https://orcid.org/0000-0003-1112-831X
Figures
Abstract
This study proposes and analyses a revised predator-prey model that accounts for a twofold Allee impact on the rate of prey population expansion. Employing the Caputo fractional-order derivative, we account for memory impact on the suggested model. We proceed to examine the significant mathematical aspects of the suggested model, including the uniqueness, non-negativity, boundedness, and existence of solutions to the noninteger order system. Additionally, all potential equilibrium points for the strong and weak Allee effect are examined under Matignon’s condition, along with the current state of conditions and local stability analysis. Analytical results are also provided for the necessary circumstances for the Hopf bifurcation initiated by the fractional derivative order to occur. We also demonstrated the global asymptotic stability for the positive equilibrium point in both the strong and weak Allee effect cases by selecting an appropriate Lyapunov function. This study’s innovation is its comparative investigation of the stability of the strong and weak Allee effects. To conclude, numerical simulations validate the theoretical findings and provide a means to investigate the system’s more dynamical behaviours.
Citation: K. R, Kumar G. R, Khan A, Abdeljawad T (2025) Analysis of the stability of a predator-prey model including the memory effect, double Allee effect and Holling type-I functional response. PLoS ONE 20(1): e0305179. https://doi.org/10.1371/journal.pone.0305179
Editor: Mattia Miotto, Italian Institute of Technology, ITALY
Received: February 19, 2024; Accepted: May 26, 2024; Published: January 3, 2025
Copyright: © 2025 K. et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Investigating the complex nature of the predator-prey framework could supply valuable insights into numerous processes which take place during interactions between predators and their prey. The Allee influence is a significant ecological phenomenon that affects the per capita expansion of predator or prey populations of predator or prey populations. It is associated with the situation in which at low levels of population, the population’s per-capita increase pace is positively correlated with its density. As far as Allee effects are concerned, there are two varieties: strong and weak. The strong Allee impact results in the population to decline below a particular point, known as the Allee level, at the moment when the per capita expansion pace starts to decline [1,2]. More specifically, in low-density populations, the conservation biology-related risk of extinction increases as the Allee threshold becomes bigger. Conversely, in situations when there are few individuals in a population, the rate of increase per individual always remains positive in the weak Allee effect [3–5]. When multiple processes that cause the Allee influence have an impact on the same population at the same time, it’s called the twofold (or multiple) Allee phenomenon [6,7].
For a given population, the most prevalent continuous increase function is given by the following equation when considering a single mechanism and the multiplicative Allee effect:
(1)
with x1 standing for population density, ρ for carrying capacity and r for the population’s inherent per capita expansion rate. An adjustment to the logistic model, the term (x1−α) shows that when x1<α, the population density drops to zero and
, and when x1>α, the population expands to ρ, with
. If a certain population level (α) is greater than zero or if −ρ<α≤0, then the impact of Allee in Eq (1) may be specified as strong or weak. Predator-prey interaction models may include multiple processes of Allee influences within the same population, ensuing in a demographic Allee influence [7]. The collective influence of these factors is referred to as a double or multiple Allee effect. The prevalence of two or more component Allee effects was shown by a recent assessment by Berec et al. [7] of the findings for these effects, which included instances from both terrestrial and aquatic ecosystems, as well as from plants, invertebrates, and vertebrates, also well as from both naturally occurring and artificially manipulated populations. The literature on prey-predator interaction models that include the double Allee effect is few, with just a handful of publications covering the topic over the last several decades [8,9]. If the multiplicative double Allee effect were to impact a single population’s growth function, the following equation would control one of its mathematical forms
(2)
for any auxiliary parameters β>0 resulting in α>−β. Both the factor) (x1−α) representing the Allee effect, is discussed above for Eq (1), and the hyperbolic function
representing the other Allee effect, which influences the species’ inherent rate of increase due to external challenges (with the supplementary parameter β>0 quantifies the intensity of the Allee effect), are present in Eq (2) and both have an impact on the same population growth. The primary goal of the aforementioned research initiatives is to explore the influence of the Allee factor on the occurrence of distinct behaviours in the predator-prey system. Despite evidence suggesting the twofold Allee effect may be seen in predator population increase, the majority of research have only examined its impact on prey population growth [10].
