R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. 
[13] 
S. J. McNaughton, Serengeti migratory wildebeest: Facilitation of energy flow by grazing, Science, 191 (1976), 9294. doi: 10.1126/science.191.4222.92. 
[14] 
S. J. McNaughton, Grazing as an optimization process: Grassungulate relationships in the Serengeti, American Naturalist, 113 (1979), 691703. doi: 10.1086/283426. 
[15] 
M. Miyaki and K. Kikuzawa, Dispersal of Quercus mongolica acorns in a broadleaved deciduous forest. 2. Scatterhoarding by mice, Forest Ecology and Management, 25 (1988), 916. doi: 10.1016/03781127(88)901302. 
[16] 
E. M. Molvar and R. T. Bowyer, Costs and benefits of group living in a recently social ungulate: The Alaskan moose, Journal of Mammalogy, 75 (1994), 621630. doi: 10.2307/1382509. 
[17] 
J. D. Murray, "Mathematical Biology," Second edition, Biomathematics, 19, SpringerVerlag, Berlin, 1993. 
[18] 
C. Neuhauser and J. Fargione, A mutualismparasitism continuum model and its application to plantmycorrhizae interactions, Ecological Modelling, 177 (2004), 337352. doi: 10.1016/j.ecolmodel.2004.02.010. 
[19] 
I. M. PérezRamos and T. Maran, Factors affecting postdispersal seed predation in two coexisting oak species: Microhabitat, burial and exclusion of large herbivores, Forest Ecology and Management, 255 (2008), 35063514. doi: 10.1016/j.foreco.2008.02.032. 
[20] 
Y. Wang, H. Wu and S. Ruan, Periodic orbits near heteroclinic cycles in a cyclic replicator system, J. Math. Biol., in press. doi: 10.1007/s0028501104353. 
[21] 
Y. Wang, D. L. DeAngelis and J. N. Holland, Unidirectional consumerresource theory characterizing transitions of interaction outcomes, Ecological Complexity, 8 (2011), 249257. doi: 10.1016/j.ecocom.2011.04.002. 
[22] 
Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a unidirectional consumerresource system, J. Theoretical Biology, 280 (2011), 4349. doi: 10.1016/j.jtbi.2011.03.038. 
[23] 
Y. Wang and H. Wu, A mutualismcompetition model characterizing competitors with mutualism at low density, Mathematical and Computer Modeling, 53 (2011), 16541663. doi: 10.1016/j.mcm.2010.12.033. 
[24] 
K. M. Stewart, R. T. Bowyer, R. W. Ruess, B. L. Dick and J. G. Kie, Herbivore optimization by North American elk: Consequences for theory and management, Wildlife Monographs, 167 (2006), 124. 
[25] 
V. Volterra, Fluctuation in the abundance of a species considered mathematically, Nature, 118 (1926), 558560. doi: 10.1038/118558a0. 
show all references
References:
[1] 
A. J. Belsky, Does herbivory benefit plants? A review of the evidence, American Naturalist, 127 (1986), 870892. doi: 10.1086/284531. 
[2] 
R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumerresource dynamics, Discrete and Continuous Dynamical Systems Ser. B, 4 (2004), 527546. doi: 10.3934/dcdsb.2004.4.527. 
[3] 
M. I. Dyer, M. L. Turner and T. R. Seastedt, Herbivory and its consequences, Ecological Applications, 3 (1993), 1016. doi: 10.2307/1941781. 
[4] 
A. J. Lotka, "Elements of Physiological Biology," Dover Publications, New York, 1956. 
[5] 
J. Herrera, Acorn predation and seedling production in a lowdensity population of cork oak (Quercus suber L.), Forest Ecology and Management, 76 (1995), 197201. doi: 10.1016/03781127(95)03566S. 
[6] 
D. W. Hilbert, D. M. Smith, J. K. Detling and M. I. Dyer, Relative growth rates and the grazing optimization hypothesis, Oecologia, 51 (1981), 1418. doi: 10.1007/BF00344645. 
