# American Institute of Mathematical Sciences

## Macroalgal allelopathy in the emergence of coral diseases

 1 Department of Wildlife and Fisheries Sciences, Texas A & M University, College Station, TX-77840, USA, United States 2 Department of Mathematics, University of Kalyani, Kalyani 741235, India

* Corresponding author: Samares Pal

Received  September 2016 Revised  November 2016 Published  April 2017

Fund Project: The second author is supported by SERB, New Delhi, India Ref.No.SR/S4/MS:863/13

Microbial disease in corals associated with the proliferation of benthic macroalgae are the major contributors to the decline of coral reefs over the past few decades. Several benthic macroalgae species produce allelopathic chemical compounds that negatively affect corals. The emergence of microbial diseases in corals occurs simultaneously with the elevated abundance of benthic macroalgae. The release of allelochemicals by toxic-macroalgae enhances microbial activity on coral surfaces via the release of dissolved compounds. Proliferation of benthic macroalgae in coral reefs results in increased physical contacts between corals and macroalgae, triggering the susceptibility of coral disease. The abundance of macroalgae changes the community structure towards macroalgae dominated reef ecosystem. We investigate coral-macroalgal phase shift in presence of macroalgal allelopathy and microbial infection on corals by means of an eco-epidemiological model under the assumption that the transmission of infection is mediated by the pathogens shed by infectious corals and under the influence of macroalgae in the environment. We perform equilibrium and stability analysis on our non-linear ODE model and found that the system is capable of exhibiting the existence of two stable configurations of the community under the same environmental conditions by allowing saddle-node bifurcations that involves in creation and destruction of fixed points and associated hysteresis effect. It is shown that the system undergoes a sudden change of transition when the transmission rate of the infection crosses some certain critical thresholds. Computer simulations have been carried out to illustrate different analytical results.