It is impossible for prey and predator interactions to take place in nature without memory, because the paces of prey and predator growth for a specific moment are dependent on the history of the variables at all past moments, rather than just the local state at that moment [11–15]. Determining the fractional-order derivative perfectly captures all of the above requirements. It may represent the memory and genetic features inherent in different processes, which is an advantage over the integer order derivative. Recent years have seen fractional derivatives in fractional calculus emerge as a powerful tool for characterising the memory and genetic features of a wide range of materials and processes [16], in fields as diverse as biology, economics, engineering, physics, and many more [17–21]. A number of researchers have shown that memory-based systems are superior at representing and explaining real-world occurrences [22–28].
In light of all of this, the purpose of this investigation is to analyse the fractional-order predator-prey system that takes the double Allee phenomena into account. To account for the influence of memory on prey and predator growth rates, the suggested framework includes the Caputo fractional-order derivative. It is our understanding that no one else has suggested or explored the complexities of the model we have provided, which includes the memory impact using the Caputo non-integer order derivative and the double Allee influence in the prey increase. It is possible that this study may shed light on hitherto unknown aspects of the multiple Allee effect and point the way for further investigation in a wide range of research methods, including theoretical modelling, field investigations, and laboratory trials.
Here is how the rest of this paper is structured. The system construction, which covers the existence, boundedness, non-negativity, and uniqueness of our system’s solutions, is stated in Section 2. Next, we examine the model’s dynamic behaviours for both weak and strong Allee effects at every possible equilibrium point. The Hopf bifurcation analysis is then provided in section 3. Furthermore, the adequate requirements for the inner equilibrium points’ global stability are met. Section 4 displays numerical simulations that both validate our analytical results and numerically investigate the effects of the fractional-order system’s order, Allee threshold on both strong and weak of our model’s dynamics. We wrap up our findings under Section 5.
2. Model formulation
This research takes a look at a prey-predator scenario in which the predator feeds on prey based on its functional response and the prey population grows in response to a twofold Alleeeffect of type (2). Following that, the predator-prey paradigm transforms into
(3)
such that x1>0, x2>0. In system (3) x1(t) and x2(t) represents the prey and predator densities at time t. Table 1 displays the descriptions and symbols of the aforementioned parameters. To capture the whole time condition of population expansion, we consider non-integer derivative order on the left side of the traditional derivative model (3), as shown below:
(4)
The symbol symbolizes the Caputo fractional-order derivative of a real valued function f which is defined as follows:
where Γ(.) is the Gamma function and ζ∈(n−1,n],n∈ℕ [16]. In system (4), all of the parameters are identical to those in system (3).
2.1 Existence, uniqueness, non-negativity, and boundedness
Theorem 1. For each , then the initial value problem of system (4) possesses unique solution Z(t)∈Ω that is specified for all t≥0.
Proof. We need to consider the possibility whether the dynamical system having a unique solution in the region [0,∞)×Ω, where Ω = {(x1,x2)∈R2:max{|x1|,|x2|}≤Q} if we want to verify this claim. Suppose that
and now consider a mapping V(Z) = (V1(Z),V2(Z)), where
and H2(V) = bx1x2−cx2.Then, for any
we have
where L = max{L1,L2} and
As a result, the mapping V(Z) fulfils the Lipschitz requirement. Hence, with the given starting condition , where
and
there is a unique solution to the system of differential Eq (4), as determined by the existence and uniqueness theorem of [29].
Theorem 2. All solutions to the differential Eq (4) that originate in the region are uniformly bounded and non-negative.
Proof. To begin, we look at the region to see whether the solution is non-negative. Consequently, for any t≥0, we have to establish that x1(t)≥0 and x2(t)≥0. Let’s say that for any t≥0, the condition x1(t)≥0 is false. Then, there exist t1>0 such that x1(t)>0 for t∈[0,t1),x1(t1) = 0 and x1(t+1)<0.Then, using the first equation in (4), we have
Lemma 1 of [30] suggests that x1(t+1) = 0, which defies the assertion that x1(t+1)<0. As a result, we may conclude that x1(t)≥0 for any t≥0. Using the same approach, it becomes clear that there is a non-negative solution to Eq (4) with a beginning point in area
. Consequently, for any t≥0, it provides x2(t)≥0 in the same way. After that, we need to prove that the solutions to Eq (4), which begin in the area
, are uniformly bounded.
