A delayed fractional order food chain model with fear effect and prey refuge

doi:10.1016/j.matcom.2020.06.015

Mathematics and Computers in Simulation

Volume 178, December 2020, Pages 218-245

https://doi.org/10.1016/j.matcom.2020.06.015 Get rights and content

Abstract

A delayed fractional-order prey–predator system with fear (felt by prey) effect of predator on prey population incorporating prey refuge has been proposed. We consider a three species food chain system with Holling type I functional response for the predator population including prey refuge. The existence and uniqueness of the system is studied along with non-negativity and boundedness of the solutions of proposed system. Next, local stability of the equilibria has been studied for both delayed and non-delayed systems. We have also established that the non delayed system is globally stable under some parametric restrictions. Finally we have discussed the Hopf bifurcation due to time delay and other parameters both theoretically and numerically by the help of MATLAB and MAPLE.

Introduction

Food chain models are very popular in the field of biomathematics. Several researchers have analyzed ecological models based on at least three trophic levels [23], [27]. The study of a mathematical model should be based on at least three trophic levels such that more focus can be made on complex behavior demonstrated by deterministic models comprised of three or more trophic levels [37]. Practically, different three species systems have become the center of significant attention in its own right. Researchers have drawn their attention on food chain system of waste-bacteria-ciliates in waste treatment process [23]. Recently, many researchers are also working on three species system in fractional order system [2], [16]. In this work, we have concentrated on fractional order system on three species model.

Fractional differential equation is a modern concept and this is a part of abstract mathematics. Fractional order system has the effect of existence of time memory but this effect is absent in integer order system. Fractional order derivative is correlated to the entire time domain for a biological process while integer order derivative indicates a change of certain trait at particular time [35]. There are few works on fractional order delay in recent years [16], [38], [47], so we have drawn our attention to develop a delayed fractional order system with different prey–predator interaction.

The fear of predator on prey affects the anti-predator defenses and reduces the reproduction rate of prey. Many researchers propose the effect of predation based on direct intake of prey by predators, there is a straightforward corroboration that the fear of predators is as important as direct consumption [3]. Many creatures have faced predation threat and have shown different kinds of anti-predator responses like change in habitat usage, foraging behaviors, vigilance and physiological changes [3], [5], [36], [45], [48]. Nowadays many researchers are developing mathematical models based on fear effect [3], [7], [28], [31], [49], [51].

A refuge is renowned for protecting a constant fraction of prey from predation. The behavior of hiding of prey may stabilize the predator–prey dynamics and the prey refuges are protected from predation. A refuge is an idea in biology and ecology, where an organism acquires protection from predation by hiding themselves in a region where they are isolated or inaccessible for predation. The existence of refuges has an important role on the coexistence of prey and predators [29], [41], [42], [46]. There are some empirical and theoretical works that analyze the influence of prey refuge [6], [8], [24], [39], [43], [44].

Motivation: The ecology of fear admits that predators behave both ways in affecting prey populations with knock-on effects down the food chain. Predators kill prey, which affects prey populations. However, predators also terrify prey who exhibit a variety of anti-predator defenses to avoid being predated on the other hand prey refuges create food crisis for predators [50]. There are many works that have been done on prey–predator dynamics incorporating prey refuge and fear effect [19], [51]. The effect of fear vastly impacts on prey’s feeding which has effects down the food chain strongly. The previous works on prey–predator interaction incorporating prey-refuge and fear effect have been based on classical integer order system. In spite of enormous amount of works focused on fractional differential equation and dynamical systems, there are remain many challenging open problems. Motivated by the recent works [2], [6], [8], [19], [38], [47], [51], we have constructed our model by considering the both responses fear and refuge in a delayed Caputo fractional order food chain model. This is due to that fact that fractional order system is more precise for the explanation of memory and hereditary properties of the system compared with an integer order system [13], [20], [34]. The objectives of this paper are as follows:

1.
To investigate the impacts of prey refuge in food chain system.
2.
To investigate the impacts fear effect in food chain system.
3.
To study the impacts of gestation time delay in food chain system.

