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2012, 9(2): 431-444. doi: 10.3934/mbe.2012.9.431

A mutualism-parasitism system modeling host and parasite with mutualism at low density

Yuanshi Wang 1, and  Donald L. DeAngelis 2,

1. 

School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China
2. 

US Geological Survey/Biological Resources Division and Department of Biology, University of Miami, Coral Gables, FL 33124, United States

Received  June 2011 Revised  October 2011 Published  March 2012

A mutualism-parasitism system of two species is considered, where mutualism is the dominant interaction when the predators (parasites) are at low density while parasitism is dominant when the predators are at high density. Our aim is to show that mutualism at low density promotes coexistence of the species and leads to high production of the prey (host). The mutualism-parasitism system presented here is a combination of the Lotka-Volterra cooperative model and Lotka-Volterra predator-prey model. By comparing dynamics of this system with those of the Lotka-Volterra predator-prey model, we present the mechanisms by which the mutualism improves the coexistence of the species and production of the prey. Then the parameter space is divided into six regions, which correspond to the four outcomes of mutualism, commensalism, predation/parasitism and neutralism, respectively. When the parameters are varied continuously among the six regions, it is shown that the interaction outcomes of the system transition smoothly among the four outcomes. By comparing the dynamics of the specific system with those of the Lotka-Volterra cooperative model, we show that the parasitism at high density promotes stability of the system. A novel aspect of this paper is the simplicity of the model, which allows rigorous and thorough analysis and transparency of the results.
Keywords: population dynamics, coexistence., parasitism, mutualism, Lotka-Volterra.
Mathematics Subject Classification: Primary: 34C37, 92D25; Secondary: 37N2.
Citation: Yuanshi Wang, Donald L. DeAngelis. A mutualism-parasitism system modeling host and parasite with mutualism at low density. Mathematical Biosciences & Engineering, 2012, 9 (2) : 431-444. doi: 10.3934/mbe.2012.9.431
References:
[1]

A. J. Belsky, Does herbivory benefit plants? A review of the evidence, American Naturalist, 127 (1986), 870-892. doi: 10.1086/284531.

[2]

R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete and Continuous Dynamical Systems Ser. B, 4 (2004), 527-546. 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), 10-16. 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 low-density population of cork oak (Quercus suber L.), Forest Ecology and Management, 76 (1995), 197-201. doi: 10.1016/0378-1127(95)03566-S.

[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), 14-18. 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, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecological Letters, 12 (2009), 1357-1366.

[9]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 5 (2010), 1286-1295. doi: 10.1890/09-1163.1.

[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), 25-30. 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 plant-herbivore dynamics, Journal of Dynamics and Differential Equations, 18 (2006), 1021-1024. doi: 10.1007/s10884-006-9029-y.

[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), 92-94. doi: 10.1126/science.191.4222.92.

[14]

S. J. McNaughton, Grazing as an optimization process: Grass-ungulate relationships in the Serengeti, American Naturalist, 113 (1979), 691-703. doi: 10.1086/283426.

[15]

M. Miyaki and K. Kikuzawa, Dispersal of Quercus mongolica acorns in a broad-leaved deciduous forest. 2. Scatterhoarding by mice, Forest Ecology and Management, 25 (1988), 9-16. doi: 10.1016/0378-1127(88)90130-2.

[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), 621-630. doi: 10.2307/1382509.

[17]

J. D. Murray, "Mathematical Biology," Second edition, Biomathematics, 19, Springer-Verlag, Berlin, 1993.

[18]

C. Neuhauser and J. Fargione, A mutualism-parasitism continuum model and its application to plant-mycorrhizae interactions, Ecological Modelling, 177 (2004), 337-352. doi: 10.1016/j.ecolmodel.2004.02.010.

[19]

I. M. Pérez-Ramos and T. Maran, Factors affecting post-dispersal seed predation in two coexisting oak species: Microhabitat, burial and exclusion of large herbivores, Forest Ecology and Management, 255 (2008), 3506-3514. 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/s00285-011-0435-3.

[21]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional consumer-resource theory characterizing transitions of interaction outcomes, Ecological Complexity, 8 (2011), 249-257. doi: 10.1016/j.ecocom.2011.04.002.

[22]

Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a uni-directional consumer-resource system, J. Theoretical Biology, 280 (2011), 43-49. doi: 10.1016/j.jtbi.2011.03.038.

[23]

Y. Wang and H. Wu, A mutualism-competition model characterizing competitors with mutualism at low density, Mathematical and Computer Modeling, 53 (2011), 1654-1663. 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), 1-24.

[25]

V. Volterra, Fluctuation in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. 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), 870-892. doi: 10.1086/284531.

[2]

R. S. Cantrell, C. Cosner and S. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete and Continuous Dynamical Systems Ser. B, 4 (2004), 527-546. 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), 10-16. 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 low-density population of cork oak (Quercus suber L.), Forest Ecology and Management, 76 (1995), 197-201. doi: 10.1016/0378-1127(95)03566-S.

[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), 14-18. 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, Consumer-resource theory predicts dynamic transitions between outcomes of interspecific interactions, Ecological Letters, 12 (2009), 1357-1366.

[9]

J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism, Ecology, 5 (2010), 1286-1295. doi: 10.1890/09-1163.1.

[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), 25-30. 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 plant-herbivore dynamics, Journal of Dynamics and Differential Equations, 18 (2006), 1021-1024. doi: 10.1007/s10884-006-9029-y.

[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), 92-94. doi: 10.1126/science.191.4222.92.

[14]

S. J. McNaughton, Grazing as an optimization process: Grass-ungulate relationships in the Serengeti, American Naturalist, 113 (1979), 691-703. doi: 10.1086/283426.

[15]

M. Miyaki and K. Kikuzawa, Dispersal of Quercus mongolica acorns in a broad-leaved deciduous forest. 2. Scatterhoarding by mice, Forest Ecology and Management, 25 (1988), 9-16. doi: 10.1016/0378-1127(88)90130-2.

[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), 621-630. doi: 10.2307/1382509.

[17]

J. D. Murray, "Mathematical Biology," Second edition, Biomathematics, 19, Springer-Verlag, Berlin, 1993.

[18]

C. Neuhauser and J. Fargione, A mutualism-parasitism continuum model and its application to plant-mycorrhizae interactions, Ecological Modelling, 177 (2004), 337-352. doi: 10.1016/j.ecolmodel.2004.02.010.

[19]

I. M. Pérez-Ramos and T. Maran, Factors affecting post-dispersal seed predation in two coexisting oak species: Microhabitat, burial and exclusion of large herbivores, Forest Ecology and Management, 255 (2008), 3506-3514. 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/s00285-011-0435-3.

[21]

Y. Wang, D. L. DeAngelis and J. N. Holland, Uni-directional consumer-resource theory characterizing transitions of interaction outcomes, Ecological Complexity, 8 (2011), 249-257. doi: 10.1016/j.ecocom.2011.04.002.

[22]

Y. Wang and D. L. DeAngelis, Transitions of interaction outcomes in a uni-directional consumer-resource system, J. Theoretical Biology, 280 (2011), 43-49. doi: 10.1016/j.jtbi.2011.03.038.

[23]

Y. Wang and H. Wu, A mutualism-competition model characterizing competitors with mutualism at low density, Mathematical and Computer Modeling, 53 (2011), 1654-1663. 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), 1-24.

[25]

V. Volterra, Fluctuation in the abundance of a species considered mathematically, Nature, 118 (1926), 558-560. doi: 10.1038/118558a0.

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