Dynamics Of Predator-Prey Model With Disease In
Prey And Nonlinear Harvesting Predator
Chandrali Baishya
Abstract: In this paper, we investigate the dynamics of a predator-prey model having disease in prey in the presence of quadratic harvesting. The boundedness and permanence of solutions are studied. We examine the dynamical behavior of the system at points of equilibrium. Sufficient conditions are derived for the system at axial equilibria, disease free equilibria and predator extinct equilibria. We analyse the global stability of the interior equilibrium point by using Lyapunov function. In support of the theoretical results, we perform numerical simulations.
Keywords: Predator-prey model, Infectious disease, Predator harvesting, Lyapunov function. 2000 Mathematics Subject Classification: 34A34, 34C11, 34D05, 34D20, 34D23.
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1.
INTRODUCTION
Since the pioneering work of Lotka [21] and Volterra [30], the modelling of interaction of biological species have reached a new milestone and involvement of mathematicians in it opened a new door to a discipline called mathematical biology[23, 22, 24]. In the past few decads, there have been massive interest among the researchers to design and study the mathematical/statistical models of population interactions. As a result of this the models like Holling Type II,III (1959), Leslie model(1945), Rosenzweig-macArthur model (1963), Yodzis model(1989) are evolved. Some of these models have prey dependent interaction and some have ratio-dependent interaction which indeed carried on lots of debate [20, 28, 2, 11, 8, 9]. In 1927, Kermach and Mackendrick [18] proposed the classical susceptible, infectious, recovered model which has attracted many reserachers to contribute in this area [3, 5, 10, 13]. Ecology and epidemiology are two different and independent fields for years. However, as time goes by, these two fields have come more and more closer and emerged into a new cross field called eco-epidemiology emerges. Anderson and May [1] investigated the dynamics of eco-epidemiology model for the first time where the predator interacts with infected prey population. This area may be broadly classified into three categories. The first one is the predator-prey model with disease in prey population [12, 29, 36, 4, 6, 19, 27]. Sharma et.al.[27] studied a two prey one predator model with disease in first prey population. The second category of model is the predator-prey model with disease in predator [33, 16, 35]. Recently Auger et al. have studied the effects of a disease affecting a predator on the dynamics of a predator-prey system, and they have observed two possible asymptotic behaviours: either the predator population dies out and the prey tends to its carrying capacity,or the predator and prey coexist. In this latter case, the predator population tends either to a disease-free or to a disease-endemic state. The third one in this line is the predator-prey model with disease in both the population [14, 7, 17].
In population management, many species are harvested or mined to balance the eco-system. For example, in 1995 at McGrath area of Alaska, 80 percent of the wolves were removed to bring the moose population to normal level [34]. Biswas et al.[4] studied an eco-epidemiological system with weak Allee effect and harvesting in prey population, and found that optimal harvesting policy is dependent on both gestation delay and Allee effect. Chattopadhy et.al. [6] investigated a harvesting predator-prey model with infection in the prey population and concluded that harvesting on infected prey prevents he limit cycle oscillations. Khan et al. [19] investigated a predator-prey model with density constraints for susceptible prey population, and considered harvesting to both susceptible and infected prey species. Huang et al. [15] presented a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, and obtained various bifurcations, such as saddle-node bifurcation, Hopf-bifurcation, repelling and attracting Bogdanov-Takens bifurcation. In [31, 32], the authors considered the non-delayed system with constant harvesting rate. Zhang et.al. [37] performed Bifurcation analysis of the modified Leslie-Gower model with nonlinear prey harvesting. Peng Feng [25] and Saleh [26] investigated predator-prey model with ratio-dependent functional response and quadratic predator harvesting. Main objective of this paper is to perform qualitative analysis of a predator-prey model with disease in prey in presence of quadratic predator harvesting. The paper is organized as follows. In Section 2 we describe the model formulation. In Section 3 and 4, we study respectively boundedness and permanence of the solutions. Section 5 and 6 deal with the derivation of sufficient conditions for stability of various types of equlibrium points. In Section 7, we performed some numerical simulation to srengthen our results.
2. MODEL FORMULATION
A Predator-prey model having disease in prey with quadratics predator harvesting takes the form
(1)
(1)
subject to the positive initial conditions
____________________
Department of Studies and Research in Mathematics Tumkur University, Karnataka, India
. Here is susceptible prey density,
is infected prey density and is the predator density.
Model 1 is defined on the set
with all the parameters
being positive. Biological meaning of the parameters are given in the Table 1.
Table 1: Meaning of symbols
r intrinsic growth rate of prey K environmental carrying capacity of
prey
B transmission rate of disease from infected prey to susceptible prey
predation rate of susceptible prey predation rate of infected prey Mortality rate of infected prey mortality rate of predator
conversion factor of predator after consuming susceptible prey
conversion factor of predator after consuming susceptible prey
half saturation constant harvesting effort
To formulate the eco-epidemological model we have made the following assumptions:
1. The total biomass of prey consists of two classes (a) susceptible prey (b) infected prey .
2. Speading of disease among prey population is horizontal(i.e. by contact only) not vertical. The infected preys donot recover.
3. Both susceptible and infected preys are captured by predators. Since susceptible preys are stronger than infectious preys, so later one is more vulnerable to predation and hence
4. In absence of the predator, susceptible prey population grow logistically.
5. Death of infected prey occurs naturally as well as by predation.
6. The interaction between prey and predator is expressed by a Holling Type-II functional response. 7. Since the prey population is reducing by infection as well as predation, in order to save the prey population from extinction, we have introduced the nonlinear harvesting
in the system 1. In order to simplify the system 1, we nondimensionalize it by the following substitutions:
and . This yields the system of equations
(2) (2) where
are positive constants subject to the initial condition
3. BOUNDEDNESS
Lemma 3.1 If and with
then
Lemma 3.1 is equvalent to the following Lemma 3.2. Lemma 3.2 If and with
, then for all
, .
