Biological control via “ecological” damping: an approach that attenuates non-target effects. (English) Zbl 1364.92061

Summary: In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population, that functions as a top predator, in a three-species food chain. We allow for finite time blow-up in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach “disastrous”, and uncontrollable population levels, in a finite time. We next improve the mathematical model and incorporate controls that are shown to drive down the invasive population growth and, in certain cases, eliminate blow-up. Hence, the population does not reach an uncontrollable level. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as “ecological damping”, as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in [the first author et al., ibid. 254, 83–102 (2014; Zbl 1323.92182)]. Lastly, we confirm the existence of spatiotemporal chaos.


92D40 Ecology
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35B35 Stability in context of PDEs
92D25 Population dynamics (general)


Zbl 1323.92182
Full Text: DOI arXiv


[1] Arim, M.; Abades, S.; Neill, P.; Lima, M.; Marquet, P., Spread dynamics of invasive species, Proceedings of the National Academy of Sciences, vol. 103, 374-378, (2006)
[2] Averill, I.; Lou, Y., On several conjectures from evolution of dispersal, J. Biol. Dyn., 6, 117-130, (2012)
[3] Akakaya, H. R.; Mills, G.; Doncaster, C., The role of metapopulations in conservation, (Macdonald, D. W.; Service, K., (2007), Blackwell Publishing), 64-84
[4] Aziz-Alaoui, M. A., Study of a Leslie-gower type tri-trophic population model, Chaos Solitons Fractals, 14, 1275-1293, (2002) · Zbl 1031.92027
[5] Hastings, A.; Powell, T., Chaos in a three-species food chain, Ecology, 72, 896-903, (1991)
[6] Kuznetsov, Y.; Rinaldi, S., Remarks on food chain dynamics, Math. Biosci., 134, 1-33, (1996) · Zbl 0844.92025
[7] Bampfylde, C. J.; Lewis, M. A., Biological control through intraguild predation: case studies in pest control, invasive species and range expansion, Bull. Math. Biol., 69, 1031-1066, (2007) · Zbl 1298.92108
[8] Beauregard, M. A.; Sheng, Q., Solving degenerate quenching-combustion equations by an adaptive splitting method on evolving grids, Comput. Struct., 122, 33-43, (2013)
[9] Berryman, A., The theory and classification of outbreaks, Insect Outbreaks, 3-30, (1987), Academic Press San Diego, CA
[10] Bryan, M. B.; Zalinski, D.; Filcek, K. B.; Libants, S.; Li, W.; Scribner, K. T., Patterns of invasion and colonization of the sea lamprey in north America as revealed by microsatellite genotypes, Mol. Ecol., 14, 3757-3773, (2005)
[11] Clark, J. S.; Lewis, M.; Horvath, L., Invasion by extremes: population spread with variation in dispersal and reproduction, Am. Nat., 157, (2001)
[12] Cai, D.; McLaughlin, D.; Shatah, J., Spatiotemporal chaos in spatially extended systems, Math. Comput. Simulat., 55, 329-340, (2001) · Zbl 0988.37093
[13] Choh, Y.; Ignacio, M.; Sabelis, M.; Janssen, A., Predator-prey role reversals, juvenile experience and adult antipredator behaviour, Sci. Rep., 2, 1-6, (2012)
[14] Don, W.; Solomonoff, A., Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J. Sci. Comp., 16, 1253-1268, (1995) · Zbl 0840.65010
[15] Dorcas, M. E.; Willson, J. D.; Reed, R. N.; Snow, R. W.; Rochford, M. R.; Miller, M. A.; Mehsaka, W. E.; Andreadis, P. T.; Mazzotti, F. J.; Romagosa, C. M.; Hart, K. M., Severe mammal declines coincide with proliferation of invasive burmese pythons in everglades national park, Proc. Natl. Acad. Sci. USA, 109, 2418-2422, (2012)
[16] Finke, D.; Denno, R., Spatial refuge from intraguild predation: implications for prey suppression and trophic cascades, Oecologia, 149, 265-275, (2006)
