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
References:
[1]

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

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

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