Dynamics of asymmetric intraguild predation with time lags in reproduction and maturation

A three dimensional (3D) stage-structured predator–prey 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 of delay differential forms of the ODE model are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a Hopf bifurcation when the time lags cross certain 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

Dynamics of the interaction of plankton and planktivorous fish with delay

This paper is devoted to the study of a plankton–fish ecosystem model. The model represents the interaction between phytoplankton, zooplankton, and fish with Holling II functional response consisting of carrying capacity and constant intrinsic growth rate of phytoplankton. It is observed that if the carrying capacity of phytoplankton population crosses a certain critical value, the system enters into Hopf bifurcation. We have introduced discrete time delay due to gestation in the functional response term involved with the growth equation of planktivorous fish. We have studied the effect of time delay on the stability behavior. In addition, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Existence of Hopf bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter. It is observed that constant intrinsic growth rate of phytoplankton and mortality rate of planktivorous fish play an important role in changing one steady state to another steady state and oscillatory behavior of the system. Computer simulations illustrate the results

Dynamics of a Ratio-Dependent Marine Bacteriophage Infection Model with Delay

A three dimensional mathematical model is considered to investigate the effect of virus infection in bacterial community. Due to virus infection the bacteria population is divided into susceptible class and infected class. The model is formulated by considering logistic growth of the susceptible class and a standard incidence function governing viral infection. The model is modified by considering an explicit time delay in virus production. The basic demographic reproductive number Rd, the basic epidemiological reproductive number R0 are used in qualitative analysis. We show that when R0 > 1, there exists a biologically meaningful steady state and the stability of this steady state is dependent upon both Rd and R0. Comparative study of models with delay and without delay are carried out both analytically and numerically

Potential effects of invasive Pterois volitans in coral reefs

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

Dynamics of a Ratio-Dependent Marine Bacteriophage Infection Model with Delay

A three dimensional mathematical model is considered to investigate the effect of virus infection in bacterial community. Due to virus infection the bacteria population is divided into susceptible class and infected class. The model is formulated by considering logistic growth of the susceptible class and a standard incidence function governing viral infection. The model is modified by considering an explicit time delay in virus production. The basic demographic reproductive number Rd, the basic epidemiological reproductive number R0 are used in qualitative analysis. We show that when R0 > 1, there exists a biologically meaningful steady state and the stability of this steady state is dependent upon both Rd and R0. Comparative study of models with delay and without delay are carried out both analytically and numerically

Herbivore harvesting and alternative steady states in coral reefs

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