[17] Holbrook, J.; Chesnes, T., An effect of burmese pythons (python molurus bivittatus) on mammal populations in southern florida, Florida Sci., 74, 17-24, (2011)
[18] Follet, P.; Duan, J., Nontarget Effects of Biological Control, (2000), Kluwer Academic Publishers Dortrecht/Boston/London
[19] Friedman, A., Partial Differential Equations of Parabolic type, (1964), Prentice Hall Englewood Chiffs, NJ · Zbl 0144.34903
[20] Gakkhar, S.; Singh, B., Complex dynamic behavior in a food web consisting of two preys and a predator, Chaos Solitons Fractals, 24, 789-801, (2005) · Zbl 1081.37060
[21] Gomez-Lopez, J., Global existence versus blow-up in superlinear indefinite parabolic problems, Sci. Math. Jpn. Online, 61, 493-516, (2005) · Zbl 1082.35086
[22] Gomez-Lopez, J.; Molina-Meyer, M., In the blink of an eye, Prog. Nonlinear Differ. Equ. Appl., 64, 291-327, (2005) · Zbl 1284.92088
[23] Gomez-Lopez, J.; Quittner, P., Complete and energy blow-up in indefinite superlinear parabolic problems, Discrete Cont. Dyn. Syst. A, 14, 169-186, (2006) · Zbl 1114.35093
[24] Grinn, L.; Hermann, P.; Korotayev, A.; Tausch, A., History and Mathematics: Processes and Models of Global Dynamics, (2010), Volgograd ’Uchitel’ Publishing House
[25] White, K. A.J.; Gilligan, C. A., Spatial heterogeneity in three species, plant-parasite-hyperparasite, systems, Phil. Trans. R. Soc. Lond. Ser. B: Biol. Sci., 353, 543-557, (1998)
[26] Henry, D., Geometric Theory of Semi-linear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, (1984), Springer-Verlag New-York
[27] Hesthaven, J.; Gottlieb, S. D.; Gottlieb, D., Spectral Methods for Time-dependent Problems, (2007), Cambridge University Press New York · Zbl 1111.65093
[28] Hillen, T.; Painter, K., A users guide to PDE models for chemotaxis, J. Math. Biol., 57, 183-217, (2009) · Zbl 1161.92003
[29] Chen, L.; Jungel, A., Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differ. Equ., 224, 39-59, (2006) · Zbl 1096.35060
[30] Chen, L.; Jungel, A., Analysis of a multi-dimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36, 301-322, (2004) · Zbl 1082.35075
[31] Kar, T. K., Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simulat., 10, 681-691, (2004) · Zbl 1064.92045
[32] Letnic, M.; Webb, J.; Shine, R., Invasive cane toads (bufo marinus) cause mass mortality of freshwater crocodiles (crocodylus johnstoni) in tropical Australia, Biol. Conserv., 141, 1773-1782, (2008)
[33] J. Hood, Asian carp: State’s fish kill in chicago sanitary and ship canal yield only 1 Asian carp, www.articles.chicagotribune.com, 2009.
[34] Medvinsky, A.; Petrovskii, S.; Tikhonova, I.; Malchow, H.; Li, B., Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44, 311-370, (2002) · Zbl 1001.92050
[35] Kumari, N., Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator, ISRN Biomath., 2013, (2013) · Zbl 1269.92069
[36] Letellier, C.; Aziz-Alaoui, M. A., Analysis of the dynamics of a realistic ecological model, Chaos Solitons Fractals, 13, 95-107, (2002) · Zbl 0977.92029
[37] Loda, S. M.; Pemberton, R. W.; Johnson, M. T.; Follet, P. A., Nontarget effects-the achilles heel of biological control? retrospective analyses to reduce risk associated with biocontrol introductions, Ann. Rev. Entomol., 48, 365-396, (2003)
[38] Lou, Y.; Munther, D., Dynamics of a three species competition model, Discrete Cont. Dyn. Syst. A, 32, 3099-3131, (2012) · Zbl 1255.35046
[39] Ludwig, D.; Jones, D.; Holling, C., Qualitative analysis of insect outbreak systems: the spruce budworm and forest, J. Anim. Ecol., 47, 315-332, (1978)
[40] Myers, J. H.; Simberloff, D.; Kuris, A. M.; Carey, J. R., Eradication revisited: dealing with exotic species, Trends Ecol. Evol., 15, 316-320, (2000)
