Stability analysis of a three species food chain model with linear functional response via imprecise and parametric approach

Highlights

• We have proposed a tri-topic food chain model with one prey and two predators such as- prey, first predator and second predator.

• All the biological parameters (except the conversion efficiency of the species) are considered as interval-valued.

• Using linear parametric representation of interval, it is shown that the system in interval form is equivalent with the system in parametric form.

• The positivity, boundedness, stability and transcritical bifurcation conditions of the proposed model are discussed.

• Finally, the dynamics of the proposed model has been illustrated with the help of some numerical simulations.

Abstract

Currently, proper modeling of a biological system in an uncertain situation is a challenging task for researchers. Facing this challenge, in this article, a new concept of interval representation named as linear parametric representation is introduced by which a biological system in an uncertain situation can be formulated mathematically in a precise way. The aim of this work is to formulate a three species imprecise food chain model with one prey and two competing predators in an interval environment. Here, all the biological parameters except the conversion efficiency of the species are considered interval-valued. And interactions among the species are taken as Holling Type I functional response. Using the linear parametric representation of interval, a system in the interval form has been converted in the parametric form. Positivity and boundedness of the solutions of the proposed imprecise system are verified. Then, conditions of local stability, global stability and transcritical bifurcation of equilibrium points of the proposed system are established. Thereafter, all the theoretical results of this work are validated by a numerical example with interval-valued parametric data and results are presented pictorially. Finally, some biological implications of the obtained results are discussed and a fruitful conclusion has been made.

Introduction

With the development of civilization, it becomes a vital challenge to human beings to keep the ecological balance of the ecosystem. Therefore, the importance of a dynamical system (especially, ecological system) increases with time. As preys and predators are the important components in the food chain of an ecosystem, studies of prey-predator models are unavoidable. In a prey-predator model, different types of dynamical behaviour are studied under different situations. At first, Lotka [1] studied the dynamical behaviour of a simple prey-predator model. From that time onwards several authors, MacDonald [2], Beretta and Kuang [3], Kar [4], Cheng et al. [5], Din [6], Sk et al. [7] and references therein have contributed there works to modify the Lotka-Volterra model. The general form of a prey-predator model with one predator and one prey can be represented in the following form:

$$\begin{array}{cc}\frac{{\mathrm{dX}}_1}{\mathrm{dt}}\hfill & =X_{1}g(X_1)-f(X_1,X_2)X_2,\hfill \\ \frac{{\mathrm{dX}}_2}{\mathrm{dt}}\hfill & =\mathrm{kf}(X_1,X_2)X_2-d_1{X}_2,\hfill \end{array}$$

where, X₁ and X₂ are the densities of prey and predator, respectively at time t, g(X₁) defines the growth rate of prey, f(X₁, X₂) measures the relation between prey and predator named as a functional response, the parameters k and d₁ denote the efficiency of converting the prey biomass into predator's biomass and mortality rate of predator, respectively.

In a prey-predator model, the relation between prey and predator plays a vital role and this relation is addressed by the functional response. Therefore, the selection of appropriate functional responses in the formulation of prey-predator model is an important part of modeling. There are several types of functional responses in the existing literature, viz. Holling Type I [1], Holling Type II [8], Holling Type III [9] and Holling Type IV [10].

On the other hand, the analyses of three-dimensional or greater than three-dimensional non-linear dynamical system (n ≥ 3) quite difficult than two-dimensional, because in those systems, the involved parameters as well as equilibrium points are large in number and hence the other theoretical analyses (both local and global stability) create some difficulties. In particular, a three-dimensional prey-predator model is of two types: (i) two prey and one predator (ii) one prey and two predators. The researchers, Liu et al. [11], Lv and Zhao [12], Meng et al. [13], Panja and Mondal [14], Sen et al. [15], Wang [16] and reference therein are studied the dynamic behaviours such as local stability, global stability, bifurcation, limit cycle, chaos etc. of the n−dimensional (n > 2) prey-predator model. It is observed from the existing literature, that most of the researchers analysed the biological models in which all the biological parameters are considered as precise in nature.

However, because of the arising of randomness and inexactness, the parameters involved in several real-life problems such as optimization problems, engineering design problems, ecological and epidemiological problems cannot be explained completely using precise formulation. In mathematical modeling (especially, biomathematical modeling), most of the parameters become imprecise due to climate change, natural disaster, deforestation etc. In these situations, the population densities of the species of a food chain are not homogeneously distributed. To study the prey-predator model in heterogeneous regions the researchers use the concept of diffusion and took the prey and predator in diffusive in nature. In this area, several researchers, Djilali [17], [18], [19], [20], [21], Souna et al. [22], [23], and Guin et al. [24] are accomplished their work on the diffusive prey-predator model.

Also, the impreciseness of the species of a food chain model can be handled by representing imprecise parameters using various approaches like fuzzy approach, stochastic approach and interval approach. In the fuzzy approach, the imprecise parameter is expressed in the form of either a fuzzy set or a fuzzy number. There are a lot of works in the existing literature with fuzzy-valued parameters on the dynamical systems. Some of those were accomplished by Sadhukhan et al. [25], Paul et al. [26], Pal et al. [27], Tudu et al. [28], Meng et al. [29] and reference therein. In the stochastic approach, the imprecise parameter is represented by the random variable with a proper probability distribution. In this area, Rudnicki and Pichor [30], Liu and Fan [31], Das and Samanta [32], Hening and Nguyen [33], and reference therein contributed their works. But the formulation of the problem, in either case, depends on some independent approach. In the fuzzy approach, it depends on selecting a proper membership function and the stochastic approach depends on a probability distribution. Also, these functions make the model more complicated. So, the earlier two approaches are not easy always for most of the mathematical models.

Alternatively, in the interval approach, the imprecise parameter can be presented in the form of interval number [34]. This interval approach is more prominent and easier than the previous two. The concept of the interval was first introduced by Pal et al. [35] in the area of mathematical biology as a recovery of the difficulties of fuzzy and stochastic approaches. They used the concept of a parametric functional representation of interval in the classical Lotka-Volterra model. In the classical Lotka-Volterra model, they replaced the precise parameters by imprecise with parametric functional forms. In this area, Pal et al.[36] also studied an imprecise three species food chain model with two prey and one predator in a fuzzy and interval environment. In their article, they considered prey-prey and predator-preys interactions linearly. After that very few works [36], [37] were done with this concept. In their work, the parametric representation of interval was taken in exponential form (highly non-linear). However, in the existing literature, there is a simple and nice parametric representation of interval in the linear form [34].

In this work, an imprecise three species prey-predator model with one prey and two competing predators in an interval environment is considered. In the proposed model, biological parameters: growth rate of prey, self-interaction of the species, the consumption rate of predator and death rates of the predators are taken as interval-valued. The positivity, boundedness, local and global stability, transcritical bifurcation of the imprecise system are established with the help of the parametric approach of interval. It is clear from the theoretical and numerical discussions that introduction of impreciseness gives rich dynamics compare to precise model.

The manuscript is organised in the following manner: In Section 2, gaps of the research and our contributions are presented. In Section 3, the basic concept of interval and its mathematics are discussed whereas, in Section 4, the formulation of the proposed model in both precise and imprecise environments is presented. Then positivity and boundedness of the solution of the proposed system are discussed in Section 5, while the existence of the equilibrium points and their stability analyses are presented in Sections 6 and 7, respectively. After that, in Section 8, the global stability of the interior equilibrium point of the proposed system is presented. In Section 9, the numerical simulation of the proposed model is discussed. Finally, conclusion regarding the findings are made in the last section.