α | H∗ | P∗ |
0.20 | 0.526316 | 1.268630 |
0.37 | 0.526316 | 1.181450 |
0.43 | 0.526316 | 1.150680 |
0.50 | 0.526316 | 1.114780 |
1.25 | 0.526316 | 0.730165 |
1.70 | 0.526316 | 0.499396 |
This case describes a 30-year-old Hispanic male who presented with a significant headache that started after a period of weightlifting and squatting. The patient was diagnosed with a basilar artery dissection. His only complaint was a headache that was exacerbated with exertion and sexual activity; there were no neurologic deficits. The diagnosis of basilar artery dissection was established and supported by findings on the CT angiogram of his head and neck. Basilar artery dissections are rarely seen, as they are likely underrecognized due to their varying clinical presentations; however, it is important to consider these phenomena due to the risk of progression and high morbidity rates.
Citation: Sahibjot Bhatia, Nimrit Gahoonia, Jeffrey Stenger, Forshing Lui. A rare case of basilar artery dissection[J]. AIMS Neuroscience, 2023, 10(2): 109-117. doi: 10.3934/Neuroscience.2023008
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This case describes a 30-year-old Hispanic male who presented with a significant headache that started after a period of weightlifting and squatting. The patient was diagnosed with a basilar artery dissection. His only complaint was a headache that was exacerbated with exertion and sexual activity; there were no neurologic deficits. The diagnosis of basilar artery dissection was established and supported by findings on the CT angiogram of his head and neck. Basilar artery dissections are rarely seen, as they are likely underrecognized due to their varying clinical presentations; however, it is important to consider these phenomena due to the risk of progression and high morbidity rates.
The intricate ecosystems found in the natural world are mostly the result of predator-prey interactions. Effective mathematical models are developed to address the problems in predator-prey interaction. A basic model for two species communities was formulated by Lotka in 1925 and Volterra in 1931. The predator's prey feeding rate is one essential component of predator-prey interaction. In literature, many interaction functions have been modeled, for example, Holling type Ⅰ, type Ⅱ or type Ⅲ [1], ratio-dependent [2], Crowley-Martin [3] and Beddington-DeAngelis models [4,5]. In recent years, the predator-prey model of Holling type Ⅱ with different biological situations has been considered in the literature. For instance, in [6], the authors considered such a model with prey refuge and showed that the model parameters have crucial importance in the coexistence, stability and oscillation of their considered model. In reality, a predator consumes prey at different rates, depending on the species. Leslie and Gower [7] proposed a predator-prey model in which the predator's growth function is given by the ratio of the sizes of predator and prey. Differential equations describe the evolution of systems in continuous and discrete time. Further, discrete models are more realistic if the populations are small or if births and deaths occur at discrete times or within certain time intervals, such as a generation [8,9].
To model real ecosystems, the term "Allee effect'' is unavoidable and meaningful. Thus, many researchers have paid much attention to studying the dynamics of predator-prey models with the Allee effect [10,11]. There are two types of Allee effects: weak and strong. In the weak Allee effect, for the low population size, the per capita growth rate is positive and there is no threshold to grow [12]. But, in the case of the strong Allee effect, the per capita growth is negative and there is a population threshold introduced by the Allee effect, so the population must surpass this threshold to grow [13]. The Allee effect can be introduced in different ways, namely, multiplicative [14] and additive [13]. Discrete-time predator-prey model with Allee effects was studied in [15,16], and some more recent developments in predator-prey models can be found in [17,18,19]. Based on the above discussions, the Allee effect term is incorporated into prey growth for this study.
The study of stability analysis, bifurcation behavior, namely, flip bifurcation and Neimark-Sacker bifurcation, and control of the chaos of the discrete-time model has long been an issue of significant interest for researchers [20,21]. Among the mathematics, physics and engineering communities, chaos control has been extensively studied [22], whereas the problem of controlling chaos in ecology has been given in [24]. In some nonlinear systems, there is a major route to chaos via period doubling. In [25], the authors proposed a new hybrid control method to control the chaotic orbits by using state feedback and parameter perturbation for the period-doubling bifurcation. The chaotic attractor can be converted into one of the possible attracting time-periodic motions by giving small perturbations to the system parameters [26]. Recently, the work related to controlling chaos in the predator-prey model has been of greater interest to researchers. For instance, a predator-prey model with the Allee effect and cannibalism in the discrete-time model has been found in the literature [27], where the authors showed that the considered system undergoes flip and Neimark-Sacker bifurcations. Further, they discussed the various methods to control the chaos. A discrete-time classical Lotka-Volterra model obtained by applying the method of a piecewise constant argument for differential equations has been proposed in [28]. The authors studied the stability, bifurcation and chaotic behavior of their proposed model. Also, they implemented hybrid control and the Ott-Grebogi-Yorke (OGY) method to control bifurcation and chaotic behavior.
Since it was an interesting topic for the researchers, we are motivated by the above literature. To the best of our knowledge, the study of stability, bifurcation and chaos control analysis for a discrete-time prey-predator model with the additive Allee effect has not been studied. In this article, we aim to study the discrete-time Holling type Ⅱ predator-prey model with the additive Allee effect in prey growth. The main highlights of this paper are listed as follows:
1) We use the method of piecewise constant arguments for the differential equations to obtain the discrete-time model. The model is a good representation of the populations' interaction, which has non-overlapping generations.
2) We study the impact of the Allee effect in terms of the existence and local stability properties of all positive equilibrium points of the discrete model.
3) We drive the conditions for the existence of flip and Neimark-Sacker bifurcations near the interior equilibrium point for the discrete model by taking the Allee parameter as a bifurcation parameter. Further, we discuss the properties of both bifurcations with the help of the center manifold (CM) theorem and normal form theory.
4) We utilize the state and OGY feedback and hybrid control methods to control the bifurcating and chaotic behavior of the discrete model. We provide extensive numerical simulations to show the rich dynamics, including periodic windows, invariant circles and chaos in the discrete model.
The rest of the article is arranged as follows: The discrete-time Holling type Ⅱ predator-prey model with the additive type Allee effect is given, and the existence and local stability of equilibrium points are discussed in Section 2. Furthermore, an analysis of the model's dynamics is presented, including the identification of key parameters and their influence on the system's behavior. In Section 3, the bifurcation behavior of the proposed model is thoroughly examined. Two types of bifurcations, such as flip and Neimark-Sacker bifurcations, are investigated to gain a comprehensive understanding of the system's response to parameter changes. In Section 4, the state and OGY feedback and hybrid control techniques for the considered predator-prey model are discussed. The effectiveness of these control approaches is highlighted, emphasizing their ability to regulate and stabilize system dynamics. To ensure the validity and reliability of the mathematical results, extensive numerical simulations are presented in Section 5. Lastly, in Section 6, a comprehensive conclusion and a brief discussion are presented.