[41] New York Sea Grant. Policy Issues., Zebra mussel clearinghouse information review, 5, 6-7, (1994)
[42] Okubo, A.; Maini, P. K.; Williamson, M. H.; Murray, J. D., On the spatial spread of the grey squirrel in britain, Proc. R. Soc. Lond. Ser. B, 238, 113-125, (1989)
[43] Parshad, R. D.; Upadhyay, R. K., Investigation of long time dynamics of a diffusive three species aquatic model, Dyn. Partial Differ. Equ., 7, 217-244, (2010) · Zbl 1227.35082
[44] Parshad, R. D.; Abderrahmanne, H.; Upadhyay, R. K.; Kumari, N., Finite time blow-up in a realistic food chain model, ISRN Biomath., 2013, (2013)
[45] Parshad, R. D.; Kumari, N.; Kasimov, A. R.; Abderrahmane, H. A., Turing patterns and long time behavior in a three-species model, Math. Biosci., 254, 83-102, (2014) · Zbl 1323.92182
[46] Parshad, R. D.; Kumari, N.; Kouachi, S., A remark on “study of a Leslie-gower-type tritrophic population model [chaos, solitons and fractals 14 (2002) 1275-1293]”, Chaos Solitons Fractals, 71, 22-28, (2015) · Zbl 1352.92130
[47] Pazy, A. S., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, (1983), Springer-Verlag New York · Zbl 0516.47023
[48] Philips, B.; Shine, R., Adapting to an invasive species: toxic cane toads induce morphological change in Australian snakes, Proc. Natl. Acad. Sci. USA, 101, 17150-17155, (2004)
[49] Pimentel, D.; Zuniga, R.; Morrison, D., Update on the environmental and economic costs associated with alien-invasive species in the united states, Ecol. Econ., 52, 273-288, (2005)
[50] Quittner, P.; Souplet, P., Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, (2007), Birkhauser Verlag Basel · Zbl 1128.35003
[51] Ackermann, N.; Bartsch, T.; Kaplicky, P.; Quittner, P., A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems, Trans. Am. Math. Soc., 360, 3493-3539, (2008) · Zbl 1143.37049
[52] Rothe, F., Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, vol. 1072, (1984), Springer-Verlag Berlin · Zbl 0546.35003
[53] Shen, J., Efficient spectral-Galerkin method II. direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comp., 16, 74-87, (1995) · Zbl 0840.65113
[54] Shigesada, N.; Kawasaki, K., Biological Invasions: Theory and Practice, (1997), Oxford University Press Oxford
[55] D. Simberloff, Introduced species: The threat to biodiversity and what can be done, http://www.actionbioscience.org/biodiversity/simberloff.html, 2000.
[56] Smith, J. G.; Phillips, B. L., Toxic Tucker: the potential impact of cane toads on Australian reptiles, Pacific Conserv. Biol., 12, 40-49, (2006)
[57] Sun, G.; Zhang, G.; Jin, Z.; Li, L., Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dyn., 58, 75-84, (2009) · Zbl 1183.92084
[58] Smoller, J., Shock Waves and Reaction-Diffusion Equations, (1983), Springer-Verlag New York · Zbl 0508.35002
[59] Straughan, B., Explosive Instabilities in Mechanics, (1998), Springer-Verlag Heidelberg · Zbl 0911.35002
[60] Upadhyay, R. K.; Iyengar, S. R.K.; Rai, V., Chaos: an ecological reality?, Int. J. Bifurc. Chaos, 8, 1325-1333, (1998) · Zbl 0935.92037
[61] Upadhyay, R. K.; Rai, V., Why chaos is rarely observed in natural populations?, Chaos Solitons Fractals, 8, 1933-1939, (1997)
[62] Upadhyay, R. K.; Iyengar, S. R.K.; Rai, V., Stability and complexity in ecological systems, Chaos Solitons Fractals, 11, 533-542, (2000) · Zbl 0943.92033
[63] Upadhyay, R. K.; Naji, R. K.; Kumari, N., Dynamical complexity in some ecological models: effects of toxin production by phytoplanktons, Nonlinear Anal.: Model. Control, 12, 123-138, (2007)
[64] Upadhyay, R. K.; Iyengar, S. R.K., Effect of seasonality on the dynamics of 2 and 3 species prey-predator systems, Nonlinear Anal.: Real World Appl., 6, 509-530, (2005) · Zbl 1072.92058