Brief overview: We have organized our work in several stages. Initially, we have formulated a three species food chain model incorporating prey refuge and fear effect with gestation time delays in Section 2. We have also considered the case without gestation time delay. Section 3 deals with the introductory theories. Section 4.1 contains the equilibrium points of both non-delayed and delayed systems and their feasibility criteria. In Section 4.2, we have studied the existence and uniqueness of the non-delayed system which assures the existence and uniqueness of the delayed system. Further, non-negativity and boundedness have also been established theoretically in Section 4.3. Next, we have analyzed local and global stability of the non-delayed system in Sections 4.4 Local stability analysis of non delayed system, 4.5 Global asymptotic stability respectively. In Section 5, we have studied delayed system and established local stability criterion and also studied the Hopf bifurcations with respect to the gestation time delay. The remaining sections (6–7) contain numerical simulations and conclusions of the whole work respectively.

Section snippets

Formulation of model

Our aim is to formulate a food chain model incorporating fear effect and prey refuge. Before demonstrating the fear effect in our model, let us assume that the prey population ( $X$ ) maintains a logistic growth in absence of predation and effect of fear. The logistic growth of prey ( $X$ ) can be split up into three portions; reproductive rate (R), natural death rate (D), and density dependent death rate due to intraspecific competition $(I)$ . We have the following ODE with respect to the time T: $\frac{d X}{d T} = R X$

Introductory theories

In this section we have recalled some important theories which are helpful for our further analysis. Our model is based on Caputo fractional differential equations and we have recalled some basic definitions, Mittag-Leffler function and some conditions for stability of equilibrium in this section.

Definition 1

[33], [35]

The fractional order derivative in Caputo sense with order $ε > 0$ for a function $f \in C^{n} ([t_{0}, \infty +), R)$ is denoted and defined as: $_{t_{0}}^{C} D_{t}^{ε} f (t) = \frac{1}{Γ (n - ε)} \int_{t_{0}}^{t} \frac{f^{(n)} (s)}{{(t - s)}^{^{ε - n + 1}}} d s, when ε \in (n - 1, n), n \in N$ $_{t_{0}}^{C} D_{t}^{ε} f (t) = \frac{d^{n}}{d t^{n}} f (t)$

Analysis of non delayed model (2.4)

In this section, we have discussed about the non delayed system (2.4). The various equilibria of systems (2.3), (2.4) are mentioned in Section 4.1. The uniqueness and boundedness of the solutions of systems (2.3), (2.4) have been discussed in Sections 4.2 Existence and uniqueness, 4.3 Non negativity and boundedness of solutions respectively. The local stability and global stability of our proposed non-delayed system (2.4) have been performed in Sections 4.4 Local stability analysis of non

Analysis of delayed system (2.3)

The system (2.3) and system (2.4) have same set of equilibrium points. The conditions for existence, uniqueness, boundedness, nonnegativity obtained for system (2.4) also hold for system (2.3). The stability criterion is different for delayed system (2.3) due to gestation time delay $τ$ and we have observed Hopf bifurcations in certain interval of $τ$ and outside this region the solutions are asymptotically stable. The following lemma is required before analyzing the delayed fractional order

Numerical simulations

In this section, numerical simulations have been performed using Adams–Bashforth–Moulton Predictor–corrector method [11], [12] and the scheme developed by Ivo Petras to validate the theoretical results established in the previous sections. In addition, the effects of fractional order, time delay and prey refuge on the stability of the equilibrium points have been presented. We are more interested in the stability of coexistence equilibrium $E_{3}^{*}$ rather than trivial $E_{0}^{*}$ , axial $E_{1}^{*}$ or predator free

Conclusions

In this study, we have investigated a fractional order food chain model with refuge in presence of time delay and fear effect on the prey. First, we have discussed the fractional order model without time delay. It is observed that the system is locally and globally asymptotically stable under restricted parametric conditions. Numerically, it is found that no bifurcation occurs for our chosen parametric values. It is also observed that periodic nature of the solutions increases when the order of

Acknowledgments

The authors are grateful to the anonymous referees, Prof. Laura Gardini, Editor-in-Chief, for their careful reading, valuable comments and helpful suggestions, which have helped them to improve the presentation of this work significantly.