First, we describe the continuous-time Holling type Ⅱ prey-predator model with the additive Allee effect [29], as follows:
{dH(t)dt=H(t)(r(1−H(t)k)−αβ+H(t)−a1P(t)1+a1a2H(t)),dP(t)dt=P(t)(γa1H(t)1+a1a2H(t)−d), | (2.1) |
where H(t) and P(t) are the population sizes of prey and predator respectively at time t; r and k represent the growth rate and carrying capacity of prey, respectively; γ and d represent the consumption rate of the predator on prey and per capita death rate of the predator, respectively. The most common type of functional response is Holling type Ⅱ, which is based on the idea that, if prey species is low, the rate of predation is proportional to prey density and is of the form a1H(t)1+a1a2H(t), where a1 and a2 represent the predator capture rate and handling time per prey, respectively.
In general, the term αβ+H(t) is known as the additive Allee effect, where α and β represent the severity of the Allee effect and the prey population size at which fitness is half of its maximum value, respectively. Many studies have been conducted to understand the dynamics of prey-predator system models with the additive Allee effect [30,31,32].
To reduce the complexity, and, for our convenience, after redefining the parameters, the model (2.1) takes the form
{dH(t)dt=H(t)(a−bH(t)−αβ+H(t)−cP(t)e+H(t)),dP(t)dt=P(t)(fH(t)e+H(t)−d), | (2.2) |
where H(t)≤0, P(t)≤0, a=r, b=rk, c=a1a1a2, e=1a1a2 and f=γa1a1a2. Note that, in the sense of [33], if α<aβ (α>aβ), the Allee effect is the weak (strong) one.
Next, based on the appropriate modification of overlapping generations, one can get the difference equations for modeling a population with non-overlapping generations. Euler approximations, non-standard finite difference schemes [34] and piecewise constant arguments [35,36] are the most common methods to obtain discrete versions from the continuous models. In this way, differential equations with piecewise constant arguments have been useful. Let us assume that the populations have no overlap between the successive generations and the population growth occurs in discrete steps t∈[n,n+1), n=0,1,2,3,…. Also, consider that the variables and constants in (2.2) change in regular time intervals and incorporate this idea; then the corresponding discrete-time model for (2.2) is obtained by the method of piecewise constant arguments for differential equations, as follows:
{H(n+1)=H(n)exp[a−bH(n)−αβ+H(n)−cP(n)e+H(n)],P(n+1)=P(n)exp[fH(n)e+H(n)−d], | (2.3) |
where H(n+1) and P(n+1) denote the populations in generation n+1 that are related to the sizes H(n) and P(n) of the populations in the preceding generation n and a, b, c, d, e, f, α and β are all positive constants.
Note, in the absence of a predator P and the Allee effect, the above model (2.3) becomes a one-dimensional model similar to the Ricker model [37], namely,
H(n+1)=H(n)ea−bH(n). |
This system represents the relationship between the current and previous population size H. In a logistic map, the population's rate of per capita growth falls as the population size approaches a limit set by the availability of resources. On the Ricker map, population growth is almost exponential; however, as population size increases, the instantaneous growth rate declines linearly; and, eventually, population size reaches a plateau and oscillates around a mean.
The main aim of this paper is to study the local stability, bifurcation behavior and various chaos control analyses for the discrete time prey-predator model (2.3) with the additive Allee effect in the case of the weak one in R={(H,P)|H>0,P>0}.
We need to solve the following equations to find the equilibrium points of the model (2.3):
{a−bH−αβ+H−cPe+H=0,fHe+H−d=0. | (2.4) |
By direct substitution, from (2.4), we have three positive equilibrium points: The trivial equilibrium point (0,0), the predator-free equilibrium point (ˉH,0), where ˉH is the positive root of the equation
bˉH2+(bβ−a)ˉH+α−aβ=0, | (2.5) |
and the interior equilibrium point (H∗,P∗), which is given by
(H∗,P∗)=(def−d,(a−bH∗)(β+H∗)(e+H∗)−α(e+H∗)c(β+H∗)). |
Then from [29], we have the following lemma.
Lemma 1. Consider α<aβ; then, the model (2.3) has a unique boundary equilibrium (ˉH,0)=((a−bβ)+√(a−bβ)2+4b(aβ+α)2b,0). Also, the model (2.3) has a unique interior equilibrium (H∗,P∗)=(def−d,(a−bH∗)(β+H∗)(e+H∗)−α(e+H∗)c(β+H∗)) if α<(a−bH∗)(β+H∗) and max(β,def−d)<ab<β+def−d.
Next, the Jacobian matrix for the model (2.3) at arbitrary equilibrium point (H,P) is calculated to investigate the local stability property:
J=[(1−bH+αH(β+H)2+cHP(e+H)2)A1−(cHe+H)A1(efP(e+H)2)A2A2], | (2.6) |
where A1=exp(a−bH−αβ+H−cPe+H) and A2=exp(fHe+H−d).
The Jacobian matrix of (2.3) derived at (0,0) is
J=[exp(a)00exp(−d)], | (2.7) |
and we have the Jacobian matrix at (ˉH,0) as
J=[a1a20a3], | (2.8) |
where a1=(1−bˉH+αˉH(β+ˉH)2)exp(a−bˉH−αβ+ˉH), a2=−(cˉHe+ˉH)exp(a−bˉH−αβ+ˉH) and a3=exp(fˉHe+ˉH−d).
The Jacobian matrix of (2.3) evaluated at (H∗,P∗) is
J∗(H∗,P∗)=[1−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2−cH∗e+H∗efP∗(e+H∗)21]. | (2.9) |
The characteristic equation of J∗ is given by
λ2−Tλ+D=0, | (2.10) |
where
T=2−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2,D=1−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3. |
From [38], let λ1 and λ2 be the eigenvalues of the Jacobian matrix (2.6) for some arbitrary equilibrium point (H,P). We recall some topological classifications of the equilibrium points. The equilibrium point (H,P) is a sink (locally asymptotically stable) if |λ1|<1 and |λ2|<1. (H,P) is a source (locally unstable) if |λ1|>1 and |λ2|>1. (H,P) is a saddle if |λ1|>1 and |λ2|<1 (or |λ1|<1 and |λ2|>1). And, (H,P) is non-hyperbolic if either |λ1|=1 or |λ2|=1. As we know, the eigenvalue of the Jacobian matrix plays an important role in establishing the stability properties of the equilibrium points. Using the eigenvalues, we discuss the stability of the equilibrium points in the following lemmas:
Lemma 2. The trivial equilibrium (0,0) is always a saddle and the predator-free equilibrium (ˉH,0) is a
(1) sink if |a1|<1 and |a3|<1;
(2) source if |a1|>1 and |a3|>1;
(3) saddle if |a1|>1 and |a3|<1 (or |a1|<1 and |a3|>1).
Lemma 3. The coexisting equilibrium point
(1) (H∗,P∗) is a sink if D<1 and |T|<D+1;
(2) (H∗,P∗) is a source if D>1 and |T|<D+1 or |T|>D+1;
(3) (H∗,P∗) is a saddle if 0<|T|+D+1<2|T|;
(4) (H∗,P∗) is non-hyperbolic if |T|=|D+1|, or D=1 and |T|≤2.