Citation: Joydeb Bhattacharyya, Samares Pal. Macroalgal allelopathy in the emergence of coral diseases. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2017146
##### References:
 [1] T. D. Andras, T. S. Alexander, A. Gahlena, R. M. Parry, F. M. Fernandez, J. Kubanek, M. D. Wang and M. E. Hay, Seaweed allelopathy against coral: Surface distribution of seaweed secondary metabolites by imaging mass spectrometry, Journal of Chemical Ecology, 38 (2012), 1203-1214. doi: 10.1007/s10886-012-0204-9. [2] A. M. Bate and F. M. Hilker, Complex Dynamics in an Eco-epidemiological Model, Bull. Math. Biol., 75 (2013), 2059-2078. doi: 10.1007/s11538-013-9880-z. [3] D. R. Bellwood, T. P. Hughes, C. Folke and M. Nystrom, Confronting the coral reef crisis, Nature, 429 (2004), 827-833. doi: 10.1038/nature02691. [4] J. Bhattacharyya and S. Pal, Hysteresis in coral reefs under macroalgal toxicity and overfishing, Journal of Biological Physics, 41 (2015), 151-172. doi: 10.1007/s10867-014-9371-y. [5] J. Bhattacharyya and S. Pal, Microbial disease in coral reefs: An ecosystem in transition, Discrete and Continuous Dynamical Systems: Series B, 21 (2016), 373-398. doi: 10.3934/dcdsb.2016.21.373. [6] 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), 25-63. [7] 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), 129-137. [8] J. C. Blackwood, A. Hastings and P. J. Mumby, The effect of fishing on hysteresis in Caribbean coral reefs, Theoretical Ecology, 5 (2012), 105-114. doi: 10.1007/s12080-010-0102-0. [9] R. M. Bonaldo and M. E. Hay, Seaweed-coral interactions: Variance in seaweed allelopathy, coral susceptibility, and potential effects on coral resilience PLOS ONE 9 (2014), e85786. [10] 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), 139-149. doi: 10.3354/meps342139. [11] 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), 1478-1484. doi: 10.1890/08-1781.1. [12] A. Conversi, V. Dakos, A. Gardmark, S. Ling, C. Folke, P. J. Mumby, C. Greene, M. Edwards, T. Blenckner, M. Casini, A. Pershing and C. Mollmann, A holistic view of marine regime shifts, Philosophical Transactions of the Royal Society B: Biological Sciences, 370 (2015), 1-8. doi: 10.1098/rstb.2013.0279. [13] T. J. Done, Phase shifts in coral reef communities and their ecological significance, Hydrobiologia, 80 (1992), 121-132. doi: 10.1007/978-94-017-3288-8_13. [14] 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), 201-216. doi: 10.3354/meps08751. [15] T. Elmhirst, S. R. Connolly and T. P. Hughes, Connectivity, regime shifts and the resilience of coral reefs, Coral Reefs, 28 (2009), 949-957. doi: 10.1007/s00338-009-0530-8. [16] D. Harvell, R. Aronson, N. Baron, J. Connell, A. Dobson, S. Ellner, L. Gerber, K. Kim, A. Kuris, H. McCallum, K. Lafferty, B. McKay, J. Porter, M. Pascual, G. Smith, K. Sutherland and J. Ward, The rising tide of ocean diseases: unsolved problems and research priorities, Frontiers in Ecology and the Environment, 2 (2004), 375-382. [17] M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nat Rev Microbiol., 8 (2010), 15-25. doi: 10.1038/nrmicro2259. [18] 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), 25-39. doi: 10.1016/S0022-0981(02)00040-0. [19] D. Lirman, Competition between macroalgae and corals: Effects of herbivore exclusion and increased algal biomass on coral survivorship and growth, Coral Reefs, 19 (2001), 392-399. doi: 10.1007/s003380000125. [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), 400-417. doi: 10.1007/s003380000129. [21] J. W. McManus and J. F. Polsenberg, Coral-algal phase shifts on coral reefs: Ecological and environmental aspects, Progress in Oceanography, 60 (2004), 263-279. doi: 10.1016/j.pocean.2004.02.014. [22] P. J. Mumby, N. L. Foster and E. A. G. Fahy, Patch dynamics of coral reef macroalgae under chronic and acute disturbance, Coral Reefs, 24 (2005), 681-692. doi: 10.1007/s00338-005-0058-5. [23] P. J. Mumby, A. Hastings and H. J. Edwards, Thresholds and the resilience of Caribbean coral reefs, Nature, 450 (2007), 98-101. doi: 10.1038/nature06252. [24] P. J. Mumby, Phase shifts and the stability of macroalgal communities on Caribbean coral reefs, Coral Reefs, 28 (2009), 761-773. doi: 10.1007/s00338-009-0506-8. [25] 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), 75-86. doi: 10.3354/meps315075. [26] M. M. Nugues, L. Delvoye and R. P. M. Bak, Coral defence against macroalgae: Differential effects of mesenterial filaments on the green alga Halimeda opuntia, Marine Ecology Progress Series, 278 (2004), 103-114. doi: 10.3354/meps278103. [27] M. M. Nugues, G. W. Smith, R. J. van Hooidonk, M. I. Seabra and R. P. M. Bak, Algal contact as a trigger for coral disease, Ecology Letters, 7 (2004), 919-923. doi: 10.1111/j.1461-0248.2004.00651.x. [28] L. Perko, Differential Equations and Dynamical Systems Third Edition, Springer, New York, 2001. [29] D. B. Rasher, E. P. Stout, S. Engel, J. Kubanek and M. E. Hay, Macroalgal terpenes function as allelopathic agents against reef corals, Proceedings of the National Acadademy of Sciences, 108 (2011), 17726-17731. doi: 10.1073/pnas.1108628108. [30] J. A. Sanchez, S. Herrera, R. Navas-Camacho, A. Rodríguez-Ramírez, P. Herron, V. Pizarro, A. R. Acosta, P. A. Castillo, P. Montoya and C. Orozco, White plague-like coral disease in remote reefs of the Western Caribbean, Rev. Biol. Trop., 58 (2010), 145-154. doi: 10.15517/rbt.v58i1.20031. [31] K. H. Sharp and K. B. Ritchie, Multi-partner interactions in corals in the face of climate change, The Biological Bulletin, 223 (2012), 66-77. doi: 10.1086/BBLv223n1p66. [32] I. Siekmann, H. Malchow and E. Venturino, An extension of the Beretta-Kuang model of viral diseases, Mathematical Biosciences and Engineering, 5 (2008), 549-565. doi: 10.3934/mbe.2008.5.549. [33] 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), 621-631. [34] S. Sunagawa, T. Z. DeSantis, Y. M. Piceno, E. L. Brodie, M. K. DeSalvo, C. R. Voolstra, E. Weil, G. L. Andersen and M. Medina, Bacterial diversity and White Plague Disease-associated community changes in the Caribbean coral, Montastraea Faveolata, The ISME Journal, 3 (2009), 512-521. doi: 10.1038/ismej.2008.131. [35] M. J. Sweet, J. C. Bythell and M. M. Nugues, Algae as reservoirs for coral pathogens PLoS One 8 (2013), e69717.

