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March  2016, 21(2): 373-398. doi: 10.3934/dcdsb.2016.21.373

Microbial disease in coral reefs: An ecosystem in transition

Joydeb Bhattacharyya 1, and  Samares Pal 1,

1. 

Department of Mathematics, University of Kalyani, Kalyani-741235, India, India

Received  January 2015 Revised  September 2015 Published  November 2015

Infectious disease outbreaks are considered an important factor for the degradation of coral reefs. Reef-building coral species are susceptible to the influences of black band disease (BBD), characterized by cyanobacteria-dominated microbial mat that migrates rapidly across infected corals, leaving empty coral skeletons behind. We investigate coral-macroalgal phase shift in presence of BBD infection by means of an eco-epidemiological model under the assumption that the transmission of BBD occurs through both contagious and non-contagious pathways. It is observed that in presence of low coral-recruitment rate on algal turf, reduced herbivory and high macroalgal immigration, the system exhibits hysteresis through a saddle-node bifurcation and a transcritical bifurcation. Also, the system undergoes a supercritical Hopf bifurcation followed by a saddle-node bifurcation if BBD-transmission rate crosses certain critical value. We examine the effects of incubation time lag of infectious agents develop in susceptible corals after coming in contact with infected corals and a time lag in the recovery of algal turf in response to grazing of herbivores by performing equilibrium and stability analyses of delay-differential forms of the ODE model. Computer simulations have been carried out to illustrate different analytical results.
Keywords: transcritical bifurcation, saddle-node bifurcation, Coral disease, phase shifts, Hopf bifurcation..
Mathematics Subject Classification: Primary: 92B05, 92D25; Secondary: 92D4.
Citation: Joydeb Bhattacharyya, Samares Pal. Microbial disease in coral reefs: An ecosystem in transition. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 373-398. doi: 10.3934/dcdsb.2016.21.373
References:
[1]

G. S. Aeby and D. L. Santavy, Factor affecting susceptibility of the coral Montastrea faveolata to black-band disease,, Mar. Ecol. Prog. Ser., 318 (2006), 103.   Google Scholar

[2]

P. L. Antonelli, Nonlinear allometric growth. I. Perfectly cooperative systems,, Mathematical Modelling, 4 (1983), 367.  doi: 10.1016/0270-0255(83)90061-1.  Google Scholar

[3]

A. M. Bate and F. M. Hilker, Complex dynamics in an eco-epidemiological model,, Bull. Math. Biol., 75 (2013), 2059.  doi: 10.1007/s11538-013-9880-z.  Google Scholar

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D. R. Bellwood, T. P. Hughes, C. Folke and M. Nystrom, Confronting the coral reef crisis,, Nature, 429 (2004), 827.  doi: 10.1038/nature02691.  Google Scholar

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E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144.  doi: 10.1137/S0036141000376086.  Google Scholar

[6]

J. C. Blackwood, A. Hastings and P. J. Mumby, The effect of fishing on hysteresis in Caribbean coral reefs,, Theor. Ecol., 5 (2012), 105.  doi: 10.1007/s12080-010-0102-0.  Google Scholar

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C. L. Birrell, L. J. McCook, B. L. Willis and G. A. Diaz-Pulido, Effects of benthic algae on the replenishment of corals and the implications for the resilience of coral reefs,, Oceanography and Marine Biology: An Annual Review, 46 (2008), 25.   Google Scholar

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C. L. Birrell, L. J. McCook, B. L. Willis and L. Harrington, Chemical effects of macroalgae on larval settlement of the broadcast spawning coral Acropora millepora,, Marine Ecology Progress Series, 362 (2008), 129.  doi: 10.3354/meps07524.  Google Scholar

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S. J. Box and P. J. Mumby, Effect of macroalgal competition on growth and survival of juvenile Caribbean corals,, Marine Ecology Progress Series, 342 (2007), 139.  doi: 10.3354/meps342139.  Google Scholar

[10]

J. F. Bruno, H. Swetman, W. F. Precht and E. R. Selig, Assessing evidence of phase shifts from coral to macroalgal dominance on coral reefs,, Ecology, 90 (2009), 1478.  doi: 10.1890/08-1781.1.  Google Scholar

