![International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 13, Issue 5 (May 2017), PP.34-36
34
Qualitative Analysis of Prey Predator System
With Immigrant Prey
Habtay Ghebrewold and Goteti V. Sarma
Department of Mathematics, Eritrea Institute of Technology Mainefhi, Asmara Eritrea
ABSTRACT: The predator prey system with immigrant prey is introduced and studied through a suitable
mathematical model. Existence conditions for interior equilibrium point and their stability is studied under
suitable ecological restrictions. Global stability of the system around equilibrium point is also discussed.
I. INTRODUCTION
Interaction between the predators and prey is always influenced by various ecological factors like
climatic conditions, distribution of prey, predators ability to detect the prey etc. These interactions caught the
attention of several authors and well studied by constructing suitable mathematical models. Some authors
studied the affect of providing additional food to the predator, the effect of securing prey in refugee camps in
achieving the ecological equilibrium. The element of immigration in any ecological model exerts a lot influence
on the existing system which can make or mar the equilibrium in the existing system. By studying
systematically the role of immigration in any prey predator system one can understand the inherent inter
dependency of the system which enables ecological planners to set rules to permit immigration in to the system.
This article is organized as follows. In section 2 a mathematical model is constructed which
incorporates the immigrant prey in the existing predator prey models. Section 3 studies the existence of
equilibrium points and establishes the boundedness of the modeal. Section 4 elucidates the dynamical behavior
of the system around equilibrium points is analyzed. In section 5 global behavior of the system is discussed. In
section 6 scope for further research is mentioned.
II. MATHEMATICAL MODEL FORMULATION:
Let N and T represent the amount of biomass of prey and predator. Let A be the amount of immigrant
prey entering into the system. Let r represents the natural growth rate of prey and k is the maximum sustainable
prey populations without predator and immigration. We assume that the predation is according to Holings type
II functional response with c as the prey to predator conversion coefficient, e1 as predation ability of the
predator and h1 being the ability of predator to detect immigrant prey.
With the above ecological parameters we propose the following basic mathematical model to
understand the effect of immigration in prey predator system
𝑁/
= 1 − 𝐴 𝑟𝑁 1 −
𝑁
𝐾
−
𝑒1 𝐴𝑁𝑃
1+𝑒1ℎ1 𝐴𝑁
𝑃/
= 𝑃 −𝑚 + 𝑐
𝑒1 𝐴𝑁
1+𝑒1ℎ1 𝐴𝑁
(1)
Without loss of generality we can restrict A in [0, 1].
III. EQUILIBRIUM ANALYSIS :
For equilibrium points consider the stable state equations of (1)
1 − 𝐴 𝑟𝑁 1 −
𝑁
𝐾
−
𝑒1 𝐴𝑁𝑃
1+𝑒1ℎ1 𝐴𝑁
= 0
𝑃 −𝑚 + 𝑐
𝑒1 𝐴𝑁
1+𝑒1ℎ1 𝐴𝑁
= 0
Clearly the above system has 3 solutions only which gives the equilibrium points of the dynamical system (1)
namely
i. E0 = (0, 0) a trivial equilibrium point
ii. E1 = (k, 0) an axial equilibrium point and
iii. The equilibrium point of coexistence E2 = ( N*
, P*
) where
𝑁∗
=
𝑚
𝑒1 𝐴(𝑐−𝑚ℎ1)
𝑎𝑛𝑑 𝑃∗
= 1 − 𝐴
𝑐𝑟
𝑚
𝑁∗
1 −
𝑁∗
𝑘
For the existence E2 it is essential that c > mh1 which can be justified in perspective of ecological parameters
and also essentially 𝐴 >
𝑚
𝑒1 𝑘(𝑐−𝑚ℎ1)
.](https://image.slidesharecdn.com/d13513436-170718084250/85/Qualitative-Analysis-of-Prey-Predator-System-With-Immigrant-Prey-1-320.jpg)
![Qualitative Analysis Of Prey Predator System With Imigrant Prey
36
Hence the proof completed.
