Simulation of Optimal Harvesting of Three Species Ecological Model with Closed Interval of Biological Parameter using by using First Order Nonlinear Ordinary Differential Equations

The paper presents the study of three species ecological model with Prey N1, predator N2 and competitor to the Predator N3 and neutral with the predator N2 with imprecise biological parameters. The model is characterized by a set of first order nonlinear ordinary differential equations. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. Equilibrium points of the model are identified, the local stability is discussed using Routh Hurwitz criteria and global stability by Liapunov function. The existence of bionomic equilibrium of the system has been discussed and optimal harvesting policy is given using Pontryagin’s maximum principle. The stability analysis is supported by Numerical simulation using Mat lab.


INTRODUCTION 1. Background
Mathematical modeling of ecosystems is a field of study which helps us to understand the interactions between different species and the mechanisms that influence the growth of species and their existence and stability. Mathematical models have been used to study the dynamics of prey-predator systems since Lotka (1925) and Volterra (1927). They proposed the simple mathematical model which describes the interaction between prey and the predator. Since then, many mathematical models have been constructed based on more realistic explicit and implicit biological assumptions.
Mathematical modeling is a frequently evolving process, to gain a deep understanding of the mathematical aspects of the problem and to yield non trivial biological insights; one must carefully construct biologically meaningful and mathematically tractable population models. Some of the aspects that need to be critically considered in a realistic and plausible mathematical model include; carrying capacity which is the maximum number of prey that the ecosystem can sustain in the absence of predator, competition among prey and predators which can be intraspecific or inter specific, harvesting of prey or predators and functional responses of the predators.
In this research work, a mathematical model to study the ecological dynamics of prey and predator system is proposed and analyzed. And also as an example some of the prey and predator system in some areas be studied.
∈ Classical arithmetic defines operations on individual numbers; interval arithmetic defines a set of operations on intervals.
| there is some in , and some in , such that }. The basic operations of interval arithmetic are, for two intervals [ , b] and [c, d] , # , !" # , # , # , # , # , # , # , # , when 0 is not in , Division by an interval containing zero is not defined under the basic interval arithmetic. Instead of working with an uncertain real we work with the two ends of the interval [ , b] which contains x such that x lies between and b, or could be one of them. Similarly a function % when applied to x is also uncertain. Instead, in interval arithmetic % produces an interval [ , b] which is all the possible values for % for all ∈ [ ].

1.1.3
Pontryagin's Maximum Principle Pontryagin's maximum principle is a powerful method for the computation of optimal controls, which has the  crucial advantage that it does not require prior evaluation of the informal cost function. Let , and 7 are differentiable function in 8 and with continuous derivatives, and that the stopping set D is a hyper plane, thus for some ∈ 9 : and some vector subspace Σ of  9 : . Define for ; ∈ 9 : the Hamiltonian function as: Η 8, , =, ; ; > 8, , = 8, , = Pontryagin's maximum principle states that if  ? , = ? @ is optimal, then there exist adjoint in 9 with the following properties for all Moreover the following transversality conditions hold: ? , = ? @ i) Note that, in the time-unconstrained case, if , and 7 are time-independent functions, then The Hamiltonian serves as a way of remembering the first four statements, which could be expressed alternatively as: The condition AB A& 0 is not always correct. For example in cases where the set of actions is an interval and where the maximum is achieved at an endpoint

MODEL FORMULATION AND ANALYSIS
In section deals with the mathematical modeling of the prey-predator dynamics where there are two predators which compete for the same limited resources. In addition, the section deals with the stability analysis of the equilibrium points and the numerical simulation of the model. Parameter Parameter Definition R net economic rent C D , ! 1,2,3 harvesting efforts H instantaneous annual rate of discount I D , ! 1,2,3 catch ability coefficients D , ! 1,2,3 harvesting cost per unit effort Table 3.1 Definition of some parameters

