Interaction among toxic phytoplankton with viral infection and zooplankton in presence of multiple time delays

Abstract

We consider an eco-epidemiological model consisting with toxin producing phytoplankton and zooplankton. In this model, only phytoplankton population is infected by some viral infection. So, the phytoplankton species is divided into two sub-species namely susceptible phytoplankton and infected phytoplankton with the assumption that viral infection is removable. Since infected phytoplankton is more weaker than susceptible phytoplankton, the capturing rate of susceptible phytoplankton is smaller than that of infected phytoplankton. As a result, zooplankton’s functional response for susceptible phytoplankton may be taken as Holling Type-II while the interaction between infected phytoplankton and zooplankton is governed by Holling type-I response. Also, zooplankton are always free from infection due to predation. Moreover, toxin producing phytoplankton are not only the food source for zooplankton but also depend on other food sources like bacterioplankton, small zooplankton etc. So, consideration of zooplankton as generalised predator is very significant ecological impacts in this work. Further, we assume that toxin liberation is not an instantaneous process, it is mediated by some time lag. We analyze the resulting model through various mathematical characteristics such as positivity, uniform boundedness, local stability and global stability. The system undergoes through different types of local bifurcations such as saddle-node, transcritical, Hopf which are of codimension-1 and cusp point, Zero Hopf, Bogdanov–Takens bifurcation which are of codimension-2 for the variation of suitable parameters. The model is incorporated with two different time lags due to toxin liberation and it is established that the time lags can destabilize system through the Hopf Bifurcation. We verify our analytical findings through numerical simulations using MATLAB.