Theorem 3. All the solutions to the system begin with , are uniformly bounded and enter the region
.
Proof. For given initial conditions x1(0)>0,x2(0)>0, let (x1(t),x2(t)) be the solution of the framework (4) at any time t>0.
Consider the function and the non-integer time derivative of this is
,
Now, for μ>0 we have
where
Following Lemma 1 of [31,32], we get
as t→∞. As a result, the region
contains all of the solutions to framework (4) that begin in
. This concludes the proof.
3. Equilibrium points and stability analysis
In this part, we get the equilibrium points and existence requirements for the weak (α<0) and strong (α>0) Allee effects, and we use the Matignon condition [16] to examine their local stability.
In the presence of strong Allee effect (α>0), the following biologically feasible equilibrium points are present in system (4): (i) , the extinction equilibrium, (ii)
, the two axial equilibrium points (iii) If
then interior equilibrium point is
exists, where
.
Theorem 4. For any possible equilibrium point with strong Allee effect (α>0), the following stability of the framework (4) is given:
- a) The extinction equilibrium point
is remains constant at all times.
- b) The axial equilibrium point
is stable if
.
- c) The axial equilibrium point
is stable if
.
- d) The interior equilibrium point
is stable if
.
Proof. This theorem may be proven by defining the community matrix for framework (4), which is shown below
(5)
where
, s12 = −ax1,
a) The Jacobian matrix (5) at is defined as
(6)
The eigen values are with
. Consequently,
is locally asymptotically stable according to the Matignon criteria [16].
b) The Jacobian matrix (5) at is defined as
(7)
The eigen values are if α<ρ and λ2 = bρ−c<0 if
. Therefore
whenever
. Hence
is locally asymptotically stable by the Matignon condition [16].
c) The Jacobian matrix (5) at is defined as
(8)
The eigen values are if ρ<α and λ2 = bα−c<0 if
. Therefore
whenever
. Hence
is locally asymptotically stable by the Matignon condition [16].
d) The Jacobian matrix at is defined in (5). Then
and
. Substituting
in above results, we obtain following results.
if
and
if
, this is always true for
exists. Therefore
whenever
. Hence
is locally asymptotically stable.
The model system (4) possess the following biologically feasible equilibrium points, for weak Allee effect (α<0) (i) , the extinction equilibrium, (ii)
the axial equilibrium point (iii) If
then interior equilibrium point is
where
.
Theorem 5. The stability of the system (4) of each feasible equilibrium points for weak Allee effect (α<0) are as follows:
- a) The trivial equilibrium point
is always a saddle point.
- b) The axial equilibrium point
is stable if
.
- c) The positive equilibrium point
is stable if
.
Proof.
- (a) The community matrix for system (4) may be obtained by replacing (5) with
, which is represented by
(9)
The eigen values are with
and
Hence
is always a saddle point according to the Matignon criteria [16].
- (b) On using
to (5), we calculate the community matrix for the system (4)
The eigen values are and λ2 = bρ−c<0 if
. Therefore
whenever
. Hence
is locally asymptotically stable by the Matignon condition [16].
- (c) The Jacobian matrix at
is defined in (5). Then
and
. Substituting
in above results, we obtain following results.
if
and
if
, this is always true for
exists. Therefore
whenever
. Hence
is locally asymptotically stable.
A Hopf bifurcation will occur in a non-integer order system if its resilience changes and the community matrix computed at the equilibrium point contains a pair of complex conjugate eigenvalues. The criteria initially proposed in [33] are utilised in this instance to determine the presence of a Hopf bifurcation. The order of the non-integer derivative (ζ) affects the resilience of the positive equilibrium points for both mild and strong Allee effects, as Theorems 4 and 5 establish. This lets us demonstrate the next theorem, which says that when ζ passes over the crucial value ζ*, a Hopf bifurcation has to happen at the positive equilibrium point.