Now we are going to establish the global stability of the system through the following theorem:
Theorem 4.2 : For each A, 0 ≤ 𝐴 ≤ 1, any solution of the ecological system with positive initial conditions
enters the bounded region R in finite time and remains there thereafter.
Proof: Take the curve 𝑚𝑃 = 𝑐 1 − 𝐴 𝑟 𝑁 1 −
𝑁
𝐾
Then clearly 𝑚𝑃 ≤ 𝐹 𝑁 where 𝐹 𝑁 = 𝑐𝑟𝑁 1 −
𝑁
𝑘
Here F(N) is a parabola with downward opening which has maximum at 𝑁 =
𝐾
2
and its maximum value is
𝑐𝑟𝑘
4
.
Also 𝐹 𝑁 = 0 has a unique positive solution at 𝑁 = 𝑘
Now define 𝐹∗
𝑁 = 𝑐𝑟𝑁 1 −
𝑁
𝑘
+ 𝜀 where 𝜀 can be choosen so that 𝐹∗
(N) = 0 has a positive root.
Now choose 𝑙∗
such that the line 𝑐𝑁 + 𝑃 = 𝑙∗
either tangent to 𝐹∗
(N) or passes through the positive root of
𝐹∗
𝑁 = 0.
Then for (N, P) on 𝐹∗
(N), 𝑐 1 − 𝐴 𝑟 𝑁 1 −
𝑁
𝐾
− 𝑚𝑃 ≤ −𝜀
Now consider 𝐼 = 𝑐𝑁 + 𝑃
Then 𝐼′
= 𝑐𝑁′
+ 𝑃′
= 𝑐 1 − 𝐴 𝑟 𝑁 1 −
𝑁
𝐾
− 𝑚𝑃 ≤ −𝜀 for every 𝐼 > 𝑙∗
Thus for any initial conditions in the first quadrant the solution of the ecological system reaches to the line
𝑐𝑁 + 𝑃 = 𝑙∗
in finite time.
Hence the solution enters the region R bounded by the lines 𝑁 ≥ 0, 𝑃 ≥ 0, 𝑐𝑁 + 𝑃 ≤ 𝑙∗
and never leaves it.
V. GLOBAL STABILITY ANALYSIS:
In this section we will study the global stability of the system around the interior equilibrium point E2= (N*
, P*
).
First let us construct a suitable Lyapunov function as
𝐿 𝑁, 𝑃 = 𝑐1 𝑁 − 𝑁∗
𝑙𝑛𝑁 + 𝑐2 𝑃 − 𝑃∗
𝑙𝑛𝑃 where 𝑐1 , 𝑐2 are positive constants.
Then clearly 𝐿 𝑁, 𝑃 > 0 i.e L is a positive definite function
Now
𝐿/
𝑁, 𝑃 = 𝑐1
𝑁/
𝑁
𝑁 − 𝑁∗
+ 𝑐2
𝑃/
𝑃
𝑃 − 𝑃∗
= 𝑐1 1 − 𝐴 𝑟 1 −
𝑁
𝐾
−
𝑒1 𝐴𝑃
1+𝑒1ℎ1 𝐴𝑁
𝑁 − 𝑁∗
+𝑐2 −𝑚 + 𝑐
𝑒1 𝐴𝑁
1+𝑒1ℎ1 𝐴𝑁
𝑃 − 𝑃∗
Now it can be easily shown that 𝐿/
𝑁, 𝑃 ≤ 0 for a proper choice of the constants 𝑐1 , 𝑐2 which depends on the
parameters 𝐴, 𝐾, 𝑚 𝑎𝑛𝑑 𝑒1.