References (51)

AhmedE. et al.
On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems
Phys. Lett. A
(2006)
AhmedE. et al.
Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models
J. Math. Anal. Appl.
(2007)
DasM. et al.
Stability analysis of a prey-predator fractional order model incorporating prey refuge
Ecol. Genet. Genomics
(2018)
KexueL. et al.
Laplace transform and fractional differential equations
Appl. Math. Lett.
(2011)
LiY. et al.
Mittag–Leffler stability of fractional order non linear dynamic systems
Automatica
(2009)
LiY. et al.
Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability
Comput. Math. Appl.
(2010)
LiB. et al.
Simple food chain in a chemostat with distinct removal rates
J. Math. Anal. Appl.
(2000)
MondalS. et al.
Dynamics of an additional food provided predator–prey system with prey refuge dependent on both species and constant harvest in predator
Physica A
(2019)
OdibatZ. et al.
Generalized Taylors formula
Appl. Math. Comput.
(2007)
RuanS. et al.
Coexistence in competition models with density-dependent mortality
C. R. Biol.
(2007)

SarwardiS. et al.

Analysis of a competitive prey–predator system with a prey refuge

Biosystems

(2012)

SihA.

Prey refuges and predator–prey stability

Theoret. Popul. Biol.

(1987)

ZhangH. et al.

Impact of the fear effect in a prey-predator model incorporating a prey refuge

Appl. Math. Comput.

(2019)

BrownJ. et al.

The ecology of fear: optimal foraging, game theory, and trophic interactions

J. Mammal.

(1999)

ChoiS.K. et al.

Stability for caputo fractional differential systems

Abstr. Appl. Anal.

(2014)

CresswellW.

Predation in bird populations

J. Ornithol.

(2011)

DasA. et al.

Modelling the fear effect on a stochastic prey-predator system with additional food for predator

J. Phys. A

(2018)

DasM. et al.

A prey-predator fractional order model with fear effect and group defense

Int. J. Dyn. Control

(2020)

DelavariH. et al.

Stability analysis of Caputo fractional-order non linear system revisited

Nonlinear Dyn.

(2012)

DengW. et al.

Stability analysis of linear fractional differential system with multiple time delays

Nonlinear Dynam.

(2007)

DiethelmK.

Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)mE Methods, Vol. 71, No. 4

(2003)

DiethelmK. et al.

A predictor-corrector approach for the numerical solution of fractional differential equations

Nonlinear Dynam.

(2002)

DuM. et al.

Measuring memory with the order of fractional derivative

Sci. Rep.

(2013)

GarrappaR.

Predictor–corrector PECE method for fractional differential equations

(2011)

HauboldH.J. et al.

Mittag–Leffler functions and their applications

J. Appl. Math.

(2011)

Cited by (59)

Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge
2024, Mathematics and Computers in Simulation
The prey–predator interaction between organisms living in an ecosystem is greatly affected by the cooperation among the species as well as immature prey refuge in presence of time delay. This study has developed and analysed a stage-structured prey–predator model that includes refuge behaviour for prey and two types of cooperative behaviours, namely cooperative behaviour among prey as well as cooperative behaviour among predators, in addition to gestational delay. The presence of Saddle–node, Transcritical, and Hopf-bifurcations is explained by analytically and numerically. We examine the model parameter’s sensitivity to determine the species’ favourable and unfavourable impacts and to identify the most significant parameters. The dynamics of delay and non-delay systems are compared, and we recognise that a consistent increase in time delay causes chaotic oscillations with period-doubling fluctuations which can be regulated by the refuge for immature prey. However, we demonstrate that the immature prey refuge destabilises the system in absence of time delay while it acts as a control parameter in presence of delay. Finally, we wrap up the paper by highlighting the key biological implications of the numerical and analytical findings.
Plants and age-structured pest dynamics with natural enemy as control strategy: A fractional differential equations model
2023, Results in Control and Optimization
In recent years, plant pests have drawn much attention from researchers. Biological control proved an efficient replacement for chemical and other pest control techniques. The present study proposes the cognition of an immature pest population unaffected by the natural enemy and grows exponentially by the harvesting effort of the mature pest population to plants. The fractional calculus environment proposes A four-compartment: plant, immature and mature pest, and natural enemy model. Equilibrium points and the system’s stability are studied theoretically and numerically. Simulation study for different values of fractional order generates the optimum solution. The suggested method is much more benignant than the existing method of ordinary differential equations as we can figure out the infrequent changes in the variable using a fractional derivative. Creating the memory effect in the system using fractional derivatives makes it more realistic.
Quasi-projective synchronization analysis of discrete-time FOCVNNs via delay-feedback control
2023, Chaos, Solitons and Fractals
In this article, the quasi-projective synchronization (QPS) issues about discrete-time fractional-order complex-valued neural networks (DFOCVNNs) are discussed. To realize QPS, the delay-feedback controllers are designed. The main advantage of this controller is that it can search for control gain parameters over a wider range. Applying inequality techniques, Lyapunov method and some lemmas related to discrete fractional calculus, some criteria for QPS are inferred. Meanwhile, the error bound is effectively estimated. Eventually, we display two examples to verify the obtained criteria.
New insight into bifurcation of fractional-order 4D neural networks incorporating two different time delays
2023, Communications in Nonlinear Science and Numerical Simulation
Citation Excerpt :
Here we must point out that the works of [16,17] are merely concerned with the integer-order dynamical models. Recent exploration manifests that fractional-order dynamical model is usually deemed as a more valuable way to portray lots of objective natural phenomena in the real world than the integer-order counterparts since it has good memory peculiarity and hereditary function in many materials and evolution process [28–37]. Presently, fractional-order differential system has witted tremendous application foreground in lumping areas of natural science and social science such as numerous physical waves, control engineering, biological medical treatment, economics, neural networks and so on [28–30,38–40].
Delay has a vital influence on the dynamics of neural networks. Exploring the effect of time delay on the dynamics of neural networks has become a hot issue in mathematics and engineering fields. In this current manuscript, on the basis of the earlier publications, we put forward a new fractional-order 4D neural networks incorporating two different time delays. First of all, the existence and uniqueness, boundedness of the solution of the fractional-order 4D neural networks incorporating two different time delays. are analyzed by applying contraction mapping principle, construct of an adaptive function, respectively. Next, the stability and the emergence of Hopf bifurcation are explored by making use of the stability and bifurcation theory of fractional-order dynamical system. A series of novel stability criteria and bifurcation conditions guaranteeing the stability and the emergence of Hopf bifurcation of the considered fractional-order 4D neural networks under the different delay cases are built. What $^{,}$ s more, the impact of delay on stabilizing neural networks and controlling the emergence of Hopf bifurcation of neural networks is adequately uncovered. At last, Matlab simulation figures are presented to confirm scientificness of the derived prime conclusions. The derived prime conclusions of this manuscript are perfectly innovative and own momentous theoretical reference value in the control issue and design aspect of neural networks.
Time-dependent and Caputo derivative order-dependent quasi-uniform synchronization on fuzzy neural networks with proportional and distributed delays
2023, Mathematics and Computers in Simulation
Citation Excerpt :
Fractional calculus has genetic and memory characteristics [16,23], which make the fractional-order systems more accurate than the traditional integer-order systems to describe the complex relationship between signal input and output and the dynamical behaviours of systems. Recently, fractional calculus have been applied to biological systems [8,39] to study the food chain model and predator–prey system, where fractional order derivative is correlated to the entire time domain for biological processes. It is worth pointing out that many results have been reported with regard to the dynamics of fuzzy NNs [1,4,9,20,28–30,38,46] including the integer-order fuzzy NNs in [20,30,36,46] and fractional order fuzzy NNs in [1,4,9,28,29,38].
This paper focuses on studying the quasi-uniform (Q-U) synchronization on fractional-order fuzzy neural networks (FOFNNs) with the proportional and distributed delays. The impacts of the Caputo derivative order, coefficients of network system and control gain constants on the synchronization performance are taken into account. Two novel time-dependent and Caputo derivative order-dependent algebraic criteria on the Q-U synchronization of FOFNNs are established by using Cauchy–Schwartz inequality, Minkowski inequality, $C_{p}$ inequality and Hölder inequality, which reveal the internal influences of the time and fractional derivative order on Q-U synchronization with the derivative order interval $(0, 2)$ . The simulation examples further confirm the effectiveness and practicability of the algebraic criteria in terms of the MATLAB toolbox.
Bifurcation control of a delayed fractional-order prey-predator model with cannibalism and disease
2022, Physica A: Statistical Mechanics and its Applications
Citation Excerpt :
However, these existing bifurcation results rarely take into account the effect of time delays. There are only a few researches on bifurcation and control of delayed fractional-order models [32–36]. Rihan et al. [37] concluded that Hopf bifurcation will occur when the time delay reaches a critical value.
In this paper, the bifurcation control for a fractional-order delayed prey-predator system with disease and cannibalism is investigated. Firstly, the existence, uniqueness and non-negativity of the solutions are studies, and the stability of equilibrium points are discussed. Next, taking time delay as the bifurcation parameter, the conditions of creation for Hopf bifurcation are confirmed. Then, two feedback controllers are introduced, which successfully control the Hopf bifurcation, and bifurcation control of the two controllers are simply compared theoretically. Finally, numerical simulations show that cannibalism has an significant influence in controlling the stability of the model, and when the parameters are appropriate, the disease can be removed. Meanwhile, the feedback controllers can control the bifurcation well. In general, the control effect of time-delay feedback controller should be better than that of traditional feedback controller. However, we found that the control effect of the traditional feedback controller is better than the time-delay feedback controller, the control effect completely depends on the selection of parameters.

