Study of Fractional Order Tri-Tropic Prey-Predator Model with Fear Effect on Prey Population ()
1. Introduction
Due to its worldwide existence and domination, the progressive connection between predator and prey has been studied. Fear of predators on prey has been shown to have a significant impact on anti-predator defenses and lower prey reproduction in recent zoological studies on terrestrial vertebrates. Prey that is afraid forage less, which may lower fertility, and they live by famine [1] [2]. Furthermore, the fear effect has a negative impact on the physiological state of young prey, which has a negative impact on their adult survival. All organisms in taxonomy face the risk of predation, and they display a range of anti-predator behaviours, including as adjustments to habitat utilisation, feeding behaviour, alertness, and physiological changes [3] [4] [5]. However, there is clear evidence that predator fear is just as important as direct eating. Fear is induced by such behavioral changes, which can lead to stress-related physiology and have a poor impact on physical health [6] [7] [8], which can affect reproductive health and survival. As a result, predator fear has long-term implications for population dynamics and ecosystems. Fear of predators on prey causes people to spend more time being watchful instead of foraging and looking for better food and reduced habitats [9] [10] [11]. As a result, impact of fear on reproductive success and adult body mass growth are still present. Because people become more attentive when density declines, fear of predators may amplify Allee effects. Individual attentiveness rises when group size diminishes, according to field study on social animals. Predation is causing small groups of prey to become extinct [12]. In population demographics, group defense is a common phrase that describes an occurrence in which predation is reduced as prey’s capacity to protect or hide themselves increases when they are in large numbers or form a group.
The fractional order derivative, unlike the conventional derivative, has an essential trait known as the memory effect. For biological systems, fractional order derivatives are connected to the entire time domain, whereas integer order derivatives reflect a change or a specific attribute at a certain moment [13]. As a result, the fractional order derivative is more suited to simulating memory difficulties in biological systems [14]. Recent study areas including ecological modeling, epidemiology, financial mathematics, and the physical and mathematical sciences have seen an increase in popularity of the fractional order system. The difference between our conventional integer order system and fractional order systems are illustrated in a number of articles on fractional order platforms [15] - [20]. Some ideas have been offered to show our research into various stability criterions such as limit cycle, Hopf bifurcations, global stability, and so on. The effect of fear has been taken up recently in the study prey-predator models [21] [22]. The authors in [23] studied the global asymptotic stability and hopf bifurcation in a homogeneous diffusive predator-prey System with Holling type II functional response. Hopf bifurcation analysis of the repressilator model is proposed in [24]. Our major goal is to combine the fear effect with Caputo fractional order model.
1.1. Motivation and Novelties of the Article
In order to replicate real-world issues, several innovative fractional operators with various properties have been designed. Moreover, the integer derivative has a local identity, whereas the fractional derivative has a global character. Numerous varieties of fractional derivatives, both with and without singular kernels, are available today. Leibniz’s query from 1695 marks the beginning of the fractional derivative. The fractional derivative also improves in the improvement of the system’s consistency domain. We have the derivatives of Caputo, Riemann-Liouville, and Katugampola for singular kernels [25] [26]. There are two varieties of fractional derivatives without singular kernels: the Caputo-Fabrizio fractional derivative, which has an exponential kernel, and the Atangana-Baleanu fractional derivative, which has a Mittag-Leffler kernel [27]. Numerous academic articles, monographs, and novels have provided evidence to support this claim; for instance, [28] - [32]. Motivated by the abovementioned works and the advantages of Caputo derivatives, this study built a prey predator model with fear effect in Caputo sense. Because the Caputo derivative allows for the inclusion of conventional starting and boundary conditions in the derivation and because the derivative of a constant is zero, as opposed to the Riemann-Liouville fractional derivative, it is particularly helpful for describing real-world problems. Non-local operators, which may represent non-localities and certain memory effects, are typically better suited for such situations since they can account for power law, fading memory, and overlap effects.
The objective of the current work is:
● To study a tri-tropic prey predator model in Caputo environment and its stability analysis.
● Determination of all equilibrium points.
● Existence of Hopf bifurcation of the proposed model.
● Numerical Solution by Adam-Bashforth method.