VI. CONCLUSIONS AND SCOPE FOR FURTHER INVESTIGATIONS
In this research article the effect of immigrant prey to the existing prey predator model is studied. It is
observed that the global stability of the model can be achieved through the careful allowance of amount of
immigrant prey in to the system. This model also offers to study in detail for periodic solutions, occurrences of
Hopf bifurcation and optimal analysis for further investigations. Further study is also possible on replacing the
immigration prey quantity with a variable.
REFERENCES
[1]. R.Clark Robinson, An introduction to dynamical systems, 2004
[2]. Jawdat Alebraheem, Persistence of predators in a two predators one prey model with non periodic
solution, Applied mathematical sciences, Vol 6, 2012 N0:19, 943-956
[3]. S. Gakkhar, B. Sing, R.K.Naji Dynamical behavior of two predators competing over a single prey,
Biosystems, 90 , 2007, 808-817
[4]. B.Dubey, R.K.Upadhyay. Persistence and extinction of one prey two predators system, Non linear
analysis: Modeling and control, 9, 2004, 307-329
[5]. H.I.Freedman, Deterministic mathematical models in population ecology, Newyork 1980
[6]. J.N.Kapur, Mathematical modeling in biology and medicine, East west press 1981
[7]. T.K.Kar, A. Batayal, Persistence and stability of two prey and one predator systems, Int. jour.
Engg.Sci.Tech. 2(1), 2010, 174-190
[8]. T.K.Kar and S.Misra, Influence of prey reserve in a prey predator fishery, Nonlinear analysis 65, 2006,
1725-1735
[9]. J.D.Murray, Mathematical biology I, Springer](https://image.slidesharecdn.com/d13513436-170718084250/85/Qualitative-Analysis-of-Prey-Predator-System-With-Immigrant-Prey-3-320.jpg)
Qualitative Analysis of Prey Predator System With Immigrant Prey
- 1. International Journal of Engineering Research and Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 5 (May 2017), PP.34-36 34 Qualitative Analysis of Prey Predator System With Immigrant Prey Habtay Ghebrewold and Goteti V. Sarma Department of Mathematics, Eritrea Institute of Technology Mainefhi, Asmara Eritrea ABSTRACT: The predator prey system with immigrant prey is introduced and studied through a suitable mathematical model. Existence conditions for interior equilibrium point and their stability is studied under suitable ecological restrictions. Global stability of the system around equilibrium point is also discussed. I. INTRODUCTION Interaction between the predators and prey is always influenced by various ecological factors like climatic conditions, distribution of prey, predators ability to detect the prey etc. These interactions caught the attention of several authors and well studied by constructing suitable mathematical models. Some authors studied