Original articles
Studies of different types of bifurcations analyses of an imprecise two species food chain model with fear effect and non-linear harvesting

https://doi.org/10.1016/j.matcom.2021.08.019Get rights and content

Abstract

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can’t be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences.

Introduction

The analyses of dynamical features of prey–predator interactions are the most important studies of theoretical as well as computational biology. Some times prey–predator models become highly nonlinear in nature due to the consideration of more realistic assumption regarding the biological behaviour of prey and predator. Thus, to study real-life biological models, computational techniques play a significant role. On the other hand, due to the occurrence of randomness and lack of certainty, most of the parameters of real-life problems especially, bio-mathematical models are not precise. Depending upon the nature of biological parameters, a prey–predator model can be categorized into the following types:

  • Deterministic prey–predator model

  • Fuzzy prey–predator model

  • Stochastic prey–predator model

  • Interval prey–predator model

In a deterministic prey–predator model, all the biological parameters are precise. The deterministic two species prey–predator model can be presented mathematically in a more general setting as follows: dNdt=Ng(N)f(N,P),dPdt=kf(N,P)dP, where N and P denote the population densities of prey and predator, respectively at any time t, g(N) measures the growth rate of prey whereas f(N,P) defines the interaction between prey and predator, the parameters k and d denote the conversion rate of prey biomass into predator biomass and mortality rate of predator, respectively. In the nineteenth century, Lotka and Volterra [29], [54] studied the dynamical behaviour of a simple deterministic prey–predator model with linear prey–predator interaction. After that, several researchers have modified Lotka–Volterra’s model by introducing various realistic features like nonlinear functional responses, a fear effect, harvesting, cooperation, etc.

The functional response among prey and predator is a key factor of any food chain model. This response measures the consume pattern between prey and predator. Based on the nature of food habit of predators, several researchers accomplished their work by considering different types functional responses such as Holling type I–IV [3], [11], [18], [19], [21], [28], [29], [39], [58], Beddington–De-Angelis type [25], [26], Ratio-dependent [47] etc.

The fear effect is a behavioural and stress-related physiological change of prey in presence of predator. Due to the fear of predator, prey species are always wary of possible attacks. Fear of predator enhances the survivability of adult prey in the short term, but it can affect the reproduction of prey in the long run. In modelling process of the prey–predator system, the fear effect is a real phenomenon, which was first introduced by Wang et al. [55]. In that paper, the authors have shown that the direct killing of preys by predators is not the only way to control the reckless growth of prey populations. In reality, there is also an indirect way to control the prey densities which is fear of predation. In this way, it is observed that the growth rate of the prey is slightly affected by this factor and in the presence of predators, the prey populations change their behaviour with the passing of time. Thus, introducing the fear factor in the growth rate of prey population, a deterministic prey–predator model can be made more realistic. After the seminal work of Wang et al. [55], many researchers [13], [14], [46], [49], [53], [56], [60] analysed various food chain model with fear effect.

Over the last few decades, due to the public demand for biological food resource (viz. fish, birds, domestic animals etc.), markets of these phenomena are booming. Presently, the concept of harvesting is heavily used in the respective sector of business management for gaining more capital. Generally, harvesting is implemented in agriculture [9], fishery [12], poultry [59], forestry [31] etc. To prevent the extinction of biological resources (preys) and reduce the reckless growth of the predators, harvesting plays an important role. Thus, harvesting of prey or predator or both has a great impact on the dynamics of the prey–predator system. Commonly, the harvesting is applied on the predator populations and there are three types of harvesting viz. constant [20], linear [1] and nonlinear harvesting [16]. So, to study a deterministic model, from both the economic and ecological points to view, several researchers introduced the concepts of harvesting in their biological models. Among them, the works of [12], [34] are noteworthy. In most of the prey–predator models, either prey or predator or both of the species are most likely to be harvested and sold for gaining capital in business purposes. After motivating from these facts, harvesting of predator is introduced in the proposed imprecise model.

Due to the occurrence of randomness and impreciseness, there arise many real-life conflict situations in our day-to-day life in which to formulate the mathematical model is a formidable task. In the case of ecological study of a system, sometimes the biological parameters like growth rate, death rate, conversion rate, harvesting rate, etc. of the species of a system cannot be predicted accurately because of natural disasters like storms, earthquakes, forest fire, Tsunami, etc. In this circumstance, to study an ecological model in imprecise form is very interesting.

