2013, 10(5&6): 1519-1538. doi: 10.3934/mbe.2013.10.1519

Spatially heterogeneous invasion of toxic plant mediated by herbivory

1. 

Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, China

2. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

3. 

Department of Biology, University of Miami, Coral Gables, Florida 33124, United States

Received  August 2012 Revised  February 2013 Published  August 2013

Spatially homogeneous (ODE) and reaction-diffusion models for plant-herbivore interactions with toxin-determined functional response are analyzed. The models include two plant species that have different levels of toxicity. The plant species with a higher level of toxicity is assumed to be less preferred by the herbivore and to have a relatively lower intrinsic growth rate than the less toxic plant species. Two of the equilibrium points of the system representing significant ecological interests are $E_1$, in which only the less toxic plant is present, and $E_2$, in which the more toxic plant and herbivore coexist while the less toxic plant has gone to extinction. Under certain conditions it is shown that, for the spatially homogeneous system all solutions will converge to the equilibrium $E_2$, whereas for the reaction-diffusion model there exist traveling wave solutions connecting $E_1$ and $E_2$.
Citation: Zhilan Feng, Wenzhang Huang, Donald L. DeAngelis. Spatially heterogeneous invasion of toxic plant mediated by herbivory. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1519-1538. doi: 10.3934/mbe.2013.10.1519
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L. Butler, K. Kielland, S. Rupp and T. Hanley, Interactive controls of her- bivory and fluvial dynamics on landscape vegetation patterns on the Tanana River floodplain, interior Alaska,, Journal of Biogeography, 34 (2007), 1622.   Google Scholar

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Z. Feng, R. Liu and D. L. DeAngelis, Plant-herbivore interactions mediated by plant toxicity,, Theor. Pop. Biol., 73 (2008), 449.  doi: 10.1016/j.tpb.2007.12.004.  Google Scholar

[10]

Z. Feng, R. Liu, D. L. DeAngelis, J. P. Bryant, K. Kielland, F. S. Chapin III and R. K. Swihart, Plant toxicity, adaptive herbivory, and plant community dynamics,, Ecosystems, 12 (2010), 534.  doi: 10.1007/s10021-009-9240-x.  Google Scholar

[11]

Z. Feng, Z. Qiu, R. Liu and D. L. DeAngelis, Dynamics of a plant-herbivore-predator system with plant-toxicity,, Mathematical Biosciences, 299 (2011), 190.  doi: 10.1016/j.mbs.2010.12.005.  Google Scholar

[12]

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K. Kielland and J. P. Bryant, Moose herbivory in taiga: Effects on biogeochemistry and vegetation dynamics in primary succession,, Oikos, 82 (1998), 377.   Google Scholar

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K. Kielland, J. P. Bryant and R. W. Ruess, Mammalian herbivory, ecosystem engineering and ecological cascades in Alaskan boreal forests,, in, (2006), 211.   Google Scholar

[20]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[21]

W. Li and S. Wu, Traveling waves in a diffusive predator-prey model with holling type - III functional response,, Chaos, 37 (2008), 476.  doi: 10.1016/j.chaos.2006.09.039.  Google Scholar

[22]

Y. Li, Z. Feng, R. Swihart, J. Byant and H. Huntley, Modeling plant toxicity on plant-herbivore dynamics,, J. Dynam. Differential Equations, 18 (2006), 1021.  doi: 10.1007/s10884-006-9029-y.  Google Scholar

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X. Lin, C. Wu and P. Weng, Traveling wave solutions for a predator-prey system with Sigmoidal response function,, J. Dynam. Diff. Equations, 23 (2011), 903.  doi: 10.1007/s10884-011-9220-7.  Google Scholar

[24]

R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response,, J. Diff. Equations, 245 (2008), 442.  doi: 10.1016/j.jde.2007.10.034.  Google Scholar

[25]

R. M. May, Unanswered questions in ecology,, Phil. Trans. Royal. Society. London B, 354 (1999), 1951.  doi: 10.1098/rstb.1999.0534.  Google Scholar