1.2. Structure of the Paper
We discuss some important definitions and characteristics of fractional derivatives related to this article in Section 2. We have formulated a tri-tropic prey predator model with fear effect in Section 3. Section 4 consists of the existence, uniqueness, non-negativity, boundedness of solutions of the system. We have studied the local stability and global stability in Section 5. Influence of fear on population density in the model system is described in Section 6. In Section 7, we have analyzed Hopf bifurcation of the model system with respect to the fear effect parameter. In Section 8, we have used MATLAB (2018a) for numerical simulations in order to demonstrate the validity of our mathematical conclusions. Section 9 consists of the conclusion.
2. Preliminaries
The definitions and features of fractional derivatives that we offer the reader are both informative and practical.
Definition 1. [33] “The Caputo fractional derivative of order
for the function
is defined as
,
where
is a n tines continuously differentiable function and the Gamma function is defined by Γ( ) such that
”.
Theorem 1. [34] “If
is piecewise continuous, then
,
,where the Laplace transform is denoted by
”.
Theorem 2. [35] “One-parametric and two-parametric Mittag-Leffler functions are described as follows:
and
, where
”.
Lemma 1. [36] “Let
and if
is continuous in
, then
, where
,
”.
Note 1. “If
(
),
, then
is a non-decreasing (non-increasing) function for
”.
Lemma 2. “Let us consider the fractional order system as
,
with
,
and
. For calculate the equilibrium points, we have
. These equilibrium points are locally asymptotically stable iff each eigenvalue
of the Jacobian matrix
calculated at the equilibrium points satisfies
”.
Lemma 3. “Assume that
is a differentiable function. Then, for any
,
,
,
”.
3. Model Formulation
Step-1: Particular species never occupy the entire space in nature; they are either prey or predators of other species. The dynamical prey-predator model makes an assumption about the functional response, which is quantified by the amount of prey taken per predator per unit time. Let
and
represent the density of the prey population, intermediate predator, and top predator, respectively, at any timet. To formulate the model system, the following assumption is made:
1) In the absence of a predator, the prey population
increases logistically at an inherent growth rate of r, with k representing the environment’s carrying capacity.
2) According to Holling Type-I functional response, the intermediate predator
consumes the prey species of with the predation rate
. With the predation
, the top predator
also consumes the prey according to the law of mass action (Holling Type-I functional response).
3) The intermediate predator’s energy conversion coefficient and natural date rate, respectively are
and
.
4) The energy conversion coefficient of predation and the natural date rate of the top predator, respectively are
and
.
5)
is the intra-specific predation rate of top predator.
Kar and Ghosh [37] suggested the model system in the following way depending on these assumptions:
(3.1)
Setp-2: We have now changed the model by include a fear element that the intermediate predator causes in the prey. Then, we write down the model system
with the fear factor
where
represents the fear level as follows:
(3.2)
The function
, due to fear (felt by prey) of predator, the birth
rate of prey species is reduced. In biological aspects of
, it is appropriate to assume:
Step-3: We have analyzed the suggested model utilizing the Caputo derivative of order
.
(3.3)
Step-4: Now, we rewrite the parameters as follows for the sake of computation convenience:
,
,
,
,
,
,
,
,
,
.
With beginning circumstances, the improved model system (3.3) may eventually be stated as follows:
(3.4)
4. Analysis of the Model
This section investigates the existence, uniqueness, non-negativity, and boundedness of the proposed model.
4.1. Existence and Uniqueness
Theorem 4.1.1. There exists a unique solution of the proposed model (3.4) for each non-negative initial condition.
Proof: We are seeking for a sufficient condition for the presence and uniqueness of the proposed model (3.4) solutions in the region
, where,
.
The method employed in [38] is used. Consider a mapping
where
and
:
For any
:
where
and
,
,
.
As a result,
fulfils the Lipschitz requirement. As a consequence, the fractional order system (3.4) exists and is unique.
4.2. Positivity and Boundedness of Proposed Model
Theorem 4.2.1. The model system’s solutions are all non-negative.
Proof: Let us assume
be the initial solution and
, where
. Let us choose a constant
,
such that
From system (3.4), we have
at
.
Using Lemma 1, we get
, that contradicts
. So
. In similar way we have
.
Theorem 4.2.2. The model system (3.4) has bounded solutions.
Proof: Let the function
. (4.1)
Differentiating with respect to time on the above function, we have
where,
.
Therefore,
, where,
.
With the help of Laplace transform, we have,
ð
ð
(4.2)
Taking inverse Laplace transform, we have
According to Mittag-Leffler function,
.