Theorem 6. Let
, system (4) undergoes a Hopf bifurcation around the interior point
when ζ crosses ζ*.
Proof. If eigen values of system (4) at
consist of a pair of complex conjugate numbers, both of which have positive real components. We also confirm ν1,2(ζ*) = 0, where
and
. The equilibrium point
experiences a Hopf bifurcation when the parameter ζ passes the critical value ζ*, as stated in Theorem 3 in [34].
The same argument applies to the scenario of a mild Allee effect.
3.1 Global stability
Under the weak and strong Allee effects, we examine the interior equilibrium ’s global asymptotic stability in the following.
Theorem 7. The positive equilibrium point of framework (4) is globally asymptotically stable.
Proof. Assume the following positive definite Lyapunov function
(11)
Clearly θ(x1,x2) is a positive definite function. Applying the ζth order Caputo non-integer derivative on both sides of the Eq (11) we obtain,
(12)
Lemma 4 in Reference [31] provides us with a relation,
(13)
Now we construct the domain . Then, clearly
for all (x1,x2) belongs to the set D as x1,x2>0 and
specifies that
. Following that,
is the equilibrium condition. Therefore, the singleton set {E*} is the only invariant set for which
. Lemma 4.6 from Reference [35] implies that the equilibrium point
is globally stable in region
. For weak Allee effect the domain
.
4. Numerical simulations
We analyse system (4) using the below hypothetical parameter values , Adams-Bashforth-Moulton algorithm [36], and run simulations to explore the impact of the order of the non-integer derivative (ζ). In light of the parameter values mentioned earlier and the results of Theorem 6, we can establish that the positive equilibrium point is asymptotically stable while ζis less than ζ*and destabilising when ζ is more than ζ* by setting the crucial value ζ* to be 0.98. The asymptotic stability of the positive equilibrium point
is observed to be present when ζ is smaller than ζ*. However, when ζ exceeds ζ*, the stability of the positive equilibrium point
deteriorates, resulting in the convergence of all numerical solutions to a limit-cycle through a Hopf bifurcation. The Hopf bifurcation may be noticed based on the phase-portraits and trajectories shown in both Figs 1 and 2 given ζ = 0.95<ζ* and forζ = 0.984>ζ*, respectively.
We next use the following hypothetical parameter values to analyse numerically how the Allee criterion α>0 affects the behaviour of system (4), r = 0.067, ρ = 7.65, .
If we take a look at the parameter values mentioned before, we can determine a critical value α* = 0.19 that makes positive equilibrium point asymptotically stable if α<α* and not stable if α>α*. All numerical solutions converge to a limit-cycle through a Hopf bifurcation once the interior point loses its stability,
which is asymptotically stable for α<α* and unstable for α>α*. The phase-portraits for α = 0.02<α* and α = 0.31>α* in Fig 3 also demonstrates the Hopf bifurcation.
We achieved a critical value α* = −0.3868 and found the Hopf bifurcation result (shown in Fig 4). provided the influence of Allee is weak (α<0). In this case the parameter values are and corresponding to this equilibrium point is
.
5. Conclusions
This work presented and analysed the dynamics of a fractional-order predator-prey system with Holling type-I functional response and double Allee effect in the prey population. The combined impact accelerates the probability of extinction and population decline, and more theoretical research is required to support the coexistence of these varied ecosystems of vulnerable populations [37]. Therefore, when managing exploited or threatened populations, the multiple Allee effect is more important than the single Allee effect from an ecological perspective. By using Adams-Bashforth-Moulton algorithm and parametric values , we studied different scenarios and phase-portraits for the prey and predator, Fig 1. ζ = 0.95<ζ*, and ζ = 0.984>ζ* for Predator, Fig 2. ζ = 0.984, 0.95 for prey and ζ = 0.984, 0.95 for prey and predator, Fig 3. ζ = 0.98, α = 0.02,0.31 predator and prey, and Fig 4. ζ = 0.98, α = 0.02,0.31 for prey and predator.