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T. J. Done, Phase shifts in coral reef communities and their ecological significance,, Developments in Hydrobiologia, 80 (1992), 121.  doi: 10.1007/978-94-017-3288-8_13.  Google Scholar

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S. R. Dudgeon, R. B. Aronson, J. F. Bruno and W. F. Precht, Phase shifts and stable states on coral reefs,, Marine Ecology Progress Series, 413 (2010), 201.  doi: 10.3354/meps08751.  Google Scholar

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T. Elmhirst, S. R. Connolly and T. P. Hughes, Connectivity, regime shifts and the resilience of coral reefs,, Coral Reefs, 28 (2009), 949.  doi: 10.1007/s00338-009-0530-8.  Google Scholar

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T. Fung, R. M. Seymour and C. R. Johnson, Alternative stable states and phase shifts in coral reefs under anthropogenic stress,, Ecology, 92 (2011), 967.  doi: 10.1890/10-0378.1.  Google Scholar

[15]

T. Fung, R. M. Seymour and C. R. Johnson, Warning signals of regime shifts as intrinsic properties of endogenous dynamics,, The American Naturalist, 182 (2013), 208.  doi: 10.1086/670930.  Google Scholar

[16]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population dynamics,, Kluwer Academic Publishers, (1992).  doi: 10.1007/978-94-015-7920-9.  Google Scholar

[17]

J. Jompa and L. J. McCook, Effects of competition and herbivory on interactions between a hard coral and a brown alga,, Journal of Experimental Marine Biology and Ecology, 271 (2002), 25.  doi: 10.1016/S0022-0981(02)00040-0.  Google Scholar

[18]

D. Lirman, Competition between macroalgae and corals: Effects of herbivore exclusion and increased algal biomass on coral survivorship and growth,, Coral Reefs, 19 (2001), 392.  doi: 10.1007/s003380000125.  Google Scholar

[19]

H. I. McCallum, A. Kuris, C. D. Harvell, K. D. Lafferty, G. W. Smith and J. Porter, Does terrestrial epidemiology apply to marine systems?,, Trends in Ecology and Evolution, 19 (2004), 585.  doi: 10.1016/j.tree.2004.08.009.  Google Scholar

[20]

L. J. McCook, J. Jompa and G. Diaz-Pulido, Competition between corals and algae on coral reefs: A review of evidence and mechanisms,, Coral Reefs, 19 (2001), 400.  doi: 10.1007/s003380000129.  Google Scholar

[21]

J. W. McManus and J. F. Polsenberg, Coral-algal phase shifts on coral reefs: Ecological and environmental aspects,, Progress in Oceanography, 60 (2004), 263.  doi: 10.1016/j.pocean.2004.02.014.  Google Scholar

[22]

P. J. Mumby, A. Hastings and H. J. Edwards, Thresholds and the resilience of Caribbean coral reefs,, Nature, 450 (2007), 98.  doi: 10.1038/nature06252.  Google Scholar

[23]

M. M. Nugues and R. P. M. Bak, Differential competitive abilities between Caribbean coral species and a brown alga: A year of experiments and a long-term perspective,, Marine Ecology Progress Series, 315 (2006), 75.  doi: 10.3354/meps315075.  Google Scholar

[24]

L. Perko, Differential Equations and Dynamical Systems,, Third Edition, (2001).  doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[25]

L. L. Richardson, Black band disease,, Rosenberg E, 3 (2004), 325.  doi: 10.1007/978-3-662-06414-6_18.  Google Scholar

[26]

K. Rützler and D. L. Santavy, The black band disease of atlantic reef corals: I. description of the cyanophyte pathogen,, P.S.Z.N.I. Mar. Ecol., 4 (1983), 301.   Google Scholar

[27]

I. Siekmann, H. Malchow and E. Venturino, An extension of the Beretta-Kuang model of viral diseases,, Mathematical Biosciences and Engineering, 5 (2008), 549.  doi: 10.3934/mbe.2008.5.549.  Google Scholar

