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In our present paper, we formulate and study a prey-predator system with imprecise values for the parameters. We also consider harvesting for both the prey and predator species. Then we describe the complex dynamics of the proposed model system including positivity and uniform boundedness of the system, and existence and stability criteria of various equilibrium points. Also the existence of bionomic equilibrium and optimal harvesting policy are thoroughly investigated. Some numerical simulations have been presented in support of theoretical works. Further the requirement of considering imprecise values for the set of model parameters is also highlighted.

The eternal relationship between prey and predators is one of the major topics to be discussed in recent science. Scientists from various fields are currently engaged in finding out different interactions among prey populations and predator populations. With the help of mathematical modeling, one can describe the strong and competitive relationship between these two types of creatures. However the discussion on this fascinating topic, with the help of some mathematical tool, was started during the first quarter of the twentieth century, thanks to the age-breaking works of Lotka [

Harvesting is however a common and quite natural phenomenon. In fishery harvesting is used frequently as the biological resources are mostly renewable resources. On an exploited fishery system with interacting prey and predator species, researchers are considering harvesting on either prey species or predator species or harvesting on both prey and predator species. Martin and Ruan [

In this regard, in our present article, we consider a prey-predator type ecological system with harvesting on both the species. However, till now most of the models are proposed by considering only precise set of parameters but the natural world may not be precise every time. In many situations at experimental field like birth and death rate of different individuals of the same species, interaction between two different species, etc., may be imprecise. For this purpose introduction of fuzzy sets (Zadeh [

The rest of the paper is organized in the following manner: In Section

Here we give the definition of interval numbers with some operations. We use interval-valued function in lieu of interval number.

We denote interval number

A real number

The basic operations between any two interval numbers are as follows:

For an interval

In this article, we consider only two species, namely, prey species and predator species. Let

Next if we consider that both the species are harvested, this is carried out on assuming the demand in the market of both species (prey and predator). Taking

The environment and other factors including temperature and food habits caused the parameters to be imprecise. So they should be taken as interval number rather than a single value. Let

For the interval number

In this section, we describe a thorough dynamical behavior of the proposed model system. To do so, we first check the positivity of the solutions of crisp system and uniform boundedness of the solution of the same system. Now, it can also be concluded that uniform boundedness and positivity in the solutions also hold for the corresponding fuzzy systems, if these things hold in crisp system.

First we consider the corresponding crisp system in following form.

In this section we now study the uniform boundedness of the proposed imprecise system. Now from the first expression of system (

Next we are targeting to show that the biomass of predator population

Thus proceeding in the same way as prey populations and with the help of Birkhoff and Rota [

Hence the biomass density of both the population species is uniformly bounded.

The equilibrium points of this system are given below.

(1) Trivial equilibrium:

(2) Axial equilibrium:

(3) Interior equilibrium:

The interior equilibrium exists

if

In this section we state and prove the local asymptotic stability criteria at different equilibrium points. Also the corresponding conditions for which the system is stable at different equilibria are given below.

For trivial equilibrium the variational matrix at

Here

In the next theorem, we state the stability criteria of trivial equilibrium point

Trivial equilibrium point

At axial equilibrium

In the next theorem we will state the local asymptotic stability criteria of the axial equilibrium or the predator free equilibrium

The axial equilibrium

In this condition the trivial equilibrium becomes unstable.

The variational matrix for interior equilibrium

The characteristic equation of

Therefore, The system is locally asymptotically stable at

The interior equilibrium

Here we will discuss the global asymptotic stability criteria of the system around its interior equilibrium point. In next theorem we study the criteria.

The interior equilibrium

A Lyapunov function is constructed here as follows

Taking derivative with respect to

That is, the system is globally stable around its interior equilibrium

In this section we study the bionomic equilibrium of the competitive predator-prey model. Here we consider the following parameters: (1)

The ‘zero-profit line’ is given by

For the points on the equilibrium line where

These three cases may arise in bionomic equilibrium.

When fishing or harvesting of predator species is not possible, then

When harvesting of prey is not possible, then

When the bionomic equilibrium is at a point

The bionomic equilibrium (

Here both prey and predator populations are considered as fish populations. The optimal net profit is obtained from fishing. We discuss in this section the optimal harvesting policy. We consider the profit gained from harvesting taking the cost as a quadratic function and focusing on the conservation of fish population. The price assumed here is inversely proportional to the available biomass of fish (prey and predator); i.e., if the biomass increases, the price decreases (see Chakraborty et al. (2011)). Let

Here the control

There exists an optimal control

In this section, we analyze our mathematical model through some simulation works. The main difference of our proposed model compared to other models of the same type is the consideration of interval-valued parameters instead of fixed-valued parameters. Inclusion of the parameter

Stability of interior equilibrium depends on the numerical value of

Next we describe optimal control theory to simulate the optimal control problem numerically. We consider the same parametric values as above and find the solution of optimal control problem numerically. For this purpose we solve the system of differential equations of the state variables (

Variation of prey biomass both with control and without control cases.

Variation of predator biomass both with control and without control cases.

Variation of harvesting effort with time.

Variation of adjoint variables with time when harvesting control is applied optimally.

The interactions between prey species and their predator species is an important topic to be analyzed. In present era, many experts are still analyzing the different aspects on this relationship. For this purpose in our present paper, we formulate and analyze a mathematical model on prey-predator system with harvesting on both prey and predator species. Further the model system is improved with the consideration of system parameters assuming an interval value instead of considering a single value. In reality due to various uncertainty aspects in nature, the parameters associated with a model system should not be considered a single value. But often this scenario has been neglected although some recent works considered these types of phenomena (see the works of Pal et al. [

The proposed model is analyzed for both crisp and interval-valued parametric cases. Different dynamical behavior of the system, including uniform boundedness, and existence and feasibility criteria of all the equilibria and both their local and global asymptotic stability criteria, has been described. It is found that the system may possess three equilibria, namely, the vanishing equilibrium point, the predator free equilibrium point, and the interior equilibrium point. Theoretical analysis shows that all of these three equilibria may be conditionally locally asymptotically stable depending on the numerical value of the harvesting parameter

Next we study explicitly the existence criteria of bionomic equilibrium considering

Consideration of imprecise parameters set makes the model more close to a realistic system which can be well explained with the help of Figure

However, in the present work we consider only a single prey species interacting with a single predator species which makes the model a quite simple one. For our future work we preserve the option of considering more than one type of prey species interacting with more than one type of predator species with imprecise set of parameters. Further due to unavailability of real world data, to simulate our theoretical works, we consider a hypothetical set for the parameters and obtain the result. However, we mainly aim to study the qualitative behavior of the system (not quantitative behavior) which would not be hampered at all due to the consideration of a simulated parametric set.

The data used in the manuscript are hypothetical data.

The authors declare that they have no conflicts of interest.