Optimal Harvesting of Prey in Three Species Ecological Model with a Time Delay on Prey and Predator
Paparao A V^{1}, Lakshmi Narayan.K^{2}
^{1}Department of Mathematics, JNTUK, University College of Engineering,Vizianagarm535003, India
^{2}Department of Mathematics, VITS, Deshmukhi, Hyderabad  508284, India
*Corresponding Author Email: paparao.alla@gmail.com, narayan.kunderu@gmail.com
ABSTRACT:
This paper deals with the stability analysis of three species Ecological model consists of a Prey (N_{1}), a predator (N_{2}) and a competitor (N_{3}) .The competitor (N_{3}) is competing with the Predator Species (N_{2}) and neutral with the prey (N_{1}). In this model the third species is competing with the predator for food other than the prey (N_{1}).In addition to that, the death rates, carrying capacities of all three species are also considered , a delay in the interaction between the prey and the predator (gestation period of the predator) and harvesting effort of prey population is also considered. The model is characterized by a couple of integro differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function. Suitable parameter are identified for Numerical simulation which shows that this continuous time delay model exhibits rich dynamics and time delay can further stabilize or destabilize the system.
KEYWORDS: Prey, Predator, Competitor, Equilibrium points, local stability, Global stability, harvesting Numerical simulation
I. INTRODUCTION
The study of Ecology gains importance with the aid ofmodelingover the few decades. The system stability in Ecological systems describes the longterm behavior of the system. The basic population models are widely discussed by Lotka^{1} and Volterra^{2}. The various ecological models are discussed, materialized and analyzed by many authors^{36}.The delays in biological systems are playing a significant role in the system stability analysis. These delays can switch over stable equilibrium to unstable or vice versa. The elegant approach of delay differential equations and models in ecology are discussed in detail by Sreehari Rao^{7}, Gopala Swamy^{8} Yang Kuang^{9}.
A time delay can be incorporated in different ways like maturation time, gestation period etc. A detailed analysis of time delays in biological systems is studied extensively by and. Paprao^{10,11} studied the system stability of three species ecological model with different interactions among three species. The authors^{12,13} studied Optimal harvesting policy in ecological species with different interactions. Recently paparao^{14} studied the dynamics of three species ecological model with time delay interactions predator and competitor species.
Inspired from that we studied the dynamics of three species ecological model with a prey and predator and competitor model which included a time delay between the interaction in prey and predator species and carrying capacities of all three species, death rates of all populations and optimal harvesting policy on the prey species. The model is characterized by a couple of integro differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function. Suitable parameter are identified for extinct of prey, predator and competitor population and compared the result with system without delay arguments and harvesting efforts. The rich dynamics is observed when delay and harvesting efforts are incorporated in this model .The Numerical simulation shows that this continuous time delay model with harvesting of the prey exhibits rich dynamics.
2. MATHEMATICAL MODEL:
Consider Mathematical model for the system is
Consider Mathematical model for the system is
(2.1)
2.1: NOMENCLATURE:
S.No 
Parameter 
Description 
1 
,& 
Populations of the prey, predator and competitor respectively 
2 
,, 
Natural growth rates of prey, predator and competitor respectively 
3 
k_{1},k_{2},k_{3} 
Carrying capacities of prey, predator and competitor respectively 
4 

Rate of decrease of the prey due to inhibition by the predator 
5 

Rate of increase of the predator due to successful attacks on the prey 
6 

Rate of decrease of the predator due to the competition with the competitor 
7 

Rate of decrease of the competitor due to the competition with the predator 
8 
d_{1},d_{2},d_{3} 
Death rates of prey, predator and competitor respectively 
9 
q, 
catch ability coefficient of prey species (N1) 
10 
E 
Effort applied to the harvest the prey species (N1) 
11 