[65] Van Driesche, R.; Bellows, T., Biological Control, (1996), Kluwer Academic Publishers Boston, MA
[66] Van Voorn, G. A.K.; Koi, B.; Boer, M., Ecological consequences of global bifurcations in some food chain models, Math. Biosci., 226, 120-133, (2010) · Zbl 1194.92078
[67] Wittmeier, B., Mathematical biologist applies hard data to soft science, Edmonton J., (2012)
[68] Emsens, W.; Hirch, B.; Kays, R.; Jansen, P., Prey refuges as predator hotspots: ocelot (leopardus pardalis) attraction to agouti (dasyprocta punctata) dens, Acta Theriol., 59, 257-262, (2014)
[69] Ivan, J. S.; Murphy, R. K., What preys on piping plover eggs and chicks?, Wildlife Soc. Bull., 33, 113-119, (2005)
[70] Smith, R.; Pullin, A.; Stewart, G.; Sutherland, W., Is nest predator exclusion an effective strategy for enhancing bird populations?, Biol. Conserv., 144, 1-10, (2010)
[71] Hedges, K.; Abrahams, M., Hypoxic refuges, predatorprey interactions and habitat selection by fishes, J. Fish Biol., 86, 288-303, (2015)
[72] Chapmana, L.; Chapmana, C.; Nordliea, F.; Rosenberger, A., Physiological refugia: swamps, hypoxia tolerance and maintenance of fish diversity in the lake Victoria region, Comp. Biochem. Physiol. A, 133, 421-437, (2002)
[73] Rodriguez, D., Foraging ecology of the puerto rican boa (epicrates inornatus): bat predation, carrion feeding, and piracy, J. Herpetol., 30, 533-536, (1996)
[74] Hobbs, R.; Hallett, L.; Ehrlich, P.; Mooney, H., Intervention ecology: applying ecological science in the twenty-first century, BioScience, 61, 1-9, (2011)
[75] Rooker, J.; Dokken, Q.; Pattengil, C.; Holt, G., Fish assemblages on artificial and natural reefs in the flower garden banks national marine sanctuary, USA, Coral Reefs, 16, 83-92, (1997)
[76] Isaksson, D.; Wallander, J.; Larsson, M., Managing predation on ground-nesting birds: the effectiveness of nest exclosures, Biol. Conserv., 136, 136-142, (2007)
[77] Chapman, M.; Kramer, D., Gradients in coral reef fish density and size across the barbados marine reserve boundary: effect of reserve protection and habitat characteristics, Marine Ecol. Prog. Ser., 181, 81-96, (1999)
[78] Robinson, O.; Fefferman, N.; Lockwood, J., How to effectively manage invasive predators to protect their native prey, Biol. Conserv., 165, 146-153, (2013)
[79] Goldberg, J.; Hebblewhite, M.; Bardsley, J., Consequences of a refuge for the predator-prey dynamics of a wolf-elk system in Banff national park, Alberta, Canada, PLOS One, 9, 1-10, (2014)
[80] Xie, Z., Cross diffusion induced Turing instability for a three species food chain model, J. Math. Anal. Appl., 388, 539-547, (2012) · Zbl 1243.35093
[81] Ylonen, H.; Sundell, J.; Tiilikainen, R.; Eccard, J.; Horne, T., Weasels (mustela nivalis nivalis) preference for olfactory cues of the vole (clethrionomys glareolus), Ecology, 84, 1447-1452, (2003)
[82] Zheng, S., Nonlinear Evolution Equations, Monographs and Surveys in Pure and Applied Mathematics, vol. 133, (2004), Chapman & Hall/CRC Boca Raton
[83] Leslie, P., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 213-245, (1948) · Zbl 0034.23303
[84] Banerjee, M.; Banerjee, S., Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-tanner model, Math. Biosci., 236, 64-76, (2012) · Zbl 1375.92077
[85] Pal, P. J.; Mandal, P. K., Bifurcation analysis of a modified Leslie-gower predator-prey model with beddington-deangelis functional response and strong allee effect, Math. Comput. Simulat., 97, 123-146, (2014)
[86] Aziz-Alaoui, M.; Daher-Okiye, M., Boundedness and global stability for a predator-prey model with modified lesliegower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075, (2003) · Zbl 1063.34044
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