Previously, we discussed the local stability properties of the equilibrium point (H∗,P∗). Now, we study the bifurcation behavior, namely, flip and Neimark-Sacker bifurcations, of the model (2.3) at (H∗,P∗) by varying the parameter α and keeping all model parameters fixed. It should be noted that further analyses can hold for other parameters also. Also, stability properties of the existing period-two orbit and the invariant closed curve are discussed with the help of the CM theorem and normal form theory as in [38,39]. The conditions for the model (2.3) to undergo both types of bifurcation are discussed below.
We know that the model can undergo a flip bifurcation if one of the eigenvalues of J∗ is −1 and the other not 1 or −1. Hence, we assume one eigenvalue as −1, then, from (2.10),
4−2bH∗+2αH∗(β+H∗)2+2cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3=0, | (3.1) |
α=(H∗+β)22H∗(−4+2bH∗−cefH∗P∗(e+H∗)3−2cH∗P∗(e+H∗)2)=αf, |
where α=αf is the critical point of α that satisfies (3.1), and it is necessary for the flip bifurcation near the equilibrium point (H∗,P∗) for the model (2.3).
Let ΩF={(a,b,c,d,e,f,α,β):α=αf,a,b,c,d,e,f,β>0}; at the critical parameter value α=αf, the model exhibits flip bifurcation at (H∗,P∗), when α=αf varies in the neighborhood of ΩF.
Also, the model (2.3) admits the Neimark-Sacker bifurcation if the Jacobian matrix J∗ has complex conjugate eigenvalues with a modulus value of one, and also satisfies the following conditions
(T(H∗,P∗))2−4D(H∗,P∗)<0 and D(H∗,P∗)=1, | (3.2) |
which gives
A1(α)=(2−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2)2−4(1−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3)<0,A2(α)=−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3. |
Now A2(α)=0 gives
α=(b−cefP∗(e+H∗)3−cP∗(e+H∗)2)(H∗+β)2=αh. |
Let ΩNS={(a,b,c,d,e,f,α,β):α=αh,A1(α)<0,a,b,c,d,e,f,β>0}; at the critical parameter value α=αh, the model (2.3) can exhibit Neimark-Sacker bifurcation near the equilibrium point (H∗,P∗), when α=αh changes in the neighborhood of ΩNS. Then, we conclude the above results in the following theorem:
Theorem 1. (i) If ΩF={(a,b,c,d,e,f,α,β):α=αf,a,b,c,d,e,f,β>0} exists and satisfies (3.1) at critical parameter value α=αf, then the model (2.3) undergoes flip bifurcation.
(ii) If ΩNS={(a,b,c,d,e,f,α,β):α=αh,A1(α)<0,a,b,c,d,e,f,β>0} exists and satisfies (3.2) at critical parameter value α=αh, then the model (2.3) undergoes Neimark-Sacker bifurcation.
Now we investigate the possible flip bifurcation of the model (2.3) at (H∗,P∗). From the above discussion the model (2.3) undergoes flip bifurcation at α=αf, where α varies in ΩF. Given a perturbation |α1|≪1 of αf, then perturbation of model (2.3) is described as
{Hn+1=Hnexp[a−bHn−αf+α1β+Hn−cPne+Hn],Pn+1=Pnexp[fHne+Hn−d]. | (3.3) |
Next by shifting (H∗,P∗) to the origin of (2.3) by using the transform un=Hn−H∗ and zn=Pn−P∗, we have
{un+1=γ1un+γ2vn+γ3α1+γ4u2n+γ5v2n+γ6α12+γ7unvn+γ8unα1+γ9vnα1+O((|un|+|vn|+|α1|)2),vn+1=ρ1un+ρ2vn+ρ3α1+ρ4u2n+ρ5v2n+ρ6α12+ρ7unvn+ρ8unα1+ρ9vnα1+O((|un|+|vn|+|α1|)2), | (3.4) |
where
γ1=1−bH∗+cP∗H∗(e+H∗)2+αfH∗(β+H∗)2, γ2=−cH∗e+H∗, γ3=−H∗β+H∗,γ4=H∗2(−b+cP∗(e+H∗)2+αf(β+H∗)2)2−b+ceP∗(e+H∗)3+αfβ(β+H∗)3,γ5=c2H∗2(e+H∗)2, γ6=H∗2(β+H∗)2, γ7=−cH∗e+H∗(−b+cP∗(e+H∗)2+αf(β+H∗)2)−ce(e+H∗)2,γ8=−H∗β+H∗(−b+cP∗(e+H∗)2+αf(β+H∗)2)−β(β+H∗)2, γ9=cH∗(e+H∗)(β+H∗),ρ1=feP∗(e+H∗)2, ρ2=1, ρ3=0, ρ4=−feP∗(e+H∗)3+f2e2P∗2(e+H∗)4,ρ5=0, ρ6=0, ρ7=fe(e+H∗)2, ρ8=0, ρ9=0. |
Let us assume that the eigenvalues are λ1=−1 and λ2=3−bH∗+cH∗P∗(e+H∗)2+αfH∗(β+H∗)2 for the matrix J with |λ1|=1, |λ2|≠1.
Next, we construct the non-singular matrix L as follows
L=(γ2γ2−1−γ1λ2−γ1), |
and we use the translation (unvn)=L(UnVn); then, (3.4) can be written as
{Un+1=−Un+F1(un,vn,α1)+O((|un|+|vn|+|α1|)2),Vn+1=λ2Vn+(un,vn,α1)+O((|un|+|vn|+|α1|)2), | (3.5) |
where
F1(un,vn,α1)=M1α1+M2u2n+M3v2n+M4α12+M5unvn+M6unα1+M7vnα1,F2(un,vn,α1)=N1α1+N2u2n+N3v2n+N4α12+N5unvn+N6unα1+N7vnα1, |