[7] 
J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, 1998. 
[8] 
J. N. Holland and D. L. DeAngelis, Consumerresource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecological Letters, 12 (2009), 13571366. 
[9] 
J. N. Holland and D. L. DeAngelis, A consumerresource approach to the densitydependent population dynamics of mutualism, Ecology, 5 (2010), 12861295. doi: 10.1890/091163.1. 
[10] 
K. Kielland, J. P. Bryant and R. W. Ruess, Moose herbivory and carbon turnover of early successional stands in interior Alaska, Oikos, 80 (1997), 2530. doi: 10.2307/3546512. 
[11] 
Y. Li, Z. L. Feng, R. Swihart, J. P. Bryant and N. Huntly, Modeling the impact of plant toxicity on plantherbivore dynamics, Journal of Dynamics and Differential Equations, 18 (2006), 10211024. doi: 10.1007/s108840069029y. 
[12] 
R. M. May, "Stability and Complexity in Model Ecosystems," Princeton University Press, Princeton, 1974. 
[13] 
S. J. McNaughton, Serengeti migratory wildebeest: Facilitation of energy flow by grazing, Science, 191 (1976), 9294. doi: 10.1126/science.191.4222.92. 
[14] 
S. J. McNaughton, Grazing as an optimization process: Grassungulate relationships in the Serengeti, American Naturalist, 113 (1979), 691703. doi: 10.1086/283426. 
[15] 
M. Miyaki and K. Kikuzawa, Dispersal of Quercus mongolica acorns in a broadleaved deciduous forest. 2. Scatterhoarding by mice, Forest Ecology and Management, 25 (1988), 916. doi: 10.1016/03781127(88)901302. 
[16] 
E. M. Molvar and R. T. Bowyer, Costs and benefits of group living in a recently social ungulate: The Alaskan moose, Journal of Mammalogy, 75 (1994), 621630. doi: 10.2307/1382509. 
[17] 
J. D. Murray, "Mathematical Biology," Second edition, Biomathematics, 19, SpringerVerlag, Berlin, 1993. 
[18] 
C. Neuhauser and J. Fargione, A mutualismparasitism continuum model and its application to plantmycorrhizae interactions, Ecological Modelling, 177 (2004), 337352. doi: 10.1016/j.ecolmodel.2004.02.010. 
[19] 
I. M. PérezRamos and T. Maran, Factors affecting postdispersal seed predation in two coexisting oak species: Microhabitat, burial and exclusion of large herbivores, Forest Ecology and Management, 255 (2008), 35063514. doi: 10.1016/j.foreco.2008.02.032. 
[20] 
Y. Wang, H. Wu and S. Ruan, Periodic orbits near heteroclinic cycles in a cyclic replicator system, J. Math. Biol., in press. doi: 10.1007/s0028501104353. 
[21] 
Y. Wang, D. L. DeAngelis and J. N. Holland, Unidirectional consumerresource theory characterizing transitions of interaction outcomes, Ecological Complexity, 8 (2011), 249257. doi: 10.1016/j.ecocom.2011.04.002. 
[22] 
Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a unidirectional consumerresource system, J. Theoretical Biology, 280 (2011), 4349. doi: 10.1016/j.jtbi.2011.03.038. 
[23] 
Y. Wang and H. Wu, A mutualismcompetition model characterizing competitors with mutualism at low density, Mathematical and Computer Modeling, 53 (2011), 16541663. doi: 10.1016/j.mcm.2010.12.033. 
[24] 
K. M. Stewart, R. T. Bowyer, R. W. Ruess, B. L. Dick and J. G. Kie, Herbivore optimization by North American elk: Consequences for theory and management, Wildlife Monographs, 167 (2006), 124. 
[25] 
V. Volterra, Fluctuation in the abundance of a species considered mathematically, Nature, 118 (1926), 558560. doi: 10.1038/118558a0. 
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