×

Herbivore harvesting and alternative steady states in coral reefs. (English) Zbl 07332697

Summary: Coral reefs can undergo relatively rapid changes in the dominant biota, a phenomenon referred to as phase shift. Degradation of coral reefs is often associated with changes in community structure towards a macroalgae-dominated reef ecosystem due to the reduction in herbivory caused by overfishing. We investigate the coral-macroalgal phase shift due to the effects of harvesting of herbivorous reef fish by means of a continuous time model in the food chain. Conditions for local asymptotic stability of steady states are derived. We have shown that under certain conditions the system is uniformly persistent in presence of all the organisms. Moreover, it is shown that the system undergoes a Hopf bifurcation when the carrying capacity of macroalgae crosses certain critical value. Computer simulations have been carried out to illustrate different analytical results.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34D20 Stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bender, D.; Diaz-Pulido, G.; Dove, S., Effects of macroalgae on corals recovering from disturbance, J. Exp. Mar. Biol. Ecol. 429 (2012), 15-19
[2] Bhattacharyya, J.; Pal, S., Coexistence of competing predators in a coral reef ecosystem, Nonlinear Anal., Real World Appl. 12 (2011), 965-978 · Zbl 1203.92059
[3] Box, S. J.; Mumby, P. J., Effect of macroalgal competition on growth and survival of juvenile Caribbean corals, Mar. Ecol. Prog. Ser. 342 (2007), 139-149
[4] Bruno, J. F.; Precht, W. F.; Vroom, P. S.; Aronson, R. B., Coral reef baselines: How much macroalgae is natural?, Marine Pollution Bull. 80 (2014), 24-29
[5] Bruno, J. F.; Sweatman, H.; Precht, W. F.; Selig, E. R.; Schutte, V. G. W., Assessing evidence of phase shifts from coral to macroalgal dominance on coral reefs, Ecology 90 (2009), 1478-1484
[6] Cantrell, R. S.; Cosner, C., On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 257 (2001), 206-222 · Zbl 0991.34046
[7] Cheal, A. J.; MacNeil, M. A.; Cripps, E.; Emslie, M. J.; Jonker, M.; Schaffelke, B.; Sweatman, H., Coral-macroalgal phase shifts or reef resilience: Links with diversity and functional roles of herbivorous fishes on the Great Barrier Reef, Coral Reefs 29 (2010), 1005-1015
[8] Cui, J.; Takeuchi, Y., Permanence, extinction and periodic solution of predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl. 317 (2006), 464-474 · Zbl 1102.34033
[9] Evans, R. D.; Wilson, S. K.; Field, S. N.; Moore, J. A. Y., Importance of macroalgal fields as coral reef fish nursery habitat in north-west Australia, Mar. Biol. 161 (2014), 599-607
[10] Feng, J.; Wang, H.; Huang, D.; Li, S., Alternative attractors in marine ecosystems: A comparative analysis of fishing effects, Ecological Modelling 195 (2006), 377-384
[11] Gupta, R. P.; Chandra, P., Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl. 398 (2013), 278-295 · Zbl 1259.34035
[12] Harriott, V. J.; Banks, S. A., Latitudinal variation in coral communities in eastern Australia: A qualitative biophysical model of factors regulating coral reefs, Coral Reefs 21 (2002), 83-94
[13] Hughes, T. P.; Graham, N. A. J.; Jackson, J. B. C.; Mumby, P. J.; Steneck, R. S., Rising to the challenge of sustaining coral reef resilience, Trends Ecol. Evol. 25 (2010), 633-642
[14] Jessen, C.; Roder, C.; Lizcano, J. F. V.; Voolstra, C. R.; Wild, C., In-situ effects of simulated overfishing and eutrophication on benthic coral reef algae growth, succession, and composition in the Central Red Sea, Plos One 8 (2013), Article ID e66992, 13 pages
[15] Kramer, D. B., Adaptive harvesting in a multiple-species coral-reef food web, Ecol. Soc. 13 (2007), Article ID 17, 25 pages
[16] Lenzini, P.; Rebaza, J., Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., Ruse 4 (2010), 791-803 · Zbl 1189.37098
[17] Lirman, D., Competition between macroalgae and corals: Effects of herbivore exclusion and increased algal biomass on coral survivorship and growth, Coral Reefs 19 (2001), 392-399
[18] Littler, M. M.; Littler, D. S.; Brooks, B. L., Harmful algae on tropical coral reefs: Bottom-up eutrophication and top-down herbivory, Harmful Algae 5 (2006), 565-585
[19] McCook, L. J.; Jompa, J.; Diaz-Pulido, G., Competition between corals and algae on coral reefs: A review of evidence and mechanisms, Coral Reefs 19 (2001), 400-417
[20] McManus, J. W.; Polsenberg, J. F., Coral-algal phase shifts on coral reefs: Ecological and environmental aspects, Progress in Oceanography 60 (2004), 263-279
[21] Mumby, P. J., Phase shifts and the stability of macroalgal communities on Caribbean coral reefs, Coral Reefs 28 (2009), 761-773
[22] Mumby, P. J.; Foster, N. L.; Fahy, E. A. G., Patch dynamics of coral reef macroalgae under chronic and acute disturbance, Coral Reefs 24 (2005), 681-692
[23] Mumby, P. J.; Steneck, R. S., Coral reef management and conservation in light of rapidly evolving ecological paradigms, Trends Ecol. Evol. 23 (2008), 555-563
[24] Murray, G. P. D.; Stillman, R. A.; Gozlan, R. E.; Britton, J. R., Experimental predictions of the functional response of a freshwater fish, Ethology 119 (2013), 751-761
[25] Poore, A. B., On the theory and application of Hopf-Friedrichs bifurcation theory, Arch. Ration. Mech. Anal. 60 (1976), 371-393 · Zbl 0358.34005
[26] Ruan, S., Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling, J. Math. Biol. 31 (1993), 633-654 · Zbl 0779.92021
[27] Siekmann, I.; Malchow, H.; Venturino, E., An extension of the Beretta-Kuang model of viral diseases, Math. Biosci. Eng. 5 (2008), 549-565 · Zbl 1158.92320
[28] Smith, J. E.; Hunter, C. L.; Smith, C. M., The effects of top-down versus bottom-up control on benthic coral reef community structure, Oecologia 163 (2010), 497-507
[29] J. E. Smith, M. Shaw, R. A. Edwards; D. Obura; O. Pantos; E. Sala; S. A. Sandin; S. Smriga; M. Hatay; F. L. Rohwer, Indirect effects of algae on coral: Algae-mediated, microbe-induced coral mortality, Ecol. Lett. 9 (2006), 835-845
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.