# American Institute of Mathematical Sciences

May  2017, 22(3): 741-762. doi: 10.3934/dcdsb.2017036

## Macroalgal allelopathy in the emergence of coral diseases

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

* Corresponding author: Samares Pal

Received  November 2015 Revised  April 2016 Published  January 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, 2017, 22 (3) : 741-762. doi: 10.3934/dcdsb.2017036
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Schematic representation of the model
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 $f_1(\lambda), f_2(\lambda)$ and $\phi(\lambda)$ 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)
Relation between $H(k^{2})$ (a) and $\Re( \lambda )$ (b) for $\det( \Phi_w(w^{*}) ) >0$
 Parameter Description 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] $a$ Macroalgal vegetative growth rate on turf algae 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 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 L cell$^{-1}$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 L cell$^{-1}$yr$^{-1}$ -
 Parameter Description 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] $a$ Macroalgal vegetative growth rate on turf algae 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 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 L cell$^{-1}$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 L cell$^{-1}$yr$^{-1}$ -
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