232 A DYNAMIC EFFECT OF INFECTIOUS DISEASE ON PREY PREDATOR SYSTEM AND HARVESTING POLICY BIOSCIENCE BIOTECHNOLOGY RESEARCH COMMUNICATIONS
Rachna Soni and Usha Chouhan
INTRODUCTION
Mathematical models have become important tools in
analyzing the dynamical relationship between predator
and their prey. The predator prey system is one of the
well-known models which have been studied and dis-
cussed a lot. The Lotka-Volterra predator prey system
has been proposed to describe the population dynam-
ics of two interacting species of a predator and its prey
(Lotka, 1925, Volterra, 1931, Arb Von et al., 2013),
Lotka–Volterra equation are of form
(1.1)
Where x and y are the prey and predator respectively; a
is the growth rate of the prey (species) in the absence of
interaction with the predator (species), b is the effect of
the predation of species to species, c is the growth rate
of species in perfect conditions: abundant prey and no
negative environmental impact and d is the death rate
of the species in perfect conditions: abundant prey and
no negative environmental impact from natural cause.
One of the unrealistic assumptions in the Lotka-Volt-
erra model is that the growth of the prey populations is
unbounded in the absence of the predator. Murray (Mur-
ray, 1989) modi ed the Lokta-Volterra model and the
model were based on assumptions that the prey popula-
tion exhibits logistic growth in the absence of predators,
then the model obtained:
(1.2)
where a, b, c, d, k
1
, k
2
are all positive constants. This
model was investigated and the conditions for stabil-
ity obtained.Ecological populations suffer from various
types of diseases. These diseases often play signi cant
roles in balancing the population sizes. Most impor-
tant models for the transmission of infectious diseases
descend from the classical SIR model (Kermack and
McKendrick,1927). In the past decades, several epidemic
models with disease in prey have been extensively stud-
ied in various forms and contexts, for example, by Heth-
cote, (2000), Hethcote et al., (2004), Johri et al., (2012),
Nandi et al., (2015), Sujatha et al., (2016), Mbava, (2017)
and Yang, (2018).
In particular, a predator-prey model with disease in
the prey and analyzed a model of a three species eco-epi-
demiological system, namely, susceptible prey, infected
prey and predator (Chattopadhyay and Arino,1999).
Another prey-predator model with harvesting activity of
prey which has been observed is that when the harvest-
ing activity of prey is taken into consideration, then the
population size of predator decreases and the naturally
stable equilibrium of the model becomes unstable (Singh
and Bhatti, 2012). A mathematical model to study the
response of a predator-prey model to a disease in both
the populations and harvesting of each species (Das,
2014), the model with two-stage infection in prey, the
early stage of infected prey is more vulnerable to preda-
tion by the predator and the later stage of infected pests
is not eaten by the predator (Nandi el al., 2015), har-
vested prey – predator model with SIS epidemic disease
in the prey population (Sujatha et al., 2016). The preda-
tor–prey model with disease in super-predator are inves-
tigated and obtained the results that in the absence of
additional mortality on predator by a super - predator,
the predator species survives extinction (Mbava, 2017).
A diffusive predator-prey model with herd behavior has
been developed and the local and global stability of the
unique homogeneous positive steady state is obtained
(Yang, 2018).
A compartmental mathematical model based on the
dynamics of the infection and apply vaccination strate-
gies with herd immunity to reduce the intensity of dis-
ease spread in the prey-predator ecosystem (Bakare et al.,
2012). We considered the work proposed by E.A. Bakare,
because sometimes vaccination strategies become inef-
fective, in that case dynamic changes developed in the
system, which we were investigated in the present work.
We are trying to demonstrate the effect of vaccination
when it failed to recover from the disease One of the
purposes of this article is to explore the complex effect
of the prey predator model in epidemiological system
due to failure of vaccination strategies. The proposed
model is characterized by a pair of rst order nonlinear
differential equations and the existence of the possible
equilibrium points along with their stability is discussed.
And nally, some numerical examples are discussed.
MATERRIAL AND METHODS
We shall consider the following prey predator system for
analyzing it mathematically,
(2.1)