the affect of providing additional food to the predator, the effect of securing prey in refugee camps in achieving the ecological equilibrium. The element of immigration in any ecological model exerts a lot influence on the existing system which can make or mar the equilibrium in the existing system. By studying systematically the role of immigration in any prey predator system one can understand the inherent inter dependency of the system which enables ecological planners to set rules to permit immigration in to the system. This article is organized as follows. In section 2 a mathematical model is constructed which incorporates the immigrant prey in the existing predator prey models. Section 3 studies the existence of equilibrium points and establishes the boundedness of the modeal. Section 4 elucidates the dynamical behavior of the system around equilibrium points is analyzed. In section 5 global behavior of the system is discussed. In section 6 scope for further research is mentioned. II. MATHEMATICAL MODEL FORMULATION: Let N and T represent the amount of biomass of prey and predator. Let A be the amount of immigrant prey entering into the system. Let r represents the natural growth rate of prey and k is the maximum sustainable prey populations without predator and immigration. We assume that the predation is according to Holings type II functional response with c as the prey to predator conversion coefficient, e1 as predation ability of the predator and h1 being the ability of predator to detect immigrant prey. With the above ecological parameters we propose the following basic mathematical model to understand the effect of immigration in prey predator system 𝑁/ = 1 − 𝐴 𝑟𝑁 1 − 𝑁 𝐾 − 𝑒1 𝐴𝑁𝑃 1+𝑒1ℎ1 𝐴𝑁 𝑃/ = 𝑃 −𝑚 + 𝑐 𝑒1 𝐴𝑁 1+𝑒1ℎ1 𝐴𝑁 (1) Without loss of generality we can restrict A in [0, 1]. III. EQUILIBRIUM ANALYSIS : For equilibrium points consider the stable state equations of (1) 1 − 𝐴 𝑟𝑁 1 − 𝑁 𝐾 − 𝑒1 𝐴𝑁𝑃 1+𝑒1ℎ1 𝐴𝑁 = 0 𝑃 −𝑚 + 𝑐 𝑒1 𝐴𝑁 1+𝑒1ℎ1 𝐴𝑁 = 0 Clearly the above system has 3 solutions only which gives the equilibrium points of the dynamical system (1) namely i. E0 = (0, 0) a trivial equilibrium point ii. E1 = (k, 0) an axial equilibrium point and iii. The equilibrium point of coexistence E2 = ( N* , P* ) where 𝑁∗ = 𝑚 𝑒1 𝐴(𝑐−𝑚ℎ1) 𝑎𝑛𝑑 𝑃∗ = 1 − 𝐴 𝑐𝑟 𝑚 𝑁∗ 1 − 𝑁∗ 𝑘 For the existence E2 it is essential that c > mh1 which can be justified in perspective of ecological parameters and also essentially 𝐴 > 𝑚 𝑒1 𝑘(𝑐−𝑚ℎ1) .