and
M1=(λ2−γ1)γ3−γ2ρ3γ2(1+λ2),M2=(λ2−γ1)γ4−γ2ρ4γ2(1+λ2),M3=(λ2−γ1)γ5−γ2ρ5γ2(1+λ2),M4=(λ2−γ1)γ6−γ2ρ6γ2(1+λ2),M5=(λ2−γ1)γ7−γ2ρ7γ2(1+λ2),M6=(λ2−γ1)γ8−γ2ρ8γ2(1+λ2),M7=(λ2−γ1)γ9−γ2ρ9γ2(1+λ2),N1=(1+γ1)γ3+γ2ρ3γ2(1+λ2),N2=(1+γ1)γ4+γ2ρ4γ2(1+λ2),N3=(1+γ1)γ5+γ2ρ5γ2(1+λ2),N4=(1+γ1)γ6+γ2ρ6γ2(1+λ2),N5=(1+γ1)γ7+γ2ρ7γ2(1+λ2),N6=(1+γ1)γ8+γ2ρ8γ2(1+λ2),N7=(1+γ1)γ9+γ2ρ9γ2(1+λ2). |
Now, let us assume Gc to be the CM; then, by using the CM theorem, we approximate the CM Gc of (3.5) at the origin for small changes in α1=0:
Gc(0,0)={(Un,Vn):Vn=h(Un,α1)}={(Un,Vn):Vn=c1α1+c2U2n+c3α1Un+c4α12+O((|Un|+α1)2)}. | (3.6) |
By substituting (3.5) on both sides of Vn=h(Un,α1), we have
λ2Vn+F2(un,vn,α1)=c1α1+c2(−Un+F1(un,vn,α1))2+c3α1(−Un+F1(un,vn,α1))+c4α12+O((|Un|+α1)2), |
where
un=γ2(Un+Vn)=γ2(Un+h(Un,α1)),vn=(−1−γ1)Un+(λ2−γ1)Vn=(−1−γ1)Un+(λ2−γ1)h(Un,α1),c1=N11−λ2, c2=11−λ2[N2γ22+N3(1+γ1)2−N5γ2(1+γ1)],c3=11+λ2[−2c2M1−N6γ2+N7(1+γ1)−2c1N2γ22+2c1N3(1+γ1)(λ2−γ1)−c1N5γ2(ρ2−γ1)],c4=11−λ2[c21N2γ22+c21N3(λ2−γ1)2+c21N5γ2(λ2−γ1)+c1N6γ2+c1N7(λ2−γ1)−c2M21+N4−M1c3]. |
Accordingly, on the CM Gc at the origin, we have
u2n=γ22(U2n+2UnVn+V2n),unvn=−γ2(1+γ1)U2n+γ2(ρ2−γ1)UnVn+γ2(λ2−γ1)V2n,v2n=(1+γ1)2U2n−2(1+γ1)(λ2−γ1)UnVn+(λ2−γ1)2V2n, |
where
UnVn=c1α1Un+c2U3n+c3α1U2n+c4α12Un+O((|Un|+|α1|)3),V2n=c21α12+2c1c2α1U2n+2c1c4α13+O((|Un|+|α1|)3). |
Moreover, the map confined to the CM Gc(0,0) takes the form
G∗(Un)=−Un+F1(un,vn,α1)=−Un+d1α1+d2U2n+d3Ung∗+d4α12+d5U2nα1+d6Unα12+d7U3n+d8α13+O((|Un|+|α1|)3), |
where
d1=M1,d2=M2γ22+M3(1+γ1)2−M5γ2(1+γ1),d3=2c1M2γ22−2c1M3(1+γ1)(λ2−γ1)+c1M5γ2(ρ2−γ1)+M6γ2−M7(1+γ1),d4=c21M2γ22+c21M3(λ2−γ1)2+M4+c21M5γ2(λ2−γ1)+c1M6γ2+c1M7(λ2−γ1),d5=2c3M2γ22+2c1c2M2γ22−2c3M3(1+γ1)(λ2−γ1)+2c1c2M3(λ2−γ1)2+c3M5γ2(ρ2−γ1)+2c1c2M5γ2(λ2−γ1)+c2M6γ2+c2M7(λ2−γ1),d6=2c4M2γ22+2c1c3M2γ22−2c4M3(1+γ1)(λ2−γ1)+2c1c3M3(λ2−γ1)2+c4M5γ2(ρ2−γ1)+2c1c3M5γ2(λ2−γ1)+c3M6γ2+c3M7(λ2−γ1),d7=2c2M2γ22−2c2M3(1+γ1)(λ2−γ1)+c2M5γ2(ρ2−γ1),d8=2c1c4M2γ22+2c1c4M3(λ2−γ1)2+2c1c4M5γ2(λ2−γ1)+c4M6γ2+c4M7(λ2−γ1). |
Finally, from [40], we define Δ1 and Δ2 as follows:
Δ1=(G∗Unα1+12G∗α1G∗UnUn)|(Un,α1)=(0,0)=d3+d1d2, | (3.7) |
Δ2=(16G∗UnUnUn+(12G∗UnUn)2)|(Un,α1)=(0,0)=d7+d22. | (3.8) |
Therefore, we have the following findings about flip bifurcation from the aforementioned study.
Theorem 2. If Δ1≠0 and Δ2≠0, the model (2.3) exhibits a flip bifurcation at (H∗,P∗) while changing the parameter Δ nearby Δ1. Moreover, if Δ2>0 (or Δ2<0) then the existing period two orbits from (H∗,P∗) are stable (or unstable).
Now we analyze the properties of possible Neimark-Sacker bifurcation around (H∗,P∗) for model (2.3) if, for instance, (3.2) holds for some αh. Given a perturbation |α2|≪1 of αh, perturbation of model (2.3) is described as
{Hn+1=Hnexp[a−bHn−αh+α2β+Hn−cPne+Hn]Pn+1=Pnexp[fHne+Hn−d]. | (3.9) |
Let us use the transforms un=Hn−H∗ and vn=Pn−P∗ and shift (H∗,P∗) to (0,0); the model (3.9) takes the form
{un+1=(un+H∗)exp[a−bHn−αh+α2β+Hn−cPne+Hn]−H∗,vn+1=(vn+P∗)exp[fHne+Hn−d]−P∗. | (3.10) |
Then, the Taylor expansion of (3.10) at the origin, up to order three, is
{un+1=δ1un+δ2vn+δ3u2n+δ4unvn+δ5v2n+δ6u3n+δ7u2nvn+δ8unv2n+δ9v3n+O((|un|+|vn|)3),vn+1=ρ1un+ρ2vn+ρ3u2n+ρ4unvn+ρ5v2n+ρ6u3n+ρ7u2nvn+ρ8unv2n+ρ9v3n+O((|un|+|vn|)3), | (3.11) |
where
δ1=1−bH∗+cP∗H∗(e+H∗)2+αhH∗(β+H∗)2,δ2=−cH∗e+H∗,δ3=H∗2(−b+cP∗(e+H∗)2+αh(β+H∗)2)2−b+ceP∗(e+H∗)3+αhβ(β+H∗)3,δ4=−cH∗e+H∗(−b+cP∗(e+H∗)2+αh(β+H∗)2)−ce(e+H∗)2, δ5=c2H∗2(e+H∗)2,δ6=H∗6(−b+cP∗(e+H∗)2+αh(β+H∗)2)3+12(−b+cP∗(e+H∗)2+αh(β+H∗)2)2−(cH∗P∗(e+H∗)3+αhH∗(β+H∗)3)(−b+cP∗(e+H∗)2+αh(β+H∗)2)−ceP∗(e+H∗)4−αhβ(β+H∗)4,δ7=−cH∗2(e+H∗)(−b+cP∗(e+H∗)2+αh(β+H∗)2)2−ce(e+H∗)2(−b+cP∗(e+H∗)2+αh(β+H∗)2)+cH∗e+H∗(cP∗(e+H∗)3+αh(β+H∗)3)+ce(e+H∗)3,δ8=c2H∗2(e+H∗)2(−b+cP∗(e+H∗)2+αh(β+H∗)2)+c2e−c2H∗2(e+H∗)3, δ9=−c3H∗6(e+H∗)3,ρ1=feP∗(e+H∗)2, ρ2=1, ρ3=−feP∗(e+H∗)3+f2e2P∗2(e+H∗)4,ρ4=fe(e+H∗)2,ρ5=0, ρ6=fe(e+H∗)4−f2e2P∗(e+H∗)5+f3e3P∗6(e+H∗)6,ρ7=−fe(e+H∗)3+f2e22(e+H∗)4, ρ8=0, ρ9=0. |