[26]

I. S. Myers-Smith, B. C. Forbes, M. Wilmking, M. Hanninger, T. Lantz, D. Blok, K. D. Tape, M. Macias-Fauria, U. Sass-Klaassen, E. Lvesque, S. Boudreau, P. Ropars, L. Hermanutz, A. Trant, L. Siegwart Collier, S. Weijers, J. Rozema, S. A. Rayback, N. M. Schmidt, G. Schaepman-Strub, S. Wipf, C. Rixen, C. B. Mnard, S. Venn, S. Goetz, L. Andreu-Hayles, S. Elmendorf, V. Ravolainen, J. Welker, P. Grogan, H. E. Epstein and D. S. Hik, Shrub expansion in tundra ecosystems: Dynamics, impacts and research priorities,, Environmental Research Letters, 6 (2011).  doi: 10.1088/1748-9326/6/4/045509.  Google Scholar

[27]

R. P. Nielson, Transient ecotone response to climatic change: Some conceptual and modeling approaches,, Ecological Applications, 3 (1993), 385.   Google Scholar

[28]

J. Olofsson, L. Oksanen, T. Callaghan, P. E. Hulme, T. Oksanen and P. Suominen, Herbivores inhibit climate-driven shrub expansion on the tundra,, Global Change Biology, 15 (2009), 2681.  doi: 10.1111/j.1365-2486.2009.01935.x.  Google Scholar

[29]

J. Pastor and R. J. Naiman, Selective foraging and ecosystem processes in boreal forests,, Am. Nat., 139 (1992), 690.  doi: 10.1086/285353.  Google Scholar

[30]

E. Post and C. Pedersen, Opposing plant community responses to warming with and without herbivores,, Proceedings of the National Academy of Sciences, 105 (2008), 12353.  doi: 10.1073/pnas.0802421105.  Google Scholar

[31]

E. Ranta and V. Kaitala, Traveling waves in vole population dynamics,, Nature, 390 (1997).   Google Scholar

[32]

J. D. Speed, M. G. Austrheim, A. J. Hester and A. Mysterud, Elevational advance of alpine plant communities is buffered by herbivory,, Journal of Vegetation Science, 23 (2012), 617.  doi: 10.1111/j.1654-1103.2012.01391.x.  Google Scholar

[33]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[34]

G.-R. Walther, E. Post, P. Convey, A. Menzel, C. Parmesan, T. J. C. Beebee, J.-M. Fromentin, O. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change,, Nature, 416 (2002), 389.  doi: 10.1038/416389a.  Google Scholar

[35]

Q. Yu, H. E. Epstein, D. A. Walker, G. V. Frost and B. C. Forbes, Modeling dynamics of tundra plant communities on the Yamal Peninsula, Russia, in response to climate change and grazing pressure,, Environmental Research Letters, 6 (2011).  doi: 10.1088/1748-9326/6/4/045505.  Google Scholar

show all references

References:
[1]

J. P. Bryant and F. S. Chapin III, Browsing-woody plant Interactions during boreal forest plant succession,, in, (1986), 213.   Google Scholar

[2]

L. Butler, K. Kielland, S. Rupp and T. Hanley, Interactive controls of her- bivory and fluvial dynamics on landscape vegetation patterns on the Tanana River floodplain, interior Alaska,, Journal of Biogeography, 34 (2007), 1622.   Google Scholar

[3]

C. Castillo-Chavex, Z. Feng and W. Huang, Global dynamics of a plant-herbivore model with toxin-determined functional response,, SIAM J. Appl. Math., 74 (2012), 1002.  doi: 10.1137/110851614.  Google Scholar

[4]

F. S. Chapin III, M. W. Oswood, L. W. Viereck and D. L. Verbyla, Successional processes in the Alaskan boreal forest,, in, (2006).   Google Scholar

[5]

K. Clay and J. Hola, Fungal endophyte symbiosis and plant diversity in successional fields,, Science, 285 (1999), 1742.   Google Scholar