Hence,
.
Thus
. (4.3)
And hence the model (3.4) is bounded above by
.
As a result, all of the system’s (3.4) solutions will be bounded in
.
5. Stability Analysis
In this part, we’ll identify all of system (3.4)’s trivial and non-trivial equilibrium points, as well as their existence conditions.
5.1. Equilibrium Points and Existence Criteria
There are five different types of equilibrium points in system (3.4).
1)
: Trivial equilibrium,
2)
: Axial equilibrium,
3)
: Planar equilibrium where,
and
is to be obtained from the equation
Hence
is positive if
.
4)
: Planar equilibrium points where,
,
.
Hence
is positive if
.
5) The interior equilibrium is
, here
are the positive roots of the following system of equations
(5.1)
After solving the above equations, we get:
(5.2)
Hence
and
is positive if
. (5.3)
And
is to be determined from the following equation
. (5.4)
The Equation (5.4) has a positive root for
5.2. Local stability
The Jacobian matrix of the model (3.4) at
can be represent as
.
where,
,
,
Theorem 5.2.1. The system (3.4) always exhibits unstable behavior at
.
Proof: The eigenvalues of the Jacobian matrix are given by
,
,
. It follows that
and
(
). So, the system (3.4) is unstable at
.
Theorem 5.2.2. The system (3.4) is locally stable at
if
.
Proof: The eigenvalue is
. To solve the equation
, (5.5)
we find another two eigenvalues.
Therefore, the condition of negative roots of Equation (5.5) is
So the system (3.4) is locally stable if
.
Theorem 5.2.3. The model system (3.4) is locally stable at
if
.
Proof: The matrix(J) at
can be written as:
.
One of the eigenvalue of Jacobian matrix is
and the other two eigenvalues are obtain from of the equation
. So the system (3.4) is locally stable if
.
Theorem 5.2.4. The model system (3.4) is locally asymptotically stable at
if
, for
and
.
Proof: The characteristic equation of the Jacobian matrix at
is given by
. (5.6)
where,
,
,
.
The roots of (5.6) are negative or have a negative real component, according to Rowth-Hurwitz criterion if
and
.
5.3. Global Stability
In this section we discuss the global stability of the system (3.4).
Theorem 5.3.1. The system (3.4) is globally asymptotically stable at
if
.
Proof: Let the function L at
:
(5.7)
Taking time derivative of the above equation is
. (5.8)
Using Lemma 3, then
So
, if
.
6. Influence of Fear on Population Density in the Proposed Model
The main objective of this section is to investigate the impact of fear all populations. To look at this, we must first separate each component of the fear factor
at
.
Now
,
. Hence
and
is positive if
, and
is the solution of the equation
, (6.1)
where
,
and
.
Differentiating (6.1) w.r. to
, we have
, where
and
, with
,
.
As a result, the sign of
is always negative, suggesting that as the fear level
rises,
decreases. Differentiating equations (5.2) with respect to fear level
, we have
As a result, we observed that as fear levels rise, the density of intermediate predators decreases.
7. Hopf-Bifurcation
Theorem 7.1. Let us consider
, where
. A fractional order Hopf bifurcation is proposed in which the model (3.4) undergoes a Hopf bifurcation through
at the value
if
● The Jacobian matrix has two complex-conjugate eigenvalues
.
●
.
●
, where
.
Proof: For
, the characteristic Equation (5.6) becomes
.
i.e.,
.
i.e.,
.
Let
.
For
, where
, then we have
,
,
.
Now the condition
at
, is verify below.
Substitute
into the characteristic equation and taking derivative with respect to
, we get
,
.
where
,
,
,
.
Noticing that,
.
We have,
,
,
,
.
Therefore,
when,
, and
.
8. Numerical Discussions
To validate the theoretical findings presented in the earlier portions of this study, numerical simulations have been carried out using the modified Predictor-corrector approach [39] [40] in the Matlab framework. The Adams-Bashforth-Moulton scheme is employed for the numerical study.
,
, (8.1)
,
where
,
and in the Caputo interpretation,
is equivalent to the popular Volterra integral equation.
,
. (8.2)
We established model parameter values for numerical simulation purposes based on information from appropriate journal articles (see Table 1). This section is divided into four parts. The stability of our proposed model is discussed at
,
and
in Part 1. Part 2 delves into the dynamical behavior of all population of various fractional orders. Part 3 is to explore the Hopf bifurcation of the model system (3.4). Graph of mean density of all population with respect to the variation of
is discussed in Part 4.