Initially, it has been shown that the solution exists, is unique, non-negativity, and is bounded. The local and global stability features of each potential non-negative equilibrium point were then determined. The extinction of prey and predator points continues to be stable in strong Allee effect situations, but unstable in weak Allee influence scenarios. When , the predator extinction threshold remains stable in the strong Allee effect scenario; however, when
, it remains stable under the weak Allee effect case. On top of that, we established that there exists a Hopf bifurcation governed by the order of the non-integer derivative (ζ) surrounding the interior point and we analytically determined the critical ζ* of this bifurcation. We confirm that the Hopf bifurcation occurs in our numerical simulations. In addition, computer simulations have shown that Hopf bifurcation occurs in both strong and weak Allee effects. In these cases, the double Allee effect-affected predator population may be able to reside in our system (4).
According to the results shown above, the dynamical behaviour of this system can be controlled and predicted by paying close attention to the sensitivity of the extinction, coexistence, and oscillation phenomena in the two populations that make up the biological model that incorporates the double Allee effect.
Acknowledgments
Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and support through TAS research lab.
References
- 1. Cushing J.M. Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations. J. Biol. Dyn. 2014, 8, 57–73. pmid:24963977
- 2. Buffoni G.; Groppi M.; Soresina C. Dynamics of predator-prey models with a strong Allee effect on the prey and predator dependent trophic functions. Nonlinear Anal. Real World Appl. 2016, 30, 143–169.
- 3. Sasmal S.K.; Chattopadhyay J. An eco-epidemiological system with infected prey and predator subject to the weak Allee effect. Math. Biosci. 2013, 246, 260–271. pmid:24427786
- 4. Saifuddin M.; Biswas S.; Samanta S.; Sarkar S.; Chattopadhyay J. Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator. Chaos Solitons Fractals 2016, 91, 270–285.
- 5. Pal S.; Sasmal S.K.; Pal N. Chaos control in a discrete-time predator-prey model with weak Allee effect. Int. J. Biomath. 2018, 11, 1–26.
- 6. Arancibia-Ibarra C.; Flores J.; Bode M.; Pettet G.; Van Heijster P. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discret. Contin. Dyn. Syst. 2021, 26, 943–962.
- 7. Berec L.; Angulo E.; Courchamp F. Multiple Allee effects and population management. Trends Ecol. Evol. 2007, 22, 185–191. pmid:17175060
- 8. Angulo E, Roemer G, Berec L, Gascoigne J, Courchamp F. Double Allee effects and extinction in the island fox. Conserv Biol 2007;21(4):1082–91. pmid:17650257
- 9. González-Olivares E, González-Yanez B, Mena Lorca J, Rojas-Palma A, Flores J. Consequences of double Allee effect on the number of limit cycles in a predator–prey model. Comput Math Appl 2011;62(9): 3449–63.
- 10. Courchamp F.; Clutton-Brock T.; Grenfell B. Multipack dynamics and the Allee effect in the African wild dog, Lycaon pictus. Anim. Conserv. 2000, 3, 277–285.
- 11. El-Shahed M.; Ahmed A.M.; Abdelstar I.M.E. Dynamics of a plant-herbivore model with fractional order. Progr. Fract. Differ. Appl. 2017, 3, 59–67.
- 12. Suryanto A.; Darti I.; Panigoro H.S.; Kilicman A. A fractional-order predator–prey model with ratio-dependent functional response and linear harvesting. Mathematics 2019, 7, 1100.
- 13. Khoshsiar Ghaziani R.; Alidousti J.; Eshkaftaki A.B. Stability and dynamics of a fractional order Leslie–Gower prey–predator model. Appl. Math. Model. 2016, 40, 2075–2086.
- 14. Panigoro H.S.; Suryanto A.; Kusumawinahyu W.M.; Darti I. Continuous threshold harvesting in a Gause-type predator-prey model with fractional-order. AIP Conf. Proc. 2020, 2264, 040001.
- 15. Panigoro H.S.; Suryanto A.; Kusumawinahyu W.M.; Darti I. Dynamics of an eco-epidemic predator–prey model involving fractional derivatives with power-law and Mittag–Leffler kernel. Symmetry 2021, 13, 785.
