Effect of Holling type-II function on dynamics of discrete Prey-Predator system with scavenger

Discrete Prey-Predator System with Scavenger

Sudipa Chauhan and Sumit Kaur Bhatia Department of Mathematics (AIAS)

Amity University Noida, Sector-125, U.P

schauhan96@amity.edu, skbhatia@amity.edu

Surbhi Gupta Department of Mathematics (AIAS)

Amity University Noida, Sector-125, U.P

surbhi.gupta0027@yahoo.in

Abstract—A discrete prey predator model with scavengers is proposed. The model is build up by the interaction between three species prey predators and scavengers using difference equation. The stability analysis for the boundary fixed points is carried out showing the survival of prey population and prey-predator population both in the presence of scavenger.

Keywords—Prey-Predator; Scavenger; Holling Type-II; Fixed Point; Stability.

I. INTRODUCTION In mathematics, a dynamical system is a concept which describes the time dependence of a point in a geometrical space through a fixed point. Examples of dynamical system include the mathematical models which describe the swinging of a clock pendulum and the flow of water in a pipe. A dynamical system contains two elements: a set of possible states (represented by one or more variables) and a deterministic rule that describes the present state from past states. In the modeling of biological phenomena, two kinds of dynamical systems are being used which include discrete dynamical system and continuous dynamical system. In discrete dynamical system, the time domain is on fixed intervals whereas in continuous dynamical system, it becomes difficult to study the qualitative behavior of the solutions. To analyze and model many different types of real world problems like population growth, radioactive decay, and population control and medication dosages, discrete dynamical systems is preferred over continuous dynamical system. For example, situations like when a population changes at discrete moments in time such as each spring when new members of the population are born. The situations in which population at a certain stage is being determined by the population at a previous stage are described by a discrete dynamical system. A general algebraic approach was discussed in [8, 9] in which biological networks was modeled as continuous dynamical systems for the detection and analysis of stability of real equilibria.

In this paper, we are discussing a discrete dynamical system of prey predator model with scavengers. The scientific study of interactions between organisms and their environment is known as Ecology, which studies interactions of organisms among each other and with their abiotic environment. Lotka and Volterra introduced the first mathematical model which described the interaction of populations [4] and their system concludes qualitative behavior of various systems. We have considered three species in our model which includes prey, predator and scavengers. Scavengers play an important role in food web and consume decaying meat. These species are capable of breaking down the organic material, which includes bodies of dead animals, and recycling it into the ecosystems as nutrients [3, 10]. Examples of scavengers are vultures, insects like blowflies, mammals like Hyenas and many animals like lions, wolves and many others. There is no effect of scavenger species on prey or predator population. For example, introduction of carrion beetle in a standard hare/lynx system neither bene�t nor inhibits the hare or lynx. In the present model, effect of Holling type-II functional response is studied on a discrete prey-predator system with scavenger.

II. FORMULATION OF MATHEMATICAL MODEL

A. Mathematical Model

1. The prey population, which is assumed to be “x=x(t)” in the model, is growing exponentially at rate “r” when the predator is absent.

2. Presence of predators in the environment always has

a negative effect on the prey growth rate. If y=y(t) is the predator population at time t, then the rate with which predators feed the prey population has a negative impact on prey population. Thus, the

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evolutionary equation of prey population is as follows:

Where ‘a’ is the rate of decrease of prey population due to consumption by predator population.

3. The predator population dies if there is no food supply or no prey in the system and is proportional at rate “b” to its own size.

4. In presence of prey, predator population is

proportional to deaths of prey,

5. In this paper we have assumed that the scavenger

population, in the absence of prey and predator, disappear exponentially.

6. Scavengers get benefit from the natural death of both

prey and predators as well as from the predation of prey and predator population . Therefore, the equation becomes:

where “d” represents the natural death rate of the scavengers, “c” represents the rate at which the scavengers eats the dead bodies of the prey killed by the predator, “f” represents the rate of consumption of dead prey population who die naturally by scavengers and “e” represents the rate consumption of dead predator population who die naturally by scavengers. Thus, the mathematical proposed mathematical model is:

……..... (1)

…..……. (2)

……….. (3)

The above mathematical model comprises of differential equations which can be changed into difference equations by following method:

, ,

Considering the differential equation (1) of the above model and difference equation method, we get,

which gives

Similarly, we get

and

Therefore, the mathematical model (A) with difference equation is:

Where r, a, b, d, c, f, e > 0

III Stability Analysis For the above discrete mathematical model we are considering following three fixed points:

, and when .

The jacobian J(x,y,z) for the system is as follows:

………….. (4) LEMMA 1 Let ……… (5)

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be the characteristic equation for the above matrix then the following statements are true:

1. The fixed points of the system (A) are locally asymptotically stable and the equilibrium point is called a sink, if all the roots of the characteristic equation have absolute value less than one.

2. The fixed point of the system (A) is unstable and equilibrium point is saddle, if at least one of the roots of equation (5) has absolute value greater than one.

3. The fixed point of the system (A) is a source, if every

root of equation (5) has absolute value greater than one.

4. The fixed point of system (A) is hyperbolic if no root of the equation (5) has absolute value equal to one. If there exist a root of equation (5) with absolute value equal to one, then the fixed point is called non-hyperbolic.