The characteristic polynomial equation associated with the linearized system of (3.10) at the origin:
λ2+q1(α2)λ+q2(α2)=0, | (3.12) |
where
q1(α2)=(−1+bH∗−(αf+α2)H∗(β+H∗)2−cH∗P∗(e+H∗)2)A1−A2,q2(α2)=(1−bH∗+(αf+α2)H∗(β+H∗)2+cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3)A1A2, |
with A1=exp(a−bH∗−(αh+α2)β+H∗−cP∗e+H∗) and A2=exp(fH∗e+H∗−d). Now, the roots of (3.12) are expressed as a pair of complex conjugates:
λ1,2=12[−q1(α2)±i√4q2(α2)−(q1(α2))2]. |
Since (a,b,c,d,e,f,α,β)∈ΩNS, we have that |λ1,2|=√q2(α2) and
d|λ1,2|dα2|α2=0=12√q2(0){H∗(β+H∗)2−1β+H∗×(1−bH∗+αhH∗(β+H∗)2+cH∗P∗(e+H∗)2+ecfH∗P∗(e+H∗)3)}<0. | (3.13) |
Further, we assume that q1(0)=−2+bH∗−αhH∗(β+H∗)2−cH∗P∗(e+H∗)2≠0,−1 and (2.6) implies that q1(0)≠±2,0,−1, which means that λk1,λk2≠1 for k=1,2,3,4 when α2=0. We only require that q1(0)≠0,1, which we can attain if it satisfies
bH∗−αhH∗(β+H∗)2−cH∗P∗(e+H∗)2≠2,3. | (3.14) |
Letting α2=0, ξ=−q1(0)2 and θ=√4q2(0)−q21(0)2, we construct the non-singular matrix
L=(δ20ξ−δ1θ), |
and we use the translation (unvn)=L(UnVn); thus, the model (3.9) takes the form
{Un+1=ξUn+θVn+Q(Un,Vn)+O((|Un|+|Vn|)3),Vn+1=−θUn+ξVn+R(Un,Vn)+O((|Un|+|Vn|)3), | (3.15) |
where
Q(Un,Vn)=1δ2[{δ3δ22+δ4δ2(ξ−δ1)+δ5(ξ−δ1)2}U2n+{δ4δ2θ+2θδ5(ξ−δ1)}UnVn+δ5θ2V2n+{δ6δ23+δ7δ22(ξ−δ1)+δ8δ2(ξ−δ1)2+δ9(ξ−δ1)3}U3n+{δ7δ22+2θδ8δ2(ξ−δ1)+3θδ9(ξ−δ1)2}U2nVn+{θ2δ8δ2+3θ2δ9(ξ−δ1)}UnV2n+θ3δ9V3n],R(Un,Vn)=1δ2θ[{δ22(δ3(δ1−ξ)+δ2ρ3)+δ2(ξ−δ1)(δ4(δ1−ξ)+δ2ρ4)+(ξ−δ1)2(δ5(δ1−ξ)+δ2ρ5)}U2n+{θδ2(δ4(δ1−ξ)+δ2ρ4)+2θ(ξ−δ1)(δ5(δ1−ξ)+δ2ρ5)}UnVn+θ2{δ5(δ1−ξ)+δ2ρ5}V2n+{δ23(δ6(δ1−ξ)+δ2ρ6)+δ22(ξ−δ1)(δ7(δ1−ξ)+δ2ρ7)+δ2(ξ−δ1)2(δ8(δ1−ξ)+δ2ρ8)−(ξ−δ1)3(δ9(δ1−ξ)+δ2ρ9)}U3n+{θδ22(δ7(δ1−ξ)+δ2ρ7)+2θδ2(ξ−δ1)(δ8(δ1−ξ)+δ2ρ8)+3θ(ξ−δ1)2(δ9(δ1−ξ)+δ2ρ9)}U2nVn+{θ2δ2(δ8(δ1−ξ)+δ2ρ8)+3θ2(ξ−δ1)(δ9(δ1−ξ)+δ2η9)}UnV2n+θ3(δ9(δ1−ξ)+δ2ρ9)V3n]. |
Next, we require the non zero quantity a∗ to ensure that (3.9) admits Neimark-Sacker NS bifurcation.
a∗=−Re[(1−2λ)ˉλ21−λξ11ξ20]−12|ξ11|2−|ξ02|2+Re(ˉλξ21), | (3.16) |
where
ξ20=18[(QUnUn−QVnVn+2RUnVn)+i(RUnUn−RVnVn−2QUnVn)],ξ11=14[(QUnUn+QVnVn)+i(RUnUn+RVnVn)],ξ02=18[(QUnUn−QVnVn−2RUnVn)+i(RUnUn−RVnVn+2QUnVn)],ξ21=116[(QUnUnUn+QUnVnVn+RUnUnVn+RVnVnVn)+i(RUnUnUn+RUnVnVn−QUnUnVn−QVnVnVn)]. |
Finally, from [38], we can state the following findings:
Theorem 3. If (3.13) and (3.14) hold and the quantity a∗ is non zero then the model (3.1) admits Neimark-Sacker bifurcation at (H∗,P∗) when αh changes in the neighborhood of ΩNS. Additionally, the quantity a∗<0 (or resp. a∗>0); then, the stable (or resp. unstable) invariant closed curve from (H∗,P∗) starts to bifurcate.
The theory of chaos control and bifurcation is known to be an essential area of current research. In particular, it has significant uses in biological and engineering sciences. Its features have emerged in ecological models. The nature of discrete-time population models is mostly unstable and chaotic in contrast to the continuous-time models. It is known that the logistic difference equation (logistic map) and the Ricker model [37] are the discrete analogs of the logistic differential equation [9]. The logistic map is derived from the logistic differential equation by using Euler approximation, taking the finite difference quotient as Δt=1, i.e., the time step is one generation. The sequence of periodic orbits starts oscillate from stable point like period 2, period 4, period 8, .., cycles and finally enter into the chaotic regime for many difference equations; this collection of such simple difference equations have been reported in ecology [41]. The Neimark-Sacker bifurcation is similar to the Hopf bifurcation in the continuous system. Similarly, there has been a lot of research work on the presence of complex and chaotic nature in the two-dimensional discrete prey-predator system, this chaotic dynamics absent in the continuous time system [42]. Further, it is important to introduce the appropriate methods for controlling chaos to prevent unexpected scenarios.