View all citing articles on Scopus

View full text

Original articlesA delayed fractional order food chain model with fear effect and prey refuge

Abstract

Introduction

Section snippets

Formulation of model

Introductory theories

[33], [35]

Analysis of non delayed model (2.4)

Analysis of delayed system (2.3)

Numerical simulations

Conclusions

Acknowledgments

Phys. Lett. A

J. Math. Anal. Appl.

Ecol. Genet. Genomics

Appl. Math. Lett.

Automatica

Comput. Math. Appl.

J. Math. Anal. Appl.

Physica A

Appl. Math. Comput.

C. R. Biol.

Biosystems

Theoret. Popul. Biol.

Appl. Math. Comput.

The ecology of fear: optimal foraging, game theory, and trophic interactions

J. Mammal.

Stability for caputo fractional differential systems

Abstr. Appl. Anal.

Predation in bird populations

J. Ornithol.

Modelling the fear effect on a stochastic prey-predator system with additional food for predator

J. Phys. A

A prey-predator fractional order model with fear effect and group defense

Int. J. Dyn. Control

Stability analysis of Caputo fractional-order non linear system revisited

Nonlinear Dyn.

Stability analysis of linear fractional differential system with multiple time delays

Nonlinear Dynam.

Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)mE Methods, Vol. 71, No. 4

A predictor-corrector approach for the numerical solution of fractional differential equations

Nonlinear Dynam.

Measuring memory with the order of fractional derivative

Sci. Rep.

Predictor–corrector PECE method for fractional differential equations

Mittag–Leffler functions and their applications

J. Appl. Math.

Original articles
A delayed fractional order food chain model with fear effect and prey refuge