In the fuzzy prey–predator model, the imprecise biological parameters are represented in the form of fuzzy numbers or fuzzy sets with appropriate membership functions. A fuzzy prey–predator model can be presented mathematically by the system of fuzzy differential equations. In this area, several researchers have studied fuzzy prey–predator model with various dynamical behaviours like global stability, bifurcations, the chaos of the solutions, harvesting, bio-economic stability, etc. [2], [4], [32], [35], [38], [40], [45], [52], [61]. Recently, Pal et al. [36] worked on fuzzy optimal harvesting of the ecosystem of a fishery.

On the other hand, in the stochastic prey–predator model, imprecise biological parameters are presented in the form of random variables with proper probability distribution functions. Thus, a stochastic prey–predator is formulated mathematically with the help of stochastic differential equations. A lot of works with stochastic prey–predator model were accomplished by several researchers, [14], [15], [26], [27], [33], [42], [43], [44], [48]. Among them, Roy and Alam [43] studied a prey–predator model with the effect of fear. But to analyse an imprecise model in a fuzzy or stochastic environment is quite difficult. Because in these cases, some extra decisions have to take about the type of fuzzy number or fuzzy set or the type of the distribution function of a random variable.

In the interval prey–predator model, all the imprecise parameters are represented in interval form [30]. This representation is simpler and more efficient than the fuzzy and stochastic representation. An imprecise prey–predator model with interval coefficients is presented mathematically by interval differential equations. In this area, very few works were contributed by few researchers. The concept of the interval was first introduced by Pal et al. [37] in the area of mathematical biology. They have used the concept of a parametric functional representation of interval in the classical Lotka–Volterra model. And other interesting works with this concept were done by several authors [10], [13], [34], [35], [57]. In these works, the parametric functional representation of interval is in exponential (highly non-linear) form. However, there is a nice parametric representation, named as linear parametric representation [41] in the existing literature. Our goal is to use the linear parametric representation of interval in the area of bio-mathematical modelling. Reviewing all the literature and the conditions of nature, we have studied an imprecise prey–predator model.

To overcome the lack of precise numerical information of the biological parameters, Pal et al. [37] first introduced the concept of impreciseness of the prey–predator model and introduce the parametric functional form of an interval. They considered an exponential parametric representation of model parameters and studied the optimal harvesting policy. Afterwards many researchers have work on the same concept [10], [13], [34], [35], [57]. To include the effect of environmental uncertainty/randomness, Ghosh et al. [17] studied an imprecise food chain model considering the linear parametric representation of model parameters and discussed the existence of Transcritical bifurcation with respect to imprecise measuring parameter. Reviewing available literature in this field, we have studied an imprecise prey–predator model with linear parametric representation of model parameters with the effects of fear on the growth of prey population and non-linear harvesting of predator population. The contributions of this article are listed below:

  • (1)

    We have studied the combined effects of fear and harvesting in an imprecise environment.

  • (2)

    We considered some of the biological parameters in increasing order and some in decreasing order.

  • (3)

    We have studied our imprecise model for Saddle–Node, Hopf and Bogdanov–Takens bifurcation theoretically as well as numerically.

  • (4)

    We construct two parametric bifurcation diagram with respect to increasing and decreasing order parameters as bifurcation parameters.

In this work, we proposed a two-dimensional prey–predator model with two types of imprecise parameter. The dynamical behaviours of the proposed imprecise prey–predator model are formulated considering Holling type II functional response with the biological effects: (a) fear on the growth of prey population and (b) non-linear harvesting of predator population. The boundedness of solutions, local and global stability, bifurcations (Hopf bifurcation, Saddle–node bifurcation and Bogdanov–Takens bifurcation) about the interior equilibrium point of the proposed imprecise model are analysed theoretically in parametric form. All the theoretical results of the proposed imprecise model are verified by performing numerical simulations.

The organization of this work is done in the following manner: In Section 2, the basic concept of interval number, interval differential equation has been discussed. In Section 3, the model formulation of imprecise prey–predator system has been described in both interval and parametric form. In the next two respective sections, the positivity and boundedness of the system, the existence of the equilibrium points and their local stability have been discussed. The global stability of the interior equilibrium points has been shown in Section 6 whereas all the bifurcations analyses have been performed in Section 7. Finally, to verify all the theoretical results, the numerical simulations are carried out in Section 8 and a fruitful conclusion has been made in Section 9.