First, we use the state feedback control method as in [43,44] to control the chaotic system. For the model (2.3) we consider the following corresponding controlled model:
{Hn+1=Hnexp(a−bHn−αβ+Hn−cPne+Hn)−u(Hn,Pn)=f(Hn,Pn),Pn+1=Pnexp(fHne+Hn−d)=g(Hn,Pn), | (4.1) |
where u(Hn,Pn)=h1(Hn−H∗)+h2(Pn−P∗) is the feedback controlling force with feedback gains h1 and h2, and (H∗,P∗) is the unique positive interior equilibrium point of (4.1). Consider the controlled model (4.1) whose Jacobian matrix at (H∗,P∗) is given as
J(H∗,P∗)=(b11−h1b12−h2b21b22), | (4.2) |
where
b11=1−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2,b12=−cH∗e+H∗,b21=efP∗(e+H∗)2,b22=1. |
Then we have the characteristic polynomial of J(H∗,P∗) as
λ2−(b11+b22−h1)λ+b22(b11−h1)−b21(b12−h2)=0. | (4.3) |
Let λ1 and λ2 be the eigenvalues of (4.3); then, we obtain
λ1+λ2=b11+b22−h1,λ1λ2=b22(b11−h1)−b21(b12−h2). | (4.4) |
Then, the lines of marginal stability are derived by solving λ1=±1 and λ1λ2=1, assuming that |λ1,2|<<1. If λ1λ2=1, using the second part of (4.4), we have
L1:b11b22−b21b12−1=h1b22−h2b21. | (4.5) |
Next, if λ1=1, and using (4.4), we get
L2:h1(1−b22)+h2b21=b11+b22−1−b11b22+b21b12. | (4.6) |
Also, if λ1=−1, and using (4.4), we obtain
L3:h1(1+b22)−h2b21=b11+b22+1+b11b22−b21b12. | (4.7) |
Then the triangular region enclosed by lines L1, L2, L3 have stable eigenvalues.
Next, we study a chaos controlling technique based on the OGY control method as in [26]. By taking α as the control parameter, we rewrite the model (2.3) as follows:
{Hn+1=Hnexp(a−bHn−αβ+Hn−cPne+Hn)=f1(Hn,Pn),Pn+1=Pnexp(fHne+Hn−d)=f2(Hn,Pn). | (4.8) |
Further, α must lie in some interval |α−α0|<δ with δ>0, and α0 denotes the nominal value, for which the model (2.3) has chaotic dynamics. Now, we apply the state feedback control strategy to shift the trajectory to the desired state. Let (H∗,P∗) be the unstable equilibrium point of the model (2.3) in the chaotic state due to Neimark-Sacker bifurcation; then, model (2.3) can be approximated near (H∗,P∗) by the following linear map:
[Hn+1−H∗Pn+1−P∗]≈A[Hn−H∗Pn−P∗]+B[α−α0], | (4.9) |
where
A=[∂f1(H∗,P∗,α0)∂Hn∂f1(H∗,P∗,α0)∂Pn∂f2(H∗,P∗,α0)∂Hn∂f2(H∗,P∗,α0)∂Pn],B=[∂f1(H∗,P∗,α0)∂α∂f2(H∗,P∗,α0)∂α]=[−H∗β+H∗0]. |
It is easy to see that model (4.8) is controllable provided that the following matrix
C=[B : AB]=[−H∗β+H∗(∂f1(H∗,P∗,α0)∂Hn)−H∗β+H∗0(∂f2(H∗,P∗,α0)∂Hn)−H∗β+H∗] | (4.10) |
has rank 2. Furthermore, assume that −H∗β+H∗≠0 and ∂f2(H∗,P∗,α0)∂Hn≠0; then, model (4.8) is controllable. Next, we assume that [α−α0]=−K[Hn−H∗Pn−P∗], where K=[s1s2]; then, model (4.8) can be written as
[Hn+1−H∗Pn+1−P∗]≈[A−BK][Hn−H∗Pn−P∗]. | (4.11) |
Then, the controller model is given by
{Hn+1=Hnexp(a−bHn−α0−s1(Hn−H∗)−s2(Pn−P∗)β+Hn−cPne+Hn),Pn+1=Pnexp(fHne+Hn−d). | (4.12) |
Moreover, equilibrium point (H∗,P∗) is locally asymptotically stable if and only if both eigenvalues of the matrix A−BK, say, λ1 and λ2, lie in an open unit disk. Then, the matrix A−BK can be written as
A−BK=[b11−θs1b12−θs1b21b22], | (4.13) |
where
b11=1−bH∗+α0H∗(β+H∗)2+cH∗P∗(e+H∗)2,b12=−cH∗e+H∗,b21=efP∗(e+H∗)2,b22=1,θ=−H∗β+H∗. |
Then, the characteristic polynomial of (4.13) is given by
λ2−(b11+b22−θs1)λ+b22(b11−θs1)+b21(θs2−b12)=0. | (4.14) |
Then, the lines of marginal stability can be obtained, that is,
L1:b22(b11−θs1)+b21(θs2−b12)=1, | (4.15) |
L2:b11+b22=1+θs1+b22(b11−θs1)+b21(θs2−b12), | (4.16) |
L3:θs1=b11+b22+1+b22(b11−θs1)+b21(θs2−b12). | (4.17) |
Then, stable eigenvalues lie within the triangular region in the s1,s2-plane bounded by the straight lines L1,L2,L3 for the model's parametric values.
Next, we apply a hybrid control feedback methodology [21,25,45] for the control of flip and Neimark-Sacker bifurcations of the model near the equilibrium point (H∗,P∗); then, the controlled model can be written as
{Hn+1=ϵHnexp(a−bHn−αβ+Hn−cPne+Hn)+(1−ϵ)Hn,Pn+1=ϵPnexp(fHne+Hn−d)+(1−ϵ)Pn, | (4.18) |
where 0<ϵ<1 is the controlled strategy of a combination of both feedback control and parameter perturbation. The Jacobian matrix evaluated for the model (4.18) at (H∗,P∗) is given by
[ϵ(1−bH∗+αH∗(β+H∗)2+cH∗P∗(e+H∗)2)+1−ϵ−ϵcH∗e+H∗ϵefP∗(e+H∗)21]. | (4.19) |
Note, one can select appropriate values for ϵ to ensure that all eigenvalues of the above matrix satisfy |λ1,2|<1.
In this section, we provide a numerical simulation to ensure our theoretical analysis. The numerical iteration used to validate our results is similar to the processes described in [27,28,36,44,46]. The mathematical software Matlab (2018a) has been used to plot the numerical graphs. The computational approach provides fascinating details about how ecological systems function. Analytical approaches are exact and wide, but they only operate with the most fundamental models. The parameter values used here are only for illustrative purposes; neither experiments nor field research was used to obtain them [29,30,31]. Here, all parameters are considered positive values for which the model (2.3) has at least one positive interior equilibrium point (coexistence of both populations). Also, the chosen parameter values for the considered model (2.3) are biologically feasible.