[6]

K. Clay, J. Holah and J. A. Rudgers, Herbivores cause a rapid increase in hereditary symbiosis and alter plant community composition,, Proc. Nat. Acad. Sci., 102 (2005), 12465.  doi: 10.1073/pnas.0503059102.  Google Scholar

[7]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations,, J. Math. Biology, 17 (1983), 11.  doi: 10.1007/BF00276112.  Google Scholar

[8]

S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $\mathbbR^4$,, Trans. Amer. Math. Society, 286 (1984), 557.  doi: 10.2307/1999810.  Google Scholar

[9]

Z. Feng, R. Liu and D. L. DeAngelis, Plant-herbivore interactions mediated by plant toxicity,, Theor. Pop. Biol., 73 (2008), 449.  doi: 10.1016/j.tpb.2007.12.004.  Google Scholar

[10]

Z. Feng, R. Liu, D. L. DeAngelis, J. P. Bryant, K. Kielland, F. S. Chapin III and R. K. Swihart, Plant toxicity, adaptive herbivory, and plant community dynamics,, Ecosystems, 12 (2010), 534.  doi: 10.1007/s10021-009-9240-x.  Google Scholar

[11]

Z. Feng, Z. Qiu, R. Liu and D. L. DeAngelis, Dynamics of a plant-herbivore-predator system with plant-toxicity,, Mathematical Biosciences, 299 (2011), 190.  doi: 10.1016/j.mbs.2010.12.005.  Google Scholar

[12]

Z. Feng, J. A. Alfaro-Murillo, D. L. DeAngelis, J. Schmidt, M. Barga, Y. Zheng, M. H. B. Ahmad Tamrin, M. Olson, T. Glaser, K. Kielland, F. S. Chapin III and J. P. Bryant, Plant toxins and trophic cascades alter fire regime and succession on a boreal forest landscape,, Ecological Modeling, 244 (2012), 79.  doi: 10.1016/j.ecolmodel.2012.06.022.  Google Scholar

[13]

P. C. Fife, "Mathematical Aspects of Reacting and Diffusing Systems,", Lecture Notes in Biomath., 28 (1979).   Google Scholar

[14]

N. Fisichelli, L. E. Frelich and P. B. Reich, Sapling growth responses to warmer temperatures cooled by browse pressure,, Global Change Biology, 18 (2012), 3455.  doi: 10.1111/j.1365-2486.2012.02785.x.  Google Scholar

[15]

R. A. Gardner, Review on traveling wave solutions of parabolic systems by A.I. Volpert, V.A. Volpert,, Bull. Amer. Math. Soc., 32 (1995), 446.  doi: 10.1090/S0273-0979-1995-00607-5.  Google Scholar

[16]

W. Huang, Traveling wave solutions for a class of predator-prey systems,, J. Dyn. Diff. Equations, 22 (2012), 633.  doi: 10.1007/s10884-012-9255-4.  Google Scholar

[17]

E. Kaarlejärvi, E., R. Baxter, A. Hofgaard, H. Hytteborn, O. Khitun, U. Molau, S. Sjgersten, P. Wookey and J. Olofsson, Effects of warming on shrub abundance and chemistry drive ecosystem-level changes in a forest-tunda ecotone,, Ecosystems, 15 (2012), 1219.  doi: 10.1007/s10021-012-9580-9.  Google Scholar

[18]

K. Kielland and J. P. Bryant, Moose herbivory in taiga: Effects on biogeochemistry and vegetation dynamics in primary succession,, Oikos, 82 (1998), 377.   Google Scholar

[19]

K. Kielland, J. P. Bryant and R. W. Ruess, Mammalian herbivory, ecosystem engineering and ecological cascades in Alaskan boreal forests,, in, (2006), 211.   Google Scholar

[20]

B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Math. Biosci., 196 (2005), 82.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[21]

W. Li and S. Wu, Traveling waves in a diffusive predator-prey model with holling type - III functional response,, Chaos, 37 (2008), 476.  doi: 10.1016/j.chaos.2006.09.039.  Google Scholar