show all references

##### References:
 [1] T. D. Andras, T. S. Alexander, A. Gahlena, R. M. Parry, F. M. Fernandez, J. Kubanek, M. D. Wang and M. E. Hay, Seaweed allelopathy against coral: Surface distribution of seaweed secondary metabolites by imaging mass spectrometry, Journal of Chemical Ecology, 38 (2012), 1203-1214. doi: 10.1007/s10886-012-0204-9. [2] A. M. Bate and F. M. Hilker, Complex Dynamics in an Eco-epidemiological Model, Bull. Math. Biol., 75 (2013), 2059-2078. doi: 10.1007/s11538-013-9880-z. [3] D. R. Bellwood, T. P. Hughes, C. Folke and M. Nystrom, Confronting the coral reef crisis, Nature, 429 (2004), 827-833. doi: 10.1038/nature02691. [4] J. Bhattacharyya and S. Pal, Hysteresis in coral reefs under macroalgal toxicity and overfishing, Journal of Biological Physics, 41 (2015), 151-172. doi: 10.1007/s10867-014-9371-y. [5] J. Bhattacharyya and S. Pal, Microbial disease in coral reefs: An ecosystem in transition, Discrete and Continuous Dynamical Systems: Series B, 21 (2016), 373-398. doi: 10.3934/dcdsb.2016.21.373. [6] 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), 25-63. [7] 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), 129-137. [8] J. C. Blackwood, A. Hastings and P. J. Mumby, The effect of fishing on hysteresis in Caribbean coral reefs, Theoretical Ecology, 5 (2012), 105-114. doi: 10.1007/s12080-010-0102-0. [9] R. M. Bonaldo and M. E. Hay, Seaweed-coral interactions: Variance in seaweed allelopathy, coral susceptibility, and potential effects on coral resilience PLOS ONE 9 (2014), e85786. [10] 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), 139-149. doi: 10.3354/meps342139. [11] 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), 1478-1484. doi: 10.1890/08-1781.1. [12] A. Conversi, V. Dakos, A. Gardmark, S. Ling, C. Folke, P. J. Mumby, C. Greene, M. Edwards, T. Blenckner, M. Casini, A. Pershing and C. Mollmann, A holistic view of marine regime shifts, Philosophical Transactions of the Royal Society B: Biological Sciences, 370 (2015), 1-8. doi: 10.1098/rstb.2013.0279. [13] T. J. Done, Phase shifts in coral reef communities and their ecological significance, Hydrobiologia, 80 (1992), 121-132. doi: 10.1007/978-94-017-3288-8_13. [14] 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), 201-216. doi: 10.3354/meps08751. [15] T. Elmhirst, S. R. Connolly and T. P. Hughes, Connectivity, regime shifts and the resilience of coral reefs, Coral Reefs, 28 (2009), 949-957. doi: 10.1007/s00338-009-0530-8. [16] D. Harvell, R. Aronson, N. Baron, J. Connell, A. Dobson, S. Ellner, L. Gerber, K. Kim, A. Kuris, H. McCallum, K. Lafferty, B. McKay, J. Porter, M. Pascual, G. Smith, K. Sutherland and J. Ward, The rising tide of ocean diseases: unsolved problems and research priorities, Frontiers in Ecology and the Environment, 2 (2004), 375-382. [17] M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson, Bacterial competition: Surviving and thriving in the microbial jungle, Nat Rev Microbiol., 8 (2010), 15-25. doi: 10.1038/nrmicro2259. [18] 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), 25-39. doi: 10.1016/S0022-0981(02)00040-0. [19] D. Lirman, Competition between macroalgae and corals: Effects of herbivore exclusion and increased algal biomass on coral survivorship and growth, Coral Reefs, 19 (2001), 392-399. doi: 10.1007/s003380000125. [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), 400-417. doi: 10.1007/s003380000129. [21] J. W. McManus and J. F. Polsenberg, Coral-algal phase shifts on coral reefs: Ecological and environmental aspects, Progress in Oceanography, 60 (2004), 263-279. doi: 10.1016/j.pocean.2004.02.014. [22] P. J. Mumby, N. L. Foster and E. A. G. Fahy, Patch dynamics of coral reef macroalgae under chronic and acute disturbance, Coral Reefs, 24 (2005), 681-692. doi: 10.1007/s00338-005-0058-5. [23] P. J. Mumby, A. Hastings and H. J. Edwards, Thresholds and the resilience of Caribbean coral reefs, Nature, 450 (2007), 98-101. doi: 10.1038/nature06252. [24] P. J. Mumby, Phase shifts and the stability of macroalgal communities on Caribbean coral reefs, Coral Reefs, 