Next, the conditions for the occurrence of two possible bifurcation behaviors of the model (2.3) were examined by varying the Allee parameter, namely, flip and Neimark-Sacker bifurcation verified numerically. It is also shown with the help of bifurcation diagrams. For this, we take two sets of parameter values assuming α<aβ, and described them in the following cases:
Case (i): Show the model (2.3) exhibits flip bifurcation with the fixed parametric values of a=5.5, b=5.5, c=20, d=0.1, e=10, f=2, β=0.5, while varying α=(0,2]. We obtained the unique interior positive equilibrium points (H∗,P∗) for the various values of α given in Table 1. At the critical point α=αf=1.5833188, the model (2.3) has the unique equilibrium point (H∗,P∗)=(0.526316,0.559233), and (2.10) becomes
λ2+0.0504708λ−0.949529=0, |
α | H∗ | P∗ |
0.20 | 0.526316 | 1.268630 |
0.37 | 0.526316 | 1.181450 |
0.43 | 0.526316 | 1.150680 |
0.50 | 0.526316 | 1.114780 |
1.25 | 0.526316 | 0.730165 |
1.70 | 0.526316 | 0.499396 |
which shows that, λ1,λ2=0.949529,−1 are the eigenvalues of (2.9), Then, it satisfies the condition. It ensures the occurrence of flip bifurcation near the interior equilibrium point for the model (2.3). Also, for α=0.20, the model (2.3) shows chaotic behavior, and by further increasing, it shows period-16 at α=0.37, period-8 at α=0.43, period-4 at α=0.50, period-2 at α=1.25, and stable equilibrium point at α=1.70. The different states of the model (2.3) are plotted for different values of α in the time plot and phase portrait in Figures 1 and 2, respectively. It indicates that the model becomes stable from chaos by the reverse period-doubling phenomenon. So, the existence of flip bifurcation for the model (2.3) is shown in Figure 3(a). Also, the existence of chaos is confirmed by plotting the corresponding largest Lyapunov exponent in Figure 3(b). Further, from (3.7) and (3.8), we obtain Δ1=0.415148≠0, Δ2=−2.22965≠0 and the property of the existing flip bifurcation which is illustrated in Theorem 2.
Case (ii): Show that the model (2.3) exhibits Neimark-Sacker bifurcation with the fixed parametric values of a=3.71, b=4.8, c=10, d=1, e=10, f=29, β=0.5, and varying values of α=(0,0.5]. We obtained the unique interior positive equilibrium points (H∗,P∗) for the various values of α given in Table 2. At the critical point α=αh=0.4135275, the model (2.3) has the unique equilibrium point (H∗,P∗)=(0.357143,1.56731) and the characteristic Eq (2.10) becomes
λ2+0.538916λ+1=0, |
which shows that λ1,λ2=0.269458±0.963012i are the eigenvalues of (2.9). Then, it satisfies the condition (3.2). It ensures the occurrence of Neimark-Sacker bifurcation near the interior equilibrium point for the model (2.3). Also, for α=0.015, the model (2.3) shows chaotic orbit. Further, for α=0.045, the result is 13-period; for α=0.048, it is 13 invariant circles; for α=0.130, it is 9-period; for α=0.410, it is a closed invariant circle and there is a asymptotically stable equilibrium point at α=0.450. The different nature of the model around the interior equilibrium point for the model (2.3) is shown in the time plots and phase portraits in Figures 4 and 5, correspondingly. The figures show the model undergoes Neimark-Sacker bifurcation. The bifurcation diagram and its largest Lyapunov exponent are shown in Figure 6(a), (b), correspondingly. Since a∗=−0.151225 from (3.16), its results are explained in Theorem 3.
α | H∗ | P∗ |
0.015 | 0.357143 | 2.04886 |
0.045 | 0.357143 | 2.01261 |
0.048 | 0.357143 | 2.00899 |
0.130 | 0.357143 | 1.90991 |
0.410 | 0.357143 | 1.57157 |
0.450 | 0.357143 | 1.52324 |
In order to show the effect of Allee parameter α on the growth rate parameter a for the values in both Cases (i) and (ii), we have plotted the largest Lyapunov exponents in Figure 7(a), (b). This figures clearly shows the chaotic and stable dynamics of the proposed model.
For the particular choice of parameter values a=3.71, b=4.8, c=10, d=1, e=10, f=29, β=0.5, and α=0.015, we get the positive unstable equilibrium point (H∗,P∗)=(0.357143,2.0488647) for the model (2.3) and for its phase portrait (see Figure 5(a)). So, we chose to use the state feedback control method and want to shift the unstable equilibrium point into stable dynamics. For this, taking α0=0.015 and the corresponding controlled model yields
{Hn+1=Hnexp(3.71−4.8Hn−0.0150.5+Hn−10Pn10+Hn)−u(Hn,Pn),Pn+1=Pnexp(29Hn10+Hn−1), | (5.1) |
where u(Hn,Pn)=u(Hn,Pn)=h1(Hn−H∗)+h2(Pn−P∗), and h1 and h2 are feedback gains. Furthermore, the Jacobian matrix of the controlled model is given as follows
J=(−0.63878−h1−0.344828−h25.5391). |
Then, we obtain the characteristic equation as
λ2−(0.36122−h1)λ+1.27122−h1+5.539h2=0. |
Moreover, the lines of marginal stability for the model (5.1) are computed as
L1:0.27122=h1−5.539h2, L2:5.539h2=−1.91, L3:2h1−5.539h2=2.63244. |
Then, the triangular region enclosed by the lines L1,L2,L3 has stable eigenvalues for the controlled model (5.1), as shown in Figure 8(a). Clearly, if h2=−0.3, then (H∗,P∗)=(0.357143,2.0488647) is locally stable if and only if h2∈[−0.642,0.4853]. Taking h1=−0.3 and h2=0.3, then the time-series plots for the controlled model (5.1) are shown in Figure 8(b), (c).
Second, for a=3.71, b=4.8, c=10, d=1, e=10, f=29, β=0.5 and α=0.015, the model (2.3) yields (H∗,P∗)=(0.357143,2.0488647) and its phase portrait is shown in Figure 5(a). So, we chose to use the OGY feedback control method to move the strange chaotic dynamics into the stable dynamics. For this, taking α=0.015 and the corresponding controlled model yields
{Hn+1=Hnexp(3.71−4.8Hn−0.015−s1(Hn−H∗)−s2(Pn−P∗)0.5+Hn−10Pn10+Hn),Pn+1=Pnexp(29Hn10+Hn−1), | (5.2) |
where s1 and s2 are feedback gains. Furthermore, in this case, we have
A=(−0.63878−0.3448285.5391), B=(−0.4166670), C=(−0.4166670.2661580−2.30792), |
and the Jacobian matrix of the controlled model is given as follows
A−BK=(−0.63878+0.416667s1−0.344828+0.416667s25.5391). |
Then we obtain the characteristic equation
λ2+(0.36122+0.416667s1)λ+1.27122+0.416667s1−2.30792s2=0. |
Moreover, the lines of marginal stability for the model (5.2) are computed as
L1:1.27122+0.416667s1−2.30792s2=1,L2:1.91=2.30792s2,L3:−0.416667s1=2.63244+0.416667s1−2.30792s2. |
Then the triangular region enclosed by the lines L1,L2,L3 has stable eigenvalues for the controlled model (5.2) as shown in Figure 9(a). Clearly, if s1=0.1, then (H∗,P∗)=(0.357143,2.0488647) is locally stable if and only if s2∈[0.55128,0.82758]. Taking s1=0.1 and s2=0.7, then the time-series plots for the controlled model (5.2) are depicted in Figure 9(b), (c).