Section snippets

Basic concepts of interval

In this section, to represent the imprecise model more precisely, the concept of interval number, interval arithmetic, interval-valued function and interval differential equation are discussed briefly. All these discussions are presented in parametric form.

An interval number (D) is defined as D=[dL,dU]={x:dLxdUanddL,dUR}and the parametric form of D be defined as (a)D={d(q):d(q)=dL+q(dUdL),q[0,1]}(Increasing form)or(b)D={d(q):d(q)=dUq(dUdL),q[0,1]}(Decreasing form).

Model formation

In every sector of the real world, there arise fluctuations of different parameters involved in various real-life problems due to the uncertainty. In the case of the ecological system, values of different biological parameters viz. growth rate, death rate, conversion rate, harvesting rate, etc cannot be determined exactly due to the uncertainty. Sometimes, such parameters fluctuate in a certain range. In that cases, the fluctuated/imprecise parameters can be presented by closed bounded

Positivity and boundedness

In this section, positivity and boundedness conditions of the solutions of system (5) are discussed in parametric form.

Theorem 2

The solutions of system (5) in parametric form are positive p,q[0,1].

Proof

Since, the systems (4), (5) are equivalent, to show the positivity of the solutions of system (4), it is enough to show the solutions of system (5) are positive.

Integrating equations of system (5) over 0 to t, we have N(t,p,q)=N(0,p,q)exp0tr(p)1+k(q)P(s,p,q)d1a(q)N(s,p,q)b(q)P(s,p,q)ds,P(t,p,q)=P(0,p,q)ex

Equilibrium points and their local stability

In this section, we found the equilibrium points of the proposed system (5) and studied their local stability. The equilibrium points of system (5) will be obtained by solving the following equations r(p)N(t,p,q)1+k(q)P(t,p,q)d1N(t,p,q)a(q)N2(t,p,q)b(q)N(t,p,q)P(t,p,q)=0,cb(q)N(t,p,q)P(t,p,q)d2P(t,p,q)q1E(p)P(t,p,q)m1E(p)+m2P(t,p,q)=0. The possible equilibrium points of the system (5) are listed below:

  • (1)

    The trivial equilibrium point E00,0, which always exists. This equilibrium point

Global stability

The global stability of an interior equilibrium point of a system is an important study to examine the existence of that system. So, in this section, we have discussed the sufficient condition for global stability of the interior equilibrium point.

Theorem 7

The interior equilibrium point EN(p,q),P(p,q) of system (5) is globally asymptotically stable (GAS) if r(p)M+d1N(p,q)+a(q)MN(p,q)+b(q)MN(p,q)+d2P(p,q)+q1P(p,q)m1<r(p)N(p,q)+cb(q)u1P(p,q).

Proof

First, we choose a suitable Lyapunov function in

Local bifurcation analyses

In this section, the conditions of local bifurcations, viz. Saddle–node, Hopf and Bogdanov–Takens bifurcations will be established for the proposed imprecise model. Among the previous mentioned bifurcations, Saddle–node and Hopf bifurcations are of co-dimension one, whereas the Bogdanov–Takens bifurcation is of co-dimension two. The conditions of said local bifurcation are derived in detail in the following subsection.

Numerical simulation

To validate the theoretical results obtained so far in this work, numerical examples have been considered with the following imprecise hypothetical data of the biological parameters [rL,rU]=[1,2],[kL,kU]=[0.01,0.35],[EL,EU]=[0.1,1.5],[aL,aU]=[0.5,0.9],[bL,bU]=[0.1,4.5],c=0.38,m1=0.01,m2=0.3,d1=0.1,d2=0.06,q1=0.093. To perform numerical simulations, the values of interval valued parameters [rL,rU] and [EL,EU] are represented in the increasing form as r(p)=rL+p(rUrL)=1+p,E(p)=EL+p(EUEL)=0.1+1.4p

Conclusions

In this work, the stability and local bifurcations analyses about equilibrium points of an imprecise prey–predator model with fear factor and nonlinear harvesting of predator have been carried out. To analyse the imprecise system mathematically, the concept of interval number and its parametric form, interval differential equations are used. The theoretical derivation of the local and global stability conditions is presented in terms of parametric form of the model parameters. We have

Acknowledgements

The authors thank the anonymous reviewers for their valuable components, which contributed to the improvement in the presentation of the paper. The author Bapin Mondal would like to thank UGC, Government of India, New Delhi, for the financial assistance. Funders and grant ID is -16-6(DEC. 2018)/2019(NET/CSIR).

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