Prey-Predator Model
The ecological model is as follows. There is one prey and two predators, where the two predators are competes with each other for the use of common recourse i.e. food. But the two predators cannot eat each other (one is not eaten by the other). By assuming that the predator and competitor to the predator have alternative food in addition to prey population (but the competitor to the predator can't eat the prey population), then the model for one Prey and two Predator and harvesting on the both species is given by the following system of first order ordinary differential equations employing the following notation: Let N 1 denotes the size of the prey population, N 2 denotes the size of the predator population and N 3 denotes the size of the competitor to the predator population, lets assuming that there is demand for all species in the market so the harvesting of both species are carried out. Let prey, predator and competitor to the predator species are subjected to harvesting efforts (effort applied to the harvest the prey, predator and competitor species) E 1, E 2 and E 3 respectively. Then the dynamics of the prey-predator is described by: where M, S and ℓ are natural growth rate of prey, predator and competitor to the predator species respectively. Whereas: 1 -is rate of decrease of the prey population due to inter species competition Pis rate of decrease of the prey population due to inhibition by the predator population, 1 O is rate of decrease of the predator population due to inter species competition, P O is rate of increase of the predator population due to successful attacks on the prey population, His rate of decrease of the predator population due to the competition with the third species(competitor), 1 T is rate of decrease of the competitor population to the predator population due to inter species competition, 13 P T is rate of decrease of the competitor population due to the competition with the third species(predator population). Where all the parameter values 1 -, 1 O , 1 T , P -, P O , P T and Hare non-negative real numbers. It is assumed that the prey reproduction is influenced by the predator only while the predator reproduction is limited by the amount of prey caught. It is also assumed that the prey population grows exponentially with the rate r in absence of predator and also predator population growth exponentially in the absence of prey by alternative food with a rate S. But the competitor to the predator can't change in the absence of the prey population. Where I -, I O , I T are the catch ability coefficients of three species and strictly positive. The catch-rate function: I -Ε -N -, I O Ε O N O , I T Ε T N T are based on CPUE (catch-per unit-effort).

Imprecise Prey -Predator Model
By the construction of the prey-predator model the parameters such as prey population growth rate r, predator population growth rate s, competitor to the predator growth rate ℓ and predation coefficients 1 -, 1 O , 1 T , P -, P O , P T and Hare positive in nature and are considered precise. Intuitively if any of the parameters are imprecise, furthermore when any parameter of the right hand side of equations (3.1) -(3.3) are interval number rather than a single value, then it is not so straight forward to convert equations to the standard form like (3.1), (3.2) and (3.3).
For an imprecise coefficient we present the problem with an interval coefficient.

2.2.1
Prey -Predator Model with Interval Coefficient Let M, S, ℓ X , 1 Y -, 1 Y O , 1 Y T , P Z -, P Z O , P Z T and H Zbe the interval counterparts of, M, S, ℓ, 1 -, 1 O , 1 T , P -, P O , P T and Hrespectively, then the prey-predator model with combined harvesting efforts E 1 , E 2 and E 3 can be written in the following form:

Prey-Predator Model with Parametric Interval Valued Function Coefficient
The parametric form of the equations (3.4), (3.5) and (3.6) are: Also M , ℓ , S , 1 , P , @ , ^ , _ , H and ` (are all > 0) are provided interval valued functional form of coefficient by the differential equations: Following the interval arithmetic operation and properties, equations (3.16), (3.17) and (3.18) reduces to: For fixed " , let us consider the interval-valued function * : , 0 ∈ p and interval 1 : ∈ : , : . Since * : / is a strictly increasing and continuous functions, then the above equation reduces to: Therefore the parametric form of the differential equations (3.10) -(3.12) is given by:

Dynamic Behavior of the Harvesting Model 2.3.1 Equilibrium States of Prey-Predator Model with Parametric Interval Coefficient
The system under investigation has eight equilibrium states given by: (3.28) IV. The state in which both the prey and the predators washed out and competitor to the predator survive: That is: (3.29) V. The state in which both the prey and the predators stay alive and competitor to the predator vanishes: That is: (3.30) Solving for Ν dand Ν d O from the 2 nd and 3 rd equations that given in equation (3.30) yields: Also assuming that I -Ε -) 0, then these equilibrium states' exist only when: VI. The state in which both prey and competitor to the predator exist and predator extinct: The equilibrium state exists when: I -Ε -) 0 and * I T Ε T ) 0

VII. The state in which both Predator and Competitor to the Predator exist and Prey washed out:
That is: (3.32) Solving for N d O and N d T from the 2 nd and 3 rd equations that given in equation (3.32) yields:

Stability Analysis
To investigate the stability of the equilibrium states we consider small perturbations u 1 , u 2 and u 3 in N 1 , N 2 and N 3 over Ν d -, Ν d O and Ν d T respectively, so that

Local Stability Analysis
The local and global stability of the equilibrium states I, II, III and IV are found to be unstable. But the reaming is stable. We restricted our study to the equilibrium states V, VI, VII and VIII.
i. Stability of the Equilibrium State } d~, } d • , € : The variational matrix at the trivial equilibrium point will become: The Characteristic equation of the above variational matrix is given by: iii.

Stability of the equilibrium state at €, } d • , } d • :
The variational matrix at the trivial equilibrium point (state) is: Now we prove that the function V is a Liapunov function. For this we need to show that: i. V is continues and positive definite function ii. negative semi definite. Therefore, N d -, N d O , N d T is globally asymptotically stable. Theorem 3: The systems (3.1) -(3.3) cannot have any limit cycle in the interior of the positive quadrant.