28 (2009), 761-773. doi: 10.1007/s00338-009-0506-8. [25] 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), 75-86. doi: 10.3354/meps315075. [26] M. M. Nugues, L. Delvoye and R. P. M. Bak, Coral defence against macroalgae: Differential effects of mesenterial filaments on the green alga Halimeda opuntia, Marine Ecology Progress Series, 278 (2004), 103-114. doi: 10.3354/meps278103. [27] M. M. Nugues, G. W. Smith, R. J. van Hooidonk, M. I. Seabra and R. P. M. Bak, Algal contact as a trigger for coral disease, Ecology Letters, 7 (2004), 919-923. doi: 10.1111/j.1461-0248.2004.00651.x. [28] L. Perko, Differential Equations and Dynamical Systems Third Edition, Springer, New York, 2001. [29] D. B. Rasher, E. P. Stout, S. Engel, J. Kubanek and M. E. Hay, Macroalgal terpenes function as allelopathic agents against reef corals, Proceedings of the National Acadademy of Sciences, 108 (2011), 17726-17731. doi: 10.1073/pnas.1108628108. [30] J. A. Sanchez, S. Herrera, R. Navas-Camacho, A. Rodríguez-Ramírez, P. Herron, V. Pizarro, A. R. Acosta, P. A. Castillo, P. Montoya and C. Orozco, White plague-like coral disease in remote reefs of the Western Caribbean, Rev. Biol. Trop., 58 (2010), 145-154. doi: 10.15517/rbt.v58i1.20031. [31] K. H. Sharp and K. B. Ritchie, Multi-partner interactions in corals in the face of climate change, The Biological Bulletin, 223 (2012), 66-77. doi: 10.1086/BBLv223n1p66. [32] I. Siekmann, H. Malchow and E. Venturino, An extension of the Beretta-Kuang model of viral diseases, Mathematical Biosciences and Engineering, 5 (2008), 549-565. doi: 10.3934/mbe.2008.5.549. [33] 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), 621-631. [34] S. Sunagawa, T. Z. DeSantis, Y. M. Piceno, E. L. Brodie, M. K. DeSalvo, C. R. Voolstra, E. Weil, G. L. Andersen and M. Medina, Bacterial diversity and White Plague Disease-associated community changes in the Caribbean coral, Montastraea Faveolata, The ISME Journal, 3 (2009), 512-521. doi: 10.1038/ismej.2008.131. [35] M. J. Sweet, J. C. Bythell and M. M. Nugues, Algae as reservoirs for coral pathogens PLoS One 8 (2013), e69717.
Schematic representation of the model
$(a)$ Bifurcation diagram of $g$ versus the equilibrium value of coral cover. $(b)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $g$. $(c)$ The relative positions of $f_1(g), f_2(g)$ and $\phi(g)$ showing that a Hopf bifurcation occurs when the two curves intersect at $g^*=0.5442$. $(d)$ Bifurcation diagrams of $g$ versus the equilibrium value of coral cover for $\lambda=0.05$
$(a)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $g$ for $\lambda=0.05$. $(b)$ The relative positions of $f_1(g), f_2(g)$ and $\phi(g)$ showing that a Hopf bifurcation occurs when the two curves intersect at $g=0.342$. $(c)$ Two parameter bifurcation diagram with $g$ and $\lambda$ as active parameters. (The saddle-node curve is in blue, Hopf curve is in red and codimension one bifurcation curve with $\lambda=0.05$ is in green)
Bifurcation diagrams of $g$ versus the equilibrium value of coral cover for $(a)$ $\lambda=0$ and $(b)$ $\eta=0.5$.
Bifurcation diagram of $\gamma$ versus the equilibrium value of coral cover. $(b)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $\gamma$. $(c)$ The relative positions of $f_1(\gamma), f_2(\gamma)$ and $\phi(\gamma)$ showing that a Hopf bifurcation occurs when the two curves intersect at $\gamma=1.001$.
Bifurcation diagrams of $\gamma$ versus the equilibrium value of coral cover for $(a)$ $g=0.4$, $(b)$ $g=1$, $(c)$ $\lambda=0$ and $(d)$ $\lambda=1$.
Coexistence regions in $(a)$ $\gamma-g$ parameter space, $(b)$ $\gamma-\lambda$ parameter space and $(c)$ $\eta-g$ parameter space (Blue indicates coral-dominated $E^*$, red indicates macroalgae-dominated $E^*$ and green indicates coral-dominated $E_2$).
$(a)$ Bifurcation diagram of $\lambda$ versus the equilibrium value of coral cover. $(b)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $\lambda$.
