Impact of fear-effect on a delayed eco-epidemiological model with
standard incidence rate, Holling type-II functional response and fading
This study explores the impact of fear of predators among prey
populations in an eco-epidemiological model where an infectious disease
infects prey. An incidence delay is introduced for the transition of the
susceptible population into the infected population. Further, the
dynamical behavior of the non-delay system is studied by modifying
Holling type II functional response incorporating the fading memory.
This is based on the concept that the predator’s growth rate not only
depends on a single moment from the past but also over the whole past
(chiefly, on recent history). The conditions for the existence of all
the biologically feasible equilibrium points are established. The
criterion for the local stability about equilibrium points of both
(non-delay and delay) systems as well as for the global stability around
the coexistence equilibria of the non-delay system are established here.
Sufficient conditions for the existence of Hopf-bifurcation by taking
the force of infection and delay as bifurcation parameters are derived.
Numerical simulations are performed to verify the analytical results and
illuminate the system’s dynamicity. The system’s complex dynamical
behaviors are demonstrated using the bifurcation diagram, phase diagram,
and spectrum. It is observed that fears reduce predator density and also
convert an unstable (periodic or chaotic) system into a stable one. It
is recommended that past influence over a short time interval and a
higher value of the cost of fears are necessary to persist a sustainable
and stable ecosystem. An effort is made to search for the correlation
between the cost of fears and other biologically related parameters
(viz., the growth rate of prey, the carrying capacity for prey, the
force of infection, fading memory, and incidence delay) to understand
the dynamicity of the system. The semi-relative as well as logarithmic
sensitivities of the system are applied to the proposed model to observe
how much change of a state variable occur due to perturbation of the
parameters, like, the force of infection and cost of fears.