weight factors to give the influence at ‘t’ of N_{1,} N_{2 }of time u 
Notations: Assume throughout the analysis
By normalizing the kernels k_{1} and k_{2} such that with the conditions
(2.2)
Using the above relations the normalized system of equations is
(2.3)
3. EQUILIBRIUM STATES:
The possible equilibrium points are
I.E_{1:} The extinct state (3.1)
II.E_{2:} The state in which only the predator survives and the prey and competitor to the predator are extinct. (3.2)
III. E_{3}: The state in which both the prey and the predators extinct and competitor to the predator alone survive. (3.3)
IV. E_{4:} The state in which both the predator and competitor to the predator extinct and prey survive. (3.4)
V. E_{5 }: The state in which both the prey and the predators exist and competitor to the predator extinct
(3.5)
This case arise only when (3.5.a)
VI. E_{6 }: The state in which both prey and competitor to the predator exist and predator extinct,
(3.6)
VII. E_{7}: The state in which both predator and competitor to the predator exist and prey extinct,
(3.7)
The equilibrium state exist only when
(3.7.a)
VIII. E_{8 }: The state in which prey, predator and competitor to the predator exist
(3.8)
The equilibrium state exist only when
(3.8.a)
4. LOCAL STABILITY OF THE EQUILIBRIUM POINT E_{8}:
Theorem: The positive equilibrium point is locally asymptotically stable If
Proof: The variational matrix is given by
(4.1)
(4.2)
The characteristic equation of (4.2) be (4.3)
Where
(4.4)
By RouthHurwitz criteria, the system is stable if, and.
The algebraic calculations of gives that
(4.5)
From the equation (4.5) if
Therefore if
Hence the positive equilibrium point is locally asymptotically stable if
5. GLOBAL STABILITY:
Theorem: The positive equilibrium point is globally asymptotically stable
Proof: Let the Lyapunov function be (5.1)
The time derivate of V along the solutions of equations 2.1 is
(5.2)
Using the system of equations (2.2) and the relations
By proper choice of
(5.3)
Using the inequality
Where
Hence the system is globally stable at positive equilibrium point
6. NUMERICAL EXAMPLE:
The systems of equations (2.1) are simulated using Matlab using ode45. The system of equations without delay is of the following form and solve with the same package we get the following results illustrated by the graphs 1(A), 1(B) for the following parametric values
S.No 
Figures 
Description 
1 
The figures(A) 
Shows the variation of N_{1}, N_{2} and N_{3} with respect to Time (t) for system of Eq (2.2). 
2 
The figures(B) 
The phase portrait of N_{1}, N_{2} and N_{3 }for system of Eq (2.2) 
Example 1: Let a_{1}=1.5, a_{2 }=2.5, a_{3}=3.5, α_{12}=0.4, α_{21}=0.5, α_{23}=0.3, α_{32}=0.2, k_{1} = 50, k_{2} = 50, k_{3} = 50, d_{1} = 0.02, d_{2} = 0.02, d_{3} = 0.03, N_{1 }=10, N_{2 }=15 and N_{3 }=20.
For the above set of parametric values, the system exhibits oscillatory behaviour, the predator population id decreasing from its initial strength, hence the competitor population increases from its initial population. The system becomes unstable. The time series and phase portrait analysis is shown in the figures 1(A) & 1(B) respectively .
Fig 1. (A) Fig 1. (B)
For the different kernels for the system (2.1)For different values of a&b and different harvesting efforts , the solution graphs are illustrated with the same parametric values shown in example 1.
The delay kernels for different values of a , b, q, E the system of equations (2.1) are plotted
S.No 
Parameters values a&b and Converging equilibrium point E 
Nature of system 
1 
a=1. 5, b=1.5, q=0.1,E=1 E (34, 2, 46).Graph (2A &2B) 
The predator population is almost extinct In the interval[0,10] , later on the it increases and stabilize at 2 , the initial growth rates of prey and competitor populations are increasing due to the delay kernels and harvesting effort on the prey populations makes system stable. 
2 
a=1. 5, b=1.5, q=0.1,E=5 E (33, 0, 50).Graph (3A &3B) 
The predator population is extinct and the populations of prey and competitor are increasing due to the delay kernels and harvesting effort on the prey populations makes system neutrally stable. 
3 
α=0.5,β=0.5, q=0.1,E=5 E(5,0,46) Graph (4A &4B) 
The competitor population is high and the system is oscillating with unbounded oscillations hence the system becomes un stable 
4 
α=0.5,β=1.5, q=0.1,E=1 E(0,50,0) Graph (5A &5B) 
The prey and competitor populations extinct , hence the predator population is high and the system becomes unstable 
5 
α=0.25,β=0.25, q=0.5,E=10 E(0,0,50) Graph (6A &6B) 
Both the prey and predator populations extinct , hence the system is unstable .the competitor population is increasing from its initial strength . 
GRAPHS PLOTTED FOR THE ABOVE PARAMETRIC VALUES WITH DIFFERENT DELAY KERNELS AND DIFFERENT HARVESTING EFFORTS
Fig 2. (A) Fig 2. (B)
Fig 3. (A) Fig3.(B)
Fig 4. (A) Fig 4. (B)
Fig 5. (A) Fig 5. (B)
Fig 6. (A) Fig 6. (B)
7. CONCLUSIONS:
We study the stability analysis of three species Ecological model consists of a Prey (N_{1}), predator (N_{2}) and a competitor (N_{3}) .The competitor (N_{3}) is competing with the Predator Species (N_{2}) and neutral with the prey (N_{1}). In this model the third species is competing with the predator for food other than the prey (N_{1}). In addition to that, the death rates, carrying capacities of all three species and harvesting effort of prey species are also considered for investigation. In addition to that a delay in the interaction between the prey and the predator (gestation period of the predator) is also considered. The model is characterized by a couple of integro differential equations. All the eight equilibrium points of the model are identified and their local stability is discussed for interior equilibrium point. The global stability is studied by constructing a suitable Lyapunov’s function.
The numerical simulation is carried out for different parametric values of the system shown in equation (2.1). The simulation shows that for the specified parameters in examples from 1 for the different delay kernels shown in the graphs from 2 to 6 the system exhibits rich dynamics and these kernels and harvesting efforts play a significant role in system stability analysis. The system is becomes stable, neutrally stable and unstable for the specified parameters shown in the table. Using numerical simulation Parameters of the model are identified such that extinct of prey, predator and competitor populations. The delay kernels along with harvesting coefficients and efforts play a significant role.
8. REFERENCES:
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2. Volterra, V.,1931, Leconssen la theoriemathematique de la leitte pou lavie,GauthierVillars, Paris.
3. Kapur J.N., 1988, Mathematical Modeling, WileyEatern.
4. Kapur,J.N., 1985 ,Mathematical Models in Biology and Medicine, Affiliated Eastwest,.
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6. Paul Colinvaux., 1986: Ecology, John Wiley and Sons Inc., New York.
7. SreeHari Rao V. and Raja Sekhara Rao P. Dynamic Models and Control of Biological Systems. Springer Dordrecht Heidelberg London New York, 2009.
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14. Paparao AV and Lakshmi Narayan K, Dynamics of a prey predator and competitor model with time delay. International Journal of Ecology and Development. 32(1); 2017: 7586.
Received on 14.05.2017 Modified on 12.08.2017
Accepted on 20.09.2017 ©A&V Publications All right reserved
Research J. Science and Tech. 2017; 9(3):368376.
DOI: 10.5958/23492988.2017.00064.X