[22]

Y. Li, Z. Feng, R. Swihart, J. Byant and H. Huntley, Modeling plant toxicity on plant-herbivore dynamics,, J. Dynam. Differential Equations, 18 (2006), 1021.  doi: 10.1007/s10884-006-9029-y.  Google Scholar

[23]

X. Lin, C. Wu and P. Weng, Traveling wave solutions for a predator-prey system with Sigmoidal response function,, J. Dynam. Diff. Equations, 23 (2011), 903.  doi: 10.1007/s10884-011-9220-7.  Google Scholar

[24]

R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response,, J. Diff. Equations, 245 (2008), 442.  doi: 10.1016/j.jde.2007.10.034.  Google Scholar

[25]

R. M. May, Unanswered questions in ecology,, Phil. Trans. Royal. Society. London B, 354 (1999), 1951.  doi: 10.1098/rstb.1999.0534.  Google Scholar

[26]

I. S. Myers-Smith, B. C. Forbes, M. Wilmking, M. Hanninger, T. Lantz, D. Blok, K. D. Tape, M. Macias-Fauria, U. Sass-Klaassen, E. Lvesque, S. Boudreau, P. Ropars, L. Hermanutz, A. Trant, L. Siegwart Collier, S. Weijers, J. Rozema, S. A. Rayback, N. M. Schmidt, G. Schaepman-Strub, S. Wipf, C. Rixen, C. B. Mnard, S. Venn, S. Goetz, L. Andreu-Hayles, S. Elmendorf, V. Ravolainen, J. Welker, P. Grogan, H. E. Epstein and D. S. Hik, Shrub expansion in tundra ecosystems: Dynamics, impacts and research priorities,, Environmental Research Letters, 6 (2011).  doi: 10.1088/1748-9326/6/4/045509.  Google Scholar

[27]

R. P. Nielson, Transient ecotone response to climatic change: Some conceptual and modeling approaches,, Ecological Applications, 3 (1993), 385.   Google Scholar

[28]

J. Olofsson, L. Oksanen, T. Callaghan, P. E. Hulme, T. Oksanen and P. Suominen, Herbivores inhibit climate-driven shrub expansion on the tundra,, Global Change Biology, 15 (2009), 2681.  doi: 10.1111/j.1365-2486.2009.01935.x.  Google Scholar

[29]

J. Pastor and R. J. Naiman, Selective foraging and ecosystem processes in boreal forests,, Am. Nat., 139 (1992), 690.  doi: 10.1086/285353.  Google Scholar

[30]

E. Post and C. Pedersen, Opposing plant community responses to warming with and without herbivores,, Proceedings of the National Academy of Sciences, 105 (2008), 12353.  doi: 10.1073/pnas.0802421105.  Google Scholar

[31]

E. Ranta and V. Kaitala, Traveling waves in vole population dynamics,, Nature, 390 (1997).   Google Scholar

[32]

J. D. Speed, M. G. Austrheim, A. J. Hester and A. Mysterud, Elevational advance of alpine plant communities is buffered by herbivory,, Journal of Vegetation Science, 23 (2012), 617.  doi: 10.1111/j.1654-1103.2012.01391.x.  Google Scholar

[33]

H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755.  doi: 10.1007/BF00173267.  Google Scholar

[34]

G.-R. Walther, E. Post, P. Convey, A. Menzel, C. Parmesan, T. J. C. Beebee, J.-M. Fromentin, O. Hoegh-Guldberg and F. Bairlein, Ecological responses to recent climate change,, Nature, 416 (2002), 389.  doi: 10.1038/416389a.  Google Scholar

[35]

Q. Yu, H. E. Epstein, D. A. Walker, G. V. Frost and B. C. Forbes, Modeling dynamics of tundra plant communities on the Yamal Peninsula, Russia, in response to climate change and grazing pressure,, Environmental Research Letters, 6 (2011).  doi: 10.1088/1748-9326/6/4/045505.  Google Scholar

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