[28]

S. H. Sokolow, P. Foley, J. E. Foley, A. Hastings and L. L. Richardson, Disease dynamics in marine metapopulations: Modelling infectious diseases on coral reefs,, Journal of Applied Ecology, 46 (2009), 621.   Google Scholar

[29]

J. N. Underwood, L. D. Smith, M. J. H. Oppen and J. P. Gilmour, Ecologically relevant dispersal of corals on isolated reefs: Implications for managing resilience,, Ecological Applications, 19 (2009), 18.  doi: 10.1890/07-1461.1.  Google Scholar

[30]

S. H. Yee, D. L. Santavy and M. G. Barron, Assessing the effects of disease and bleaching on Florida Keys corals by fitting population models to data,, Ecological Modelling, 222 (2011), 1323.  doi: 10.1016/j.ecolmodel.2011.01.009.  Google Scholar

show all references

References:
[1]

G. S. Aeby and D. L. Santavy, Factor affecting susceptibility of the coral Montastrea faveolata to black-band disease,, Mar. Ecol. Prog. Ser., 318 (2006), 103.   Google Scholar

[2]

P. L. Antonelli, Nonlinear allometric growth. I. Perfectly cooperative systems,, Mathematical Modelling, 4 (1983), 367.  doi: 10.1016/0270-0255(83)90061-1.  Google Scholar

[3]

A. M. Bate and F. M. Hilker, Complex dynamics in an eco-epidemiological model,, Bull. Math. Biol., 75 (2013), 2059.  doi: 10.1007/s11538-013-9880-z.  Google Scholar

[4]

D. R. Bellwood, T. P. Hughes, C. Folke and M. Nystrom, Confronting the coral reef crisis,, Nature, 429 (2004), 827.  doi: 10.1038/nature02691.  Google Scholar

[5]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144.  doi: 10.1137/S0036141000376086.  Google Scholar

[6]

J. C. Blackwood, A. Hastings and P. J. Mumby, The effect of fishing on hysteresis in Caribbean coral reefs,, Theor. Ecol., 5 (2012), 105.  doi: 10.1007/s12080-010-0102-0.  Google Scholar

[7]

C. L. Birrell, L. J. McCook, B. L. Willis and G. A. Diaz-Pulido, Effects of benthic algae on the replenishment of corals and the implications for the resilience of coral reefs,, Oceanography and Marine Biology: An Annual Review, 46 (2008), 25.   Google Scholar

[8]

C. L. Birrell, L. J. McCook, B. L. Willis and L. Harrington, Chemical effects of macroalgae on larval settlement of the broadcast spawning coral Acropora millepora,, Marine Ecology Progress Series, 362 (2008), 129.  doi: 10.3354/meps07524.  Google Scholar

[9]

S. J. Box and P. J. Mumby, Effect of macroalgal competition on growth and survival of juvenile Caribbean corals,, Marine Ecology Progress Series, 342 (2007), 139.  doi: 10.3354/meps342139.  Google Scholar

[10]

J. F. Bruno, H. Swetman, W. F. Precht and E. R. Selig, Assessing evidence of phase shifts from coral to macroalgal dominance on coral reefs,, Ecology, 90 (2009), 1478.  doi: 10.1890/08-1781.1.  Google Scholar

[11]

T. J. Done, Phase shifts in coral reef communities and their ecological significance,, Developments in Hydrobiologia, 80 (1992), 121.  doi: 10.1007/978-94-017-3288-8_13.  Google Scholar

[12]

S. R. Dudgeon, R. B. Aronson, J. F. Bruno and W. F. Precht, Phase shifts and stable states on coral reefs,, Marine Ecology Progress Series, 413 (2010), 201.  doi: 10.3354/meps08751.  Google Scholar

[13]

T. Elmhirst, S. R. Connolly and T. P. Hughes, Connectivity, regime shifts and the resilience of coral reefs,, Coral Reefs, 28 (2009), 949.  doi: 10.1007/s00338-009-0530-8.  Google Scholar

[14]