Bionomic equilibrium of the imprecise prey-predator model
The bionomic equilibrium is nothing but the combination of the concepts of biological equilibrium as well as economic equilibrium. The biological equilibrium is given by equation (3.25). It is the study of the dynamics of living resources using economic models. Economic equilibrium is said to be achieved when the total revenue obtained by selling the harvested biomass (TR) equals to the total cost for the effort devoted to the harvesting (TC). To discuss the bionomic equilibrium of the imprecise prey, predator and competitor to the predator model, we consider the following parameters. Letbe the harvesting cost per unit effort for prey species, + -be the price per unit biomass of the prey, O be the harvesting cost per unit effort for predator species, + O be the price per unit biomass of the predator, T be the harvesting cost per unit effort for competitor to the predator species and + T be the price per unit biomass of the competitor to the predator species, Then the net economic rent (net revenue) for the prey, predator and competitor to the predator at any time is given by: O and T represent the net revenues for the prey, predator and competitor to the predator species respectively. The bionomic equilibrium Ν -' , Ν O ' , Ν T ' , E -' , E O ' , E T ' is given by the following simultaneous equations.
In order to determine the bionomic equilibrium we come across the following cases.

Qualitative Analysis Of Optimal Harvesting Policy
In commercial exploitation of renewable resources the fundamental problem from the economic point of view, is to determine the optimal trade-off between present and future harvests. If we look at the problem it is observed that the marine fishery sectors become more important not only for domestic demand but also from the imperatives of exports.
In this section we study optimal harvesting policy of the system of equation (3.1) -(3.3); and also our objective is to maximize, the objective functional form of the harvesting model, with the instantaneous annual rate of discount δ is as follows: 8 (3.51) Subject to the state constraints (3.1) -(3.3) with control constraints (variables): 0 Ε D 8 Ε D -eš , ! 1, 2, 3 Firs we construct the following Hamiltonian function for the problem by: where λ λ 2 1 , and λ3 are additional unknown functions called the adjoint variables. Now by differentiating Η with respect to Ε -, Ε O and Ε T respectively, we obtain: The optimal control Ε D 8 must satisfy the condition: , 1 = i and the optimal control cannot be determined by the above procedure. It is then called a singular control Ε D * 8 , 0 4 Ε D * 8 4 Ε D -eš 8 . Hence the optimal harvesting policy is The aim is to find an optimal equilibrium Ν -™ , Ν O ™ , Ν T ™ , Ε -™ , Ε O ™ , Ε T ™ to maximize Hamiltonian Η . Since Hamiltonian Η is linear in the control variablesΕ -, Ε O and Ε T the optimal control can be extreme controls or the singular controls.
Thus for singular control  where ' - The above equation is linear in λ2 and its solution is given by: And also from (3.59) and (3.67), we obtain a singular path: Similarly, from (3.60) and (3.70) we obtain a singular path:

4.CONCLUSIONS AND RECOMMENDATIONS 4.1 Conclusions
Prey-predator (competitor) harvesting model has undergone different development in theoretical and practical applications in the field of biomathematics. Most of the researchers have developed the prey, predator and competitor to the predator harvesting model based on the assumption that the biological parameters are precisely known but the scenario is different in real life situation. In this paper, we developed a method to find the biological equilibrium points, bio-economic equilibrium points and optimal harvesting policy when some biological parameters are imprecise in nature. Here we develop the concepts imprecise parameters to the prey, predator and competitor to the predator harvesting model by considering the prey population growth rate, predator population growth rate, competitor to the predator growth rate and predation coefficients are imprecise in nature for the lack of precise numerical information. The ability of calculating the biological equilibrium points, bio-economic equilibrium points and optimal harvesting policy developed in this paper might help to develop more realistic mathematical models in the area of mathematical biology. Before ending this article we would like to mention that one may consider Lotka-Volterra model with logistic growth under imprecise biological parameters. Impreciseness of the harvesting cost and price of the biomass of the species of the harvesting model are also important characteristic to be considered.

Recommendation
Basing on the results of qualitative analysis and numerical simulation of the model, we recommend that; i. Prey-predator (competitor) should not be harvested at a rate higher than their growth rate. However optimal harvesting of the prey-predator (competitor) at a rate much lower than their growth rate is permissible, since this would not lead to collapse of the system in the long term. ii. The population density of the predator can be increased drastically by increasing the growth rate of the prey species e.g. regular recruiting more prey into the area. Since regular recruiting of prey may not be realistic, the best alternative is to minimize or stop poaching of the preys so as to greatly increase the number of their population in that area, which will in turn result in an increase in the population of the predator. But the number of population of competitor to the predator does not dependent on the number of prey population; it depends on the number of predator population that competes' with them for common resource. This common resource may be additional resources for predator population. iii. The population density of the predator depends mainly on the biomass of the prey than that of