Finally, we discuss the hybrid control strategy for some parametric values. In this case, the controlled model can be written as
{Hn+1=ϵHnexp(5.5−5.5Hn−0.200.5+Hn−20Pn10+Hn)+(1−ϵ)Hn,Pn+1=ϵPnexp(2Hn10+Hn−1)+(1−ϵ)Pn. | (5.3) |
Moreover, the Jacobian matrix of the model (5.2) at (H∗,P∗)=(0.526316,1.26863) is given by
[1−2.6748ϵ−ϵ0.228987ϵ1]. |
The eigenvalues of the above matrix lie inside the open disk if and only if 0<ϵ<0.773477. For ϵ∈(0,1] and (H∗,P∗)=(0.526316,1.26863), the bifurcation diagram for the controlled model (5.3) are depicted in Figure 10(a), (b).
Remark 1. The model (2.3) is the discrete counterpart of the continuous-time model (2.1) investigated in [29]. The method of using a piecewise constant argument for the differential equations has been is adopted to obtain the discrete time model, similar to those in [36,46]. In the absence of predators, the resultant model transforms into the classical Ricker stock recruitment model [37]. It is worth mentioning here that this type of Ricker-type model (2.3) has not been investigated previously. For instance, in [47], the authors studied a basic discrete predator model with the effect of using Ricker map for the prey growth by replacing the classical logistic map; they showed the existence of flip and Neimark-Sacker bifurcations. A Ricker-type predator-prey model with hunting cooperation was investigated by the authors of [48], who studied the existence of discrete Hopf bifurcation. The authors of [29] demonstrated that the model (2.1) experiences Hopf bifurcation, while the model (2.3) undergoes flip and Neimark-Sacker bifurcations and also exhibits chaotic dynamics. The direction and stability properties of both bifurcations can be discussed by using the results form [38,39]. Also, the bifurcation and chaos control analyses have been carried out using the methods described in [27,28,44].
This paper deals with the discrete-time predator-prey model of Holling Ⅱ-type interaction with a weak additive Allee effect. In the first step of this study, we derived the discrete-time model from the corresponding continuous model in [29] by using the method of piecewise constant arguments for the differential equations [35]. This discrete form of Lotka-Volterra equations is due to May [9]. The model (2.3) without the Allee effect or predator population has been reduced to a one-dimensional model similar to the model in [37]. The discrete model (2.3) considered in this present work is a good representation of population interaction, as it has a non-overlapping generation, i.e., the birth and death rates occur in certain time intervals. From a biological point of view, the positive equilibrium points for the continuous-time model described in [29] and the discrete-time model (2.3) in this present study are the same, as we showed in Lemma 1. For model (2.3), we showed that the density of the prey population remains unchanged, and that the density of the predator population decreases by increasing the Allee parameter α; see Tables 1 and 2. The local stability of a coexisting equilibrium (H∗,P∗) is described via Lemma 3, which ensures the long-time survival of both species with the impact of α. In the contrasting continuous-time model [29] of (2.3), for a small value the of Allee parameter, the model exhibits a periodic solution and becomes locally asymptotically stable via Hopf bifurcation, when the Allee parameter crosses some critical value. Moreover, this is similar to the Neimark-Sacker bifurcation for the discrete model (2.3) in this present study.
Next, we derived the conditions for the existence of flip and Neimark-Sacker bifurcations for model (2.3) around the positive interior equilibrium point by taking the Allee parameter as a bifurcation parameter. We showed the properties of both bifurcations with the help of the CM theorem and normal form theory in Theorems 2 and 3. In the simulation part, we verified all the analytical results numerically. We showed that model (2.3) undergoes flip bifurcation at αh in Case (i) and Neimark-Sacker bifurcation at αh in Case (ii), for a different set of parameter values. We notice that model (2.3) has strange dynamics for smaller values of α; see Figures 2(a) and 5(a). The strange behavior of the model (2.3) for both cases is shown clearly in the one-parameter bifurcation diagrams in Figures 3(a) and 6(a). We notice that the proposed model reduces complex dynamical behavior, and we say that the sensitivity of the population dynamics to initial conditions has been reduced in a sense by the Allee effect. Further, we verified this sensitivity to the initial conditions by plotting the largest Lyapunov exponents in Figures 3(b) and 6(b). In this study, we successfully implemented state and OGY feedback and hybrid control methods by constructing the corresponding controlled models (5.1), (5.2) and (5.3). The lines of stable margins for the controlled models using state feedback control and the OGY feedback control method have been derived and plotted in Figures 8(a) and 9(a). The choice of feedback value inside the stability triangle helps to shift the desired chaotic orbit into the area of stable dynamics, see Figures 8(b), (c) and 9(b), (c). In the case of the hybrid control method, for the control parameter 0<ϵ<0.773477, the model (5.3) is stable, and the one-parameter bifurcation diagram is shown in Figure 10. A different possibility for showing the existence of chaos in real populations has been provided in the theory of chaos control [26]. The need for this control supports the adoption of the OGY technique for the three fish food chain, i.e., stabilization for the desired periodic orbit. This control condition goes against empirical attempts for lakes and rivers [23].
This study concludes that increasing the Allee parameter, which leads to converting the strange attractor into a proper pattern, i.e., period-16, period-8, period-4, period-2, period-13, period-9 and a closed invariant circle, means that the population oscillates periodically and finally reaches stable dynamics (long-term coexistence of both species) in the considered model. Also, this periodic window and chaotic orbits cannot occur in the counter continuous-time model. This stabilization takes place via both the flip and the Neimark-Sacker bifurcations. In one word, the Allee effect plays a positive role in stabilizing the proposed model, i.e., keeping the population persistent for a long time. However, there will be more complexity in predicting the future population size under chaotic dynamics because a small perturbation in the initial population size can cause massive divergence in population size. For smaller values of the Allee parameter, the presence of bifurcation and chaos in the proposed model can cause both species to have a higher risk of extinction due to unpredictability.
However, it is important to study the discrete-time predator-prey model with the Allee effect in both prey and predator populations. It is interesting to note that one may study the model's relationship to other ecological phenomena like prey refuge, cannibalism and fear impact by appropriately adjusting the strength of the Allee effect simultaneously. Also, considering the other functional responses such as ratio-dependent, Crowley-Martin and Beddington-DeAngelis functional responses, may lead to complex models and cause strange behaviors; we will leave this for future research.
The authors declare that they have not used artificial intelligence tools in the creation of this article.
This work was partially supported by the Center for Nonlinear Systems, Chennai Institute of Technology, India, with funding number CIT/CNS/2023/RP/016.
Prof. Dimplekumar Chalishajar is the Guest Editor of special issue "Recent advances in differential and partial differential equations and its applications'' for AIMS Mathematics. Prof. Dimplekumar Chalishajar was not involved in the editorial review or the decision to publish this article.
All authors declare no conflict of interest this paper.
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α | H∗ | P∗ |
0.20 | 0.526316 | 1.268630 |
0.37 | 0.526316 | 1.181450 |
0.43 | 0.526316 | 1.150680 |
0.50 | 0.526316 | 1.114780 |
1.25 | 0.526316 | 0.730165 |
1.70 | 0.526316 | 0.499396 |
α | H∗ | P∗ |
0.015 | 0.357143 | 2.04886 |
0.045 | 0.357143 | 2.01261 |
0.048 | 0.357143 | 2.00899 |
0.130 | 0.357143 | 1.90991 |
0.410 | 0.357143 | 1.57157 |
0.450 | 0.357143 | 1.52324 |