T. Fung, R. M. Seymour and C. R. Johnson, Alternative stable states and phase shifts in coral reefs under anthropogenic stress,, Ecology, 92 (2011), 967.  doi: 10.1890/10-0378.1.  Google Scholar

[15]

T. Fung, R. M. Seymour and C. R. Johnson, Warning signals of regime shifts as intrinsic properties of endogenous dynamics,, The American Naturalist, 182 (2013), 208.  doi: 10.1086/670930.  Google Scholar

[16]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population dynamics,, Kluwer Academic Publishers, (1992).  doi: 10.1007/978-94-015-7920-9.  Google Scholar

[17]

J. Jompa and L. J. McCook, Effects of competition and herbivory on interactions between a hard coral and a brown alga,, Journal of Experimental Marine Biology and Ecology, 271 (2002), 25.  doi: 10.1016/S0022-0981(02)00040-0.  Google Scholar

[18]

D. Lirman, Competition between macroalgae and corals: Effects of herbivore exclusion and increased algal biomass on coral survivorship and growth,, Coral Reefs, 19 (2001), 392.  doi: 10.1007/s003380000125.  Google Scholar

[19]

H. I. McCallum, A. Kuris, C. D. Harvell, K. D. Lafferty, G. W. Smith and J. Porter, Does terrestrial epidemiology apply to marine systems?,, Trends in Ecology and Evolution, 19 (2004), 585.  doi: 10.1016/j.tree.2004.08.009.  Google Scholar

[20]

L. J. McCook, J. Jompa and G. Diaz-Pulido, Competition between corals and algae on coral reefs: A review of evidence and mechanisms,, Coral Reefs, 19 (2001), 400.  doi: 10.1007/s003380000129.  Google Scholar

[21]

J. W. McManus and J. F. Polsenberg, Coral-algal phase shifts on coral reefs: Ecological and environmental aspects,, Progress in Oceanography, 60 (2004), 263.  doi: 10.1016/j.pocean.2004.02.014.  Google Scholar

[22]

P. J. Mumby, A. Hastings and H. J. Edwards, Thresholds and the resilience of Caribbean coral reefs,, Nature, 450 (2007), 98.  doi: 10.1038/nature06252.  Google Scholar

[23]

M. M. Nugues and R. P. M. Bak, Differential competitive abilities between Caribbean coral species and a brown alga: A year of experiments and a long-term perspective,, Marine Ecology Progress Series, 315 (2006), 75.  doi: 10.3354/meps315075.  Google Scholar

[24]

L. Perko, Differential Equations and Dynamical Systems,, Third Edition, (2001).  doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[25]

L. L. Richardson, Black band disease,, Rosenberg E, 3 (2004), 325.  doi: 10.1007/978-3-662-06414-6_18.  Google Scholar

[26]

K. Rützler and D. L. Santavy, The black band disease of atlantic reef corals: I. description of the cyanophyte pathogen,, P.S.Z.N.I. Mar. Ecol., 4 (1983), 301.   Google Scholar

[27]

I. Siekmann, H. Malchow and E. Venturino, An extension of the Beretta-Kuang model of viral diseases,, Mathematical Biosciences and Engineering, 5 (2008), 549.  doi: 10.3934/mbe.2008.5.549.  Google Scholar

[28]

S. H. Sokolow, P. Foley, J. E. Foley, A. Hastings and L. L. Richardson, Disease dynamics in marine metapopulations: Modelling infectious diseases on coral reefs,, Journal of Applied Ecology, 46 (2009), 621.   Google Scholar

[29]

J. N. Underwood, L. D. Smith, M. J. H. Oppen and J. P. Gilmour, Ecologically relevant dispersal of corals on isolated reefs: Implications for managing resilience,, Ecological Applications, 19 (2009), 18.  doi: 10.1890/07-1461.1.  Google Scholar

[30]

S. H. Yee, D. L. Santavy and M. G. Barron, Assessing the effects of disease and bleaching on Florida Keys corals by fitting population models to data,, Ecological Modelling, 222 (2011), 1323.  doi: 10.1016/j.ecolmodel.2011.01.009.  Google Scholar

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