- 2. Qualitative Analysis Of Prey Predator System With Imigrant Prey 35 It is interesting to observe that the interior equilibrium point is always dependent on the parametric value of biomass of immigrant prey entered in to the system which in turn dependent on other ecological parameters of the system. Therefore we can determine the amount of biomass of immigrant prey based on required stable levels of biomasses of the prey and predator of the system under investigation. IV. DYNAMICAL BEHAVIOR OF THE SYSTEM To understand the dynamics and local stability of the equilibrium points we use the variational principle. The variational matrix for the system (1) is J (N, P) = 1 − 𝐴 𝑟 1 − 2𝑁 𝐾 − 𝑒1 𝐴𝑃 (1+𝑒1ℎ1 𝐴𝑁)2 −𝑒1 𝐴𝑁 (1+𝑒1ℎ1 𝐴𝑁) 𝑐𝑒1 𝐴𝑃 (1+𝑒1ℎ1 𝐴𝑁)2 −𝑚 + 𝑐𝑒1 𝐴𝑁 (1+𝑒1ℎ1 𝐴𝑁) Now let us analyze each of the equilibrium points 𝐸1 , 𝐸2 𝑎𝑛𝑑 𝐸3 1. At the equilibrium point E1 (0, 0) the variational matrix becomes J (0, 0) = 1 − 𝐴 𝑟 0 0 −𝑚 whose eigenvalues are (1-A) r and - m which are having opposite signs hence the equilibrium point E1 (0, 0) is always a unstable saddle point. 2. At the equilibrium point E2 (k, 0) the variational matrix becomes J (k, 0) = − 1 − 𝐴 𝑟 −𝑒1 𝐴𝑘 (1+𝑒1ℎ1 𝐴𝑘) 0 −𝑚 + 𝑐𝑒1 𝐴𝑘 (1+𝑒1ℎ1 𝐴𝑘) whose eigenvalues are - (1-A) r and −𝑚 + 𝑐𝑒1 𝐴𝑘 (1+𝑒1ℎ1 𝐴𝑘) which are having opposite signs under natural assumption 𝑚 < 𝑐𝑒1 𝐴𝑘 (1+𝑒1ℎ1 𝐴𝑘) hence the equilibrium point (k, 0) is also always a saddle point. Now we are going to discuss the nature of the interior equilibrium point E2 = ( N* , P* ) where 𝑁∗ = 𝑚 𝑒1 𝐴(𝑐−𝑚ℎ1) 𝑎𝑛𝑑 𝑃∗ = 1 − 𝐴 𝑐𝑟 𝑚 𝑁∗ 1 − 𝑁∗ 𝑘 and c > mh1 and 𝐴 > 𝑚 𝑒1 𝑘(𝑐−𝑚ℎ1) Theorem 4.1: The interior equilibrium point E2= (N* , P* ) is locally asymptotically stable if 𝐴 < 𝑚 𝑒1 𝑘(𝑐−𝑚ℎ1) 2−𝑚 𝑐−𝑚ℎ1 1−𝑚 𝑐−𝑚ℎ1 Proof: The variational matrix at the interior equilibrium point is V (N* , P* ) = 𝑣11 𝑣12 𝑣21 𝑣22 Where 𝑣11 = 1 − 𝐴 𝑟 1 − 2𝑁∗ 𝑘 − 𝑚 1− 𝑁∗ 𝑘 1+𝑒1ℎ1 𝐴𝑁∗ , 𝑣12 = − 𝑚 𝑐 , 𝑣21 = 𝑃∗ 𝑁∗ 𝑚 1+𝑒1ℎ1 𝐴𝑁∗ and 𝑣22 = 0 For asymptotical stability of E2, by Routh Hurwitz criteria 𝑣11 + 𝑣22 < 0 and 𝑣12 + 𝑣21 > 0 For this 1 − 2𝑁∗ 𝑘 − 𝑚 1− 𝑁∗ 𝑘 1+𝑒1ℎ1 𝐴𝑁∗ < 0 ⇒ 2𝑁∗ 𝑘 + 𝑚 1− 𝑁∗ 𝑘 1+𝑒1ℎ1 𝐴𝑁∗ > 1 ⇒ 𝑚 1− 𝑁∗ 𝑘 1+𝑒1ℎ1 𝐴𝑁∗ > 1 − 2𝑁∗ 𝑘 On substituting 𝑁∗ and rearranging gives the restriction 𝐴 < 𝑚 𝑒1 𝑘(𝑐−𝑚ℎ1) 2−𝑚 𝑐−𝑚ℎ1 1−𝑚 𝑐−𝑚ℎ1 . Surprisingly when this restriction holds, it satisfies 𝑣12 + 𝑣21 > 0 also.
- 3. Qualitative Analysis Of Prey Predator System With Imigrant Prey 36 Hence the proof completed. Now we are going to establish the global stability of the system through the following theorem: Theorem 4.2 : For each A, 0 ≤ 𝐴 ≤ 1, any solution of the ecological system with positive initial conditions enters the bounded region R in finite time and remains there thereafter. Proof: Take the curve 𝑚𝑃 = 𝑐 1 − 𝐴 𝑟 𝑁 1 − 𝑁 𝐾 Then clearly 𝑚𝑃 ≤ 𝐹 𝑁 where 𝐹 𝑁 = 𝑐𝑟𝑁 1 − 𝑁 𝑘 Here F(N) is a parabola with downward opening which has maximum at 𝑁 = 𝐾 2 and its maximum value is 𝑐𝑟𝑘 4 . Also 𝐹 𝑁 = 0 has a unique positive solution at 𝑁 = 𝑘 Now define 𝐹∗ 𝑁 = 𝑐𝑟𝑁 1 − 𝑁 𝑘 + 𝜀 