Part 1:
The stability of our suggested model is discussed in this section. The parameter values used for the numerical simulations in Part 1 is provided in Table 1. Instead of the trivial
and axial
, we are more concerned in the stability of the coexistence equilibrium
,
and
. System (3.4) is shown in Figures 1-3 to be asymptotically stable for
.
Part 2:
The parameter values in Table 1 are used to examine the dynamical behaviour
Table 1. Parameter values for numerical study.
(a)(b)
Figure 1. The system’s time series and phase portrait are consistent with Table 1 at
.
(a)(b)
Figure 2. The system’s time series and phase portrait are consistent with Table 1 at
.
(a)(b)
Figure 3. The system’s time series and phase portrait are consistent with Table 1 at
.
of the entire population. Figures 4(a)-(c) depict all populations’ behavior over time for different fractional orders α. Figure 4(a) depicts that the number of prey population increases when
changes from 0.85 to 0.95. We see in Figure 4(b) that number of intermediate predator population increases when α increases. Figure 4(c) depicts that the number of top predator population increases with time when α changes from 0.95 to 0.8.
Part-3:
In this part, it is explored whether the model system (3.4) exhibits a Hopf bifurcation with fractional order
. The relevance of Hopf bifurcation is discussed using the following set of parametric variables.
The bifurcation analysis is investigated using the values of the parameters in Table 2. The model system’s unique endemic equilibrium
is obtained using the parameters in Table 2. Figure 5 shows the Hopf bifurcation diagram of the model system (3.4) with respect to the parameter
taking
. Also Figure 5 depicts that the model remains stable until
crosses its threshold value
, when
crosses its threshold value the model become unstable.
Part 4:
Figure 6 shows the change in mean density of prey, intermediate predator and top predator with respect to the variation of
. Figure 6(a) depicts that mean density of prey population increases for
, decreases for
and oscillates for
. Figure 6(b) shows that mean density of intermediate predator population increases for
, decreases for
and oscillates for
. Figure 6(c) depicts that mean density of top predator population decreases for
.
(a)(b)(c)
Figure 4. Variation of
with respect to time of the model (3.4) corresponds to Table 1.
(a)(b)(c)
Figure 5. Bifurcation diagram of the system (3.4) corresponds with Table 2.
(a)(b)(c)
Figure 6. Graph of mean density of all population with respect to the variation of
.
Table 2. Parameter values for study of Hopf bifurcation.
9. Conclusion
A fractional order prey-predator system with a fear impact on the prey population has been suggested. The non-negativity, boundedness, and uniqueness of the model system’s solutions have been demonstrated and investigated. The equilibrium points have been determined, and the stability of all of the model system’s possible equilibrium points has been studied both analytically and numerically. The system’s local and global stability are both established. Under constrained parametric circumstances, the system is found to be locally and globally asymptotically stable. The system is found to have Hopf bifurcations with respect to the parameter
. Local stability and Hopf bifurcation analysis have been the focus of our research. Fractional derivatives and integrals are a difficult idea to convey since they are derived from pure mathematics. The influence of predator anxiety on the population of prey is proposed in this research. A higher order indexing can be associated with weak memory, but a lower order classification can be associated with distant memory since the fractional order and memory are coupled. As a result, our research suggests that weak memories can help to improve the predator-prey system’s ability to cohabit peacefully, but powerful memories can actually make this situation worse. The discretization technique and FDE12 based on Adams-Bashforth-Moulton scheme are used to perform simulation studies. The concept of fractional calculus has nothing to do with any major geometrical meaning, such as function trend or convexity. Finally, we translate our mathematical findings into ecological terms as follows: a large number of prey refuges are produced in the system as a result of the prey species’ profound recollection of the exogenous effects that fear has on their life cycles. However, as we gradually lower the model system’s order, especially in the case of a low amount of predator-induced fear, the dynamics of the model system change away from its unstable behavior and toward stability. Consequently, our extensive mathematical findings show that weak memory might contribute to the stable existence of the predator-prey system whereas excessive memory of the species degrades the stability of the model system. In the suggested model (3.4), there is more work to be done, such as determining the species’ maturity to release harmful compounds into the environment. The model is now more realistic and intriguing as a result of these changes. We will leave this for further study.