- 16.
Petras I. Fractional-Order Nonlinear Systems; Springer: London, UK, 2011.
- 17. Kumar S.; Kumar R.; Cattani C.; Samet B. Chaotic behaviour of fractional predator-prey dynamical system. Chaos Solitons Fractals 2020, 135, 109811.
- 18. Singh J.; Ganbari B.; Kumar D.; Baleanu D. Analysis of fractional model of guava for biological pest control with memory effect. J. Adv. Res. 2021, in press. pmid:34484829
- 19. El-Ajou A.; Al-Smadi M.; Oqielat M.N.; Momani S.; Hadid S. Smooth expansion to solve high-order linear conformable fractional PDEs via residual power series method: Applications to physical and engineering equations. Ain Shams Eng. J. 2020, 11, 1243–1254.
- 20. Ali H.M.; Habib M.A.; Miah M.M.; Akbar M.A. Solitary wave solutions to some nonlinear fractional evolution equations in mathematical physics. Heliyon 2020, 6, e03727. pmid:32322721
- 21. Shi G.; Gao J. European option pricing problems with fractional uncertain processes. Chaos Solitons Fractals 2021, 143, 110606.
- 22. Khan H., Rajpar A.H., Alzabut J., Aslam M., Etemad S., Rezapour S. On a Fractal–Fractional-Based Modeling for Influenza and Its Analytical Results, Qual. Theory Dyn. Syst. 23, 70 (2024).
- 23. Alfwzan WF., Khan H., Alzabut J. Stability analysis for a fractional coupled Hybrid pantograph system with p-Laplacian operator Results in Control and Optimization 14, 100333 (2024).
- 24. Ahmed S., Azar AT., Abdel-Aty M., Khan H., Alzabut J. A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy Ain Shams Engineering Journal, 102566 (2024).
- 25. Tajadodi H., Khan H., Alzabut J., Gómez-Aguilar JF., An optimization method for solving fractional oscillation equation, Results in Physics, 57, 107403 (2024).
- 26. Sarijul Islam Md, Sarwardi S., Analysis of an Eco-Epidemic Predator-Prey Model with Nonlinear Prey Refuges and Predator Harvesting, Journal of Applied Nonlinear Dynamics, 12(3): 465–483 (2023).
- 27. Sarijul Islam, Md., Hossain S., Sarwardi S., Dynamical Analysis of an Eco-epidemic System with Different Forms of Prey Refuges and Predator Harvesting, The interdisciplinary journal of Discontinuity Nonlinearity and Complexity, 13(1): 95–112 (2024).
- 28. Sarif N., Sarwardi S., Complex dynamical study of a delayed prey-predator model with fear in prey and square root harvesting of both species, Chaos, 33(3):033112 (2023). pmid:37003791
- 29. Li Y, Chen Y, Podlubny I (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput Math Appl 59(5):1810–1821.
- 30. Odibat ZM, Shawagfeh NT. Generalized Taylor’s formula. Appl Math Comput. 186(1): 286–293 (2007).
- 31. Hong LL, Long Z, Cheng H, Jiang YL, Teng Z. Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. J Appl Math Comput. 2017, 54: 435–449.
- 32.
Kilbas A, Srivastava H, Trujillo J. Theory and Application of Fractional Differential Equations. New York, NY: Elsevier; 2006.
- 33. Li X.; Wu R. Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system. Nonlinear Dyn. 2014, 78: 279–288.
- 34. Li X.; Wu R. Hopf bifurcation analysis of a new commensurate fractional-orderfractional order hyperchaotic system. Nonlinear Dyn. 78, 279–288 (2014).
- 35. Huo J, Zhao H, Zhu L. The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal Real-World Appl.2015, 26: 289–305.
- 36. Diethelm K., Ford NJ., and Freed AD. (2002) ‘A predictor-corrector approach for the numerical solution of fractional differential equations’, Nonlinear Dyn, Vol. 29, pp. 3–22.
- 37. Pal PJ, Saha T. Qualitative analysis of a predator-prey system with double Allee effect in prey. Chaos Solit Fract 2015, 73: 36–63.