A three-dimensional stage-structured predator-prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.
Macroalgae and corals compete for the available space in coral reef ecosystems.While herbivorous reef fish play a beneficial role in decreasing the growth of macroalgae, macroalgal toxicity and overfishing of herbivores leads to proliferation of macroalgae. The abundance of macroalgae changes the community structure towards a macroalgae-dominated reef ecosystem. We investigate coral-macroalgal phase shifts by means of a continuous time model in a food chain. Conditions for local asymptotic stability of steady states are derived. It is observed that in the presence of macroalgal toxicity and overfishing, the system exhibits hysteresis through saddle-node bifurcation and transcritical bifurcation. We examine the effects of time lags in the liberation of toxins by macroalgae and the recovery of algal turf in response to grazing of herbivores on macroalgae by performing equilibrium and stability analyses of delay-differential forms of the ODE model. Computer simulations have been carried out to illustrate the different analytical results.
Allelopathic species can alter biodiversity. Using simulated assemblages that are characterised by neutrality, lumpy coexistence and intransitivity, we explore relationships between within-assemblage competitive dissimilarities and resistance to allelopathic species. An emergent behaviour from our models is that assemblages are more resistant to allelopathy when members strongly compete exploitatively (high competitive power). We found that neutral assemblages were the most vulnerable to allelopathic species, followed by lumpy and then by intransitive assemblages. We find support for our modeling in real-world time-series data from eight lakes of varied morphometry and trophic state. Our analysis of this data shows that a lake's history of allelopathic phytoplankton species biovolume density and dominance is related to the number of species clusters occurring in the plankton assemblages of those lakes, an emergent trend similar to that of our modeling. We suggest that an assemblage's competitive power determines its allelopathy resistance.
A two-dimensional single-species stage-structured model is proposed and analyzed to study the effect of intraguild predation with harvesting of the adult species. Time lags in reproduction and maturation of the organism are introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. It is found that there are stability switches for time delays, and Hopf bifurcations when time delays cross through some critical values. The stability and direction of the Hopf bifurcations are determined by applying the normal form method and the center manifold theory. Computer simulations have been carried out to illustrate various analytical results.
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 cyanobacteriadominated 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 coralrecruitment 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.2010 Mathematics Subject Classification. Primary: 92B05, 92D25; Secondary: 92D40.
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