$(a)$ The location of emerging oscillations for changes in $\lambda$. $(b)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $\lambda$. $(c)$ The relative positions of $f_1(g), f_2(g)$ and $\phi(g)$ showing that Hopf bifurcation occurs when the two curves intersect at $\lambda_*=0.43326$ and $\lambda^*=0.621632$.
$(a)$ Bifurcation diagram of $\eta$ versus the equilibrium value of coral cover. $(b)$ Eigenvalues for the interior equilibrium $E^*$ as functions of $\eta$.
$(a)$ Changes in the resilience of the system with $\eta$ as an active parameter for $(a)$ $g=0.5$, $(b)$ $g=0.45$ and $(c)$ $g=0.4$ (Stability at $E^*$ is indicated in blue, stability at $E_1$ and $E_2$ are shown in black and cyan respectively and unstable $E^*$ is shown in red).
$(a)$ Bifurcation diagram of $\nu_1$ versus the equilibrium value of coral cover. $(b)$ Bifurcation diagram of $\nu_1$ versus the equilibrium value of coral cover for different values of $g$ (The saddle-node curve is in green, Hopf curve is in red and codimension one bifurcation curves are in blue).
Parameters used in the model (1).
 Parameters Description of Parameters Value Reference $\alpha$ Rate of macroalgal direct overgrowth over coral 0.1 yr$^{-1}$ [8,15,19] $r$ Recruitment rate of corals on turf algae 0.55 yr$^{-1}$ [10,15] $a$ Rate of macroalgal vegetative spread over algal turfs 0.77 yr$^{-1}$ [15,22] $b$ Immigration rate of macroalgae on algal turf 0.005 yr$^{-1}$ [15] $d_1$ Mortality rate of macroalgae 0.1 yr$^{-1}$ [4,22] $d_2$ Natural mortality rate of corals 0.24 yr$^{-1}$ [4,10] $\gamma$ Toxin-induced death rate of corals 0.1 yr$^{-1}$ [4] $\nu_1$ Rate of release of pathogens by toxic-macroalgae 0.1 yr$^{-1}$ - $\nu_2$ Pathogen-shedding rate by infectious corals 0.3 yr$^{-1}$ - $\frac{1}{d_3}$ Average time pathogens exist in environment 100 yrs [16] $\lambda$ Rate of infection 0.2 yr$^{-1}$ - $\eta$ Disease induced death rate of infected corals 0.01 yr$^{-1}$ [4] $g$ The maximal grazing rate of herbivorous fish 0.6 yr$^{-1}$ [15] $\delta$ Crowding parameter 0.01 yr$^{-1}$ -
 Parameters Description of Parameters Value Reference $\alpha$ Rate of macroalgal direct overgrowth over coral 0.1 yr$^{-1}$ [8,15,19] $r$ Recruitment rate of corals on turf algae 0.55 yr$^{-1}$ [10,15] $a$ Rate of macroalgal vegetative spread over algal turfs 0.77 yr$^{-1}$ [15,22] $b$ Immigration rate of macroalgae on algal turf 0.005 yr$^{-1}$ [15] $d_1$ Mortality rate of macroalgae 0.1 yr$^{-1}$ [4,22] $d_2$ Natural mortality rate of corals 0.24 yr$^{-1}$ [4,10] $\gamma$ Toxin-induced death rate of corals 0.1 yr$^{-1}$ [4] $\nu_1$ Rate of release of pathogens by toxic-macroalgae 0.1 yr$^{-1}$ - $\nu_2$ Pathogen-shedding rate by infectious corals 0.3 yr$^{-1}$ - $\frac{1}{d_3}$ Average time pathogens exist in environment 100 yrs [16] $\lambda$ Rate of infection 0.2 yr$^{-1}$ - $\eta$ Disease induced death rate of infected corals 0.01 yr$^{-1}$ [4] $g$ The maximal grazing rate of herbivorous fish 0.6 yr$^{-1}$ [15] $\delta$ Crowding parameter 0.01 yr$^{-1}$ -
 [1] Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 [2] Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923 [3] Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 [4] Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 [5] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [6] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [7] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [8] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [9] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [10] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209 [11] Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71 [12] R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147 [13] Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197 [14] Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 [15] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [16] Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911 [17] Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21 [18] W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 [19] Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069 [20] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

2016 Impact Factor: 0.994