where 𝜀 can be choosen so that 𝐹∗ (N) = 0 has a positive root. Now choose 𝑙∗ such that the line 𝑐𝑁 + 𝑃 = 𝑙∗ either tangent to 𝐹∗ (N) or passes through the positive root of 𝐹∗ 𝑁 = 0. Then for (N, P) on 𝐹∗ (N), 𝑐 1 − 𝐴 𝑟 𝑁 1 − 𝑁 𝐾 − 𝑚𝑃 ≤ −𝜀 Now consider 𝐼 = 𝑐𝑁 + 𝑃 Then 𝐼′ = 𝑐𝑁′ + 𝑃′ = 𝑐 1 − 𝐴 𝑟 𝑁 1 − 𝑁 𝐾 − 𝑚𝑃 ≤ −𝜀 for every 𝐼 > 𝑙∗ Thus for any initial conditions in the first quadrant the solution of the ecological system reaches to the line 𝑐𝑁 + 𝑃 = 𝑙∗ in finite time. Hence the solution enters the region R bounded by the lines 𝑁 ≥ 0, 𝑃 ≥ 0, 𝑐𝑁 + 𝑃 ≤ 𝑙∗ and never leaves it. V. GLOBAL STABILITY ANALYSIS: In this section we will study the global stability of the system around the interior equilibrium point E2= (N* , P* ). First let us construct a suitable Lyapunov function as 𝐿 𝑁, 𝑃 = 𝑐1 𝑁 − 𝑁∗ 𝑙𝑛𝑁 + 𝑐2 𝑃 − 𝑃∗ 𝑙𝑛𝑃 where 𝑐1 , 𝑐2 are positive constants. Then clearly 𝐿 𝑁, 𝑃 > 0 i.e L is a positive definite function Now 𝐿/ 𝑁, 𝑃 = 𝑐1 𝑁/ 𝑁 𝑁 − 𝑁∗ + 𝑐2 𝑃/ 𝑃 𝑃 − 𝑃∗ = 𝑐1 1 − 𝐴 𝑟 1 − 𝑁 𝐾 − 𝑒1 𝐴𝑃 1+𝑒1ℎ1 𝐴𝑁 𝑁 − 𝑁∗ +𝑐2 −𝑚 + 𝑐 𝑒1 𝐴𝑁 1+𝑒1ℎ1 𝐴𝑁 𝑃 − 𝑃∗ Now it can be easily shown that 𝐿/ 𝑁, 𝑃 ≤ 0 for a proper choice of the constants 𝑐1 , 𝑐2 which depends on the parameters 𝐴, 𝐾, 𝑚 𝑎𝑛𝑑 𝑒1. VI. CONCLUSIONS AND SCOPE FOR FURTHER INVESTIGATIONS In this research article the effect of immigrant prey to the existing prey predator model is studied. It is observed that the global stability of the model can be achieved through the careful allowance of amount of immigrant prey in to the system. This model also offers to study in detail for periodic solutions, occurrences of Hopf bifurcation and optimal analysis for further investigations. Further study is also possible on replacing the immigration prey quantity with a variable. REFERENCES [1]. R.Clark Robinson, An introduction to dynamical systems, 2004 [2]. Jawdat Alebraheem, Persistence of predators in a two predators one prey model with non periodic solution, Applied mathematical sciences, Vol 6, 2012 N0:19, 943-956 [3]. S. Gakkhar, B. Sing, R.K.Naji Dynamical behavior of two predators competing over a single prey, Biosystems, 90 , 2007, 808-817 [4]. B.Dubey, R.K.Upadhyay. Persistence and extinction of one prey two predators system, Non linear analysis: Modeling and control, 9, 2004, 307-329 [5]. H.I.Freedman, Deterministic mathematical models in population ecology, Newyork 1980 [6]. J.N.Kapur, Mathematical modeling in biology and medicine, East west press 1981 [7]. T.K.Kar, A. Batayal, Persistence and stability of two prey and one predator systems, Int. jour. Engg.Sci.Tech. 2(1), 2010, 174-190 [8]. T.K.Kar and S.Misra, Influence of prey reserve in a prey predator fishery, Nonlinear analysis 65, 2006, 1725-1735 [9]. J.D.Murray, Mathematical biology I, Springer