The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents.
Chemical contamination from agricultural runoff associated with fish morbidity and mortality have increased in frequency and severity over the past several decades. We investigate the effects of exposure of insecticide on a zooplankivorous reef-fish and its prey by means of a continuous time model of the dynamics of organisms in a food chain in presence of a constant rate of flow of an insecticide. The insecticide is sub-lethal to the fish but is lethal to zooplankton. Our analysis leads to different thresholds in terms of the model parameters acting as conditions under which the species associated with the system cannot thrive even in absence of predation. Also, we prove that under certain conditions the system is permanent in presence of all the organisms and obtain conditions for permanence under which one of the two organisms is extinct. Local stability of the system is obtained in absence of sensitive species and in presence of both sensitive and resistant species. Computer simulations have been carried out to illustrate different analytical results.
The invasion of predatory lionfish (Pterois volitans) represents a major threat to the western Atlantic coral reef ecosystems. The proliferation of venomous, fast reproducing and aggressive P. volitans in coral reefs causes severe declines in the abundance and diversity of reef herbivores. There is also widespread cannibalism amongst P. volitans populations. A mathematical model is proposed to study the effects of predation on the biomass of herbivorous reef fishes by considering two life stages and intraguild predation of P. volitans population with harvesting of adult P. volitans. The system undergoes a supercritical Hopf bifurcation when the invasiveness of P. volitans crosses a certain critical value. It is observed that cannibalism of P. volitans induces stability in the system even with high invasiveness of adult P. volitans. The dynamic instability of the system due to higher invasiveness of P. volitans can be controlled by increasing the rate of harvesting of P. volitans. It is also proven that P. volitans goes extinct when the harvest rate is greater than some critical threshold value. These results indicate that the dynamical behaviour of the model is very sensitive to the harvesting of P. volitans, which in turn is useful in the conservation of reef herbivores.
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