In Particular, for all . From suceptible prey equation,
therefore as Again, from infected prey equation ,
This implies
This yields . If then for ,
From the predator equation,
Therefore and hence
4. PERMANENCE
Permanence: From biological point of view, permanence of a system means the survival of all populations of the system in future time. Mathematically, permanence of a system is defined as follows:
Definition 4.1 System 2 with given initial conditions is permanent if there are positive constants and
such that each positive solution of the system satisfies
From boundedness we have
with
From susceptible prey equation
or, Let So, Therefore, From the predator equation,
From boundedness
, . If and are positive then this shows that the solutions are permanent.
5. STABILITY ANALYSIS
To find the equilibrium points of the model 2, we study the zeroth growth isoclines and the points of interaction. The equilibrium points of the system 2 are obtained by solving
,
and
Theorem 5.1
(i) The trivial equlibrium in unstable. (ii) Axial equlibrium point is stable if
and .
(iii) Predator extinct equilibrium
exist if and is stable if
(iv) Disease free equilibrium points are stable if ,
and
(v) Interior equilibrium exists if
Proof. (ii) At the equilibrium point the eigenvalues of the Jacobian matrix are {
} . For
stability it must satisfies the conditions
and
. Both together
implies that
.
(iii) Predator extinct equilibrium point is . For
exixtence it must satisfy . The eigenvalues of
the Jacobian matrix are √ √ and
For stability eigenvalues must be negative or complex congugate with negative real parts.
implies that .
We know that
implies
.
Also, since .
This implies that are always negative or have negative real parts.
(iv) Disease free equilibrium points are roots of the cubic equation
( ) .
And since and , there is
atlease one positive root i.e. atleast one disease free equilibrium point. For analysis of disease free equilibrium point we write the Jacobian matrix
(
)
where ; ;
;
Characteristic polynomial is
The coefficients of the characteristic polynomial are 1,
;
;
By Routh equilibrium criterion we constuct the following table
If all the elements of the first column have same sign, then the equilibrium point is stable. Since 1 is positive, therefore the sufficient condition for stability is
, , .
This implies, , and
Or, ,
and
(v) Interior equilibrium points are the root of the equation
(3)
where - - - d ;
d d - d - dp- dp;
d d d d d d d d dp
dp- dp ddp dp dp- dp
d d d d d
d d d d d d dp
dp ddp d dp ddp dp dp dp
d d d d d d ddp d dp dp dp
d d z
d d d d d d d
Clearly . Also
with an assumption that .
Therefore equation 3 has atleast one positive root.
and if
6. STABILITY OF INTERIOR EQUILIBRIUM
Theorem 6.1 The equilibrium point is globally stable if
Proof: Let us construct a Lyapunov function
(4) where, ,
and Time derivative of yield
(5)
(6)
(7)
(
)
Where,
Let
(
)
If for all and for , then the interior fixed points are globally stable. In the above expression , if the matrix M is positive definite and this needs every principal minor of M to be positive definite. ,
(
)
if
if i.e.
if
For this the sufficient condition is
Therefore
is negative definite if
7. NUMERICAL SIMULATION
The model 2 is integrated numerically by using Runge-Kutta method for different set of parameter values to verify the stability of different types of equilibrium points.
1. Axial Equilibrium: The parameters values are given as:
.
. Equilibrium point is . For the point eigenvalues of the Jacobian matrix are
. The equilibrium point is a stable focus as shown in Fig(1).
2. Predator Extinct Equilibrium: Parameters values are taken as
. These values satisfy the required condition and
. Eigenvalues of Jacobian matrix at point of Equilibrium are
. Therefore is a stable spiral which is shown in Fig(2) and Fig(3).
For the parameter values
and the feasible equilibrium point is and eigenvalues of Jacobian matrix are . Therefore is a stable focus as shown in the Fig(3)
Figure 2: Profile of Predator extinct equilibrium
Figure 3: Predator extinct equilibrium profile
3. Disease Free Equilibrium: The parameter values are taken as
. The feasible point of equilibrium point is . The conditions for stability of are satisfied: (
( ) ) and
(
) . At
eigenvalues of the Jacobian matrix are
. Therefore is a stable focus, shown in Fig(6) and Fig(7).
Figure 4: Disease free equilibrium profile
4. Interior Equilibrium: The parameter values are taken as
. Feasible point of equilibrium are
and
. At the condition for
existence is satisfied. At
condition for stability
is satisfied. It is seen that at this point eigenvalues of Jacobian matrix are
Figure 5: Endemic state
Figure 6: Solution curves tend to disease free state
8. CONCLUSION
In this paper, we propose a predator-prey model with disease in prey incorporating predator harvesting.The functional response infused in the model is the Holling type-II. Quadratic harvesting term in the third equation of the system 2 gives a more realistic feature, because many times to protect a predator species from extinction, harvesting shall be used as an effective tool. After formulating and then nondimensionalizing the model, we have verified the boundedness and permanence of the system 2 with the help of some existing theorem in the literature. The proposed system has trivial, axial, predator extinct, disease free and interior equilibrium points. Moreover, sufficient conditions for existence of equilibrium points and local stability of them are derived. Condition for global stability of coexistence equilibrium points are derived by constructing suitable Lyapunov function. To support the analytical results, we perform numerical simulations with the help of Runge-Kutta 4th order method and Mathematica Software. In the systeml 2, the prey population tends either to a disease free state where it tends to the carrying capacity or to a endemic state. This is numerically established in Fig(5) and Fig(6).
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