Department of Mathematics, Maulana Azad National Institute of Technology Bhopal-462003, (M. P.) India

Article Publishing History

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 discussed a lot. The Lotka-Volterra predator prey system has been proposed to describe the population dynamics 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

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-Volterra model is that the growth of the prey populations is unbounded in the absence of the predator. Murray (Murray, 1989) modified the Lokta-Volterra model and the model were based on assumptions that the prey population exhibits logistic growth in the absence of predators, then the model obtained:

Materrial And Methods

We shall consider the following prey predator system for analyzing it mathematically,

Where x, y and z stand for the density of susceptible predator, susceptible prey andinfected prey populations, respectively.And the parameters ‘a’ is the natural death of the healthy susceptible predator, ‘b’ is the number of contact between susceptible prey and healthy susceptible predator, ‘c’ is the number of contact between healthy susceptible predator and infected prey, ‘e’ is the number of contact between healthy susceptible predator with infected prey and susceptible prey, ‘f’ is the number of contact between healthy susceptible prey and infected prey, ‘g’ is the harvesting rate of infected prey , h is the per capita birth rate of susceptible prey (per time) and infected prey and ö is the proportion of those successively vaccinated at birth.

The model consists of basic assumptions that we have made in formulating the model are: The relative birth rate for infected prey and that of susceptible prey remains the same.The disease is severely weakened and ultimately causing death for the predators. Once a predator is infected, it can be assumed to be dead. We will therefore consider only susceptible predator andinfectious disease spreads among the prey population by contact, and the rate of infection is proportional to the infected and the susceptible prey.The predator makes no difference between susceptible and infected members of the prey population. The predator becomes infected by consuming the infected prey. The rate of predator infection is proportional to the product of infected prey and susceptible predators.The infected prey does not recover.

To begin with, let us find the equilibrium points of the system (2.1)

The system (2.1) has the following equilibrium points:

Where x^*, y^*, z^* are given by

In the next section, let us discuss the stability of the five equilibrium points in the next which are obtained above.

Results and Discussion

Stability Analysis

In this section, we analyzed the local behavior of the system (2.1) around each equilibrium point. The Jacobian matrix of the system of state variables is as follows:

To determine the stability of the equilibrium points, we look at the most useful techniques for analyzing non-linear system is the linearized stability technique by theorem1.

Theorem 1:

1. If every root of the equation has absolute value less than one, then the equilibrium point of the system is locally asymptotically stable and equilibrium point is called a sink.
2. If at-least one of the roots of the equation has an absolute value greater than one, then the equilibrium point of the system is unstable and equilibrium point is called a saddle.
3. If every root of the equation has an absolute value greater than one, then the system is sourced.
4. The equilibrium point of the system is called hyperbolic if no root of the equation has absolute value equal to one. If there exists a root of the equation with absolute value equal to one, then the equilibrium point is called non-hyperbolic (i.e. one eigenvalue has a vanishing real part).

Let us preparefour propositions in order to discuss the local stability around each equilibrium point.

Preposition 1: For system (2.1),

The equilibrium point E₀ is locally asymptotically stable if h<1 and h<g.

Proof: The Jacobian matrix at E₀ (0,0,0) is given by

In the similar manner, we can show in the x-z plane for E1 with the condition ∆(x,z) < 0 for all x > 0 and z > 0 if h > 0, in the x-y plane for E₃ with the condition ∆(x,z) < 0 for all x > 0 and y > 0 if h,b < 0 and in the same way E₄ can be globally asymptotically stable in x-y,y-z and x-z planes.

We have performed some numerical simulation to study the role of harvesting on the prey predator system and we illustrate the dynamical and complex features of the model using MATLAB. In the starting, we fixed all parameters to ensure that the three classes of populations survive. Numerical simulations explain the effect of the parameters on the complex behavior of a given system (2.1).

(i) Let us consider following set of parameters,

a = 1.0; b = 1.5; c = 0.1; h = 0.5; e = 1.5;f=0.1;g=0.7,ö=0.91,

With initial condition x (0) =0. 8, y (0) =1. 70, z (0) =0. 75. For this set of parameter, we get the following variation of the population of the healthy predator, susceptible prey and infected prey with respect to time, which isillustrated below in figure 4 (a) and figure 4 (b).

(ii) Let us consider following set of parameters,

Figure 4a: Represents the effect of high harvesting on the population of the infected prey as time goes on

Figure 4b: Represents the effect of high harvesting on the population of the healthy predator, susceptible prey and infected prey as time goes on.

a = 1.0; b = 1.5; c = 0.1; h = 0.5; e = 1.5;f=0.1;g=0.1;ö=0.91;

With initial condition x (0) =0. 8, y (0) =1. 70, z (0) =0. 75. For this set of parameter, we get the following variation of the population of the healthy predator, susceptible prey and infected prey with respect to time, which is illustrated below in figure 4(c) and figure 4(d).

Figure 4c: Represents the effect of low harvesting on the population of the infected prey as time goes on

Figure 4d: Represents the effect of low harvesting on the population of the healthy predator, susceptible prey and infected prey as time goes on

It is observed that effective harvesting of diseased prey, increase the growth rate of the susceptible predator population. If the value of harvesting rate g≥0. 7 then the infected prey population decreases more rapidly, but if the value of g<0.7 then infected prey population decreases slowly that shown in fig. 4(a), 4(b), 4(c) and 4(d) respectively. In this analysis, we have also observed that the whole population of the susceptible predators may be wiped out due to increase in the number of the susceptible and infected preys. This result shows that the system is biologically well behaved. In another case when the diseased prey can be washed out, a rational use of the stability criterion of non-zero equilibrium point may be useful for ecological balance. In this case, the parameters of the system should be regulated in such a way that stability criterion of non-zero equilibrium is satisfied but infected prey remains low enough. Sometimes, harvesting became a suitable option for prevention of the population rather than the vaccination strategies. Therefore, effective harvesting became essential for the survival of the population.

Conclusion

A non-linear system based on the epidemic SIR model has been studied and discussed. Conditions for local and global stability at various equilibrium points were obtained. We have illustrated the effective harvesting of diseased prey in the whole system and reveal that the increases of predator population when the harvesting rate of infected prey population increases. We may conclude that effective harvesting of diseased prey may be used as a biological control for the spread of disease. And maintain balance in these species populations by preventing in the predator population to extinction. Finally, some numerical simulations illustrate and supplement our theoretical analysis by considering different parameter values. Low harvesting and high harvesting rates play an important role in this analysis. Global stability of equilibrium E₀ shows that disease free equilibrium always exists. In future other effecting condition can be used to save the predator population by introducing alternative food for predator rather than diseased prey.

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