We have the following results from following lemma: Proposition 1: The fixed point

(a) can never be a sink since r>0. (b) is saddle if r>0 or b>2 or d>2. (c) is source if r>0, b>2 and d>2. (d) is non-hyperbolic if r=0 or b=2 or d=2.

Proof: The jacobian matrix for the above model with the fixed point is given by:

The eigenvalues for the above matrix are:

and . Since, r>0 therefore (r+1) can never be less than 1 and hence we have at least one eigenvalue whose absolute value can be less than one and hence can never be a sink. can be saddle point if one of the eigenvalue has absolute value greater than 1 which is possible when either r>0 or |1-b|>1 which gives b<0 which is not possible due to assumptions of our model and b>2 or |1-d|>1 which gives d<0 which is again not possible due to assumptions of our model and d>2. can be source if absolute value of every eigenvalue is greater than 1 which is possible when r>0 and b>2 and d>2. The fixed point

can be non-hyperbolic when absolute value of the eigenvalue is equal to one which is possible when either r=0 or b=2 or d=2. Proposition 2: The fixed point

(a) is sink if 0<r<2, 0<a<2b where b>0 and f<d<2+f where f>0.

(b) is saddle if r>2 or a>2b. (c) can never be source. (d) is non-hyperbolic if r=2 or a=2b or d=f and r=2 or

a=2b-4 or d=f+2.

Proof: The Jacobian matrix for the above model with the fixed point is given by:

The eigenvalues for the above matrix are:

is sink when the absolute values of all the eigenvalues are less than one which is possible whenever |1-r|<1 which gives 0<r<2 and |1-b+a/2|<1 which gives 0<a<2b and |1-d+f|<1 which gives f<d<f+2.

is saddle point if one of the eigenvalue has absolute value greater than 1 which is possible when either |1-r|>1 or |1-b+a/2|>1 or |1-d+f|>1. Considering |1-r|>1 gives r<0 which contradicts our assumption and r>2. Also, |1-b+a/2|>1 gives a>2b and a<2b-4 and since (2b-4) can be negative which will contradict our assumption, hence we will not consider a<(2b-4). Now, |1-d+f|>1 gives f+2<d<f which is not possible. Therefore, can be saddle point when either r>2 or a>2b.

can be source if absolute value of every eigenvalue is greater than one, i.e.,|1-r|>1 and |1-b+a/2|>1 and |1-d+f|>1 which is possible when r>2, a>2b and f+2<d<f. As f>0, therefore f+2<d<f is never possible. Thus, can never be source.

can be non-hyperbolic when absolute value of the eigenvalue is equal to one that is either |1-r|=1 or |1-b+a/2|=1 or |1-d+f|=1 which gives r=2 or a=2b, or d=f or f+2. Proposition 2: The dynamics of is as follows:

a) is a sink if

1.

provided .

2. provided

and

b) is a saddle if

1. or

r provided

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2. provided

and . 3. r>0 provided .

c) It can never be a source.

d) Non-hyperbolic if either

or or r=0.

Proof: The Jacobian matrix for the above model with the fixed point is given by:

The eigenvalues for the above matrix are:

and

….. (6)

is a sink when the absolute values of all the eigenvalues are

less than one which is possible whenever <1 and

<1

which gives and

1. If then is a sink if

.

2. If and

then is sink if .

3. If which gives r<0 and hence in this case can never be a sink.

can be saddle point if one of the eigenvalue has absolute value greater than 1 which is possible when either

>1 or

>1 but (6) is always less

than one so we have conditions on first eigenvalues to be a saddle which are:

1. If then is saddle if either

or

2. If and

then is saddle if .

3. If then is saddle if r>0.

can be source if absolute value of every eigenvalue is greater than 1 which is possible when >1 and

>1

but we know (6) is always less than one therefore can never be a source.

can be non-hyperbolic when absolute value of the eigenvalue is equal to one that is either

or

=1

which gives either or

or r=0. This completes the proof. CONCLUSION AND FUTURE WORK: In this paper, the rich dynamics of the interspecies interaction of three species prey-predator and scavenger is proposed and dynamical behavior of fixed points is investigated. The Holling type-II function has a major effect on the fixed points. The fixed points was when linear interaction between prey-predator populations was considered [10]. Further, has also changed which was earlier We have also obtained interesting results for the fixed points that can never be sink and thus it is never stable i.e the trajectories will never approach to . can never be a source and can be sink if the conditions given in proposition 2 are satisfied. This concludes the survival of prey and predator population in the ecosystem in the presence of scavengers. The above model can be further extended by introducing new interacting terms like Holling Type –III functional response. We hope to investigate generalizations in future work.

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[6] T.R.Malthus, “An Essay on the Principle of Population, “Printed for J. Johnson, in St. Pauls Church-Yard, London, 1798. [7] M. Danca , S. Codreanu and B. Bako, “Detailed Analysis of a Nonlinear Prey-predator Model, “Journal of Biological Physics vol 23, pp. 11-20, 1997. [8] W. Niu and D.Wang. Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci.,1(3):507–539, 2008.

[9] D. Wang and B. Xia. Stability analysis of biological systems with real solution classification. In Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (Beijing, China, July 24–27, 2005), pages 354–361. ACM Press, New York, 2005. [10] M.R. Raj, A.G. M. Selvam and R.Janagaraj. Dynamical Behavior of Discrete Prey-Predator system with Scavenger, pp.27-33, 2013.

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